Serge Galam – Sociophysics a Physicist's Modeling Of

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Sociophysics

A Physicist’s Modeling of Psycho-political Phenomena

123 Serge Galam CREA Boulevard Victor 32 75015 Paris France

ISSN 1860-0832 e-ISSN 1860-0840 ISBN 978-1-4614-2031-6 e-ISBN 978-1-4614-2032-3 DOI 10.1007/978-1-4614-2032-3 Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011944508

© Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com) To Giuliana, my so Jewish Mom (see Fig. 1)

Fig. 1 Your only freedom is... to think!

Preface

When I was at high school, I was very much interested in philosophy, politics, and history, but I was too lazy to read all the necessary books required to sustain any sound discussion. There was too much material to learn by heart along with the hundreds of great citations that each prestigious contributor had made to those fields from the early stages of human culture, which cover a few thousand years. At odds with this, mathematics was much less demanding in terms of memory and reading, and to top it all, a really exciting game. Laziness, combined with the pleasure of resolving small challenges thus put me on the scientific highway of education without a clue of where it would take me. How could I know, or even figure it out? Coming from a Jewish family expelled from Libya in the 1950s, my parents arrived in Paris without having ever met any academic in the whole of their lives, not to mention any physicist. My birth there was like a camel being put into a strange world of Martians. And sometimes, I thought that in addition, my family was from Venus. I was only instructed that to learn was the thing to do, something supposed to be much more promising than selling clothes in a little shop, at least according to Sion, my dear late father. He kept on pushing me into learning, with the irrefutable argument that I would understand later on why it was the right thing to do. Indeed, I discovered later than it is more fun to shop around looking for good bargains rather than trying to sell them. A wise man my father. And my adorable mother started calling me Professor Nimbus, due to some of my hair that pointed skywards. So, not yet being a contrarian, I kept on learning since after all it was not such a demanding thing to do, with the mathematics being quite transparent for me. It was, sometimes, even enjoyable. I thus kept on studying one year after the other until I graduated in mathematics. But proving theorems began to fade away in front of my eyes. I started to feel a bit confined by the closed world of mathematics and at the same time I actually had to wear glasses for the first time in my life. Viewing all of a sudden the so many awful details of everything around me was a shock, which in turn made me switch to physics. Do not ask me why. It is called a chaotic bifurcation. Somehow, I thought that theoretical physics would bring me closer to the real world. I agree that this was quite a peculiar view of the real world.

vii viii Preface

Doing physics, I found the idea of playing formal games to question the laws that govern basic nature rather attractive. You can have fun playing with mathematics by ignoring up to some point, part of its rigor, but always complying with the robust and nonnegotiable constraints of nature. Moreover, it was not a lonely game like mathematics. To have nature as your partner is quite a tough challenge. Maybe the fact that electronic games did not exist at that time was a key to my gaining such a vision. I thus engaged in theoretical physics, a field that is as close to nature as a Rubik’s cube! Nevertheless, rather quickly I realized that I did not have much interest in the understanding of the laws of inert matter as such. Atoms did not excite my imagination. Human behavior did. I became determined to confront the secrets of human behavior, although I had no idea of how precisely we would be able to grasp some of its big and essential features. But I was also convinced that the power and genius of the pragmatic methodology of physics could contribute towards my “crazy” plan. Of course, to implement my endeavor I had first to learn physics and in particular how physicists operate so as to identify the secrets of inert matter. My program was set. It simply took me over 40 years to highlight what stands in common between atoms and human beings. Through a long and tortuous path, “spiced” with a lot of fights as well as a huge amount of self-will, my dream finally came true. The fundamental ingredients are in this book. At least in part. And I am no longer the only contributor to the settling down of this vision into solid equations and concepts. Today, sociophysics exists as a novel emerging field of science that involves hundreds of physicists all around the world. At present, we are on the road, an unknown road with still a long way ahead of us before sociophysics is eventually validated as a science. In the future it may even become a hard science like physics, or be dismissed as an absurd pipe dream. However, for the time being, it is surely a new and exciting adventure of human research at the very frontiers of knowledge. The consequences and accomplishments are unpredictable and even unconceivable. This is all the fun and value of being engaged in fundamental research at the edge. Nevertheless, piercing some of the secrets of human behavior brings with it certain risks. That is why, while applying some of the methodology of physics to this new field of social sciences as a physicist, simultaneously, as a responsible citizen I discuss these risks thoroughly. To always keep in perspective both the dangers and the hopes of sociophysics, together with the fact that dealing with human beings is of a different nature to that of dealing with atoms, is a cornerstone requirement to allow a robust and ethical development of sociophysics. In this book, I am presenting my personal testimony, itinerary, and contributions to the field of sociophysics, but its fate will stem from the collective construction made up of the many individual contributions of researchers all over the world. In addition, the capacity to avoid a partisan politicization in formulating and addressing some fundamental issues of the political and social arena will be the key to sociophysics becoming a success story. I think.

Paris Serge Galam Contents

The Reader’s Guide to a Unique Book of Its Kind ...... xvii References...... xxiii

Part I Sociophysics: Setting the Frame

1 What is Sociophysics About? ...... 3 1.1 Science Fiction and Asimov’s Foundation Syndrome ...... 3 1.2 What Makes Sense and What Does Not with Asimov’s Fictional Psychohistory ...... 5 1.2.1 Predicting the Future Is Not Possible ...... 5 1.2.2 Historical Vs. Ahistorical Sciences ...... 7 1.2.3 Small Groups, Large Groups, Not so Simple...... 8 1.2.4 A Mathematician Could Not Have Done It, a Physicist Could, Maybe ...... 9 1.3 A Parenthesis on the Fall of Empires ...... 10 1.4 Time for New Paradigms ...... 11 1.5 Sociophysics, a Possible Novel Hard Science: Not a Zero Risk Path...... 13 1.6 Discovering the Limits of Human Freedom Opens the path to Social Freedom ...... 16 References...... 19 2 The Question: Do Humans Behave like Atoms? ...... 21 2.1 My Basic Philosophy ...... 21 2.2 Telling the Truth About What Indeed Physics Is and What It Is Not ...... 23 2.3 Physics Does Not Care About Mathematical Rigor ...... 27 2.4 Implementing a Physics-like Approach Outside Physics ...... 28 2.5 But Indeed, Do Humans Behave Like Atoms? ...... 29 2.6 Building Up an “Atom–Individual” Connection ...... 31 2.7 Our Bare Methodology ...... 34

ix x Contents

2.8 To Sum up ...... 37 References...... 39 3 Sociophysics: The Origins ...... 41 3.1 The First Days...... 42 3.1.1 Breaking the Secret of Critical Phenomena ...... 42 3.1.2 The Physicist’s Corner...... 43 3.2 The First Days, the First Fight ...... 44 3.3 From Claim to Demonstration ...... 46 3.4 The Story Behind the Scene ...... 48 3.5 More About Academic Freedom ...... 51 3.6 Surviving Within Physics by Not Playing Tennis ...... 52 3.7 Breaking the Gap with a Social Scientist ...... 55 3.8 Changing My Strategy: Back to the World of Physics ...... 57 3.9 The Secret One Shot International Seminar ...... 58 3.10 The Rising Sun of Sociophysics ...... 58 3.11 When Too Much Is Too much ...... 60 3.12 Claiming the Paternity of Sociophysics...... 61 3.13 Reorientating My Strategy Again to Join a Social Sciences Group...... 63 References...... 66 4 Sociophysics: Weaknesses, Achievements, and Challenges ...... 69 4.1 The Essential Challenges of Sociophysics...... 69 4.2 Sociophysics: A New Field Is Emerging ...... 70 4.3 Deciding the Future of Sociophysics...... 71 4.4 Sociophysics: Epistemological Foundations...... 72 4.5 Flashback to the Origins ...... 74 4.6 The Soviet-Like Rewriting of the History of Sociophysics ...... 75 4.7 Fatherhood with a Touch of Humor ...... 77 4.8 Basic Weaknesses of Growing Sociophysics ...... 79 4.9 The Positive Achievements of Sociophysics so Far ...... 81 4.10 The First Sociophysics Successful Prediction of a Precise Event ...... 83 4.10.1 Taking Risks to Validate Sociophysics...... 84 4.10.2 When the Prediction Turns Out to be True ...... 85 4.10.3 When a Prediction Fails ...... 86 4.11 Proposal to Establish a Road Map ...... 86 4.12 What the Climatologists Did with the IPCC Should Not Be Repeated ...... 87 References...... 88 Contents xi

Part II Discovering the Wonderful (and Maybe Scary) World of Sociophysics

5 Sociophysics: An Overview of Emblematic Founding Models...... 93 5.1 In a Few Words ...... 95 5.1.1 Decision Making ...... 95 5.1.2 Bottom-up Democratic Voting ...... 96 5.1.3 Terrorism ...... 96 5.1.4 Coalitions Versus Fragmentation...... 96 5.1.5 Public Opinion ...... 97 References...... 97 6 Universal Features of Group Decision Making...... 101 6.1 The Strike Phenomenon ...... 101 6.1.1 The Model...... 102 6.1.2 The Operating Mechanism ...... 102 6.1.3 The Overlap with the Physical Model ...... 104 6.1.4 The Novel Counterintuitive Social Highlights ...... 105 6.1.5 Achievements of the Model ...... 106 6.2 How Do Groups Make Their Decisions? ...... 106 6.2.1 The Symmetrical Individual Versus the Symmetrical Group ...... 107 6.2.2 The Random Symmetry Breaking Choice ...... 108 6.2.3 Anticipating the Group Choice...... 109 6.2.4 Individuals Are Often Not Symmetrical ...... 111 6.2.5 Life Is Not a Paradise ...... 113 6.2.6 Some Emblematic Illustrations of the Model...... 114 6.2.7 The Leader Effect ...... 117 6.2.8 The Overlap with the Physical Model ...... 118 6.2.9 The Model’s Achievements ...... 118 References...... 119 7 The Dictatorship Paradox of Democratic Bottom-up Voting ...... 121 7.1 The Local Majority Rule Model ...... 123 7.2 Incorporating the Inertia Effect of Being the Ruler ...... 123 7.3 From Probabilistic to Deterministic Voting...... 125 7.4 The Magic Formula for Presidency ...... 126 7.5 A Simulation to Visualize the Dictatorship Effect of Democratic Voting ...... 127 7.6 From Two to Three Competing Parties ...... 129 7.7 The Overlap with the Physical Model ...... 132 7.8 Achievements of the Model ...... 135 References...... 137 xii Contents

8 The Dynamics of Spontaneous Coalition–Fragmentation Versus Global Coalitions...... 139 8.1 The Two-Country Problem ...... 139 8.2 The Unstable Three-Country Problem ...... 142 8.3 Superposing Current Pair Bonds to Historical Ones ...... 149 8.4 From Binary to a Multiple Coalitions...... 151 8.5 The Overlap with the Physical Model ...... 152 8.6 Achievements of the Model ...... 153 References...... 153 9 Terrorism and the Percolation of Passive Supporters ...... 155 9.1 The Geometry of Terrorism ...... 156 9.2 Local Versus Global Terrorism: A Unified Frame...... 160 9.3 What Is to Be Done? ...... 163 9.4 The Various Flags of a Terrorist Group ...... 165 9.5 The Overlap with the Physical Model ...... 167 9.6 Achievements of the Model ...... 168 References...... 168 10 The Modeling of Opinion Dynamics...... 169 10.1 An Overview ...... 169 10.2 Why Is Public Opinion Often Conservative? ...... 171 10.3 The Local Majority Model and the Existence of Biases ...... 173 10.4 The Appearance of Nonthreshold Dynamics ...... 184 10.5 Mixing the Group Sizes ...... 185 10.6 Heterogeneous Agents and the Contrarian Effect ...... 188 10.7 Thresholdless Driven Coexistence ...... 189 10.8 The One-Sided Inflexible Effect and the Global Warming Issue ...... 191 10.9 The Thresholdless Case...... 193 10.10 Extending the Competition to Three Opinions ...... 195 10.11 The Reshuffling Effect and Rare Event Nucleation ...... 196 10.12 The Overlap with Physical Systems and Other Sociophysics Models ...... 197 10.13 Achievements of the Model ...... 198 10.14 In the Meantime...... 199 References...... 200 11 By Way of Caution ...... 203

Part III Democratic Voting in Bottom-Up Hierarchical Structures: From Advantages and Setbacks to Dictatorship Paradoxes

12 Highlights of the Part ...... 209 12.1 Dictatorships Can Be Democratic...... 209 Contents xiii

12.2 What It Is About ...... 210 12.3 The Wonderful World of Democracy ...... 210 12.4 Not Ruling Is Bad for You...... 211 12.5 Big Is Better ...... 212 12.6 What Matters ...... 213 12.7 The Strategic Key ...... 214 12.8 Visualizing the Democratic-Driven Dictatorship Twist ...... 215 12.9 The Key Configurations to Infiltrate a Party ...... 215 12.10 Life Is More Risky with Three Competing Parties ...... 220 12.11 Eastern European Communist Collapse Was Not Sudden ...... 220 References...... 221 13 Basic Mechanisms for the Perfect Democratic Structure ...... 223 13.1 Starting from a Naive View of Former Communist Organizations ...... 223 13.2 Setting Up the Simplest Form of a Voting Process ...... 225 13.3 The Single Random Small Group Voting Scheme ...... 227 13.4 Fluctuations, Group Sizes, and Democratic Balance...... 229 13.5 Limits of the Single Group Voting Scheme...... 231 13.6 Including Even-Sized Voting Groups ...... 232 13.7 Setting Up the Perfect Democratic Structure ...... 234 13.8 The Dynamics Driven by Repeated Democratic Voting ...... 240 13.9 Some Comments About Zero ...... 243 14 Going to Applications ...... 247 14.1 The Practical Scheme ...... 247 14.2 The Physicist’s Corner: Trying to Be a Little More Mathematical ...... 250 14.3 The Magic Formula ...... 256 14.4 What It Means in Terms of Global Size ...... 263 14.5 The Physicist’s Corner: To Make It Simpler...... 264 14.6 Putting a Limit on the Global Size ...... 268 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human ...... 273 15.1 Phase Transitions and Critical Phenomena ...... 273 15.2 Revisiting the Practical Scheme ...... 276 15.3 Revisiting the Magic Formula ...... 277 15.4 Rare Dictatorial Events Versus Antidemocratic Ones...... 279 15.4.1 Another Viewpoint...... 282 15.4.2 The Physicist’s Corner...... 283 15.5 From Rare Antidemocratic Events to the Radical Efficiency of Geometric Nesting ...... 284 15.5.1 Monitoring the Rare Antidemocratic Bottom Configurations ...... 284 15.5.2 When the Radical Efficiency Turns Nasty ...... 286 xiv Contents

15.6 More About Hierarchies ...... 290 15.6.1 Randomness Is Sufficient at the Bottom ...... 290 15.6.2 Geography and Multisize Combination of Voting Groups...... 291 15.6.3 A Digression of About a Fifty Percent Score: What Is the Meaning of a Majority? ...... 294 References...... 295 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures ...... 297 16.1 The Inertial Effect of Being in Power ...... 297 16.2 The Dramatic Effect of Tie Break Voting in the Single...... 300 16.3 Varying the Voting Group Size ...... 302 16.4 From the Perfect Democratic Structure to the Perfect...... 305 16.5 The Dynamics Driven by Repeated Democratic Voting ...... 312 16.5.1 The Physicist’s Corner...... 313 16.6 From the Magic to the Machiavelli Formula ...... 316 16.6.1 The Physicist’s Corner...... 320 16.7 Global Size, the Practical Scheme, the Magic Formula ...... 323 16.7.1 Global Size ...... 324 16.7.2 The Practical Scheme ...... 325 16.7.3 The Physicist’s Corner...... 327 16.7.4 The Magic Formula ...... 330 16.7.5 The Physicist’s Corner...... 332 16.7.6 The Super Magic Formula...... 336 16.7.7 The Physicist’s Corner...... 337 16.8 What Happens to the Rare Antidemocratic Events? ...... 339 16.8.1 The Minimum Number of Bottom Agents to Win the Presidency...... 341 16.8.2 The Associated Number of Different Bottom Nasty Configurations ...... 346 16.8.3 The Actual Probability of a Nasty Bottom Configuration ...... 347 16.9 All Bottom Minorities and Majorities Winning the Presidency . 348 16.9.1 The Odd Case r D 3 ...... 351 16.9.2 The Even Case r D 4: The Challenging View Point ...... 356 16.9.3 The Even Case r D 4: The Running Power View Point ...... 358 16.9.4 The Physicist’s Corner...... 366 16.10 The Worrying Power of Geometric Nesting or How to Make Certain a Very Rare Event ...... 367 16.10.1 The Sudden and Unexpected Taking Over of Large Institutions ...... 368 16.10.2 The Scary Lobbying ...... 368 Contents xv

16.10.3 A Striking Idealized Illustration...... 369 16.10.4 Hint to Restore the Democratic Functioning ...... 369 16.11 Softening the Inertia Principle...... 370 16.11.1 The Physicist’s Corner...... 373 16.11.2 Three Competing Opinions: It Becomes Even More Counterintuitive ...... 374 16.12 Communist Collapse and French FN Victory ...... 374 16.12.1 Hierarchies Are Everywhere ...... 375 References...... 376

Part IV The Risky Business of Alliances in Bottom-Up Democratic Voting with Three-Choice Competition

17 Bottom-Up Democratic Voting in a Three-Choice Competition...... 379 17.1 Two Competing Parties in Short ...... 380 17.1.1 The General Frame ...... 380 17.1.2 Predicting the Results of Democratic Elections ...... 380 17.1.3 The Bottom-Up Voting Dynamics ...... 382 17.1.4 From Theoretical Principles to Reality ...... 386 17.1.5 From Reality to Implementation ...... 389 17.2 Three Competing Parties ...... 390 17.2.1 Two Competing Parties, a One-Dimensional Problem ...... 391 17.2.2 Three Competing Parties, a Two- Dimensional Problem...... 393 17.2.3 The Three-Party Bottom-Up Voting Flow ...... 395 17.2.4 The Physicist’s Corner...... 402 17.3 The Bottom-Up Voting Flow Diagram...... 404 17.3.1 The Frequent Case of Two Large Opposing Parties with a Small One in Between ...... 406 17.3.2 Some General .˛;ˇ;/Case Snapshots ...... 409 17.4 The “Golden Triangle” to Win the Presidency ...... 411 References...... 413 18 So Sorry, That’s the End of the Tour! ...... 423 19 I Thank You ...... 427 Index ...... 429

The Reader’s Guide to a Unique Book of Its Kind

While huge progress has been made in understanding and mastering inert matter as well as in manipulating biological systems, very little has been achieved with respect to human matter. The major difficulty is that we are the main part of it, making an objective investigation rather illusory. It is from this intrinsic impossibility that the use of a novel set of tools and concepts, extracted from a totally different but highly developed field of research, could turn out to be very fruitful. Accordingly, I ground the work in physics, a very successful and powerful branch of both knowledge and experimentation. Learning from a reality of a totally different nature, consisting of far less complicated entities, of “poor and limited” atoms, could be very efficient in embracing human complexity. Precisely because of this hypothesis, the work has been built without incorporating the numerous and apparently inextricable set of human attributes that we are overwhelmed with, as soon as we discuss human matter. The fact that solid regularities are observed from statistics in many human activities allows the postulation of the existence of simplicity beyond some part of complex human behavior. At least, my approach is founded on that statement. My “humans” will be very simple agents with a rather restrictive spectrum of individual features. Yet, they will be shown to exhibit complex and nontrivial behavior that provides novel insight into human complexity. The world of atoms is much more exciting and richer than previously thought up to the middle of the last century. Physics could provide an efficient framework in which to embrace a series of human puzzles. My paradoxical anchorage is that learning from atoms could provide substantial enlightenment for discovering the mechanisms behind some intriguing features of human collective organizations. Maybe human societies share common ingredients with the far less complicated entities involved in inert matter. Bridging the two extremes of known complexity, that of humans versus that of atoms, could turn out to be rather productive. Such a path provides a “trick” to be able to bypass our anthropocentric feeling of being superior and more complexly elaborated than everything else. Nevertheless, it should be emphasized that the whole approach does not pretend at a complete and precise description of human behavior. It aims at shedding

xvii xviii The Reader’s Guide to a Unique Book of Its Kind a new counterintuitive light on some basic problems of our social and political organizations. This book is therefore rather unusual in both its content and its presentation, not mentioning the style. This is why a short guide is necessary to state the general framework as well as the spirit behind the presentation. It deals with new materials, new subjects, new results, new issues, and in particular on a possible new paradigm with which to envision human behavior. The presentation is also unusual for a scientiﬁc book, in providing an initiation in carrying out research.

What It Is About

The main content is about sociophysics (Fig. 2). But, what is sociophysics? It is the use of concepts and techniques that are taken from statistical physics to investigate some social and political behavior. The topic does not aim at an exact description of the associated reality with respect to all its details. It focuses at singling out some basic mechanics which may be rather counterintuitive and in turn shedding new light on our otherwise taken for granted best sets and frameworks for political organizations. A few decades ago I envisioned the logical birth of the ﬁeld of sociophysics stemming from the recognition of the high level of development of theories in condensed matter physics combined with a predictable increase in a physicist’s frustration of being conﬁned to the study of the world of inert matter. I wrote several papers to support this statement. Most of the predictions came true and some are still waiting to do so. In particular I predicted that after the development and success

Fig. 2 Indeed, what is the book all about? The Reader’s Guide to a Unique Book of Its Kind xix of the “invasion” of other fields by physics, the phase we are entering currently, a counterreaction would form within the invaded fields to expel part of it, rather like in a decolonization process. However, I do hope that the result will be more successful than the case for countries. The paper titles are rather explicit for most of them with “Physicists as a revolutionary catalyst” [1], “Les physiciens et la frustrations des electrons”´ [2], “Sauver la nouvelle Byzance” [3], “Physicists are frustrated” [4], “Misere` des physiciens” [5], “About imperialism of physics” [6], “Should God save the queen?” [7]. I simultaneously wrote two foundational papers to establish the field as such and not just as single one shot contributions. The first one is in French “Entropie, desordre´ et liberte´ individuelle” [8]. It deals with the very ancient and puzzling paradox that arises from the principle of maximum entropy and the existence of life that is perceived as an ordered state as opposed to the ultimate fate of supposedly maximum disorder. The second one “Sociophysics: A mean behavior model for the process of strike” was written in collaboration with Yuval Gefen and Yonathan Shapir today at the Weizmann Institute and the Rochester University respectively [9]. It is a manifesto for sociophysics. In addition, it addresses the question of the occurrence of strikes in the working world, revealing some of the mechanisms behind several paradoxical facts that can be observed in strike phenomena.

Main Material from Peer-reviewed International Journals

The book is built from two different, but intricately connected parts. One is scientific and the other is philosophical. The scientific part deals with sociophysics itself, i.e., the construction of models, the discussion of their validity and their confrontation to real events. Most of the enclosed material has already been published in high ranking international research journals having international peer review procedures over a period of almost 30 years. It includes my original models as well as those made in collaboration with a few colleagues. This explains why not too many references are given to other research papers published in the last 15 years during which the field of sociophysics has gained the interest of quite a number of physicists around the world. The book also incorporates novel unpublished material, which is, however, derived directly from the published models and results. Many details and extra calculations are given which are otherwise out of the scope of a research paper.

Philosophical, Ethical, Epistemological, and Political Issues

The philosophical part of the book addresses the issues of ethics, morals, epistemol- ogy, and politics which are intimately associated to the very concept of sociophysics. xx The Reader’s Guide to a Unique Book of Its Kind

The goals and the approach of sociophysics could result in drastic changes in our lives, in particular if it turns out to be successful in its ambitious program. The question of the corresponding potential social dangers is not avoided. I am not positioning myself as an irresponsible scientist who is only excited by his ongoing research. I am fully aware that there exists a serious responsibility in engaging, implementing, and advocating the new field of sociophysics. I stand by it. Accordingly, these essential key issues for the future of sociophysics are addressed from the start of the book in the first chapter. Hints on how to neutralize possible misuses are given. No “stonewalling,” no “put-off.” These topics cover some very fundamental issues and although they have been much less developed and discussed in the literature, they are placed at the beginning of the book. At this stage, most of it is constituted from various comments and digressions, which express my personal opinion and do not reflect either a consensus among physicists working today in sociophysics or an elaborated framing for sociophysics, which indeed do not exist at present as a well delimited field. This is why several figures look like cartoons to emphasize that fragility.

The Whereabouts, the Injustices, and the Fate of the Creation of a New Field

I present a testimony about my more than 30 years of fighting and struggling to create the new field of sociophysics. I tell the whole story without any attempt to make politically correct statements and so emphasizing the numerous conflicts which have occurred. It is very rare to have access to the real story of the internal fight for the emergence of novel research, in particular within the so called hard sciences. I am undergoing this approach not just to advocate my particular case but for the sake of knowledge and the understanding of the universal character of the process, which, I think, is emblematic of most of those cases. Such real paths, which lead to the establishment of novel paradigms, are almost never known, as observed from the written idyllic logics found in most scientific textbooks. Once a breakthrough has been established and accepted, everyone, including those who have created it, wants to forget the initial tough rejection. Once the “disturbing and unacceptable” truth has been recognized as valid, its fringe and controversial scientist wants to become part of the established community. My story also demonstrates that scientists working at the challenging frontiers of cutting edge research are no better human beings than in any other field of social activity, having the same little arrangements to gain power, glory, and fame. The Reader’s Guide to a Unique Book of Its Kind xxi

Fig. 3 It sounds like real fun

Discovering the Fantastic and a Bit Scary World of Sociophysics

Once the questions surrounding the very nature of sociophysics have been clearly addressed, I can then define the guideline of the approach for setting the framework, the limits, and the first goals. To be able to understand, to describe and eventually to make predictions about social and political behavior and organizations is rather fascinating and exciting (Fig. 3). It also raises certain fears, with the real danger of becoming a “sorcerer’s apprentice.” I will first present some details of my personal itinerary in this field in creating the basis of sociophysics, so revealing the initial hostility of the physic’s community. Then I will comment on from my own experience of how paternity recognition is far from being natural in science as in other human activities. The real history of the first stages of sociophysics provides a very nice illustration of how sometimes the building of history can turn out to be misleading and even to falsely create politically correct paradigms at the price of modifying the truth.

An Initiation in Carrying Out Some Research “alafac` ¸on” of a Physicist

This book has two unique features, the first one being the subject and the second one being the manner in which the subject is presented. Instead of going through a classical presentation of the field of sociophysics, I implement it through an xxii The Reader’s Guide to a Unique Book of Its Kind initiation in carrying out some research as, using the same approach as a physicist. To develop the field of sociophysics, not only tools and concepts from physics are not just necessary, but even more important is to learn how to tackle a given problem, how to model it, step by step with an ongoing critical review of the results obtained. This method is designed for both nonphysicists, who want to become familiar with the sociophysic’s modelling of societies, and graduate physics students, who can learn how a physicist’s mind works. It is worth emphasizing that the content of the book does not require any preliminary knowledge of physics. Only very basic elementary mathematics is necessary to follow the developments presented in the book since all details are given, together with explanations on how and why.

The Physicist’s Corner

From time to time, depending on the particular problem being developed, I open a “physicist’s corner” to emphasize and enlighten what made the physicist’s thinking so speciﬁc, which in turn has allowed so many great discoveries. It contains illustrations of the “magic” and fantastic power of playing with mathematics, models, and experiments in order to reach a realistic goal.

Each Chapter can be Read Independently (Almost)

The content is divided into two parts. The first one contains no equations and focuses on explaining, positioning, arguing, telling stories, discussing epistemol- ogy, and revealing the real history of sociophysics. The second part thoroughly investigates one particular subject of sociophysics, the paradoxical effects of using majority rule voting in bottom-up democratic hierarchies. In particular a model for democratic dictatorship is set up to demonstrate the negative side effects of using some a priori very positive social procedures. Here, equations and explanations are mixed. To be more accessible, each Chapter is self-sufficient and can be read separately. Accordingly, some equations, graphs, and references are repeated from one Chapter to another, to make each Chapter independent. But a reading of all the Chapters in their natural order will be more beneficial for gaining a better comprehension of what sociophysics is, and to lead to a clearer understanding of how it works (Fig. 4). References xxiii

Fig. 4 The reader’s guide

References

1. S. Galam, “Physicists as a revolutionary catalyst”, Fundamenta Scientae 1, 351 (1980) 2. P. Pfeuty and S. Galam, “Les physiciens et la frustrations des electrons”,´ La Recherche JulyAugust, 23 (1981) 3. S. Galam, “Sauver la nouvelle Byzance”, La Recherche Lettre 127, 1320 (1981) 4. S. Galam and P. Pfeuty, “Physicists are frustrated”, Physics Today Letter April, 89 1982) 5. S. Galam, Misere` des physiciens, Pandore 18, 57 (1982) 6. S. Galam, “About imperialism of physics”, Fundamenta Scientiae 3, 125 (1982) 7. S. Galam and P. Pfeuty, “Should God save the queen?”, Physics Today Letter October, 110 (1983) 8. S. Galam, Entropie, desordre´ et liberte´ individuelle, Fundamenta Scientiae 3, 209 (1982) 9. S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A mean behavior model for the process of strike, Journal of Mathematical Sociology 9, 1 (1982)

Part I Sociophysics: Setting the Frame Chapter 1 What is Sociophysics About?

Sociophysics deals with one of the most ancient of human dreams, which is simultaneously its nightmare, i.e., the capability to predict and thus to control human behavior. As is often the case, science ﬁction creates what science will be able to achieve in the future, preceding in part the reality to come. But while succeeding in embodying some of the features of the future, it usually fails to grasp its more fundamental aspects, which is somehow comforting as regards our nonprewritten destiny, which always has many uncertainties.

1.1 Science Fiction and Asimov’s Foundation Syndrome

A few years ago, after giving a lecture about some of my work on sociophysics, once the audience’s questions were over, on my way out of the auditorium, someone came to me with big shining eyes to ask me if I knew about Hari Seldon, his psychohistory, Foundation, and Asimov’s trilogy? This scenario repeated itself a few times in a row. Each time I was surprised and my answer was polite but somehow negative. I had never heard about this Asimov and his program before, and I think I was a bit reluctant to consider that I might be practicing some science fiction-like ideas. But all these people were telling me in very convinced tones and with great enthusiasm that I should read the novels, since somehow I was doing what the novel’s main figure, Hari Seldon, does in the Asimov saga. According to my questioners, my sociophysics was identical to his psychohistory. One more step and I would have become the incarnation of this novel’s figure; this was a strange feeling for a physicist whose aim was and still is to create a solid science of social behavior. To appear as a novel’s character was thus a disturbing threat that risked jeopardizing the seriousness of my approach. Just after these repeated events I went to America to an International Conference and I decided that it was time to check out what was behind these strange and

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 3 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 1, © Springer Science+Business Media, LLC 2012 4 1 What is Sociophysics About?

Fig. 1.1 Hari Seldon is watching me

intriguing statements about both my work and my identity. I went to a large bookstore, wearing large wrap-around sunglasses to buy incognito Asimov’s first Foundation trilogy (Fig. 1.1). After an exciting read, I ended up being a bit disappointed, but secure as regards my own construction. First, I understood why these people were making the analogy with me. Then I identified what was similar, indeed very little, and what was different, the main part. However this comparison turned out to be very fruitful since it allowed me to identify the possibilities and the impossibilities associated with the IDEA of quantifying human behavior. For the sake of both curiosity and clarity, let me first review in short what The Foundation is about. Isaac Asimov is a well known and successful science fiction author who has created many interesting and intriguing worlds that could possibly come true in the future. Among the numerous books he has written stands the trilogy called “Foundation” with Foundation [1], Foundation and Empire [2], and Second Foundation [3]. The central figure is a fictional character called Hari Seldon, a mathematician living in some Galactic Empire. In the course of his research, he envisioned a new field from mathematics that he called “psychohistory.” The radical content of this novel theory was to embody the possibility of predicting the Galactic Empire’s future. Thanks to the size of its full population, consisting of billions of individuals, psychohistory was supposed to be able to make predictions using an analogy with the statistical theory of gases. At first, Seldon had only the formal proof that his program was feasible in principle, but he had no clue as to how to implement the theory in practice. He emphasized this limitation while publicizing his theory at a mathematical conference. Indeed he was pessimistic at that stage about the fact that the practical 1.2 What Makes Sense and What Does Not with Asimov’s Fictional Psychohistory 5 aspect could not be resolved in the near future. The technical difficulties were tremendous, even if the general theory was mastered. His incredible program was not yet near to becoming a real scientific field. However, already some politicians became interested and / or scared by the power of the theory. They were proven right since later in the novel, Seldon succeeded in overcoming the technical barrier that made it possible to turn the theory into an efficient predictive political tool. With the capacity to develop psychohistory into a practical set of equations, he was able to predict the Empire’s future in terms of probabilities. His prediction was the eventual fall of the Galactic Empire. The whole saga follows from that statement which appeared to be accomplished later in the novel.

1.2 What Makes Sense and What Does Not with Asimov’s Fictional Psychohistory

Of course Asimov’s psychohistory is a fiction created by a science fiction author and therefore it could seem out of place to discuss it in a “serious” scientific book. However, it is not superfluous to comment on it since sometimes science fiction, like art, is capable of picking out some features of what will become reality later on. Obviously, not every dream becomes reality, but some do, at least in part. Simultaneously, some realities produce dreams. Let us thus examine what makes sense and what does not in Asimov’s vision of the role of mathematics in the course of collective and individual human lives.

1.2.1 Predicting the Future Is Not Possible

Using an analogy with the statistical theory of gases, psychohistory is supposed to be able to predict the future of the whole population of the Galactic Empire consisting of billions of individuals. It relates to the fact that dealing with large numbers of particles results in the averaging out of the initial conditions to result in the existence of macroscopic laws which govern the collective behavior of the ensemble of particles. This is the way it works for gases. The question is then (Fig. 1.2): Given the hypothesis that it was possible to collect all the information concerning the individuals of a large population, would it become possible to describe its future using some macroscopic quantities and their associated laws of evolution? In other words, imagining it being possible to store in a huge computer at a given time all the individual characteristics of every person from the whole of the world’s population, knowing in addition their laws of interaction, does this make it possible to calculate, and thus predict, their evolution with time, i.e., their future, their history to come? 6 1 What is Sociophysics About?

Fig. 1.2 Predicting the future as against predicting an event: it is not possible to predict the future but sometimes it is feasible to predict an event

Although the question sounds right, the answer is “no,” it will never be possible to calculate their future however powerful the computer and however large is the quantity of personal information collected for each person. What is erroneous is the mere idea of predicting the future of a population, the future being taken in terms of its coming history which implies a large spectrum of levels and contents (Fig. 1.3).

Fig. 1.3 Science ﬁction versus science: it sounds right, it is almost right, but it turns out to be wrong

The analogy with a gas does not hold since in physics the control of the evolution of a gas presupposes that the gas is contained in an enclosed space and constrained by known external parameters. In contrast, an assembly of people can perform a large number of activities that are carried out under unknown external conditions. 1.2 What Makes Sense and What Does Not with Asimov’s Fictional Psychohistory 7

However, the analogy can be sustained if the range of possible activities by a group of people is reduced to just a few issues and occurring over a ﬁnite interval of time, focusing on one particular event. Then prediction becomes possible at least in principle and under certain conditions. It is precisely in this context that sociophysics could be developed. It is far less ambitious but yet an incredible challenge. A society is not a perfect gas. Even most of inert matter is not a perfect gas. Physics does not predict the evolution of all inert matter. It separates things, isolates levels, controls the external conditions, and deﬁnes orders of magnitude in time and energies to discover the appropriate laws relevant to each subcase of the incommensurate number of available atoms.

1.2.2 Historical Vs. Ahistorical Sciences

To elaborate on the respectively different natures of historical and ahistorical sciences is out our scope here. Nevertheless let us mention the essential separation between them. The first ones are concerned with unique objects, such as the universe, the planet earth, its climate, myself, you, and so on. The second group allows a hard science approach since it concerns a collection of identical objects whose past does not matter to its evolution with time. The history becomes irrelevant and thus the future cannot be defined as such. The social and political phenomenon driven by the so-called global warming is a perfect illustration of these difficulties, impossibilities, and wrong statements since although climatology is not a hard science, it is instrumented by climatologists to make unfounded and usually catastrophic predictions, claimed to be scientifically proven. I wrote a book on this “hot” subject. Unfortunately, at present (Spring 2011), it is available only in French [4]. And even more unfortunate, I have the feeling it will stay so. I also published a research paper in the international journal of physics, Physica A, about the dynamics of public opinion related to several issues driven by incomplete scientific data such as that involved in global warming, evolution theory and H1N1 pandemic influenza [5]. To conclude, I do not think history could be predicted even in principle, given our current tools of research and perception of the world. But it is worth stressing that predicting the collapse of an empire is not predicting the future. However localized in time, events may well be predicted. It is unfortunate but necessary to clearly state that what could count as a “global science,” which embraces multiple aspects of different sciences at various scales to study a given phenomenon such as the climate, still does not exist. For instance, physics is not a global science, with its well separated fields of knowledge that include high energy physics, nanophysics, condensed matter, astrophysics, and others. 8 1 What is Sociophysics About?

1.2.3 Small Groups, Large Groups, Not so Simple

Statistical physics deals with averages that make use of thermodynamic limits. Because the number of objects tends toward infinity, an essential ingredient is to ensure a correct mapping between the various theories and the corresponding experiments. Such a procedure is necessary to average out microscopic fluctuations. But in some situations fluctuations turn out to be essential, like in the so-called phase transition in which a system changes its macroscopic order from one organization into another. For our purpose, what matters is to underline the fact that increasing the number of particles reduces the importance of the initial conditions, making the evolution independent of them. For a large enough system, whatever the initial conditions the equilibrium state will be unique. This situation does not apply to the case for a small number of particles. Here stands the mathematical basis behind Seldon’s requirement for a huge population in order to get his psychohistory to work, which is indeed the case with the Galactic Empire. However, the larger a system is, the more probable is the occurrence of an extremely rare local event which can totally destroy the otherwise predictable and normally accomplished evolution. Asimov actually introduced this ingredient with the appearance of a “mule” in his saga. And as expected, this “mule” seriously shook all of Seldon’s previsions. Therefore, this feature of the Asimov panoply makes sense. I encountered this phenomenon while developing some modeling of opinion dynamics. Extending my work on democratic dictatorships [6, 7] to opinion dynamics with a group of Swiss colleagues, Bastien Chopard, Alexandre Masselot, and Michel Droz, we found that the opinion formation obeys threshold dynamics [8]. For two competing opinions A and B, an unstable threshold determines a full range of values for initial supports for which an A victory is guaranteed through a public debate and others for the B opinion victory. Later on, performing large simulations with Bastien Chopard and Michel Droz, we were surprised to discover the existence of what we called “killer geometries” [9, 10]. Given a reaction/diffusion like model it appears that some peculiar local geometries could emerge with the astonishing property of having a nonzero probability of invading the whole population. A few elements adequately organized can thus win against a whole population. They do not even constitute a small minority, being simply a few individuals. With Jan Radomski, a Polish colleague, we then extended this above finding to propose a mechanism for cancerous tumor growth. In particular, we emphasized that while such an occurrence is extremely rare, it becomes very probable after a long period of life. Our paper is entitled “Cancerous tumor: The high frequency of a rare event” [11]. From our model, natural survival from cancerous tumors, i.e., using the body’s natural defenses, becomes random after a certain age. At the opposite extreme, at younger ages, survival to cancer cells is almost certain. However, these conclusions were not extracted from real data but only from a model, which is far from being close to the real evolution of cancerous tumors over time. 1.2 What Makes Sense and What Does Not with Asimov’s Fictional Psychohistory 9

It is therefore of central importance to underline that deterministic laws can perfectly exist with the possibility of “mutants” which eventually break up the prediction. The above example of a dynamics threshold with the existence of killer geometries is a good illustration, which could also provide a hint in helping to understand lobbying effects.

1.2.4 A Mathematician Could Not Have Done It, a Physicist Could, Maybe

If the “mule” event made sense, the character of the main figure of the novel does not. Choosing a mathematician to create psychohistory is not a valid setting for the fiction, in the case where one tries to take it as a solid scenario. To make my point, and to be a bit more provocative, I would say that a mathematician could not have invented physics. By a mathematician, I mean a mathematician doing mathematics. Of course a mathematician becoming engaged in another field is a different matter, but then he or she would no longer be a mathematician. To support this surprising argument, I could mention the mathematicians working in economics, who keep playing with mathematics and not much with economic reality. If necessary, the 2008 financial and economic crisis could be used to illustrate my point. Accordingly, if a mathematician could not have created physics, and if mathematicians have created the field of economics, which is so far from the real economy, how could a mathematician come up with a psychohistory-like theory which will have a connection to real history? Moreover, designating history as a social field that can be quantified is also absurd. By its very nature, history does not allow a predictive methodology since it is the unique history of a unique system. I do not want to be misunderstood. I am not taking a contemptuous posture toward mathematicians. Without them, there would be no physics. I simply want to stress the radical gap that exists between the respective frameworks of the minds of mathematicians and those of physicists. Physicists use mathematics as a very useful and powerful tool to investigate the hidden laws of inert matter. What matters is reaching a conclusion that corroborates with experimental results. If, while building up a theoretical model to explain some phenomenon, mathematical rigor gets in the way and prevents the reaching of the required result, the physicist will have no “moral” or professional qualms in making up a mathematical hypothesis as long as it allows the bypassing of the difficulty. On the contrary, often a physical explanation will be created within which the mathematical distortion is given an explanation that has yet to be validated. To a physicist’s mind, what prevails is the result, not the mathematical rigor of the demonstration (Fig. 1.4). The experimental level always prevails over the mathematical level. I could formulate the difference in another way by stating that given a problem, for a physicist, what matters is to find some minimum conditions in order to obtain a solution. On the opposite side, for the mathematician, what matters is to prove the universality of the solution. 10 1 What is Sociophysics About?

Fig. 1.4 Who could eventually predict social and political events? Mathematicians cannot. Physicists might be able to

This is why only physicists could apply mathematics to describe another field of the real world in a quantitative way, analogous to what they do in physics. Neither chemists nor biologists could do it because both of their fields deal with the modification of the nature of their objects. In the study of human behavior, humans are assumed to remain unchanged, as with atoms. When mentioning mathematicians and physicists, I am not concerned about the education or knowledge people may have but about their cognitive framework for tackling a problem. In principle, a mathematician has a very different cognitive nature than a physicist and vice versa.

1.3 A Parenthesis on the Fall of Empires

The core of Asimov’s trilogy is the prediction by the mathematician Seldon, ahead of time, of the fall of the Galactic Empire using his psychohistory, i.e., a set of equations. Taking his prediction as a solid fact and considering that it could not be avoided, he engages in a race to save all of the existent human accomplishments in order to shrink drastically the time that would normally be required to rebuild a new and better universe. Instead of the new humanity starting from scratch, Seldon envisions having delivered to it all of the accomplishments of the previous humanity. What a noble cause, isn’t it? But for me, it is the prediction content which is the most interesting. It is amazing to see that the prediction of the fall of an empire is considered as a dangerous scoop, a subversive fact that must be kept top secret. It is indeed incredible to notice that while through known past history every and each empire has fallen, every and each new empire is convinced that it is eternal, with the self-certainty of constituting the ultimate state of history and the fate of existing forever. People tend to forget that history demonstrates that every empire’s fate, whatever the height of prosperity it achieves, is to fall, collapse and then to disappear rather quickly (Fig. 1.5). 1.4 Time for New Paradigms 11

Fig. 1.5 The destiny of any empire is to eventually fall

Any empire is going to fall at some time; that is the remarkable evidence, which is, however, always forgotten while living in any particular empire. Such a situation could result from the fact that to sustain respect toward an institution’s authority, the belief in it lasting forever is a prerequisite. Although one can be certain that one day the institution will fall, the calculation of the particular date of its fall is always unknown. Knowing the date in advance would clearly offer plenty of strategic advantages to the person possessing the information. Such a hypothetic possibility would provide tremendous consequences and advantages to the ones who knew.

1.4 Time for New Paradigms

The idea in pointing at connections between on the one hand, collective phenomena in physics, and on the other, collective behavior in human systems, appears today as being rather appealing, almost as a natural link. That was not the case 30 years ago when paradigms for physics and human societies were so far apart. Nevertheless, in the late 1970s, physics was the very unique ﬁeld that was able to tackle theoretical and experimental challenges arising from the existence of collective phenomena. A great deal of new materials yielding exotic features have been discovered and understood, thanks to the development of the appropriate concepts and techniques. The physics of collective phenomena is now a well 12 1 What is Sociophysics About?

Fig. 1.6 Our present “glasses” are insufﬁcient to see the world

established and flourishing field of research, particularly in its massive and smart use of the enormous developments in simulation capabilities triggered by the ongoing increase in the power of computers. The field of collective phenomena is a major ingredient for understanding inert matter, and in turn for discovering novel and unexpected properties such as the so-called high Tc superconductors, which opened the way to dream about the possibility of building superconductors at temperatures not too far from room temperature in the near future. The massive recent development of nanotechnologies is another example. Nowadays, globalization has made economic, financial, social, political, and religious collective phenomena the focus of the most striking world events. This state of affairs demonstrates the richness as well as the many different fears associated with essential contributions to all new developments at every level of human society. And yet many innovations are still to come. The supremacy of collective phenomena nowadays is often perceived as being negative, preventing the search for one’s own personal accomplishment. Individualism does not seem to get along with the reality of world collectiveness. You find the same shops, the same garments, and the same food all over the world. At the end of the last century, we shifted from a world of stability to a world of chaotic and probabilistic dynamics with incessant changes in the collective and individual reality of everyone’s lives. These features are usually perceived as catastrophes. Many would like to go back to the “old” world to suppress all the surrounding and internal disorder and to recover the happiness of an ordered world and life. But to go back to a past which no longer exists would be very counterproductive.Indeed, we should highlight the richness and advantages of chaos and disorder in order to exploit it in every part of our lives. However, to implement such a counterstatement, we desperately need to create new tools and new paradigms so as to apprehend collective phenomena as part of our basic resources for the future (Figs. 1.6 and 1.7). All our concepts and ideals are outdated, with the most recent ones coming from the nineteenth century while our world has drastically changed. The physics 1.5 Sociophysics, a Possible Novel Hard Science: Not a Zero Risk Path 13

Fig. 1.7 We need appropriate “glasses” to see the full world

of collective phenomena is a promising reservoir for innovative concepts and representations for addressing the tremendous new challenges facing our worldwide society. This book lays the groundwork for such a program, setting the foundations for sociophysics. It is a contribution to why and how the physics of collective phenomena can provide new keys to the global political and social world. Therefore, we address the fundamental question of “do humans behave like atoms?” to which the answer is “yes in certain parts of social and political collective behavior” and “no in other aspects of individual behavior” (Fig. 1.8).

Fig. 1.8 The book’s basic DoDo humanshumans bbehaveehave llikeike aatomstoms question

ThatThat isis thethe BIGBIG questionquestion

1.5 Sociophysics, a Possible Novel Hard Science: Not a Zero Risk Path

However, after making such a self-conﬁdent statement as stated above about the feasibility of quantifying some of the human collective behavior, one should immediately restrain the excitement with a fundamental basic note of caution. 14 1 What is Sociophysics About?

Fig. 1.9 Not a zero risk adventure

While the need for new concepts is clear, and the temptation to pick up new paradigms from the physics of collective phenomena is rather attractive, it may turn out to be both misleading and dangerous. Such an approach should indeed be carefully controlled (Fig. 1.9). To just map a physical theory built for a physical reality, onto a social reality, could at the best be a nice metaphor, but without the possibility of predicting anything, and at the worst, lead to a misleading and wrong social theory. It may as well become a false argument of authority in order to impose political views. Since it is immediate and easy to develop a series of a priori metaphoric mappings between on the one hand, magic words and pictures from physics, and on the other, problems and issues in the social world, it may be very tempting to create an artificial appearance of scientific arguments to impose subjective and political views in the name of science. This idea could lead to dangerous social manipulations. I could mention here again as an illustration the global warming phenomena about which climatologists claim to have scientific proof but which has indeed become a political view of the world. It is a mere vision, that is politically correct and fits to the desire for order to recover the illusion of understanding and controlling of the world’s development through the illusion of mastering the evolution of planet Earth (Fig. 1.10). Our task is to borrow from physics those techniques and concepts that can be used to build a collective theory of social behavior, but within the specific constraints of the psychosocial reality. Another danger for the physicist is to stay in physics, using a social terminology and a physical formalism. On this basis, one should be aware of such dangers and account for them while becoming engaged in this new and exciting approach to the description of social phenomena. 1.5 Sociophysics, a Possible Novel Hard Science: Not a Zero Risk Path 15

Fig. 1.10 A rather promising path

The contribution from physics should thus be restricted to guidelines for the modeling of social realities. Such a limitation does not make the program less ambitious. On the contrary, it opens very solid perspectives for efficiently tackling the current ongoing process of globalization. In fact, it may even oppose social manipulations. In addition, given the level of current and future world problems, our goal is twofold; firstly to account for the inherent risk to sociophysics of keeping a constant effort to maintain it at a minimum, and secondly, to proceed in the elaboration of a new pair of “glasses” to improve our view at the world scale. On top of the above risks, and once it has been decided to take responsibility for them, another difficulty arises; that of the ethical issues. These issues have to be addressed before taking the proposed path. It is rooted in the very nature of the sociophysics approach and in particular, to its fate. After some amount of research and work, sociophysics will eventually be proved to be either wrong or right. In the first case, it would have been worth trying it, and then the whole thing would have to be forgotten as being just another unfounded human dream or nightmare. That is all. Moreover, all the people who were scared of the adventure from the beginning, would be reassured and reinforced in their conviction that mankind does not obey an equation. It would be an indirect proof that human beings really do enjoy individual free will and are free to decide their collective fate. In short, human beings and societies have nothing in common with atoms and inert matter. Human superiority would thus be reinforced by the failure of the attempt. On the contrary, in the case of the second possibility, of sociophysics being proven to be well founded, the core hypothesis stating that human beings at a social level do obey “natural laws” in the same way that atoms do, would be validated. Therefore, besides the shaking of the human dream of superiority over other species, it could, at least in principle, pave the way to possible political manipulation. Instead of using wrong arguments, lies or coercion, it could be enough to apply a set of equations to control a population in a subtle and invisible manner. It would validate the myth of “Big Brother” with all its underlying fears (Fig. 1.11). That is the pessimistic conclusion. A more cheering alternative also exists. 16 1 What is Sociophysics About?

Fig. 1.11 Physics may provide new tools to help us catch up with the twenty-ﬁrst century

1.6 Discovering the Limits of Human Freedom Opens the path to Social Freedom

Within the above contradictory possibilities, on the one hand is the optimistic hypothesis of the possibility of a hard social science, and on the other, the pessimistic idea of the lack of social freedom. The question that immediately arises is, who would be the “Big Brother” and for what purpose would it be used for? The question is formulated in a unique way but the answer is plural. If the theory is proven to be true, it could be applied for the benefit of various social and political groups. It sounds indeed very frightening, but fortunately while the objection is solid and legitimate, it is effectively socially irrelevant. On the contrary, proving that social freedom is an erroneous myth would open the path to achieving an effective freedom (Fig. 1.12). Today many very sophisticated and efficient tools are available and used to influence people, such as in advertising, to customize consumers and voters. There exists a well developed empirical knowhow for activating social and economic manipulation, that will become more and more refined with ongoing improvements in performance. But this apparatus of knowledge and tools is not known to most of the people who are being influenced. Manipulating people while they are convinced of their freedom and freewill renders hopeless any attempt to oppose it. Control and conditioning are already very active, although within both an informal scheme and a wrong social representation of social freedom. 1.6 Discovering the Limits of Human Freedom Opens the path to Social Freedom 17

Fig. 1.12 Today, manipulation is very efﬁcient but everyone believes in spontaneous freewill

Singling out objective laws of human behavior will clearly set borders to the existing, although invisible, limits of human freedom. Simultaneously, it will open up our minds to the ways in which we are being manipulated by both external forces and current wrong conceptions. Understanding our self-limits and their mechanisms will produce a solid framework for overcoming the limitations, and thus create the possibility of more freedom. To illustrate my argument, let us look at another similar problem which occurs in the physical world: that of gravity and of the “freedom” for man to fly. For many centuries, men dreamt of being able to fly like birds do. Mixing all kinds of convictions, esoteric beliefs, and ad hoc devices, many attempts were undertaken to do so, with always the same dramatic failure of crashing to the ground. At some point, that dream of flying had to be given up with the discovery of the law of gravity, which yielded the scientific proof of the impossibility for men to fly. Due to the attraction from the earth, men were forced to stay stuck on the ground (Fig. 1.13). However, this restricting proof opened the way to overcome this fundamental limitation to our physical freedom of movement. Knowing exactly why man cannot get off the ground with the singling out of our limits, has allowed the design and the production of airplanes with the capability of flying. It has been the understanding of the natural limits set to our freedom of movement, which in turn has driven an increase of that freedom pushing away the corresponding limits. On the contrary, having the illusion of a freedom, which does not exist, is misleading and sometimes could turn dangerous. Along this line, it is the finding of eventual limitations to our free will that will eventually allow us to overcome them. It is my position here to assume that the singling out of the eventual social laws which constrain our free will is the prerequisite to the setting of more social freedom. The subtle assumption behind my 18 1 What is Sociophysics About?

Fig. 1.13 Science could single out what restrains our freedom, besides the physical limitations philosophy is that it is paradoxically the very existence of individual freedom which in turn produces the collective lack of freedom by implementing wrong conceptions for the building of social organizations (Fig. 1.14).

! ?

Surface of the earth

Fig. 1.14 Once the limitations to our freedom have been discovered, science could at least, in principle, shake up the framework that reduces these very limits References 19

References

1. I. Asimov, “Foundation”, ISBN 0-553-29335-4, Gnome Press (1951) 2. I. Asimov, “Foundation and Empire”, ISBN 0-553-29337-0, Gnome Press (1952) 3. I. Asimov, “Second Foundation”, ISBN 0-553-29336-2, Gnome Press (1953) 4. S. Galam, “Les scientifiques ont perdu le Nord, Refexions« sur le rechauffement« climatique”, Plon, Paris (2008) 5. S. Galam, “Public debates driven by incomplete scientific data: The cases of evolution theory, global warming and H1N1 pandemic influenza”, Physica A 389, 3619 (2010) 6. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, Journal of of Mathematical Psychology 30, 426 (1986) 7. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, Journal of Statistical Physics 61, 943 (1990) 8. S. Galam, B. Chopard, A. Masselot and M. Droz, “Competing Species Dynamics”, European Physical Journal B 4, 529 (1998) 9. B. Chopard, M. Droz and S. Galam, “An Evolution Theory in Finite Size Systems”, European Physical Journal B 16, Rapid Note 575 (2000) 10. S. Galam, B. Chopard and M. Droz, “Killer geometries in competing species dynamics”, Physica A 314, 256 (2002) 11. S. Galam and J. P. Radomski, “Cancerous tumor: the high frequency of a rare event”, Physical Review E 63, 51907 (2001) Chapter 2 The Question: Do Humans Behave like Atoms?

The analogy, if any, between men and atoms is discussed to single out what can be the contribution from physics to the understanding of human behavior. The basic assumptions of the approach of sociophysics are formulated together with the methodology used to tackle a given problem taken from the social and political worlds [1, 2].

2.1 My Basic Philosophy

Once ethical questions and feasibility have been clarified, we can proceed in making our scheme for building a new approach to human behavior more precise, including the psychological, social, economic, and political aspects. It should be strongly stressed that the goal is not to substitute a physical view to all aspects of human life. As a first basic step, we aim to bring to light the very plausible existence of quantitative laws which govern human behavior. In a second step yet to come, sociophysics could turn into a quantitative science in order to discover the actual laws of social behavior in the same way that physics did so with the laws governing inert matter. It does not mean that the respective laws are identical. We are convinced that understanding the laws of human behavior can only be a benefit to human life and to humanity. But as mentioned above, in order to avoid a dangerous misuse, it is necessary to emphasize my methodology before elaborating on my different studies of social phenomena. The same procedure will always be used whatever the problem investigated. Thus, hypotheses and associated weaknesses will be systematically and clearly stated to set the limits of the model and the corresponding relevance to reality. Sociophysics does not claim to reach an exact description of a human group, but instead aims to shed new light on human phenomena, which are otherwise so complicated that any assumed truth is by nature misleading, and eventually wrong.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 21 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 2, © Springer Science+Business Media, LLC 2012 22 2 The Question: Do Humans Behave like Atoms?

Fig. 2.1 Current paradigms of society belong to the physics of the ninetieth century

Behind the use of the word “physics” in our so-called “sociophysics” stands the way physics proceeds in seeking to understand the laws of inert matter, rather than the use of the laws of physics themselves. This particular way relies on the modeling “a` la façon” of the physicist, the physicist’s way of modeling (Fig. 2.1). At the core of our underlining philosophy is the view that facing such a complex and complicated world such as that of human societies, only an over simplified vision of it can indeed capture a substantial part of its essential features. The initial step is to constitute the first essential presuppositions and basic foundational assumptions of a science that has yet to be set out. This statement could sound like an epistemological contradiction, but only in appearance, as will be clear later in becoming acquainted with the physicist’s frame of mind (Fig. 2.2). To proceed with this working hypothesis, the first main path to take is to create “artificial” worlds in which everything is controlled, i.e., in which all the ingredients are clearly defined, the laws of interactions are explicitly written and the range of variations of all the parameters are set, together with the definition of the eventual dynamics of evolution. Once all that is done, the associated collective properties can be studied to find out how “artificial” humans behave within the given “artificial” world. Once this procedure has been carried out, we will proceed in checking out if some of the discovered features, which are active within our “artificial world,” could mimic some of the features of the real world we aim to capture. In case it does work, we will not jump to the conclusion that we have discovered some “social truth” but we will suggest that the mechanisms used to describe the “artificial world” may be very similar to the ones behind the corresponding real social feature. We believe that it is from simplicity that theoretical complexities can appear in a way that allows subsequent understanding of the real and many complex 2.2 Telling the Truth About What Indeed Physics Is and What It Is Not 23

Fig. 2.2 Physics may provide useful novel paradigms to help us tackle the challenges of the twenty-ﬁrst century aspects of the social world. The permanent challenge is to build a model to perform calculations in order to obtain data and results, which in the second step can be compared to the real counterpart of the phenomenon, either via the setting up of experiments, as in physics, or via quantitatively observed trends from the social world. On this basis, the model is either validated or invalidated, with the awareness that we are building up a model by successive layers, each layer being an additional level of complexity to mimic the real system.

2.2 Telling the Truth About What Indeed Physics IsandWhatItIsNot

It is of importance to emphasize that contrary to the general belief, physics from the start, gave up the search for an absolute global and unique truth to describe inert matter. By “the start,” I am referring to the early origins of the modern scientiﬁc method set in the ﬁfteenth and sixteenth centuries which are now used in physics. Man as a subjective body driven by beliefs was separated from the performing of an experiment, which was required to yield the same results independently of where and by whom it is conducted. The establishment of an independent theoretical corpus was a key to the development of science in parallel to the experimental checks of its predictions. The genius of physics has been its success in separating various levels of the inert world in terms of energy and time scales such that it became possible to investigate the inside of each one, gaining in turn substantial insight to make solid 24 2 The Question: Do Humans Behave like Atoms?

Fig. 2.3 The iterative path of physics-like modeling predictions. Physics does not provide one single view of all aspects of inert matter. It discriminates between high energy physics, nuclear physics, atomic physics, condensed matter physics, astrophysics, cosmology, and more. While mastering aspects of each part of the underlying truth, the possibility of the full truth is de facto abandoned although some are still dreamt of, such as in achieving the so-called “grand unification” of the four forces. This well proven and powerful operative mode articulates via the design of so called “toy models,” which are by nature extremely crude with respect to the reality they address. But indeed, although far removed from it, they have been capable of exhibiting nontrivial results which have been shown to be shared by the corresponding real world. Once, a “toy model” is defined, solved and validated, it becomes possible to enrich it with additional ingredients in order to narrow the gap with reality and cope with it in a more realistic way. It is this very “ try and check” procedure which has led to the many breakthroughs accomplished in science (Fig. 2.3). We can illustrate the method by considering some system built from hypothetical agents denoted XY3Wz, which could be identified with their different attributes as shown in Fig. 2.4. These agents evolve in assemblies. If we wished to study their 2.2 Telling the Truth About What Indeed Physics Is and What It Is Not 25

Fig. 2.4 A system constituted with hypothetical agents denoted XY3Wz behavior, since they are too complicated to be solved at once, a physicist will start by greatly simplifying the object by selecting only one of the agent’s attributes. Such a procedure could lead to several different options, each one focusing on one particular attribute. Three different possibilities are shown in Fig. 2.5.Moreare possible. The simpliﬁed agent can then be placed in an assembly whose properties are investigated. This shows why different models are often invented to describe the same problem.

Fig. 2.5 The ﬁrst step to model the XY3Wz problem is to simplify the attributes of the agent so as to obtain a simpliﬁed version of the XY3Wz agent. Three different versions are given. At this stage, they are all on the same footing, i.e., very far from the real agent

In the second step, the simpliﬁed XY3Wz agent is used to study an assembly, as in Fig. 2.6. The results obtained are then compared to the available data from the real system. Most of the time, a substantial gap is found between each model and the real system. 26 2 The Question: Do Humans Behave like Atoms?

Fig. 2.6 Three different models of an assembly of the same XY3Wz system

On this basis an improved version of the agent is required. Usually, this step forward is achieved by adding one attribute to the simpliﬁed agent in order to obtain a more elaborate model. Figure 2.6 shows an example of adding one attribute to each one of the three simpliﬁed agents. The above process can be repeated, combining results from the different models so as to eventually reach a single model which is closer to reality (Fig. 2.7).

Fig. 2.7 One attribute is added to each one of the three simpliﬁed XY3Wz. Then crossing the various models, a solid one can eventually be obtained which includes all the attributes of the XY3Wz agent 2.3 Physics Does Not Care About Mathematical Rigor 27

2.3 Physics Does Not Care About Mathematical Rigor

At this stage, we need to emphasize one additional fundamental feature of the nature of physics. While the use of modeling in physics has been tremendously powerful in establishing the field as an exact hard science, capable of building concrete and efficient experimental devices, its power comes from the empirical use of mathematics to describe real phenomena. This means that it is not the mathematical rigor that prevails but the capability to reproduce particular properties using some mathematics. It is the exact opposite of what economists have been doing for decades, who have focused on the mathematical rigor of their model rather than their ability to reproduce real features. Another essential characteristic of physics is that all the results obtained from the various models are aimed, sooner or later, at being tested against experimental data, even if it takes many years or decades or even centuries before being able to do so. Also, if an agreement between theory and experiment clearly validates a model, discrepancies are used not to invalidate the whole model but only part of the model’s hypotheses. Physics is a so-called hard science but it balances between the hard reality and the rich possibilities of inexact mathematics. The net result of these “arrangements” taken with mathematical power has led to the building of a solid and extended core of understanding of the laws governing inert matter. This is why while discussing the “reality” of Asimov’s psychohistory we emphasize that only a physicist could create it, not a mathematician. Within my sociophysics approach, I am going to follow the same procedure, modeling social reality with an empirical use of mathematics, giving up rigorous aspects, in favor of the capability of being able to reproduce some particular phenomenon. Behind such an apparent lack of rigor lies the basic difference between mathematics and physics. The former focus is on proving the universality of a result, whereas the latter one aims at finding some minimum conditions which are able to reproduce a given observation. However, at the current initial level of our novel approach, we are not yet dealing with what could constitute the associated experimental apparatus of sociophysics. We will first need to elaborate what could be the equivalent elements of the limited experiments performed in psychosociology before even considering larger scale experiments. Dealing with human beings is of course not the same as experimenting with some piece of inert matter. At the same time, this ethical prudence can turn out to be dangerous in the final assessment of a social law derived from our models. This is why we should always be cautious about the findings from a model. At this point, we are not going to state definite laws, but instead, to set out a new framework of investigation in order to be able to gain a coherent and different view about social and political behavior. In a second step in developing our approach, the model predictions have to be checked against real experimental data to validate the coherence of a model’s hypotheses. A whole new framework could thus emerge and gradually be built through the intricate combination of theoretical experiments and real life. It is a long and 28 2 The Question: Do Humans Behave like Atoms? exciting adventure to be engaged in, because so far, we have only just scratched the surface of this new field. What we have done is already a huge and very ambitious epistemological step.

2.4 Implementing a Physics-like Approach Outside Physics

We can now start embracing our main topic, i.e., why physics should be used for tackling anything else outside of inert matter. At this stage, it is worth emphasizing that it is of interest to focus on one particular ﬁeld, known as statistical physics. Indeed, thanks to its many achievements, the domain has already generated a lot of applications in many ﬁelds outside physics. However, the stakes are very different in respect to the applications in biology, economy, sociology, and politics (Fig. 2.8). In a very schematic manner, we can say that in biology there exists no major current interest in theory per se. It is essentially an experimental science. In the case of an experiment being suggested, on the basis of physics-driven ideas, or anything else, it could be performed. If a new result is obtained, the inspiration from physics takes second place and is forgotten. On the contrary, if no result is obtained, the physics is also forgotten.

Statistical Physics

Biology Who cares?

Economy and Finances Why not try?

Social and Political Sciences Very promishing Very dangerous Fig. 2.8 Applications of To be used with care statistical physics to biology, economics, social, and political sciences 2.5 But Indeed, Do Humans Behave Like Atoms? 29

With respect to economics, I would say, in a provocative manner, that nothing really works when confronted to economics data. In finance studies, a scheme derived from physics can be sometimes beneficial to certain types of investment. Many physicists have been hired in financial institutions for their general skills rather than for their use of physics. In particular, a field called “econophysics” has emerged and was developed in the late 1990s with hundreds of scientific papers published, focusing mainly on the analysis of financial data. I am inclined to say that, after serious hopes at the beginning, no solid achievement has been accomplished at all. Of course, as the recent 2008 financial crisis has shown, when inopportune use of abstract models leads to substantial losses, physicists will be blamed for some time to come. But as with biology, the intrusion of physics causes no decisive impact on the dynamics of the field. Contrary to both biology and economics, the use of physics-based ideas, concepts and techniques dealing with human beings may turn out to be very delicate in the social sciences. This statement is particularly true for models which appear to be false. The major associated epistemological contradiction lies in the fact that because physics can contribute substantially to the social sciences, it can also lead to all kinds of serious misuses. The above challenges are related to politics, philosophy, religion, and everything in which human beings are very involved. Convictions, beliefs, and prospects on what is or should be the social life and its organization could make the human cost of both errors and manipulation potentially very high. On this basis, nothing should be imposed in the name of physics. Physics can only shed new light on social phenomena but never substitute itself for the whole framework of the corresponding traditional views and knowledge of sociologists. Hence, sociophysics is a very exciting and promising field of research, but must always be dealt with care and much caution. To sum up the challenge, we could say that “to introduce formal rationality in human beliefs may produce irrationality in human behavior” establishing catastrophic paradigms that are totally artificial and false, although they could fit to some wrong representations of social reality (Fig. 2.9).

2.5 But Indeed, Do Humans Behave Like Atoms?

When dealing with social situations it is usually believed that one of the main difﬁculties arises from the rich variety of individual characteristics and features of the individuals involved. Along these lines, the richness of a group is expected to be an exponentially increasing function of its size, symbolized by the concept of “complexity” with its famous dogma “The whole is greater than the sum of its parts.” However, crowds, which contain large numbers of people, behave in some aspects like one “collective individual,” who in turn might well behave according to a simpler framework than that of a normal individual. 30 2 The Question: Do Humans Behave like Atoms?

Fig. 2.9 Discovering the existence of deterministic behavior may be the key to individual freedom

This paradox suggests that within a group, the individual complexity should decrease in parallel to the appearance of a new individual monitored by the “collective dimension” of the group. In this context, it is of particular importance to discriminate between, on the one hand, the properties associated with purely individual characteristics, and on the other, those properties which result from the existence of a collectivity or social system. This is exactly why we are not suggesting using, for instance, atomic physics to describe individual properties. On the contrary, collective properties may obey universal features which are valid in many different fields. And it so happens that the interplay between microscopic and macroscopic levels has been greatly studied in physics, a field which is very far from the social sciences. The field of statistical mechanics has been dealing with collective behavior in matter for more than 100 years with much success. However, only in the last few decades has the problem of collective phenomena been well understood. However, this achievement has been made only in the case of pure systems. Organic disorder and heterogeneous systems are still resisting full understanding although solid progress has been accomplished in recent decades with the framing of a series of new concepts and sophisticated tools but much more needs to be done. Out of equilibrium systems are only at the first stages of comprehension. Nevertheless, the modern theory of critical phenomena already represents a substantial qualitative leap toward the understanding of collective phenomena. Chaotic behavior represents another epistemological leap. The modern theory of critical phenomena is based on the fundamental concepts of universality and irrelevant variables. These two concepts mean that different physical systems, for instance, a magnet and a liquid, behave in the same way when passing from one macroscopic state to another, although the physical properties 2.6 Building Up an “AtomÐIndividual” Connection 31 associated with the respective macroscopic states of a liquid and a magnet have nothing in common. Corresponding well-known examples are the magnet becoming a paramagnet and the liquid, a gas. The discovering of the concept of universality has been a major breakthrough in our understanding of the universe. Accordingly, physical characteristics of systems, i.e., the form of microscopic interactions and their physical nature have no effect on the so-called critical behavior which produces the physical character of the transition from one state to another. Most of the microscopic properties turn out to be irrelevant for describing the macroscopic change, which in turn appears to be universal. By irrelevant, I mean that the microscopic properties have no effect on the process. While the number of physical systems undergoing phase transitions is infinite, all associated phase transitions have been shown to be described in terms of a finite number of universality classes. Only a few parameters, such as the space dimensionality, determine which universality class the system belongs to. The abstract and general nature of the statistical physics framework makes it tempting to extend such notions to nonphysical systems, and in particular to social systems for which, in many cases, there exists an interplay between microscopic properties and macroscopic features.

2.6 Building Up an “Atom–Individual” Connection

Nevertheless, the fields of the physical sciences and the social sciences are a priori rather different, both in their nature and in their applications. In order to develop some common framework, it is useful to find out what appropriate assumptions need to be made about the relevant correspondences. The first immediate possibility is to put in parallel the atom and the individual. In physics, the atom defines the basic level of investigation. Afterward, the exploration can go, on the one hand, inside the atom toward elementary particles, and on the other, toward bulk matter by grouping atoms together. In the social sciences, starting from the single human individual also allows us to consider human beings in the “bulk” form, with the existence of societies, as well as the “infra” individual at the level of elementary cells or small groups. Before implementing such an “atomÐindividual” connection, one is entitled to ask the following naive question: “what is common to an atom and a human being?” Without requiring much elaboration, the answer is, “nothing.” The same “nothing” also holds when comparing an atom to a country, a firm, a cell, a political party, a stock, a grain of sand, and many other entities. Such a negative assessment promptly raises a serious concern of what the book is all about. Why all these words for such a clear and precise hopeless conclusion? Without throwing away our nice motivation to build up a new scientific approach to tackle human behavior, we need to go ahead and consider not the singular entity but its plurality. Accordingly, we reformulate the question. Instead of “what is common to an atom and a human being?,” we ask: “what is common to an assembly 32 2 The Question: Do Humans Behave like Atoms?

Fig. 2.10 What is common WWhathat ddoo ttheyhey hhaveave iinn iinn ccommonommon ? to a human being, a stock market, an atom, a party, a grain of sand? A human

A stock 8.2331 An atom

A grain of sand A party

of atoms and an assembly of human beings?.” That already is much more attractive and sounds like deserving of a positive answer. Unfortunately, after some thought and analysis, we need to go against what was expected, with again the same answer of, “nothing.” And again, this remains valid for an assembly of countries, firms, cells, political parties, stocks, sand grains, and many other groups of entities. At this stage, it would be legitimate to get a bit annoyed and to start to complain about what seems to be a totally fake enterprise. However, the above fastidious search was aimed at shedding light on the fallacy of the immediate connections, one atom—one human being and many atoms—many human beings, which have to be dismissed from the start as philosophical traps. Once this is achieved, we can elaborate on climbing out of our despair in pursuing the discussion in order to discover the relevant path to implement the “atomÐ individual” connection. The right and powerful question is: “is there something in common with both processes of passing respectively from one atom to many atoms and from one human being to many human beings?” The answer becomes “yes, there is a lot.” This statement also holds true when going from one country, one firm, one cell, one political party, one stock, one grain, and any other entity toward many countries, firms, cells, political parties, stocks, grains, and other entities. More precisely, the hypothesis behind the present approach is that these microÐ macro mechanisms are universal and hold true beyond the true nature of the various entities involved. The above series of questions aim at clarifying which problems may be addressed and which are outside the scope of the approach. It is worth stressing that we are not claiming that our model will explain all aspects of human behavior. Like any modeling effort, it is appropriate only to a few classes of phenomena of social sciences and not to others (Figs. 2.10Ð2.18). 2.6 Building Up an “AtomÐIndividual” Connection 33

Fig. 2.11 What is common WWhathat ddoo ttheyhey hhaveave iinn ccommonommon ? to a ﬁrm, a cell, a country, or whatever?

A firm

A cell

A country

.... A whatever

Fig. 2.12 Unfortunately or fortunately there is nothing in common between all the preceding items

Fig. 2.13 What do a group WhatWhat iiss ttherehere iinn ccommonommon of human beings, stocks and betweenbetween groupsgroups of?of? shares, atoms, and political parties have in common? Humans

Stocks 8.231, 5.217, 9.3

Atoms Parties 34 2 The Question: Do Humans Behave like Atoms?

Fig. 2.14 What do a group WWhathat iiss ttherehere iinn ccommonommon of grains of sand, ﬁrms, bbetweenetween ggroupsroups oof?f? countries, and cells have in common?

Sand grains Firms

A firm

Countries A firm

Cells

.... A whatever

Fig. 2.15 As for previous series, unfortunately or fortunately there is nothing in common between all the preceding items

2.7 Our Bare Methodology

Having presented the focus of our philosophy and the appealing, but wrong paths, we have thus established a solid conviction about the well-founded basis of our approach, at least at an epistemological level. Ultimately, it will be the demonstration of the adequacy of our models in describing some part of the social world which eventually will turn our a priori toward making sense through intuitive statements into something that could resemble a hard science. At least, that is the challenge facing us. Before proceeding with the core of sociophysics, it may be very useful to lay down our general strategy, which is grounded on a single methodology that is used throughout this book for all the social and political phenomena presented. We will start by picking out a single global phenomenon or some social practice occurring in society. This choice could result from our feeling that there exists some hidden paradox. It could also be driven from a legitimate character that no one would ever think of questioning. These choices are ours and many others could have been 2.7 Our Bare Methodology 35

Fig. 2.16 What is in WWhathat isis inin ccommonommon common in going from a inin goinggoing ffromrom oonene human being, a stock market, Humans toto mmany?any? an atom, a political party, a grain of sand to a group of humans, stocks, atoms, political parties, sand grains? 5.234

Stocks 8.231, 5.217, 9.3

Parties

Atoms

Sand grains

Fig. 2.17 What is in WWhathat iiss iinn ccommonommon common in going from a inin ggoingoing fromfrom oneone ﬁrm, a cell, a country, or ttoo many?many? whatever to a group of ﬁrms, Firms cells, countries, or whatever?

A firm

A firm Cells

A firm

Countries

.... A whatever made instead. Through these choices, we aim at illustrating the powerful method of sociophysics rather than stating a series of absolute social laws. For instance, we will study in detail the use of bottom-up majority rule voting in hierarchical structures (Chap. 7), and the holding of public debates on issues for which decisions are to be taken that have a large public support (Chap. 10). 36 2 The Question: Do Humans Behave like Atoms?

Fig. 2.18 Contrary to the above previous series, still unfortunately or fortunately there is now a lot in common in going from one to many items

Once we have selected the phenomenon or the social practice of interest, we have to clearly identify the associated paradoxical features. For instance, picking up the study of democratic voting in bottom-up hierarchical structures, we emphasize the surprising stability of top leaderships against growing dissatisfaction from members at the bottom. On this basis, although the phenomenon is usually very complex and involves a large number of different ingredients, we apply a brutal simplification by neglecting most of everything. We reduce the phenomenon to a minimum but always preserve a few basic essential features capable of producing some nontrivial dynamics, keeping in mind the goal of recovering the initial “bare paradox.” In the case of democratic bottom-up voting in hierarchical structures, the outcome is a strong dictatorial stability effect in favor of the current top leadership against an eventual massive bottom rejection. The purpose of the “game” is first to check if the chosen mechanism can reproduce the corresponding selected paradox. For instance, can we stabilize a current leadership against an increasingly huge opposition from the base by a democratic procedure? How can repeating votes end up by democratically choosing to renew the current leadership although it is now rejected by the majority of people at the bottom of the hierarchy? Once the toy model is working, the second goal is to establish a logical link between the artificial model and what could be the equivalent real mechanism underlying the political life of institutions and their failure to take into account at the top leadership the eventual changes that have occurred at the bottom of the organization. Somehow, the purpose of the modeling is to create a virtual world from some basic interactions and to study all the associated properties. Once this elementary world is understood and its parameters controlled, we move on by pushing up its basic limits to embody more ingredients so as to make the overall model a bit more realistic. We also need to check the robustness of the mechanism under small changes of the model parameters since we are looking for universal features. In the foundations of sociophysics stands the a priori claim that there exists some finite number of universal mechanisms behind the many different sorts of social behavior occurring in human societies. The study ends by outlining possible extensions of the range of applications of the model. 2.8 To Sum up 37

2.8 To Sum up

As stated in the preceding sections, we will always start by considering one salient paradoxical feature of a particular phenomenon which occurs in society. Although, it is often the result of a very complex situation, we will considerably reduce it down to its simplest form by neglecting many other ingredients, even when they are quite clearly active in the making of the feature. The goal is to undress the selected feature in order to extract some well-defined “pure paradox.” Indeed, the a priori claim that there exist a few universal mechanisms behind much social behavior occurring in human societies is at the very foundation of sociophysics. On this basis, we set up the simplest possible model which can reproduce the selected social paradox using bare and precise mechanisms. The purpose of the “game” is first to check if this bare mechanism can underline the onset of the paradox. Once that is established, the second step seeks to establish a logical link to the supposed feature, in other words, to the real social or political property we started from. Nevertheless, it is worth stressing that it may well happen that the model will be eventually more relevant to another situation than the one from which it was initiated. Once everything in the dynamics of the model is controlled and understood, we move on with the possibility of pushing its basic limits, in order to embody more realistic ingredients. It is also of importance to check the robustness of the result with respect to the chosen bare mechanism, since in particular for social situations, the initial conditions are never exactly known. To grasp the phenomenon requires obtaining results which survive against small changes in the parameter settings. At this level of our investigation, we do not address the question of the quantitative applications to the studied social phenomenon. We aim at defining some qualitative tendencies, which are hopefully present in real-life phenomena, but embodied and masked by some others. Although our models are quantitative in the sense that, using physical modeling and mathematical calculations, they yield precise numbers and figures, these numbers should not be taken too seriously by themselves. Only the associated qualitative description of the considered phenomenon is to be given serious thought to eventually deal with social realities. It is on this basis that we will never take too seriously the figures we get but on the contrary, we will focus seriously on the nature and the trends of the various dynamics obtained. The overall descriptions will be put forward. It is also of central importance to stress the following observations. Firstly, the use of modeling in physics has been tremendously powerful in establishing the field as an exact hard science, capable of building concrete and efficient devices. Secondly, this power comes from the empirical uses of mathematics to describe real phenomena. It means that it is not the mathematical rigor which prevails but the capability of reproducing some particular property. Thirdly, and this point is essential, all the results obtained from the various models are aimed, sooner or later, at being tested against experimental data. Fourthly, discrepancies and agreements are used to validate or invalidate part or all of the model’s hypotheses. It is exactly 38 2 The Question: Do Humans Behave like Atoms? this backward and forward motion between theoretical and experimental research that has yielded an extended core of understanding for the laws governing inert matter. Here, within our approach, we are modeling and empirically using mathematics to only proceed up to the first two points. We give up the rigorous demonstration of the model to favor its capability of reproducing some particular phenomenon. While mathematics focuses on proving the universality of a fact, physics aims at finding some minimum conditions, which are able to reproduce the fact. At the present level of our novel approach, we put aside any experimental apparatus. This fact can be dangerous in the final assessment of a social law derived from our models. This is why one should always be cautious about our findings. At the moment, we are not going to state real and definite laws, but instead to set out a new framework for developing a coherent and different view about social and political behavior. This is why our findings are always and systematically to be taken with caution. The ongoing testing of a model’s predictions of real experimental data to validate the model’s hypotheses has to be left to the future whose proximity will be a function of our collective progress on the above level of research. In turn, an equivalence to the whole framework of physical laws that has to be constructed through the intricate combination of theoretical and experimental research has yet to be developed for sociophysics. This fact does not make the project any less ambitious. Last, but not least, all of our modeling is based on both “naive” assumptions of open minded people, with everyone having the same individual power and the existence of open and democratic social spaces. Clearly, although both these assumptions are false, they do not prevent an operative scheme from discovering the secrets of organized human behavior. Accounting for all the complexity of human differences is left to other fields of investigation of human behavior. In short, our general procedure is articulated around the following steps: 1. Choose a particular phenomenon 2. Single out one salient paradoxical feature 3. Define the phenomenon quantitatively down to its simplest form 4. Calculate the inherent “paradox” 5. Find a logical link to a real-life counterpart 6. Push the limits of the model 7. Enumerate possible extensions Our general guide is illustrated in Fig. 2.19. At the core of our fundamental strategy stands the somewhat provocative hypothesis that the conditions for sociophysics are: “Although you may have to accept drastic simplifications and crude hypotheses in order to start a wrong model, later, this could develop into an operative tool with predictions, which in turn may have a good chance of being true.” We can add our founding statement in Fig. 2.20. References 39

Fig. 2.19 Our constant methodology in seven distinct steps

Fig. 2.20 Our provocative, yet very constructive assessment

References

1. S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A mean behavior model for the process of strike, Journal of Mathematical Sociology 9, 1 (1982) 2. S. Galam, When humans interact like atoms, Understanding group behavior, Vol. I, Chap. 12, 293-312, Davis and Witte, Eds., Lawrence Erlbaum Ass., New Jersey (1996) Chapter 3 Sociophysics: The Origins

The origins of Sociophysics are discussed from personal testimony. I trace back its history to the late 1970s. The first steps of my 30 years of activities and research to establish and promote the field are reviewed. In particular, the decades long strong opposition from the physics community is emphasized, together with the almost total absence of contributors. During these years of rejection, only a very few and scarce papers were published in the field. However, in the mid-1990s, quite suddenly the number of contributions started to increase. Up to the beginning of the 1990s, any attempt to apply physics to social sciences was considered as a devaluing of physics, the noble science, “par excellence.” Today that is no longer the case; on the contrary, it is considered as a real challenge for shedding new light on social and political behavior through the prism of physics. The irony of this reversal in process is that the few authors who wrote about the origins of sociophysics had a tendency to set up “nice stories” about it. By the waving of a magic wand, the contentious aspects of the adventure of sociophysics, the one that really took place, are removed completely. Instead, an ideal version of a smooth continuity from social sciences to sociophysics is set up. What happened is reminiscent to me of the way communist regimes used to write history. Maybe indeed it is the way that most of the writing of history is implemented. While in the communist case the falsification was deliberate, for other cases, it could be the unique solution when prominent figures are no longer alive to testify. Moreover, even though still alive, these fighters for knowledge, once they have been accepted and reintegrated within the institutions they were fighting with, want to forget and erase the traces that show that they used to be heretical and marginal.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 41 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 3, © Springer Science+Business Media, LLC 2012 42 3 Sociophysics: The Origins

3.1 The First Days

3.1.1 Breaking the Secret of Critical Phenomena

To apprehend how sociophysics has emerged from the minds of a handful of physicists, it is of importance to recall what the general atmosphere was at that time. One of the most difﬁcult puzzles of condensed matter physics has been for centuries the understanding of the experimental fact that many physical systems can all of a sudden, at a precise value of some external parameter, such as temperature, undergo a drastic change in their macroscopic state. It is as though the material suddenly metamorphoses itself into another one, like in a chemical reaction or a “magic” transmutation. The problem is that the microscopic nature of the material remains unchanged, indicating that it is not a chemical reaction but a macroscopic reorganization. Moreover, these dramatic collective transformations are accompanied by the disappearance of certain macroscopic properties and the appearance of novel ones. The transitions are from one collective order to another, or to a partial or total disorder (Fig. 3.1). The enigma of the so-called critical phenomena resisted all attempts to reveal its secret. A theoretical explanation was missing, although many industrial applications existed. The challenge remained. But in the mid 1970s, a revolution occurred that was driven by the decisive work of Kenneth G. Wilson. He eventually came up with a coherent explanation in terms of universality classes and order parameters. Everything was solved using the powerful techniques of the renormalization group. He provided the magic key to the fantastic worlds of critical phenomena. A good review can be found in [1]. In a similar way to a phase transition, our theoretical understanding of the properties of inert matter jumped from a kind of disordered knowledge to an ordered understanding.

Fig. 3.1 Physicists were puzzled by all the experimental results obtained from materials undergoing a phase transition 3.1 The First Days 43

Fig. 3.2 The puzzle was solved by Kenneth G. Wilson and implemented by Wilson and Fisher. Physicists were looking at phase transitions with great clarity and a non hidden satisfaction

However, Wilson’s initial work solved the problem in principle but was not yet directly applicable to real systems. Shortly after his milestone publications in collaboration with Michael E. Fisher, he devised a scheme under the name of the “epsilon expansion,” which allowed the reaching of the real world that sits in three dimensions [2].

3.1.2 The Physicist’s Corner

Their paper title, “Critical Exponents in 3.99 Dimensions” was deliberately provocative... at that time. Dimensions were by nature integers. Mentioning a rational number for a space dimension was akin to heresy. And indeed it was. Here we have a nice illustration of how physics proceeds using “iconoclastic” statements. The Wilson renormalization group scheme did not allow the making of explicit calculations for systems in three dimensions, i.e., real physical systems. However, since the equations become much simpler in four dimensions, Wilson and Fisher had the idea of making an expansion around four dimensions as a function of the small variable epsilon deﬁned by d D 4 (Fig. 3.2). From a physical point of view, this idea was nonsense, not making an expansion a function of a small variable but solving equations for a noninteger value of the space dimension. Very typical of the physicist’s methodology, once the equations were solved as a function of epsilon, they plugged in the value D 1 to claim that the associated results corresponded to the real space dimension d D 3.Anditworks. This solution meant that the results now ﬁtted to experimental measurements. 44 3 Sociophysics: The Origins

Fig. 3.3 One of the biggest challenges of physics was just solved in an incredibly elegant and powerful framework. The world of physics was blossoming

What became more intriguing is that, as so often happens in physics, some years later, the above trick to overcome a mathematical barrier was given solid validation. Noninteger dimensions were found to exist in the real world. They are called fractal dimensions [3]. These crucial and brilliant steps led to the so-called Modern theory of phase transitions, which propelled condensed matter physics into its Golden Age. Both Wilson and Fisher were professors at Cornell University in America which became the “Silicon Valley” of this new fascinating “gold rush” of Physics. Several universities all over the world promptly created an active network to establish a very active and successful new ﬁeld of research. Young physicists were entering the ﬁeld with a great deal of excitement and enthusiasm. Being very productive, they published papers by the hundred. Notwithstanding, although I was one of them, I did not publish papers in such large quantities (Fig. 3.3).

3.2 The First Days, the First Fight

For a series of personal reasons, I arrived in Israel in 1976 after graduating from the Pierre and Marie Curie University (UPMC) in Paris in 1975 with a doctoral degree in physics. Being very excited about the blossoming field of critical phenomena, I embraced the new field by engaging in a second doctoral degree with a Ph.D. on the study of disordered systems at the department of physics and astronomy in Tel- Aviv University (TAU). It was one of the world’s “hot” centers in the field thanks 3.2 The First Days, the First Fight 45

Fig. 3.4 For everyone, to talk about the future frustration of successful physicists was simply absurd to a group of young and smart professors working hard for posterity, as well as for their own reputation. A quite impressive number of solid contributions to the field of critical phenomena and disordered systems were produced there during this period. Already having in mind the project of building sociophysics, my view was to promote a collective project. At that time, socialist ideas were all the rage in the Western world and many dreams were attached to them. Along the associated dialectical frame of mind which prevailed then, I foresaw a growing contradiction between on the one hand, the power of the concepts and tools of statistical physics, and on the other, the imminent exhaustion of possible subjects of interest to which they could be applied. From that idea, I really questioned the future of the role of physicists in the development of knowledge at that time (Fig. 3.4). My idea was that as physics would reach the limits in understanding of inert matter worth investigating, a frustration among physicists would appear. To compensate for their increasing frustration combined with their agility in using their particular methods, they would start applying physics outside of the field of physics. A kind of imperialism among physicists would lead to the invasion of the various other fields of research in science. Today the invasion is at its paroxysm as can be seen, for instance, from the incredible development of biomaterials, nanotechnologies, biomagnetism, etc. Within this frame of thinking, I was suggesting that physicists should deal with the rich variety of behavior related to human activities, including economics, sociology, psychology, and politics. 46 3 Sociophysics: The Origins

Fig. 3.5 Physics may provide some useful tools and concepts to tackle the twenty-ﬁrst century. I advocated that program at Tel-Aviv University (TAU)

I published a series of papers to justify and elaborate my suggested strategy to escape the physicist’s frustration, several coauthored with Pfeuty [4Ð10]. The focus was on the existence of some epistemological contradictions within physics and among physicists. Unfortunately, I was a bit naive and almost every physicist who read my work, leading and nonleading, young and old, strongly rejected my proposals. Daring to compare human beings with atoms was strongly condemned as a blasphemy to the pure sciences and an insult to human complexity. It was looked upon as being total nonsense, something to be rejected. And it was indeed fully rejected for more than 15 years (Fig. 3.5).

3.3 From Claim to Demonstration

While going around like a self-convinced salesman trying to sell my idea of what should be done to avoid physicist’s frustration and to enrich the scientific understanding of human behavior, I found out about two connected papers in the literature. I was rather happy to discover that some other physicists had already engaged in my program. The first paper I found was from 1974 by Callen and Shapiro. They wrote a short note in Physics Today to suggest a promising similarity between fish band imitation and Ising spins [11]. It was just a nice metaphor, which, to my knowledge, unfortunately had no follow-up. Three years earlier, a much more elaborate paper was published by Weidlich in the British Journal of Mathematical Psychology [12]. It dealt with the dynamics of opinion forming. Although more developed and dealing directly with a social phenomenon, the paper’s content was more of an application of partial differential equations in the spirit of Volterra’s work than an application of the Modern theory of phase transitions (Fig. 3.6). 3.3 From Claim to Demonstration 47

Fig. 3.6 The ﬁrst ﬁght was to advocate sociophysics

Then, while continuing to argue with physicists about the possibility of sociophysics I wrote my first contribution in 1982 [13]. Disturbed by the philosophical statement of an eventual thermal death for humanity, I took an opposite stand, arguing that applying the Carnot principle of maximum entropy to societies was a great perspective of freedom and development for mankind. Contrary to the asserted ineluctability of the ultimate thermal death of human societies, there are plenty of reasons to become optimistic about them. Nevertheless, it was of a factual contribution written in French and published in a philosophical journal. It is not surprising that it had only a virtual existence in terms of impact. Some time later, with two Ph.D. colleagues from the same department of Physics and Astronomy at TAU, Yuval Gefen and Yonathan Shapir, I published a paper to set simultaneously a global framework for Sociophysics as a new field of research and to make a first application linking the phase transition of an Ising model to the process of strikes in an industrial plant or a company [14]. The stable state of the agents, working or striking, was studied in particular as a function of an external reversing uniform field competing with nearest neighbor interacting agents. The rather subtle phenomenon of metastability combined with the hysteresis feature were used to give political and social explanations. Besides its scientific content, the paper contains a call to the creation of sociophysics. It is a manifesto about its goals, its limits, and its dangers. As such, it is the founding paper of Sociophysics [14], although it is not the first contribution per se (Fig. 3.7). We chose the Journal of Mathematical Sociology to submit our manuscript and not to a physical journal since I first thought sociophysics should be naturally published in a social sciences journal, although it had been developed by physicists. Later I changed my mind, focusing mainly on physical journals for my subsequent publications. It took over 2 years to get our paper finally accepted for publication after a series of tough exchanges with several referees. The paper appeared in 1982. 48 3 Sociophysics: The Origins

Fig. 3.7 The manifesto of sociophysics. A paper by Galam, Gefen and Shapir published in the Journal of Mathematical Sociology 9, 1 (1982)

All three of us were pursuing our physicists’ careers and went to the United States as postdoctoral students. We did not hear much about the earlier fate of the paper. It is worth stressing that at that time there were neither internet nor emails, nor cheap international telephone calls.

3.4 The Story Behind the Scene

I will now tell the story of the sociophysics manifesto, the one that otherwise would never have been written and thus never known. This story puts all the politically correct writing of the sociophysics history in big trouble. Here below is what happened at the Department of Physics and Astronomy at TAU. In these earlier “underdeveloped times,” when you wrote a research paper, you needed ﬁrst to write it by hand, with all the equations, numbers and references and without forgetting to draw the pictures, the most tedious and delicate part of the task. Once everything was completed, you gave the manuscript to the secretary in charge of typing it. At the time it was a highly specialized skill which once started, took several days. One has to imagine how rudimentary and slow the process was to type up in Latin script, plus putting in the integral signs, and so on and so forth. It was almost like a dentist operating in an open mouth. 3.4 The Story Behind the Scene 49

Fig. 3.8 In a world level university, at the end of the twentieth century an attempt is launched to muzzle a researcher and to deprive him of the right to submit a manuscript

Once the paper was eventually typed, a careful check was in order but only to correct mistakes. It was out of the question to rewrite the whole paper or part of it as everyone does nowadays, thanks to computers, great software, and Latex (Fig. 3.8). After having gone through such an amazing challenge, we were very anxious to get the typed manuscript, the first incarnation of the printed material before submitting the paper. Moreover, it was for each of us, more or less the first “real paper.” Running and jumping, we raced to the secretary to take delivery of our document but the unthinkable, a fair “Coup de the«atre,”ˆ happened. Our paper had been taken away by the Head of the Department and locked in a safe cupboard! A scientific paper written by three Ph.D. students had been literally sequestrated without notice under the authority of the Head of the Department and moreover, with the support of most of the faculty members. We were totally astounded. How could such a thing happen? The “event” provoked a big scandal within the department. We were denounced as putting at stake the department’s reputation of excellence. While we were claiming our right to academic freedom the chairman denied it using the fallacious argument that we had no tenure positions. Our manuscript was simply under arrest (Fig. 3.9). Trying to find a way out, it was settled that to satisfy academic freedom, non tenure fellows must obtain the endorsement by a tenure professor of the department. In an explosive atmosphere, a former member of the refusenik seminar from the Soviet Union, a well known experimental physicist and then full professor at TAU, came to our rescue. Alexander Voronel stood to grant us the freedom to recover our manuscript and, moreover, to submit it to an international journal. And so we did [15](Fig.3.10). 50 3 Sociophysics: The Origins

Fig. 3.9 In a world top university, at the end of the twentieth century, a manuscript had been conﬁscated without notice by the Head of the Department of physics with the support of most of the faculty members

Fig. 3.10 The muzzled researcher struggled for his freedom, and in the same department a courageous professor from the Soviet Union stood up successfully to ensure that full academic freedom was granted 3.5 More About Academic Freedom 51

To ensure academic freedom does not guarantee the publication of a manuscript. It only imposes the freedom to submit a manuscript, which then has to be refereed by peer experts from the journal to which it is submitted. This is the way that research publications have to operate. Apart from it being interesting in itself, the above story illustrates how much opposition there was to sociophysics from within physics itself, i.e., from physicists. And that was not specific to the physicists of TAU; the hostility was felt from most physicists around the world and I had the chance to meet a good number of them. Indeed, I could tell many stories of this kind, which occurred to me during these many years of a personal and lonely fight to create, develop and establish sociophysics. The opposition always came from within the general framework in which I was positioning my various contributions. It was the idea of creating a new field of research within physics in order to deal with human behavior which was deeply disturbing the physicists, not a factual contribution which could always be looked upon as a marginal and exotic isolated event with no follow up.

3.5 More About Academic Freedom

From these later years of the twentieth century in the late 1970s at the department of Physics and Astronomy at TAU, I accomplished a long and well-settled career. I held successive positions at different academic institutions including, in chronological order, the City College of the City University of New York (CUNY), the New York University (NYU), the UPMC in Paris, and the Ecole« Polytechnique. In 1984, I obtained a permanent position at the CNRS, the National Center for Scientific Research in France where I am currently a Director of Research, the equivalent of a full professor position but without teaching duties. My career was built upon my research accomplishments within traditional physics, although on a slowed down path due to my “other” activities. Not withstanding the negative view of both institutions and individuals for 20 years, sociophysics became a recognized and flourishing field of physics that involves today a large number of physicists all around the world. Amazingly, in the early twenty-first century I again went through a very similar experience of near ostracism from colleagues who feared for their reputation. In February 2007, I wrote a paper in a major French daily newspaper, Le Monde, in which I questioned the claimed human culpability with respect to the assessed global warming [16]. I stated that there existed no scientific proof of the Intergovernmental Panel on Climate Change (IPCC) climatologists’ claims. Taking such a position at the time resulted in quite an outcry against me and the newspaper, Le Monde, which had never experienced such a shaking from both outside and inside the newspaper [17]. I later wrote a book to detail my arguments but this is outside the scope of the present book [18]. 52 3 Sociophysics: The Origins

Fig. 3.11 A quasi unanimity of physicists looking at sociophysics during both the 1980s and the 1990s

The above cases demonstrate that modern man has not changed much and still relies on archaic and primitive reactions as soon as he fears that something could jeopardize his status. It applies to scientists in the same way as to anybody else, which is a good lesson to keep in mind. Novelty is still perceived as being dangerous even by those who have even contributed to it in the first place (Fig. 3.11). I have nevertheless got to put some caution on my plea for academic freedom. I am aware that it could open the path to the possibility for a faculty member to write “unacceptable” statements, in particular in the various fields of social sciences. But in this case, it should be the civil law that should be activated to eventually condemn a published paper that violates some existing laws. But it is not the role of the faculty to enforce an a priori censorship on any submitted publication. For sure, if a faculty member is prosecuted and proved guilty, for instance of racist incitement, or any other criminal act, to have them fired by the faculty makes sense. Such a statement is of course valid as long as we are dealing with a democratic country.

3.6 Surviving Within Physics by Not Playing Tennis

I was totally convinced that sociophysics was about to be established in the future but I was too much ahead of my time. Therefore, in order not to put at stake my ambitious program it was clear for me that I had to stay within both the academic world and the physics community. Only there could I eventually succeed in my project of creating a new ﬁeld for the social sciences. 3.6 Surviving Within Physics by Not Playing Tennis 53

Fig. 3.12 To survive and to become better equipped for the adventure of the building of sociophysics I had to do solid research in orthodox physics. At the same time I was presenting my sociophysics activities as a hobby. Others were playing tennis, while I was “playing” sociophysics

Accordingly, and so as not to jeopardize my academic career, I had to divert most of my energy away from doing full time research in developing sociophysics. The goal was not only survival as a physicist but also to blossom within physics. I thus did orthodox physics, which I must say was at that time a really exciting activity. It embodied the then new field of disordered systems from which I was sure that many ideas, concepts and tools could be used later on to produce sociophysics models. I also carried on working a bit on sociophysics, not in secret, but as a “hobby.” In the way others were playing tennis, I was “playing” sociophysics. It provided me with some kind of immunity against any criticism of lack of depth from scientific evaluating committees. As a matter of fact I did survive as a physicist. I even accomplished a fair career. During these years, physics was highly competitive and not too many openings were available around the world. The price of such an achievement was a heavy personal involvement in research to allow success in physics as well as novel work in sociophysics. It was only a few years later, while in New York, that I was able to produce more work to further establish the feasibility of Sociophysics (Fig. 3.12). I first published an additional paper using entropy in 1984 [19]. It was still about concepts and ideas but without a quantitative construction. My second significant contribution to sociophysics appeared in 1986. Using the fascinating power of renormalization group concepts and techniques, I studied dictatorship effects induced by the use of the democratic rule of majority voting in hierarchical bottom- up organizations [20]. Still thinking that sociophysics should be implemented within the social sciences, I submitted my paper to the Journal of Mathematical Psychology. The same scenario as with the manifesto paper occurred [14]. Two 54 3 Sociophysics: The Origins

Fig. 3.13 Still trying to convince others to engage in sociophysics. Zero success

long years of ongoing arguments were necessary with several referees and one of the journal Editors in order to have the paper accepted. Once the paper was printed, I did not get many reactions. But still being very enthusiastic and well orientated on my particular path, I tried to create a collective movement rather than to build an individual effort. I thus organized several informal seminars and kept on trying to convince other researchers of the validity of the approach (Fig. 3.13). After several years in a row without going back to Israel, I went to TAU for a short visit. At the same time, Dietrich Stauffer, an expert physicist in Monte Carlo simulations from Koln University in Germany was also visiting the department of physics and astronomy. We shared the same ofﬁce. He was then doing simulations related to physics but became interested in my approach to sociophysics. He encouraged me to submit a sociophysics paper in a physics journal. Back in France, I followed Stauffer’s suggestion and submitted a paper on voting to the Journal of Statistical Physics, which looked like a good choice to try and publish a ﬁrst publication of sociophysics for physicists. It is worth noting that although the paper was accepted in 1990, I received quite a surprising letter form the chief Editor Joel Lebowitz from Rutgers University. He made a comment on his own decision to accept my paper for publication [21]. He stressed explicitly that he was accepting the paper because the referees’ reports were positive and that the associated editor in charge of the paper recommended publication. However, he personally did not believe at all in the validity of such an approach although he did not intend to censor it. He thus respected and implemented academic freedom against his own feelings. Such an exemplary attitude has to go to Lebowitz’s credit. But once again, I did not receive much feedback after publication. Making the most of my still active enthusiasm I later published three more papers to extend my voting model [22Ð24]. For each of them, the same long and involved process ended up with no reaction to their publication. 3.7 Breaking the Gap with a Social Scientist 55

Fig. 3.14 In the late 1990s at last the hostility against sociophysics faded

This funny business went on for more than 20 years until my peculiar concept of “not playing tennis” became popular among a substantial number of other physicists around the world. Sociophysics was thus established de facto not by institutional decisions but somehow against the institution’s will, in the field of research itself. It has been the very existence of sociophysics throughout the first several dozen and then the hundreds of research papers published in leading international physics journals that has in turn validated the field (Fig. 3.14). To bring a happy ending to the story, I must mention that it was finally, in 2004, that my institution acknowledged that my unusual practice of “tennis” was not a hobby but in fact my professional activity. In other words, I was officially granted the right to carry out full time research in sociophysics. I implemented this change in status by quitting my physics laboratory to join a social sciences group at the Ecole« Polytechnique where I have been ever since, as happy as a fish in water. I must put in a word of caution, to stress that although I have been formally allowed to embrace sociophysics, the counterpart has been an end to my institutional career in terms of an eventual promotion (to Director of Research, first class). But one cannot ask too much from a national institution. And more importantly, when you engage in new spaces to break the existing frontiers, whatever their respective nature, you do not expect a straightforward career. You just hope to survive and keep on thriving on your own intuitions.

3.7 Breaking the Gap with a Social Scientist

Having taken so much effort and made so many contributions with no visible reaction from either physicists or social scientists, I thought of changing my strategy. 56 3 Sociophysics: The Origins

Fig. 3.15 An improbable meeting between two French Serge, one a psychosociologist and the other, a physicist, in Manhattan

The correct next step was maybe to take with me my new field of sociophysics and to seek to collaborate directly with a social scientist in studying a problem that social scientists had tried to solve without any success (Fig. 3.15). Instead of just publishing lonely papers in social science journals, elaborating joint work with a social scientist could trigger a drastic change in both the fields of physics and the social sciences. Researchers are often very naive. At least I am. Having decided this reorientation, everything still had to be done. I had to find someone who would not only be interested in the project, and who would not only be ready to invest his or her time (no money was required), and would not only have to do this and that, but who would also be ready to invest tremendous patience in the willingness to be at the same time both a teacher and a student, in addition to being engaged in subjects for which sociophysics could be helpful. As so often happens for improbable tasks, things moved on quite by chance. I was in New York enjoying the Big Apple way of life of drinking a good expresso in the West Village when I saw a local friend walking with somebody. Joining me for a drink it happened that the other person was Serge Moscovici, a leading French social psychologist, who was visiting New York. I immediately put forward my case. He appeared rather interested in the adventure and we decided to meet again once back in Paris. We got along well and started a very fruitful cooperation which lasted over a few years from 1991 until 1995. We came up with a novel theory of decision making, which yielded a series of papers, most of them published in the European Journal of Social Psychology [25Ð29](Fig.3.16). But again, a lot of work, a lot of brainstorming, a lot of time, a lot of excitement, a lot of effort, a lot of rewriting, a lot of arguing, a lot of... to end up with not much 3.8 Changing My Strategy: Back to the World of Physics 57

Fig. 3.16 A very fruitful collaboration created a novel theory of group decision making embed- ding both sociophysics and psycho-sociology feedback, except one invited paper in a book edited by a social scientist [30]. The papers had too many equations for social scientists and too much psycho-sociology for physicists, who anyway ignored the existence of this peer reviewed international journal. Often, people discussing interdisciplinary research point to the necessity of having researchers from different disciplines working together. They always talk about the pay-off of such collaboration, pointing to the resulting beneﬁt for the research. Unfortunately, my few experiences of real interdisciplinary work indicate that one should not mention pay-off but more the heavy price to pay for those doing the research. Of course, I am not discussing here the many “fake” collaborations where people from different disciplines coauthor a paper, which is in fact monodisciplinary and written only by the experts of that discipline. In such a case, everyone receives a return from their own community demonstrating their interdisciplinary involvement by having a coauthor from another discipline.

3.8 Changing My Strategy: Back to the World of Physics

Having ﬁrst published my papers in social science journals, I then sought to create a solid interdisciplinary theory with a social scientist. I eventually came to the conclusion that this was not yet the optimum way of going about things. Coming back to my initial postulate about sociophysics emerging from physicists, I realized that the natural logic of such a statement was to publish sociophysics papers in physics journals and not in social science journals. This is what I started to do [31, 32] in the mid 1990s (Fig. 3.17). 58 3 Sociophysics: The Origins

Fig. 3.17 Econophysics jumped onto the stage. Sociophysics was still in the shadow of loneliness

However, while I was engaged on this new path, I stayed just as lonely as before. There were not many reactions, apart from a few hostile ones from time to time. The main reaction was just an absence of reaction, a total indifference. Finally, around the same time a handful of physicists also turned “exotic,” not in joining sociophysics, but in analyzing ﬁnancial data and stock markets. It was the birth of the so-called “econophysics” as coined by H. Eugene Stanley from Boston University in the United States at the conference “Dynamics of Complex Systems” held in Kolkata in 1995 [33]. It was the ﬁrst validation of my prediction I made in the 1980s of physicists invading new grounds of investigation outside the scope of inert matter. Although it was a satisfying observation, it did not put an end to my loneliness in my obstinate construction of sociophysics to deal with the study of political and social behavior.

3.9 The Secret One Shot International Seminar

3.10 The Rising Sun of Sociophysics

Years were passing by, one after the other, which was the minimum requirement for maintaining my obstinate ﬁght in establishing sociophysics. At around the twentieth of them, all of a sudden a few papers appeared in the literature embracing the sociophysics approach. Several physicists here and there joined the club by making sporadic contributions. I was so happy to realize that I was not really crazy, or at least not the only one who was so. My dream was becoming a real and solid reality (Fig. 3.18). 3.10 The Rising Sun of Sociophysics 59

Fig. 3.18 Finally, a few physicists here and there started to publish sporadic work embracing my sociophysics approach. I was so happy at last

However, my happiness turned rather too quickly into frustration. Most of these new papers did not cite my earlier founding papers. Even worse, some were “rediscovering” my own contributions and appropriating the paternity to themselves. The emerging young community was endorsing this incredible ongoing practice. Sociophysics was blossoming almost as if neither I nor my earlier papers existed. What a disappointment! My naive belief that physicists were more honest than most others when dealing with science was shattered. The very human world of science was just as dishonest and double-faced as any other social sphere. In fact, such a phenomenon is rather common in science. The only difference with the other worlds is that science pretends to be honest at its foundation in order to ensure its progress. But of course, as usual, there exists a gap between the idyllic representation of reality and reality itself, often a big gap (Fig. 3.19). I thus began a new unexpected struggle to try to restore the truth about the paternity and chronology of the development of sociophysics. But it became rather quickly an overwhelming task due to the success of this new field which was expanding at a fast rate. Therefore, from time to time I sent occasional e-mails to the relevant authors mentioning my previous contributions. But overall, the outcome was that I felt rather at unease. I felt so much of an incredible injustice. The alternatives were and still are either to witness with frustration certain colleagues in this new field being promoted for supposedly “original” work that I in fact did many years ago or to appear as a frustrated paranoiac. The choice is hard to make. Indeed, I have oscillated between the two options at each new occurrence of this spoliation. It is unpleasant and psychologically consuming to play the role of a bitter scientist when you just want to restore the chronology that really happened against the one which is wrongly stated by powerful scientists who are building the institutional frame for the field (Fig. 3.20). 60 3 Sociophysics: The Origins

Fig. 3.19 The new sociophysicists were pretending to be the ﬁrst ones, ignoring my series of earlier pioneer contributions to the ﬁeld. My happiness turned into a feeling of injustice

Fig. 3.20 I found myself engaged in a new and totally paradoxical struggle for recognition by the new sociophysicists

3.11 When Too Much Is Too much

To top all of this manipulation and dishonesty is the writing of a history that denies the real history. Writing a past history is a difficult task due to the necessity of rebuilding a world that no longer exists by only using the often incomplete documents that are available. But writing a contemporary history is of a different 3.12 Claiming the Paternity of Sociophysics 61 nature, since usually some of the actors of the event are still alive and can testify alongside the associated data and documents. However, I have been astonished and shocked when I read some of the first papers about the history of sociophysics. They totally ignore the real history, the one which actually happened. To illustrate my case, I will cite two papers discussing the nature and origins of sociophysics [34,35]. I am not discussing this point just to restore my “battered” ego, but mainly to settle the historical truth on the origins of sociophysics. To strengthen sociophysics as a solid field of research it is a prerequisite condition to preserve its conflicting nature. It is an essential ingredient of the approach. Both of the above papers are different in their focus and style, but both adopt what could be defined as a very politically correct view of the dynamics of science. They introduce sociophysics as a natural outgrowth of sociology, and connect a series of papers that in particular ignore my own. On this basis, they build a theoretical history driven by a certain logical link among these selected papers. This “creative history” may be appealing intellectually but it is quite simply false, particularly with respect to both the scientists involved and the epistemological content. To support their idyllic view of the origins of sociophysics, they trace back its foundations to the work of Schelling [36] who according to them was already doing physics without being aware of it. It is not my intention to minimize the substantial contribution of Schelling to the social sciences but his work is not sociophysics and did not contribute to the first steps of the emergence of sociophysics. A few years after sociophysics began to spread, Schelling’s work did lead to a series of papers by a few “newborn” sociophysicists. Nothing less but nothing more.

3.12 Claiming the Paternity of Sociophysics

To claim a paternity is rather awkward in particular with respect to ideas. The educated and humble approach is to let others attribute a paternity to someone. It looks arrogant and inappropriate to do so oneself. However, my approach relies on the idea that reality must prevail against a hypothetical “savoir vivre.” At some point of observation of what was going on, I considered the necessity to claim the paternity of sociophysics (Fig. 3.21). By so doing, I am fully aware that to be a “father” does not mean to be the first one ever. One can always find in the past here and there some figures who evoked the idea or even created some local metaphor, which nevertheless did not have any follow up. It is similar with making love and having a baby; the father of a child is not necessarily the first man who had intercourse with the mother. I acknowledged this fact explicitly by giving references to the few papers I knew of, which were related to sociophysics in one manner or another [11, 12], in the paper in which I put forward as the manifesto of sociophysics [14]. It is also of vital importance to avoid the misuse of imported concepts from physics, as was well demonstrated by A. Sokal and J. Bricmont in their book entitled “Intellectual Impostures” [37]. 62 3 Sociophysics: The Origins

Fig. 3.21 I never thought about paternity or recognition during my more than 20 years of ﬁghting. But then, facing the threat of being usurpated, I decided to claim my paternity

For 20 years I was involved in promoting sociophysics, fighting without having in mind the question of paternity of the field. The awareness of being the father of sociophysics came along one evening in 2000 while drinking a beer at an international conference at Bad-Honnef in Germany with H. Eugene Stanley, a prominent world wide American physicist from Boston University. He knew me from the 1970s on my arrival in Israel and had witnessed the whole of the period up until now. The conference was entitled “International Workshop on Economic Dynamics from the Physics Point of View” and my talk’s title was “Random field Ising model for group decision making.” While discussing the new reality of physics going beyond inert matter Gene Stanley made the statement that indeed I was the father of sociophysics. This was like a shock for me since the concept of fatherhood was so far from my preoccupations. I was motivated by “love,” not by having a family! I thus kept this “revelation” hidden in my mind. Later on, with hindsight I came to the same observation: Gene Stanley was reporting a fact. But yet, I kept this fact “secret” up until another international conference in 2003 where I was just “pissed off” by a talk in Poland about sociophysics given by Dietrich Stauffer. There, my contributions were totally ignored. The day after, without notifying the organizers, I took a few minutes of my invited talk to put things straight: I was the father of sociophysics. Later, I published a paper [15] to provide my claim with more details on the why and whereabouts of such a claim. The paper was entitled “Sociophysics: a personal testimony.” (Fig. 3.22). At some point, it might be interesting to ask why the founding fathers of a particular field are given the label without having to make the claim for themselves. It could indicate that fathers have to wait quietly until their community recognizes their founding role, which in turn presupposes a wise and fair infallibility of scientific communities in recognizing the breakthroughs within their respective corpus of knowledge. Or, on the contrary it could mean that they obtained the label after an efficient but discrete lobbying within their community. However, there must be many cases of those who are never recognized and who do not make 3.13 Reorientating My Strategy Again to Join a Social Sciences Group 63

Fig. 3.22 After all, why not celebrate my paternity of sociophysics?

the claim themselves so that they just “disappear,” with their contributions maybe staying unknown forever. This is a whole subject in itself that is worthy of further investigation (Fig. 3.23).

Fig. 3.23 It is unfortunate to have to claim one’s paternity butitisevenmore unfortunate to be dethroned from one’s paternity

3.13 Reorientating My Strategy Again to Join a Social Sciences Group

Throughout the whole of the above happenings I never became bitter. I kept up my multiple ﬁghts that included the struggle for recognition, the true history and the carrying out of more sociophysics research to deepen my own work. At the same time, sociophysics was expanding in many directions (Fig. 3.24). I then decided once more to change my strategy. The next step was to move physically to join a social sciences group. But that was not an easy task. This 64 3 Sociophysics: The Origins

Fig. 3.24 At the beginning of the twenty-first century sociophysics becomes a well established field of physics. Hundreds of papers are published and several international conferences and workshops are held every year was not easy because the difficulty in principle was not to formally join such a group, although many existed. The aim was to find a social sciences group in which the social scientists were not only not afraid of equations, but rather, they could understand them! In addition, they had to be interested in the sociophysics approach. Surprisingly, such a group existed and so I joined it. The group’s name is “CREA,” i.e., Centre de Recherche en Epist« emologie« Appliquee« and it is part of the Ecole« Polytechnique in France. It is worth underlining that within French institutions, to implement such a move administratively requires the approval of the institutions themselves, which are usually rather strict with regards to the formal rules in place. If such an approval would have certainly been impossible a few years ago, it became quite natural at the beginning of the twenty first century, which is a testimony to the fact that sociophysics has become a well-established field of research in physics. This transformation is not yet directly supported by the institutions themselves but they do not oppose it. On this basis, a move to the social sciences appears as being natural in order to reestablish the formal frontiers of research which is of course an obsolete concept in modern research (Fig. 3.25). My move to my new research world was indeed very successful. I felt very stimulated by the full recognition and the interest I got from the group members. But I must add immediately that this positive support was more local within CREA than at the level of my institution. It did not pay off in terms of my career. But being optimistic in nature, I have not yet given up on the hope that one day it will. When a committee has to evaluate my work, physicists say that they cannot do it since it deals with the social sciences. And clearly it would be impossible for social scientists to evaluate a statistical physics paper. Indeed only sociophysicists could do an evaluation. However, this would require having sociophysics institutionalized 3.13 Reorientating My Strategy Again to Join a Social Sciences Group 65

Fig. 3.25 At the beginning of the twenty-ﬁrst century, the institution’s hostilities against sociophysics faded away. Some started to be sympathetic to it which will take many more years, a “Catch 22” situation. Interdisciplinary work is always a hard task to be implemented and once it is successful, it is not given the institutional credit it deserves. It is the unfortunate price to pay for bringing innovation to the world of knowledge. But it is worth the price! (Fig. 3.26).

Fig. 3.26 At the beginning of the twenty-ﬁrst century, the institutions’s hostility against sociophysics faded away

At this stage (in 2011), I am wondering what should be the next move in order to push on with the construction of sociophysics. I am also curious of what will be the next move? Whatever the eventualities, we should keep vigilant on what will happen at the political level since playing politics is also a requirement to reach the institutional level without which the ﬁeld cannot grow with respect to human power, i.e., to have speciﬁc graduate studies, to deliver Ph.D.’s, to obtain tenure positions and so on. The question is then to determine if it is the right time for this institutionalization? 66 3 Sociophysics: The Origins

References

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31. S. Galam, “Fragmentationversus stability inbimodal coalitions”, Physica A 230, 174 (1996) 32. S. Galam, “Rational group decision making: a random 0eld Ising model at TD0”, Physica A 238, 66 (1997) 33. H. E. Stanley, V. Afanasyev, L. A. N. Amaral, S. V. Buldyrev, A. L. Goldberger, S. Havlin, H. Leschhorn, P. Maass, R. N. Mantegna, C.-K. Peng, P. A. Prince, M. A. Salinger, M. H. R. Stanley, and G. M. Viswanathan, “Anomalous Fluctuations in the Dynamics of Complex Systems: From DNA and Physiology to Econophysics,” Physica A 224, 302 (1996) 34. P. Ball, “Utopia theory”, Phys. World October, 7 (2003) 35. D. Stauffer, “Introduction to statistical physics outside physics”, Physica A 336, 1 (2004) 36. T. C. Schelling, J. Math. Sociol. 1, 143 (1971) 37. A. Sokal and J. Bricmont, “Intellectual Impostures”, Proﬁle Books Ltd., London (1999) Chapter 4 Sociophysics: Weaknesses, Achievements, and Challenges

The current trend of physics becoming “global” is analyzed. Underlying the initial hostility of the physics community, I focus on sociophysics in order to enumerate the conditions to establish a new scientific paradigm for the understanding of the human world. Epistemological foundations are suggested so as to be able to provide a framework and to streamline our new emerging field. Existing attempts of Soviet- like rewriting of the history of sociophysics are criticized. Weaknesses and strengths are reviewed with an emphasis on what is intrinsic and specific to the very nature of the field dealing with human beings, from what is structural in the way that the field is developing. Featuring the recent success of sociophysics in the prediction of a few real political events, a strategy is proposed to collectively validate the robustness of the sociophysics approach. The challenge is to provide solid tools to make sociophysics a quantitative and heuristic field of research. A few words of warning: the reader may find several repetitions in this chapter from the previous one. This is done for the sake of completeness of being able to read each chapter independently.

4.1 The Essential Challenges of Sociophysics

During the first years of the 21st century, sociophysics has established itself as a solid field of statistical physics. Hundreds of papers are published in the best international journals of physics and several international conferences and workshops are held every year. Such growth raises three questions about the present status of the field and its future. • Can sociophysics be grounded on some unitary principles and hypotheses? • Can sociophysics identify some major problems to be solved? • Can sociophysics become a predictive field?

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 69 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 4, © Springer Science+Business Media, LLC 2012 70 4 Sociophysics: Weaknesses, Achievements, and Challenges

These challenging questions can only be answered by a respective demonstration of an eventual yes to each question. General analyses could be interesting for their own shake but are irrelevant to the hard science aspect we are addressing here. The future achievements of sociophysics will provide the answers. I have the conviction that the self-organization of people working in the field is a central key to gaining positive answers. But it is far from being self-evident to researchers in sociophysics. This is mainly because of the current trend in physics to be attracted to the “quick and new and jump to the next topic.” We enumerate the conditions necessary for possibly establishing a new scientific paradigm for the understanding of the world of human beings, in focusing particularly on the social and political conditions. Epistemological foundations are proposed to help provide a framework and to streamline our new emerging field. Existing attempts of Soviet-like rewriting of the sociophysics history are criticized. Setting a false framework to the emergence of sociophysics will jeopardize the chances of sociophysics in becoming a hard science. Weaknesses and strengths are reviewed with an emphasis on the features which are intrinsic and specific to the very nature of a field dealing with human beings. These are discriminated from those features that are associated to the structural whereabouts of a new emerging field. Accordingly, a strategy is proposed to collectively validate the robustness of the sociophysics approach. This should provide an operative framework for applying predictive tools in order to make sociophysics a quantitative and heuristic field. But the challenge is far from being achieved. I will now make a proposal to confront our involvement in building a solid field of sociophysics.

4.2 Sociophysics: A New Field Is Emerging

Nowadays physics is everywhere. It is becoming “omnipresent.” Or more precisely, physicists are applying their models and tools to an increasing number of different problems taken from almost all fields of knowledge. It has given rise to a series of subfields of physics including sociophysics [1], econophysics [2], networks [3], population dynamics [4], languages [5], evolution [6–8], genetics [9], terrorism [10, 11], and more, [12–16]. Such an interest of physicists toward new fields of investigation that are so distant from the usual fields within physics is a rather new and recent phenomenon. In particular, at the current large scale of activity, there are hundreds of physicists involved, at least part time, in these “exotic” activities. Twenty years ago, most of these applications of physics outside physics did not exist. Indeed, in the early 1980s, a few scarce pioneering papers were available but were looked upon as nonsense by all physicists besides a very few exceptions [17]. Everyone in statistical physics and condensed matter was excited in studying inert matter, enjoying to the full the wonderful world of critical phenomena, and later, disorder and chaos. The physics outside of physics approach remained on the fringe and was even banished up to the mid 1990s. Then, a few clusters of physicists started 4.3 Deciding the Future of Sociophysics 71 to tackle problems from finance using methods from physics under the name of econophysics [2]. Several years later, econophysics slowed down while sociophysics eventually emerged to become one of the main streams of “physics” research of the early 21st century. The change within the physics community has been huge and sudden. The former hostility vanished immediately, with hundreds of papers these days being regularly devoted to these new problems of interest. They are being published in the best international physics journals, including Physical Review E and rarely, Physical Review Letters, some of them devoting whole sections to these new topics, Physica A being a leader. In addition, more than a dozen international conferences and workshops are being held every year on these new subjects of research. Physics institutions worldwide are integrating this “everywhere physics” approach under the internal drive of the many physicists joining the trend. But it also provides a new exciting window to promote physics and counter the decline in recent years in the student attraction toward traditional physics. At the same time, modern scientific societies are in an urgent need of finding solutions, or at least some understanding of a large spectrum of novel social problems produced by globalization and the increasing complexity of the modern world.

4.3 Deciding the Future of Sociophysics

After having revisited the first earlier steps of the current trend of this interdisciplinary physics, with the initial underlying hostility of the physics community, we address the major issue of its possible future. The focus here is on determining whether or not sociophysics can become a solid predictive tool of knowledge. It is only at an early stage of development but has already reached a critical size in terms of numbers of both papers and physicists involved [5]. There is a motivation to start exploring the possible perspectives and issues which are at stake in the near future. I think that the time has come to start coordinating some of our research activities in order to single out a series of basic questions that we want to address collectively in order to articulate our field and to boost its achievements. Featuring the recent successes in the prediction of a few real political events using models from sociophysics, it is legitimate to consider that sociophysics can help to establish a new scientific paradigm for the understanding of some of the salient features of the human world. To provide an operational basis for fruitful discussion on this crucial challenge, I must enumerate some of the strengths as well as some of the weaknesses of the sociophysics approach. I emphasize and discriminate what is intrinsic and specific to the very human nature of this particular field of research from what is structural in the way the field is developing. I then outline what could be a good strategy for our community for achieving the turning of sociophysics into a new quantitative science of social systems. A proposal is suggested with a few challenges for the future. 72 4 Sociophysics: Weaknesses, Achievements, and Challenges

4.4 Sociophysics: Epistemological Foundations

The phenomenon of emergence of a new field of research is certainly of interest in itself for intrinsic reasons, but the case of sociophysics deserves particular attention for at least three specific reasons. • The first one is the very unusual and unnatural connection in sociophysics between the two fields of research that are the furthest apart: that of the most complex field concerning both social structures and dynamics of human beings, and the most primitive one, that of inert matter and atoms. • The second reason is the potential and huge danger which could result from a misuse of sociophysics in the political world. • The third reason is the potential and huge benefits, which could result from an appropriate use of sociophysics in the political world. Dealing with human behavior in the name of science may become dangerous in both the opposing cases of a valid model and a wrong model. Even if a model is totally wrong, it can always be used as a false argument to support a biased political view. At this level of misuse, the intrinsic authority of a supposedly scientific proof can turn out to be devastating since it would be impossible to be refuted by most people. This is not the case of a regular social science-like argument, which by its very nature, can always be either refuted or at least rejected as not being convincing. At the same time, if a model is eventually proven to be valid, it may provide an efficient tool for manipulating people. Therefore, in both cases of wrong and valid models, some caution is required to clearly define the content and the limits of sociophysics. It is the prerequisite condition to have it developed in a coherent and scientific frame. It is worth stressing that, contrary to sociophysics, applications of physics to biology and finance present no obvious danger. Either some model is valid, even in part, and then can be useful to some extent in its applications, or it is wrong and at most, results in some loss of money for the people who were expecting to benefit from it. So there is nothing much to be worried about. However, this is not the case in sociophysics since it addresses problems directly connected to human beings of which the consequences could be dramatic. On this basis and since the very name of sociophysics induces certain cognitive representations of what it could be, it is an essential and instrumental priority to first define what sociophysics is and even more importantly, what it is not. At least, this is my conviction. This does not imply that every sociophysicist should agree with me, but I seek to provide the first attempt to build some epistemological foundations for our new emerging field. Afterward it will be up to the community of sociophysicists to construct a stimulating and safe framework by setting the limits and the goals of the discipline. These are the prerequisites for the establishment of a solid science. I, therefore, propose ten guidelines in which five are positive and five are negative. The positive statements are about what sociophysics should be. 4.4 Sociophysics: Epistemological Foundations 73

(1) Sociophysics is the appropriation by physicists as physicists, of the study of some parts of social and political behavior. It is a novel additional framework within which to tackle problems related to human behavior. Some are identical or similar to the problems studied in the social sciences but some are new. The point of view and interest with respect to each field are different and specific. Connections and overlaps with the classical social sciences may appear and are welcome but they are neither the rule nor the ultimate goal. Sociophysics is merely the task of physicists. The physicists working in sociophysics do not have the vocation to become social scientists in the usual sense. They aim at becoming “sociophysicists.” Social scientists are requested to learn a little bit of statistical physics. (2) The novelty of sociophysics lies in the process of modeling social behavior in the same way that physics models natural phenomena. It is not bound to proving the universal nature of a solution, but focuses on finding out the simplest conditions which can reproduce a universal feature of a given phenomenon. This is the fundamental and irreducible difference from a purely mathematical approach. (3) The purpose of sociophysics is to discover particular trends in human behavior while ignoring many other aspects of it. The goal is to grasp a quantitative understanding without accounting for the full description, which would involve all the details. Sociophysics should not be taken literally at the level of its numbers although its models yield precise numbers. Trends and dynamics are the important features to rely on. (4) Sociophysics is a mosaic of models, which can be either complementary or even competing and contradictory, each class of models tackling only a very small part of reality. It is an evolutionary and partial description of the social world, but is not itself the reality. It is at its very earliest stages of development. (5) It aims to discover some empirical laws which can be tested against real data to describe social and political behavior from another viewpoint, which is not exclusive. In addition to the above five positive statements, the following five negative statements can be enumerated in order to set the landmarks for sociophysics to be able to develop. The negative statements are about what sociophysics should not be. (1) Sociophysics does not aim to establish absolute truths, which would govern all aspects of human behavior. It does not pretend, either, to have a religious-like status nor to become the ultimate theory of society. (2) Sociophysics does not focus on the search for some mathematical paradigm to the description of the social sciences. It does not emphasis the use of mathematics in describing human behavior. (3) Sociophysics does not pretend to be the first attempt ever to quantify social behavior using either mathematics or statistics. (4) Sociophysics does not aim at establishing theorems in the way that social scientists have been using mathematics for many years, as for instance in the extensive use of game theory in both the economic and political sciences. 74 4 Sociophysics: Weaknesses, Achievements, and Challenges

(5) Sociophysics does not intend to substitute itself to the classical approaches used in the social sciences. It does not replace the traditional studies by social scientists and does not claim any superiority over them. The above ten statements are proposed for establishing sociophysics as a ﬁeld of research in itself. It is clearly open to discussion and can be amended. Its achievements will certainly contribute to shape it more adequately. It is the ﬁrst explicit framework that aims to set up an epistemological foundation to our discipline. It has the vocation to be adjusted to the outcome of its own research.

4.5 Flashback to the Origins

In recent years, the physics-like modeling of social and political phenomena has been the subject of a growing number of papers from statistical mechanics and condensed matter physicists. The trend has become significant in terms of numbers of papers and international conferences and workshops. Although the field of Sociophysics is still at an early stage of development, it is creating a great deal of interest and could soon reach a critical size, at which point it could start either to decline or to turn into a new science. The perspectives and stakes are as numerous as the dangers and possible misuses. At this stage, to get a better understanding of what sociophysics is, and to optimize the chances of it succeeding in becoming a solid quantitative science, it is useful to trace back its origins within physics. I have already given [17] a personal testimony on the earlier stages of sociophysics, which was initiated in the late 1970s and beginning of the 1980s. It is worth recalling the historical context of condensed matter physics at that time. In 1971, after decades of theoretical difficulties with the ongoing failure to explain the fascinating puzzle of phase transitions, all of a sudden, Wilson, a nonprolific physicist in terms of number of publications, suddenly published two papers [18] with the so long awaited solution. One year later, Wilson and Fisher made the theory practical for calculations with the so-called -expansion [19]. The solving of the enigma of critical phenomena opened what can be described as a “golden age” of physics in terms of quality and quantity. The concepts and tools of the renormalization group techniques opened up a wide area of intense and active research, which has involved hundreds of physicists and the publication of thousands of both theoretical and experimental papers. The concepts of universality classes, irrelevant variables, and critical dimensions, lower and upper, were of a fantastic excitement in creating an incredible momentum in physics research. At that time, in the mid-seventies, my advocating of sociophysics as a natural extension of condensed matter research among physicists was considered as being pure nonsense by everyone except a very few exceptions. Moreover, such an option was perceived as a threat that could jeopardize the tremendous status and strength of physics at that time. Claiming that statistical physics could be applied to economy, 4.6 The Soviet-Like Rewriting of the History of Sociophysics 75

finance, sociology, politics, and psychology prompted a unanimous hostility from physicists. Such a reaction was driven by the elitist contempt physicists shared toward the social sciences in general. During the following years I kept on trying to spread my convictions amongst physicists, without much success. In fact, what I was doing was envisioning the tremendous perspectives of such a crossing of fields for the understanding of human society. It was obvious to me that on the one hand we had some new powerful tools and concepts, and on the other, there existed a great poverty in the understanding of social behavior. I was basing my call for the creation of this new field as a natural development of physics on both its internal limits as a field of research, and on the growing contradictions of physicists as a social community. Being too skilled to only tackle complex technical problems, they will ineluctably be tempted to play around with their so powerful “toys” in other areas. On this basis, I developed a coherent analysis with a dynamic perspective of a sociocultural phenomenon [20–22]. I foresaw the emergence of econophysics, sociophysics, and other “X” physics as an inevitable and deterministic outcome of the dialectical transformation of the internal contradictions, which would arise from the interplay between the subjects of physics, the tools of physics, and the people doing physics. I viewed it as a natural and direct outbreak of the last major and fundamental success of physics as a whole field of research. But right ideas can stay confined to a few people for a long time before they eventually propagate. In my case, they became plausible and attractive for others only in recent years.

4.6 The Soviet-Like Rewriting of the History of Sociophysics

With sociophysics being an active and recognized field of research, but not yet a mature one, only a few books [12, 15, 16] and review papers [1, 5] have been published so far. But it is significant to notice that the ones in which the history and development of sociophysics are discussed are adopting the same politically correct but false viewpoint. They are articulated around three central preconceptions, which are: (1) The refusal to admit its conflicting aspects within the physics community. (2) The discarding of the first founding papers, that were controversial by their very existence. (3) The artificial link to the social sciences with the construction of an imaginary perfect and smooth path of its emergence. Accordingly, sociophysics is given a biased positioning so as to fit a nice, linear historical chronology from some fake origin, while removing all the controversial aspects. Because sociophysics is now accepted by research institutions and by most 76 4 Sociophysics: Weaknesses, Achievements, and Challenges physicists, all of the “negative elements” are deliberately ignored. Only the harmony of the research community is underlined. In this way, it is necessary to leave to one side the perturbing physicist I have been at the earlier stages of sociophysics. I will focus on two publications which are emblematic of this Soviet-like rewriting of the history of sociophysics. One is a paper by Stauffer, “Introduction to Statistical Physics outside Physics” [23] and the other is a book by Ball, “Utopia theory” [24]. While the two address the question from a different perspective, discussing rather different aspects, they both adopt the same politically correct posture to legitimate the presentation of sociophysics by some falsely reconstructed history linking this field to some past work in the social sciences. They even argue about its relevance, acceptance, and rejection, but at the same time they totally ignore my own many founding contributions. I am not mentioning these facts to argue or to express what would be a legitimate bitterness. I am emphasizing it to illustrate how in general, writing about science history can be misleading and even wrong. What we have here is a very good example of what can happen with respect to other questions of science breaking into other fields. It is particularly because I do not take this as solely a personal matter that I comment on it at length. To build a history of the breakout of physics outside of physics by physicists, they present both a selection of isolated historical facts, which in turn are used to trace back some linear path to reproduce their own view of a happy and smooth view of the dynamics of science. Simultaneously, they deliberately ignore certain facts, seminal contributions, and instrumental events. Not only do they totally ignore the very conflicting nature of sociophysics with respect to the traditional social sciences but they also silence the initial stiff opposition of the physics community to it. Sociophysics is thus presented as a natural and smooth outgrowth from the social sciences. To ground their a priori view, and to manipulate the real history, they both trace back the phenomenon of sociophysics within the social sciences by setting an organic link to the work by Schelling in 1971 [25]. They claim he was already applying physics, although without knowing it. Along this line of thinking, a few papers have tried to show that the so-called Schelling model was somehow the Ising model [26]. But it is different from the Ising model. The Schelling result for urban segregation is based on a rudimentary simulation done-by-hand using a mixture of two species, at fixed densities. It is a multiagent-like simulation, nothing less nothing more. That does not make its finding uninteresting but neither make it a foundation of sociophysics. Although no explicit link to physics was mentioned, Schelling is given a kind of “innocent paternity” to sociophysics. Moreover, these pseudohistorians do not comment on the fact that Schelling’s work has been anecdotal in being totally ignored by physicists, including myself, up until recently. Nevertheless, it is an interesting contribution, which validates the compatibility of sociophysics with the social sciences. The above “sleight of hand” trick is remarkable in apparently legitimizing the sociophysics approach as being a natural extension from the social sciences itself. 4.7 Fatherhood with a Touch of Humor 77

By doing so, all the conflicting and controversial parts of it, with respect to the social sciences and within physics are simply erased. It is thus unsubstantiated from its necessarily conflicting and controversial features. It is important to underline that sociophysics has been developed by physicists within physics and independently of influence or link to earlier social sciences work. It is essentially an internal development among physicists, which does not preclude a qualitative change in the future. Up to now, sociophysics has grown essentially within physics. This fact is both its strength and its weakness. Another typical attempt of such a revisionist construction of the genesis of sociophysics can be found in a paper by Stauffer and Solomon [27]. Of course, pushing along this fictional kind of approach, one could state that in fact it was even Aristotle who initiated this topic of sociophysics a good few centuries ago. From the perspective of today, he could be regarded as a physicist and since he also lectured about politics, he must have been in some way a sociophysicist as well [28]. Soon we will be using physics to study atoms, which will appear as being a breakthrough from “antique” sociophysics! Besides my rather bruised ego, it is important to state the real facts even if they are crude and disturbing. This should matter especially for historians of science. I am stepping in to denounce this false reconstruction of history for mainly three reasons, which are: • It is of importance to state the real but crude facts, which should particularly matter to scientists, while not giving up the human content of the history. This is all the more so, since this human aspect reveals some fundamental questions about the very nature of sociophysics and its development. • It is crucial to oppose an epistemological falsification of sociophysics while it is still possible, since most of the few protagonists are still alive. • Claiming a wrong idealized origin to sociophysics may well lead to a total misconduct in the future applications of physics outside physics.

4.7 Fatherhood with a Touch of Humor

Bearing in mind the above discussion, I do claim to be the father of sociophysics, without any false humility. Nevertheless, I immediately add that I am fully aware and I acknowledge that: • As is well known in human reproduction, to be the father does not mean to be either the ﬁrst one nor the unique one to have had intercourse with the mother. • Moreover, as with any intellectual matter, no one is ever the ﬁrst one to start from scratch. As wisely stated by the former King Solomon “Nothing is new under the sun”. • The question of the mother, if any, is open! • Last but not least, the existence of a father presupposed the existence of a grand- father whose identity could be multiple. And so on and so forth. 78 4 Sociophysics: Weaknesses, Achievements, and Challenges

Accordingly I am not claiming to be the first one, who has thought or done something, which can be connected to sociophysics. I clearly do not ignore previous work associated to the field. I have even cited this work, that which I knew of, in the 1982 manifesto of sociophysics entitled “Sociophysics: A mean behavior model for the process of strike” [29]. The paper contains a whole argumentation about why and how the crossover to other fields should be justified. A basic discussion about the ethical implications of such research is also included. This founding paper, which I cite, was co-authored with two other young physicists, Yuval Gefen and Yonathan Shapir. This fact does not prevent me from standing as the founder of sociophysics. The reason for this is that it is not a single contribution that matters, but a series of contributions together with an explicit vision clearly framing the field. Among the earlier contributions cited in [29] is the 1971 Weidlich paper “The statistical description of polarization phenomena in society”. He used Master equations in the spirit of Lotka–Volterra-like population dynamics to study opinion dynamics [30]. Weidlich named his approach sociodynamics and stated that it was part of synergetics, a field developed by Haken [31], which has not much to do with sociophysics. Later on, Weidlich kept on developing his work, continuing to use the Master Equation approach [13]. Accordingly, it cannot be defined as a founding contribution to sociophysics. I would like to take the opportunity in this book to cite an unknown short- note by Stenflo and Wilhelmsson published in 1981 entitled “Nonlinearities and soliton-like structure in society” [32]. I was ignorant of its existence for many years. Later, Stenflo told me that at the time, it was the condemnation of some of their colleagues that prompted them to give up pursuing the investigation of their idea of a “plasmasociology.” It made their note a one shot, isolated contribution. I am the father of sociophysics not because of one or two earlier papers, or because I would have been the first one ever to think about applying physics to social behavior. I founded sociophysics as a field simultaneously from a philosophical viewpoint, an epistemological analysis, a consideration of physicists as a social body, and with the concrete contributions of building a series of models, which use some basic statistical physics models to deal with a large spectrum of social and political problems. In addition, I have been an active militant advocating the cause for 30 years among my colleagues, as well as those outside physics. During these earlier pioneering years, I thus organized a series of seminars along this line to try and get others to join in my vision. • The first series was at Tel Aviv University with Ben Jacob. We setup a “Bullshit Seminar” at the physics department from 1978–1980 to address all these questions from a “revolutionary perspective” within the physicist’s community. The issue at that time was to blow up the rigid frontiers of physics to extend our field of investigation from nonliving matter to the living worlds. Ben Jacob was advocating the use of physics in biology, and myself, its use in the human world. The seminars generated a lot of interest, as well as polemic and even attacks. It was a success in terms of public and impact. 4.8 Basic Weaknesses of Growing Sociophysics 79

• I held the second series of seminars at NYU during the years of 1981–1983, gathering together physicists, economists, political scientists, computer scientists, and psychologists. Besides having an interdisciplinary talk, we used to chat around a cheese and wine table. It yielded a great deal of interest and exchange. Maybe the good wines were to blame for its success? • I repeated the experience again in Paris during 1991 at the Pierre and Marie Curie University with most of the few people interested in the approach in France. Surprisingly, there was no cheese and no wine, only passionate discussion. So maybe the wine was, after all, “innocent” in the seminar success, although rather enjoyable, at NYU. It is thus my feeling of unfairness and injustice of not being given the legitimate credit for my vision, my fight, and my contributions to the birth and establishment of sociophysics, which drives me to take such an uncomfortable posture in claiming its paternity. The task may even appear ridiculous, given that the field is growing, with many people not citing me adequately for my series of anterior founding papers. It even made me totally paranoiac when I found papers appearing in leading journals such as the Physical Review Letters based on my exact models that were presented as being novel. For one of them, I had to exchange several emails to have the authors recognize the fact and have this corrected in a follow up paper in the Physical Review E. But who will connect the two papers within the huge inflation of the number of published papers? What is seen as mattering is the first publication in the prestigious Physical Review Letters. This is the one which is cited, giving automatically the credit to its authors. However, they are entitled to claim that they have been cleared from misconduct since they acknowledged the omission later on. It is a very efficient “technique.” which cannot be found out by just reading the papers. Somehow everyone is beyond reproach, those who missed crediting the original work since they did acknowledge the fact later in a following publication, as well as those who cite the publication that claimed to be first since it is the most visible one. The unfair benefit will thus prosper forever. I could cite names and references but what for? I do not intend to make the issue personal, although I am personally bruised and I do not aim either to hurt anyone. I am also aware that reporting on these practices may make me appear pathetic and bitter, but yet, like in the song: And I said to myself what a “wonderful” world! I decided not to keep silent even at that price, because it is of importance to make visible the hidden part of the “wonderful” world of research. I am convinced that such practices happen very often in all fields of fundamental research where there are no patents that would allow possible legal investigations.

4.8 Basic Weaknesses of Growing Sociophysics

Restoring the truth about the past history is just as important as scrutinizing the current situation of sociophysics in order to forecast possible evolutions, which are nevertheless only conjectures. 80 4 Sociophysics: Weaknesses, Achievements, and Challenges

At the beginning of the 21st century, sociophysics is a flourishing field of statistical physics. An increasing number of physicists are joining the field and all physics journals accept related papers. Every year several conferences include sociophysics topics. Summer schools and workshops contribute a good deal to the growth of this new field of research. Outside physics, several social scientists are starting to become interested, although with a certain amount of doubt and caution. Journalists are usually interested, and several articles are published regularly on associated subjects in major newspapers around the world. On this basis, the strategic question is to determine if sociophysics can really become a predictive social tool. However to figure out even the beginning of an answer it is necessary to analyze what are the respective strengths and weaknesses within the implementation of sociophysics in terms of research papers and solid achievements with respect to the real world of social behavior. Among the major weaknesses stands, as could be expected, what is currently happening in all fields of research. Namely, there is a dilute clustering of publications in a kind of fragmentation within informal lobbies, each ignoring what the other is doing. No credit is usually given to the relevant papers, and earlier work is republished as being new. Plagiarism, conscious, or unconscious, is rife. However, this aspect of research is happening in the emerging new field of sociophysics at a wider scale. At the time of the internet, if a paper is not readily available, then it does not exist. Moreover, if it was published too long ago, then it exists even less. The common argument to justify such a practice is the impossibility of keeping track of the millions of papers being published these days at very high speed. What is not delivered within the first two pages of a Google search does not exist. In addition, no sense of guilt or wrongness is considered since each author feels that they do not receive the credit that they deserve from others, and, therefore, cannot be blamed for not giving the necessary credit to the relevant papers they did not know about. Moreover, the imperative of searching and finding out about anterior work no longer seems to be a requirement for scientific activity. It is a serious change in research practice in which only very short-term and restricted memory is becoming the rule. In the long run, it can only jeopardize the solidity of scientific research. But to analyze such a new trend further is out the scope of the present book. It is unfortunately a general feature of modern electronic publishing of scientific research. Tackling this fundamental change in the way science progresses will be the major challenge of this new century. Within our field of sociophysics, other specific misconduct exists. For instance, in a given paper, once a paper is referred to from a social sciences journal, it is put forward to produce a social sciences legitimacy to the corresponding work, even if it is not connected to the work itself. Then, everyone recopies the reference without reading it. It becomes the “natural” social source of the problem treated by the physicists. Such practices will not help in making sociophysics a strong and robust field of science. Up until now, several models have been elaborated together with new concepts and many numerical simulations. While they are claimed to describe a series of 4.9 The Positive Achievements of Sociophysics so Far 81 social and political situations, most of the work, including some of mine, is indeed totally cut off from any real system. Moreover, each one focuses only on its own idea, often limited to a local update rule motivated by a qualitative analogy with a rather general social feature. In addition, the same models are often rephrased and republished as new ones. It has happened several times with my own models. But although I may appear paranoiac, as mentioned earlier, I do not believe that everyone is against me. So I do not take it personally, but on the contrary I look at it as a sign of a serious failure of our growing field of research. All these ill practices are the direct result of the fact that at the moment no unified corpus is being constructed for sociophysics, and not even a theoretical framework. There exists much research in the field, but mostly carried out by separate groups or individuals, who often totally ignore previous research. No agreed criterion is available to validate a model. No single specific social problem is clearly identified as “the problem” to be solved in the field. There are only claims of relevance to hypothetical applications that are made here and there. As mentioned above, the most “hip” thing to do is to cite some old papers by a social scientist to create the illusion of an organic legitimacy of the approach to the traditional social sciences and at the same time to show off knowledge of historical culture.

4.9 The Positive Achievements of Sociophysics so Far

First, it is worth underlining that up until now, sociophysics has not led to any negative achievements. We can then question whether there exists any positive ones? Indeed, what has sociophysics accomplished so far, if anything? The answer is yes, sociophysics can already boast of a series of achievements, although not all on the same footing. Here is a nonexhaustive list. • Several models have been elaborated together with new concepts and many numerical simulations to shed new light on several social phenomena. But these results are at a rather qualitative and formal level. • Some general qualitative features and properties of both opinion dynamics and voting have been given new points of views. However, as for the preceding item, these viewpoints are still very general. • Some past political events have been revisited with unexpected explanations. – A voting model was developed that provided a key to understanding the collapse of the communist party in the last century [33–38]. In particular, it yielded a coherent explanation to their sudden collapse following decades of frozen leaderships. Indeed, the model shows how a very long and sustained increasing opposition within the party can have no effect at all for many years before all of a sudden it turns the leadership up-side down without any warning. 82 4 Sociophysics: Weaknesses, Achievements, and Challenges

– A contrarian model was proposed to explain the fifty–fifty 2000 American and 2002 German elections after the events [39–41]. Using this model, it was then advocated that if it is valid, then more fifty–fifty outcomes should occur in the near future in democracies. – The so called Sznajd model [42] was used to explain the 1998 Brazilian elections [43]. – A social percolation model proposes the basics of a global framework concerning terrorism events such as the September 11th attacks in 2001 [10, 11]. – A spin glass-like model was used to explain the European stability during the cold war period as well as the Eastern European instabilities, which followed the dissolution of the Warsaw pact. This model applied in particular to the ex- Yugoslavia. The recent increase of EU members was also suggested [44, 45]. – The Indian elections have been investigated [46]. – The distribution of the number of votes received by candidates in proportional elections was shown to obey some universal scaling function, which was identical in different countries and years [47]. However, while coming up with an explanation to a past event is satisfying, it is not really convincing, and more so when a different model is used for each event. One step further has also been accomplished; that of predicting certain events that occur in the near future but without mentioning a date. • Using my voting model, I predicted how the extreme right party in France, the National Front, could eventually arrive to power democratically and by total surprise. No one believed it, but what was predicted did actually nearly happen. The National Front scenario did occur in part in 2002 with its leader running in the second round for president, to the total surprise of everyone, including the NF itself [48, 49]. • Using my contrarian model, I predicted that fifty–fifty elections were about to occur again and become a common feature of western democracies. No one believed fifty–fifty elections could happen again. But what was predicted did happen. Fifty–fifty elections did occur again in the 2005 German, 2006 Italian, 2006 Mexican, and Czech elections in contradiction to all the polls and analysts’ predictions [39–41]. • Elections in Germany and Bavaria were investigated by Schneider and Hirtreiter who not only explained the past, but also predicted that the CSU will rule Bavaria forever, if the current political structure remains valid [50, 51]. No one gave a serious thought to the above successful predictions, being too busy trying to recover from the psychological and political collapse which resulted from these totally unexpected political events. And, even if it sounds nice, they are still not completely convincing arguments to validate the approach since in both cases neither a date nor a precise location were given. It is always easy to say “I told you so!” after the event. Before ending this part, let me point out a worthwhile digression that highlights the existence of the remaining restrictions from physics institutions toward 4.10 The First Sociophysics Successful Prediction of a Precise Event 83 sociophysics. My contrarian paper was first submitted to PRL where it was withheld by the editor, who refused to send it to referees, arguing that it was too political. Later on, the Physica A Advisory Editor Marcel Ausloos showed a much more open mind and accepted the processing of the paper. It was refereed and was eventually accepted after some revisions.

4.10 The First Sociophysics Successful Prediction of a Precise Event

Along the same lines of a general prediction, using my minority spreading model, I made the speciﬁc recommendation about the temptation to hold a referendum in the process of the European construction within European countries. In my 2002 paper, I concluded with the following words [52]: To give some real life illustrations of our model, we can cite events related to the European Union which all came as a surprise. From the beginning of its construction there has never been a large public debate in most of the involved countries. The whole process came through government decisions though most people always have seemed to agree on this construction. At the same time European opponents have been systematically urging for public debates. Such a demand sounds absurd, knowing that a majority of people are in favor of the European Union. But anyhow, most European governments have been reluctant to hold a referendum on the issue. At odds with this position, several years ago, the French president Mitterand decided to run a referendum concerning the acceptance or not of the Maastricht agreement. While a large success of the “Yes” vote was taken for granted it indeed made it just a bit beyond the required ﬁfty percent. The more people discussed the issue, the less support there was for the proposal. It is even possible to conjecture that an additional two weeks extension of the public debate would have made the “No” vote win. I then reiterate the conclusion in 2004 stating in the conclusion of another paper that [53]: Applying our results to the European Union leads to the conclusion that it would be rather misleading to initiate large public debates in most of the countries involved. Indeed, even starting from a huge initial majority of people in favor of the European Union, an open and free debate would lead to the creation of a huge majority hostile to the European Union. This provides a strong ground to legitimize the on-going reluctance of most European governments to hold referendums on associated issues. It is of importance to underline that when these statements were made, no referendum was planned in Europe with respect to the proposal of a European constitution. 84 4 Sociophysics: Weaknesses, Achievements, and Challenges

4.10.1 Taking Risks to Validate Sociophysics

Then, at the end of 2004 in France, Jacques Chirac decided to hold a referendum about adopting the project of the European constitution. This turned out to be a great opportunity to make a well defined and precise prediction. Indeed, if it is nice to produce explanations of past opinion formation issues, and sociophysics has proven to be able to do this, it would be much more convincing to predict an outcome of a future opinion issue. This referendum was the perfect opportunity to apply my opinion model to make a prediction concerning the outcome of the referendum. The polls were giving 20% to the “No” vote, 70% to the “Yes” vote and 10% to abstentions. It was taken for granted by everyone that the “Yes” vote would win the day, including those from the “No” vote. The unique issue was the degree of participation in the vote, many abstentions being feared, which could weaken the expression of the French people’s support for Europe. In the mean time, I was able to introduce the existence of heterogeneous beliefs to make the minority opinion spreading model applicable to more fuzzy issues with different subpopulations [54]. I then made the analysis using rudimentary investigations by myself. My conclusion was that the critical threshold for the “No” vote to start to inflate as a result of public debate was located in the vicinity of 15%. I also concluded that a long time would be necessary for the “No” vote to pass over the 50% mark. Several months ahead of the vote, before the official campaign started, the “No” vote was scoring around 20% in the polls. There were five months of debate ahead. Therefore, from the model, within the given conditions of the debate during this period, the prediction was that the “No” vote would eventually win the day. This outcome was quite clearcut; the only problem was that I could not believe it myself. A huge majority of people were in favor of the “Yes” vote, almost all political leaders were in favor of the “Yes” vote, all the media were in favor of the “Yes” vote, all the stars were in favor of the “Yes” vote and still other vociferous groups were in favor of the “Yes” vote. For almost everyone, France could not say “No” to Europe; that would have been totally absurd. And I agreed with this. It happened that I was interviewed about my minority spreading model and its application to the coming referendum by a journalist from the newspaper Le Monde [55]. At the end of the interview, he asked me “Are you sure you want to have your conclusion printed in black and white? Your theory is nice, but the conclusion is nonsense and you will lose all credibility in the future”. For a minute, I became really scared, asking myself “Why announce an event that I myself do not believe in? Why exhibit myself in front of millions of people as a fool?” As the journalist told me “Printed matter lasts for a very long time and will be used whenever necessary against your approach. Think about it.” And I did. I realized that what this was all about was at the very core of the sociophysics challenge; not to try to win a personal reputation, but to build a robust theory of social behavior with a heuristic power, based on a scientific procedure, which can be tested against real-life events. It was not and should not be a personal issue. 4.10 The First Sociophysics Successful Prediction of a Precise Event 85

A solid social science could be built only by making predictions, which could be tested against reality. If a prediction turned out to be right, it would validate, at least in part, the model. If the prediction turned out to be wrong, the reasons for the failure would provide either more insights into the model, or lead to an abandoning of part or all of the model. This is how we proceed in physics. So, the article in Le Monde was published with my very clear prediction of a “No” victory [55].

4.10.2 When the Prediction Turns Out to be True

On May 29, 2005, the “No” won the vote, with a score of 55%. It was a total blow to all the French elite. Up until the last minute, no-one among the analysts could believe that it would happen. But it did. This is an important case that goes to show that it is not always the media that makes public opinion. I myself had contradictory feelings as a result, with on the one hand, the legitimate satisfaction of having forecast such an event using my opinion dynamics model, and on the other, the fear of how the European construction would find a way out of such a failure. The new simplified treaty which was set up later resolved what could have been quite a setback for Europe. Coming back to the meaning of my successful prediction of the “No” victory, it is worth emphasizing the following points: • This particular case study was the very first time that the outcome of a political vote had been successfully predicted using a model from sociophysics. • Moreover, it was a highly improbable event. It was not a heads or tails chance. It was not even a random selection with a low probability. Up until a few weeks before the vote, it was a zero probability event. • In addition, the prediction was made several months before the actual vote, in contradiction to all the polls and other predictions. • On top of this, I predicted a few years earlier that the use of a referendum to strengthen the European construction would be counterproductive, leading to a reversal of the current large public support. It is also of importance to add some points of safeguard: • A single successful prediction is not enough to conclude that the model is correct. More tests are required. • However, the peculiarity of sociophysics dealing with human behavior makes it impossible to reproduce exactly the same “experiment” or real social event. Therefore, it should be a series of successful predictions for similar events such as different referendums, which must be the criterion for solid validation. Nevertheless, this unexpected successful prediction at least validates the model in the sense that it is worth continuing investigating it. It also justifies the hope that sociophysics in the future may indeed yield real predictive tools. 86 4 Sociophysics: Weaknesses, Achievements, and Challenges

At the same time, it is of central importance to clearly note that, if it is proven to be true, i.e., sociophysics becomes a solid predictive science, it will have drastic and unknown consequences on our social and political lives. No one is currently prepared for such a possibility.

4.10.3 When a Prediction Fails

Motivated by the success of my 2004 general prediction of the increasing occurrence of fifty–fifty elections in world democracies, with several cases including Germany, Italy, Mexico, and the Czech Republic, in September 2006 I had an article published in the paper, Le Monde [56] about the possible outcome of the May 2007 French presidential election. I warned against the danger to the French social stability of a fifty–fifty election in the case of a Royal–Sarkozy second round. Nevertheless this prediction was not as precise as for the 2005 referendum. I mentioned it as a possibility since at that time the list of the running candidates was still unknown and not official. Moreover, even the programs of the respective candidates were unavailable. However, the second run did occur with a Royal–Sarkozy standoff and Sarkozy did win with a substantial margin. No fifty–fifty scenario occurred. This failure prompted me to react in the way that scientists do, examining the basic hypotheses of the model. This study indicated firstly that contrarian behavior is not systematic, but is only a function of the general state of society. Different situations may produce different levels of activation of contrarian behavior. Secondly, this misfit sheds light on the fact that other types of behavior should be included in our model of opinion formation. Accordingly, we have investigated the effect of having inflexible agents to discover some similarities and differences with contrarians [57].

4.11 Proposal to Establish a Road Map

The successes and failures mentioned above clearly legitimate the sociophysics approach and simultaneously indicate that much more research has to be carried out. The paradoxical aspect of the growth of sociophysics is that physicists doing sociophysics would like to have social scientists become interested in their work. They complain that social scientists do not read their contributions published in physics journals while they try to establish a somewhat ﬁctional organic link to the social sciences by citing one or two “legitimating” papers. Then, they have probably never read most of these papers but only recopied the references from one paper to another. At the same time, they do not read my earlier inaugural papers because either they have been published in nonphysics journals, which are not immediately at hand, or they were published too many years ago, outside of their memory that extends only to the last few months. 4.12 What the Climatologists Did with the IPCC Should Not Be Repeated 87

Moreover, while hoping and calling for a real link with social scientists, they have mostly ignored my extended work of collaboration with the social scientist, Serge Moscovici, which is a rather rare case, that deserves more attention. All these facts are given to show up some of the aspects of the real dynamics of scientific research, which are rarely given so explicitly. It underlines the current trend of the scientific community to produce papers by following the rather transient fashion of moving at quite a quick pace. With respect to sociophysics, I prefer to be optimistic and consider that all the mentioned weaknesses are the direct outcome of its childhood. The debate is open, as is the future of sociophysics. I think it is time to reduce isolated work, or more precisely, in addition to whatever work is being carried out to focus on a precise list of real events in order to try and predict them, so as to be more efficient and fruitful. I make the following proposal to the sociophysics community: I propose selecting collectively some real political issues that will take place in the next three to five years, and then everyone interested applying their respective models in order to make predictions about them. Once the results are known, each model could be evaluated according to its respective overlap with the outcomes of these real events. Both what was right and what was wrong about the outcomes and the models could then be combined to determine the next step in our research.

4.12 What the Climatologists Did with the IPCC Should Not Be Repeated

However, if it is of importance to start to coordinate the various projects carried out in sociophysics, it does not mean that we should set up an organization to monitor it. It is of the utmost importance to stress that any temptation to reproduce the “successful” precedent of climatology would certainly open a Pandora’s box towards a new kind of totalitarianism. Why? Twenty years ago, climatology was far from being in the headlines of research. Neither much money nor many jobs were allocated to what was then a minor discipline. At some point, climatologists set up several national and international structures to both coordinate and to unify their activities. Indeed they managed to come up with a very efﬁcient network which turned their discipline into a priority subject at the beginning of the 21st century. They were able to embody international institutions including the United Nations as well as major national agencies all over the world. Taking advantage of their dominant position, they went on to predict certain catastrophes, together with the corresponding policies that needed to be takeninordertoavoidthem. However, while the initial motivation was praiseworthy, the net result has become a politically oriented machinery which makes false statements in order to reach some ideological goals. The whole framework has been wrongly claimed in the name of science and scientiﬁc proof [58–61]. 88 4 Sociophysics: Weaknesses, Achievements, and Challenges

To imitate such a construction would be disastrous both for sociophysics and humanity. At first it would give definitive strength and means to sociophysics but then it would unfortunately lead to the determining of a single voice by some institutional body in order to settle the “scientific” guidelines for a perfect society. In a similar way that today, climatologists update their advice to policy makers, sociophysicists would dictate their demands to politicians. And, as has happened with the opponents of climatology, the opponents would be denied the right to contest the official truth established by the representative institution. However, as the context would be political, the chances are that rather quickly, a dictatorship would emerge, founded in the name of science. Therefore, any sociophysicist should adamantly oppose any attempt to establish a single body to deliver a unique voice for the field. Therefore, while scientists must be engaged in the various public debates, and should popularize their research and their suggestions, they must do it individually and never as an organized group. Such a practice will guarantee the free and contradictory expressions of different scientists. It will avoid the politicizing of science in which some scientists would be tempted to confiscate certain aspects of science in order to promote their ideological view and preserve their status and power. Otherwise, the social consequences could be really dramatic.

References

1. S. Galam, “Sociophysics: A review of Galam models”, Int. J. Mod. Phys. C 19 409–440 (2008) 2. R. N. Mantegna and H. E. Stanley, “An Introduction to Econophysics”, Cambridge University Press, England (2000) 3. A. L. Barabasi´ and R. Albert, “Statistical mechanics of complex networks”, Rev. Mod. Phys., 74, 47 (2002) 4. M. Droz and A. Pekalski, “Population dynamics with or without evolution: a physicist’s approach”, Physica A 336, 84 (2004) 5. C. Castellano, S. Fortunato, V. Loreto, “Statistical physics of social dynamics”, Rev. Mod. Phys. 81, 591–646 (2009) 6. S. Galam, B. Chopard, A. Masselot and M. Droz, “Competing Species Dynamics”, Eur. Phys. J. B 4, 529 (1998) 7. B. Chopard, M. Droz and S. Galam, “An Evolution Theory in Finite Size Systems”, Eur. Phys. J. B 16, Rapid Note, 575 (2000) 8. S. Galam, B. Chopard and M. Droz, “Killer geometries in competing species dynamics”, Physica A 314, 256 (2002) 9. J. Majewski, H. Li and J. Ott, “The Ising Model in Physics and Statistical Genetics”, The American Journal of Human Genetics 69, 853 (2001) 10. S. Galam, “The September 11 attack: A percolation of individual passive support”, Eur. Phys. J. B 26 Rapid Note, 269–272 (2002) 11. S. Galam and A. Mauger, “On reducing terrorism power: a hint from physics”, Physica A 323, 695–704 (2003) 12. S. Moss de Oliveira, P. M. C. de Oliveira, and D. Stauffer, “Evolution, Money, War, and Computers—Non-Traditional Applications of Computational Statistical Physics”, Teubner, Stuttgart-Leipzig (1999) References 89

13. W. Weidlich, “Sociodynamics; A Systematic Approach to Mathematical Modelling in the Social Sciences”, Harwood Academic Publishers, Amsterdam (2000) 14. Frank Schweitzer, Brownian Agents and Active Particles: On the Emergence of Complex Behavior in the Natural and Social Sciences, Springer, Berlin (2003) 15. D. Stauffer, S. Moss de Oliveira, P.M.C. de Oliveira, J.S. Sa Martins, “Biology, Sociology, Geology by Computational Physicists”, Elsevier, Amsterdam (2006) 16. B. K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), “Econophysics and Sociophysics: Trends and Perspectives”, Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim (2006) 17. S. Galam, “Sociophysics: a personal testimony”, Physica A 336, 49 (2004) 18. K. G. Wilson and J. Kogut, “The renormalization group and the epsilon expansion”, Phys. Rep. 12C, 75 (1974) 19. K. G. Wilson and M. E. Fisher, “Critical Exponents in 3.99 Dimensions”, Phys. Rev. Lett. 28, 240 (1972) 20. S. Galam, “About imperialism of physics”, Fundamenta Scientiae 3, 125 (1982) 21. P. Pfeuty and S. Galam, “Les physiciens et la frustrations des electrons”,´ La Recherche. July–August, 23 (1981) 22. S. Galam and P. Pfeuty, “Physicists are frustrated”, Physics Today Letter, April, 89 (1982) 23. P. Ball, “Utopia theory”, Phys. World October, 7 (2003) 24. D. Stauffer, “Introduction to statistical physics outside physics”, Physica A 336, 1 (2004) 25. T. C. Schelling, “Dynamic Models of Segregation”, J. Math. Sociology 1, 143 (1971) 26. D. Stauffer and S. Solomon, “Ising, Schelling and self-organising segregation”, Eur. Phys. J. B 57, 473 (2007) 27. D. Stauffer and S. Solomon, “Applications of Physics and Mathematics to Social Science”, arXiv:0801.0121 (2008) 28. J. J. Schneider, private communication (2007) 29. S. Galam, Y. Gefen and Y. Shapir, Sociophysics: A mean behavior model for the process of strike, Journal of Mathematical Sociology 9, 1 (1982) 30. W. Weidlich, “The statistical description of polarization phenomena in society”, British. J. Math. Stat. Psychol. 24, 251 (1971) 31. W. Weidlich, “Synergetic modelling concepts for sociodynamics with application to collective political formation”, J. Math. Sociology, 18 (1994) 267–291 32. H. Wilhelmsson and L. Stenflo, “Nonlinearities and soliton-like structure in society”, Specula- tions in Science and technology 4, 297 (1981) 33. S. Galam, “Majority rule, hierarchical structures and democratic totalitarianism: a statistical approach”, J. Math. Psychology 30 426 (1986) 34. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. Stat. Phys. 61, 943 (1990) 35. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191 (1991) 36. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organisa- tions”, in: Models of Self-organizationin Complex Systems (Moses), Akademie-Verlag, Berlin, vol. 64, 53 (1991) 37. S. Galam, Paradoxes de la regle majoritaire dans les systemes hirarchiques, Revue de Bibliologie 38, 62 (1993) 38. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132 (1999) 39. S. Galam, “Contrarian deterministic effect: the hung elections scenario”, Physica A 333, 453 (2004) 40. C. Borghesi and S. Galam, “Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect”, Phys. Rev. E 73 066118 (2006) 41. S. Galam, “From 2000 Bush–Gore to 2006 Italian elections: voting at fifty–fifty and the contrarian effect”, Quality and Quantity Journal 41, 579 (2007) 42. K. Sznajd-Weron and J. Sznajd, “Opinion evolution in closed community”, Int. J. Mod. Phys. C 11, 1157 (2000) 90 4 Sociophysics: Weaknesses, Achievements, and Challenges

43. A.T. Bernardes, D. Stauffer and J. Kertsz, “Election results and the Sznajd model on Barabasi network”, Eur. Phys. J. B 25, 123 (2002) 44. S. Galam, “Fragmentation versus stability in bimodal coalitions”, Physica A 230, 174 (1996) 45. R. Florian and S. Galam, “Optimizing Conflicts in the Formation of Strategic Alliances”, Eur. Phys.J.B16, 189 (2000) 46. M. C. Gonzalez, A. O. Sousa and H. J. Herrmann, “Opinion formation on a deterministic pseudo-fractal network”, Int. J. Mod. Phys. C 15, 45 (2004) 47. S. Fortunato and C. Castellano, “Scaling and Universality in Proportional Elections”, Phys. Rev. Lett. 99, 138701 (2007) 48. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 49. S. Galam, “Crier, mais pourquoi”, Liberation,´ Vendredi 17 Avril, 6 (1998) 50. J. J. Schneider and C. Hirtreiter, “The Impact of election results on the member numbers of the large parties in bavaria and germany”, Int. J. Mod. Phys. C 16 (2005) 1165–1215 51. J. J. Schneider and C. Hirtreiter, “Investigation of Election Results, Numbers of Party Members, and Opinion Polls in Germany”, Int. J. Mod. Phys. C 19, 441 (2008) 52. S. Galam, “Minority Opinion Spreading in Random Geometry”, Eur. Phys. J. B Rapid Note 25, 403 (2002) 53. S. Galam, “The dynamics of minority opinion in democratic debate”, Physica A 336, 56 (2004) 54. S. Galam, “Heterogeneous beliefs, segregation, and extremism in the making of public opinions”, Phys. Rev. E 71, 046123 (2005) 55. P. Lehir, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Le Monde, Samedi 26 Fevrier,´ 23(2005) 56. S. Galam, “Pourquoi des elections´ si serrees´ ?”, Le Monde, Mercredi 20 Septembre, 22 (2006) 57. S. Galam and F. Jacobs, “The role of inflexible minorities in the breaking of democratic opinion dynamics”, Physica A 381, 366 (2007) 58. S. Galam, “Pas de certitude scientifique sur le climat,”, Le Monde, Mercredi 07 Fvrier, 20 (2007) 59. V. Maurus, “Her´ esie”,´ Le Monde, Samedi 18 Fvrier, Chronique de la mediatrice´ (2007) 60. S. Galam, “Global warming : a social phenomena”, Complexity and Security, The NATO Science for Peace and Security Programme, Chapitre 13, J. J. Ramsden and P.J. Kervalishvili (Eds.) (2008) 61. S. Galam, Les scientifiques ont perdu le Nord, Reflexions´ sur le rechauffement´ climatique, Editions´ Plons, Paris (2008) Part II Discovering the Wonderful (and Maybe Scary) World of Sociophysics Chapter 5 Sociophysics: An Overview of Emblematic Founding Models

In this chapter, I review the series of rather different models that I have been developing over the past three decades (1980–2010). They go in several directions, and focus on five different sociopolitical phenomena. To have these subjects accepted by the physics community was a long and tedious fight [1]. The first one concerns the process of group decision making. It looks for eventual universal features behind all cases of group decision making. It may apply to small as well as to large groups and includes a large spectrum of situations such as a jury having to decide on the guilt of somebody accused of murder or workers who have to undergo a strike or even a military committee who have to choose between launching a minor or a major operation. Unexpected and deranging conclusions are obtained. The second focus is on the effects of bottom-up democratic voting in hierarchical systems. The emphasis is on the study of apparent natural biases. It shows how dictatorship mechanisms are driven by common sense rules. It provides a surprising explanation to last century’s sudden collapse of Eastern European communist parties. The third theme deals with changing the global state of a given system, which may be a society or a social group of agents. The question is to determine when and how the property of a local item succeeds in modifying the global property of the whole system. One illustration is the effect of having passive supporters for a terrorist group on its geographical capacity to act. Another illustration explains why opposite feelings can coexist within the same territory and both be legitimate, such as with the feelings of security and insecurity. The fourth item is related to the spontaneous formation of coalitions in society among people, enterprises, or countries. The same scheme applies for increasing size units. It analyzes both its stability as well as the corresponding dynamics of fragmentation. It provides an unusual view of Western European stability compared with the situation in Eastern Europe following the Warsaw pact collapse. The case of ex-Yugoslavia is studied in detail. New light is shed on the stability of the world during the cold war together with the role of coalitions.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 93 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 5, © Springer Science+Business Media, LLC 2012 94 5 Sociophysics: An Overview of Emblematic Founding Models

The last point on the agenda articulates around the key issue of opinion dynamics, which has become a central element of democratic societies. Nowadays, most issues are discussed extensively among the public and institutional decisions have often complied to take into account the corresponding public opinion. On crucial decisions, winning or losing public opinion can be instrumental in being able to implement new regulations to impose behavior changes such as the right to smoke or not. To keep within the book’s pedagogical approach for each class of models, I briefly outline the original physics model and the techniques which have been used to construct the associated sociophysics framework. I also single out the main similarities as well as the salient differences between the physics model and its sociophysics counterpart. A series of snapshots of the numerous novel and counterintuitive results obtained with respect to the sociopolitical realities is included. It gives a taste of the rich variety and the drastically different angles of viewpoints with which to tackle them. An emphasis is put on the fact that using these models, several major real political events were successfully predicted within the first few years of the 21st century. It includes the prediction of the two totally improbable victories of, on the one hand, the French extreme right party in the 2000 presidential elections where it reached the second round against all odds, and the other, the “NO” vote to the 2005 French referendum on the European constitution, which was indeed unconceivable. Fifty– fifty elections were also successfully predicted to occur as they did so in several democratic countries including Germany, Italy, and Mexico. This chapter is an overview and not an in-depth investigation. For further details, a reading of the original research papers is required. The corresponding list is given in the references. On the contrary, the rest of the book focuses in great detail on the second item concerning bottom-up democratic voting and the establishment of de facto dictatorships. Each time a model is presented, the precise overlap with the physics model from which it was inspired, is outlined. Eventual novelties in the model are emphasized. The various results obtained from each class of models are enumerated, enlightening their novel and counterintuitive aspects with respect to the associated social and political frameworks. Among others, several major real political events were successfully predicted. These include the victory of the French extreme right party in the 2000 first round of the French presidential election [2–7], the voting at fifty– fifty in several democratic countries (Germany, Italy, Mexico) [8], and the victory of the “NO” vote in the 2005 French referendum on the European constitution [9–11]. For a survey of other work and reviews, I refer the reader to the following selection of references [12–14]. 5.1 In a Few Words 95

5.1 In a Few Words

At the beginning of the 21st century, sociophysics has become a recognized field of statistical physics. In the last ten years, hundreds of papers have been printed in leading international physics journals and their number is growing at a steady rate. In addition, quite a few international conferences and workshops are held every year. It is a flourishing and expanding field of research. Nevertheless, it is still anchored almost exclusively within physics. Being the opposite of the present situation, it is worth reminding that during the 1970s when sociophysics made its first faltering steps, it was received with hostility from the physics community. Only in the mid 1990s did it begin to attract several physicists from around the world. Then, 15 years were necessary to get it established up and running. Its next challenge is to build a bridge toward the social sciences, which is a different task from trying to bridge the gap with the social sciences. Sociophysics covers a growing number of topics. Numerous problems are addressed including voting, coalition formation, opinion dynamics, social networks, the evolution of languages, population dynamics, the spreading of epidemics, and terrorism. Among these topics, the subject of opinion dynamics and social networks have become mainstream in sociophysics, producing a great deal of research papers. I present here a review of the models I have been developing over the 30 years from 1980 to 2010. Most of them make up much of the pioneering work of sociophysics. The fact that they are not always given this credit results from the various flaws of the citation dynamics as well as the personal ambition of certain leading researchers. Hopefully, with time, increasing awareness, and some fighting, things will become more ethical. My models focus in particular on five sociopolitical issues including decision making, democratic voting in bottom-up hierarchical systems, and fragmentation versus coalitions among countries, opinion dynamics, and terrorism.

5.1.1 Decision Making

The question of decision making is central in many social organizations, whatever the level of their complexity. It occurs everywhere in various frameworks, right down to firms and small committees. Using the ferromagnetic Ising spin Hamil- tonian with both external and random-quenched fields at both zero and nonzero temperatures has proven very successful in tackling the problem [15–24]. In each case, the phase diagrams are constructed to obtain the landscape of the various decisions with the eventual possibility of determining the final decision. The effect of reversing an external field on the collective equilibrium state has been studied with an emphasis on the existence of nucleation phenomena. Mean field treatment is applied. The inclusion of individual biases has also been studied. 96 5 Sociophysics: An Overview of Emblematic Founding Models

5.1.2 Bottom-up Democratic Voting

Another crucial question concerns the procedure for designating representatives in order to decide and carry out policy. Often, in democratic countries, but also in nondemocratic countries, bottom-up hierarchies operate so as to asses that goal. I investigate throughout the paradox of using bottom-up democratic voting to establish de facto a dictatorship. But it is not a repressive dictatorship, only one that uses repeated voting. The possible twisting of the democratic character of using majority rules is brought to light as well as the limits of the manipulation [25–37]. The models consider a mixed population with two groups of agents A and B. A is then built from the population using local majority rules with the possibility of some power inertia bias. Tree-like networks are thus constructed, which combine a random selection of agents at the bottom from the surrounding population with an associated deterministic outcome at the top. The scheme relates to the adapting of real space renormalization group techniques in building a social and political structure.

5.1.3 Terrorism

A rather puzzling question concerns the understanding of the connection between some local properties attached to a person and the appearance or not, at the global level of the corresponding social group, of a novel property. In the case in which such a property emerges, it is usually almost impossible to erase it since there exists no direct link between it and a given individual sharing the property. The concept of percolation is found to be quite adequate to tackle this intriguing feature of social systems. It has been used to provide a consistent explanation of how it is possible to have simultaneously a feeling of safety and a feeling of a lack of safety on the same territory, in which both feelings are indeed legitimate [5]. The effect of having passive supporters of a terrorist group has also been investigated through the prism of percolation. It determines the actual geographic extension of the territory, which is open to terrorists for free movement [38–43].

5.1.4 Coalitions Versus Fragmentation

Once the Warsaw pact was disbanded, I was quite unconvinced of the rationale proposed to explain such a historical event. In particular I disagreed with the arguments given to justify the various civil wars which burst out in Eastern Europe as well as the arguments given to justify the Western European stability. On this basis, a combination of random bond and random site spin glasses was found to References 97 provide a powerful framework with which to describe the formation of coalitions and the dynamics of fragmentation among a group of countries. External and local ﬁelds were also considered together with site dilution effects in mixtures of ferro- and antiferromagnetic spin Hamiltonians. Ising and Potts variables have been used. A consistent explanation of the European situation was obtained together with some detailed analysis of the wars, which teared the ex-Yugoslavia apart [44–48].

5.1.5 Public Opinion

Public opinion has become a sensitive ingredient in the now overconnected global world, and in particular with the spread of the use of the internet. Even nondemocratic countries are now having to face up to public opinion. One most salient feature is the minority spreading effect. How and why do minority opinions, usually either against changes or in support of prejudiced views, have the final favor of large groups of people, which discuss the issue freely? The question is solved by constructing reaction–diffusion like models which combine local majority rules and agent reshuffling. Several kinds of agents are also considered. These include floaters, agents that follow the local majority, contrarians, who oppose the majority, either local or global, and inflexibles, who never change their mind [49–64]. The competition between two and three opinions is analyzed. Techniques from the real space renormalization group approach are used to solve the equations.

References

1. S. Galam, “Sociophysics: a personal testimony”, Physica A 336, 49–55 (2004) 2. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 3. S. Galam, “Crier, mais pourquoi”, Liberation,´ Vendredi 17 Avril, 6 (1998) 4. S. Galam, “Le vote majoritaire est-il totalitaire ?”, Pour La Science, Hors serie,´ Les Mathematiques´ Sociales, 90–94 July (1999) 5. S. Galam, “Citation” in an editorial from Jean dOrmesson, front page, daily newspaper Le Figaro, Mardi 4 Juin, 1 (2002) 6. S. Galam, “Risque de raz-de-maree´ FN”, Entretien, France Soir, La Une et 3, Mercredi 5 Juin (2002) 7. S. Galam, “Le FN au microscope”, Le Minotaure 6, 88–91, Avril (2004) 8. S. Galam, “Pourquoi des elections´ si serrees´ ?”, Le Monde, Mercredi 20 Septembre, 22 (2006) 9. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Interview par P. Lehir, Le Monde, Samedi 26 Fevrier,´ 23 (2005) 10. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Le Monde, Lundi 11 Avril, 15 (2005), Reproduced in the international weekly selection of Le Monde 2005 11. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Le Monde, Lundi 11 Avril, 15 (2005), Reproduced in “TA NEA”, Greek daily newspaper March 3 (2005) 12. Stauffer, D., S. Moss de Oliveira, P. de Oliveira, and J. Sa Martins, “Biology, sociology, geology by computational physicists”, Elsevier, Amsterdam (2006) 98 5 Sociophysics: An Overview of Emblematic Founding Models

13. Econophysics and Sociophysics: Trends and Perspectives, B. K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim (2006) 14. C. Castellano, S. Fortunato and V. Loreto, “Statistical physics of social dynamics”, Rev. Mod. Phys. 81, 591–646 (2009) 15. S. Galam, Y. Gefen and Y. Shapir, “Sociophysics: A mean behavior model for the process of strike”, Math. J. of Sociology 9, 1–13 (1982) 16. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: Consensus and attitude changes in groups”, Euro. J. of Social Psy. 21, 49–74 (1991) 17. S. Galam and S. Moscovici, “Compromise versus polarization in group decision making”, in Defense Decision Making, R. Avenhaus, H. Karkar and M. Rudnianski (Eds), Springer-Verlag, Berlin, 40–51 (1991) 18. S. Galam and S. Moscovici, “A theory of collective decision making in hierarchical and non- hierarchical groups”, Russian Psy. J. 13, 93–103 (1993) 19. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: II. Conformity and power”, Euro. J. of Social Psy., 24, 481–495 (1994) 20. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: III. Conflicts and forms of power”, Euro. J. of Social Psy., 25, 217–229 (1995) 21. S. Galam, “When humans interact like atoms”, Understanding group behavior, vol. I, Chap. 12, 293–312, Davis and Witte, Eds, Lawrence Erlbaum Ass., New Jersey (1996) 22. S. Galam, “Rational group decision making: a random field Ising model at T D 0”, Physica A, 238, 66–80 (1997) 23. S. Galam and J. D. Zucker, “From Individual Choice to Group Decision Making”, Physica A 287, 644–659 (2000) 24. S. Galam, “Universality of Group Decision Making”, Traffic and Granular Flow ’99 D. Helbing et al, Eds., Springer, Berlin (2000) 25. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, J. of Math. Psychology 30, 426–434 (1986) 26. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943–951 (1990) 27. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191–200 (1991) 28. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organi- sations”, Models of self-organization in complex systems (Moses) Akademie-Verlag, Berlin V. 64, 53–59 (1991) 29. S. Galam, “Paradoxes de la regle majoritaire dans les systemes hierarchiques”,´ Revue de Bibliologie, 38, 62–68 (1993) 30. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132–139 (1999) 31. S. Galam, “Real space renormalization group and totalitarian paradox of majority rule voting”, Physica A 285, 66–76 (2000) 32. S. Galam and S. Wonczak, “Dictatorship from Majority Rule Voting”, Eur. Phys. J. B 18, 183–186 (2000) 33. S. Galam, “Democratic Voting in Hierarchical Structures”, Application of Simulation to Social Sciences, G. Ballot and G. Weisbush, Eds. Hermes, Paris, 171–180 (2000) 34. S. Galam, “Building a Dictatorship from Majority Rule Voting”, ECAI 2000 Modelling Artificial Societies, C. Jonker et al, Eds., Humboldt U. Press (ISSN: 0863-0957), 23–26 (2001) 35. S. Galam, “How to Become a Dictator”, Scaling and disordered systems. International Workshop and Collection of Articles Honoring Professor Antonio Coniglio on the Occasion of his 60th Birthday. F. Family. M. Daoud. H.J. Herrmann and H.E. Stanley, Eds., World Scientific, 243–249 (2002) 36. S. Galam, “Dictatorship effect of the majority rule voting in hierarchical systems”, Self- Organisation and Evolution of Social Systems, Chap. 8, Cambridge University Press, C. Hemelrijk (Ed.) (2005) 37. S. Galam, “Stability of leadership in bottom-up hierarchical organizations”, Journal of Social Complexity 2 62–75 (2006) References 99

38. S. Galam, “The September 11 attack: A percolation of individual passive support”, Eur. Phys. J. B 26 Rapid Note, 269–272 (2002) 39. S. Galam and A. Mauger, “On reducing terrorism power: a hint from physics”, Physica A 323, 695–704 (2003) 40. S. Galam, “Global physics: from percolation to terrorism,: guerilla warfare and clandestine activities”, Physica A 330, 139–149 (2003) 41. S. Galam, “La detection´ des reseaux´ terroristes”, Strategie´ et decision´ : La crise du 11 septembre, General Loup Francart et Isabelle Dufour, Economica, Paris (2002) 42. S. Galam, “Terrorisme et percolation”, Pour La Science 306, 90–93, Avril (2003) 43. S. Galam, “Global terrorism versus social permeability to underground activities”, in Econophysics and Sociophysics: Trends and Perspectives, B. K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), Chap. 14, Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim (2006) 44. S. Galam, “Comment on A landscape theory of aggregation”, British J. Political Sciences 28, 411–412 (1998) 45. S. Galam, “Fragmentation versus stability in bimodal coalitions”, Physica A 230, 174–188 (1996) 46. S. Galam, “Spontaneous coalitions forming: a model from spin glass”, arXiv:cond- mat/9901022 (1999) 47. R. Florian and S. Galam, “Optimizing Conﬂicts in the Formation of Strategic Alliances”, Eur. Phys. J. B 16, 189 (2000) 48. S. Galam, “Spontaneous Coalition Forming. Why Some Are Stable?”, Lecture Notes in Computer Science. 5th International Conference on Cellular Automata for Research and Industry, Springer-Verlag Heidelberg, 1–9 Vol. 2493 (2002) 49. S. Galam, B. Chopard, A. Masselot and M. Droz, “Competing Species Dynamics”, Eur. Phys. J. B 4, 529–531 (1998) 50. B. Chopard, M. Droz and S. Galam, “An Evolution Theory in Finite Size Systems”, Eur. Phys. J. B 16, Rapid Note, 575–578 (2000) 51. S. Galam, “Minority Opinion Spreading in Random Geometry”, Eur. Phys. J. B 25 Rapid Note, 403–406 (2002) 52. S. Galam, B. Chopard and M. Droz, “Killer geometries in competing species dynamics”, Physica A 314, 256–263 (2002) 53. S. Galam, “Modeling Rumors: The No Plane Pentagon French Hoax Case”, Physica A 320, 571–580 (2003) 54. S. Galam, “Contrarian deterministic effect: the hung elections scenario”, Physica A 333, 453–460 (2004) 55. S. Galam, “The dynamics of minority opinion in democratic debate” Physica A 336, 56–62 (2004) 56. S. Galam and Bastien Chopard, “Threshold Phenomena versus Killer Clusters in Bimodal Competion for Standards”, Cognitive Economics – An Interdisciplinary Approach, P. Bourgine and J.-P. Nadal, Eds, Springer, 429–440 (2004) 57. S. Galam and A. Vignes, “Fashion, novelty and optimality: an application from Physics”, Physica A 351, 605–619 (2005) 58. S. Gekle, L. Peliti, and S. Galam, “Opinion dynamics in a three-choice system”, Eur. Phys. J. B 45, 569–575 (2005) 59. S. Galam, “Heterogeneous beliefs, segregation, and extremism in the making of public opinions”, Phys. Rev. E 71, 046123-1-5 (2005) 60. S. Galam, “Local dynamics vs. social mechanisms: A unifying frame”, Europhys. Lett. 70, 705–711 (2005) 61. C. Borghesi and S. Galam, “Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect”, Phys. Rev. E 73 066118 (1–9) (2006) 62. S. Galam, “Opinion dynamics, minority spreading and heterogeneous beliefs”, in Econo- physics and Sociophysics: Trends and Perspectives, B. K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), Chap. 13, Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim (2006) 100 5 Sociophysics: An Overview of Emblematic Founding Models

63. S. Galam, “From 2000 Bush–Gore to 2006 Italian elections: voting at fifty–fifty and the contrarian effect”, Quality and Quantity Journal 41 579–589 (2007) 64. S. Galam and F. Jacobs, “The role of inflexible minorities in the breaking of democratic opinion dynamics”, Physica A 381 366–376 (2007) Chapter 6 Universal Features of Group Decision Making

Anyone familiar with the so-called Ising ferromagnetic model of modern statistical physics would certainly be tempted to make a connection with the behavior of people. Its major feature relies on its incredible universality. Many situations in the world are governed by the emergence of a global order from local interactions. It has proven very powerful in explaining the properties of many different physical systems. It is therefore a very appealing universal model, which could certainly also apply to a large spectrum of social situations. Any book on statistical mechanics will provide an introduction to the Ising model. Among them, I recommend the books by Pathria, Reif, and Ma [1–3].

6.1 The Strike Phenomenon

In the late 1970s, many people had leftist visions of society and it being reworked, making the strike a central key to implementing economic changes in the organization of society. Within this frame of mind, I was puzzled by the fact that sometimes a tiny group of activists succeeded in putting on strike thousands of workers while at other times even large unions could not achieve this. Discussing the question with Y. Shapir and Y. Gefen [4], both PhD. students like me in the same group at the Department of Physics at Tel Aviv University, we developed the idea of using an Ising ferromagnetic system to describe the collective state of an assembly of agents, each being in either one of two distinct individual states, that of working or striking. This produces two collective ordered states: a working state versus a striking state. The ferromagnetic coupling between agents was motivated by the social fact that people have the tendency to reproduce the leading choice of their neighbors, in particular in conflicting situations. We thus implemented the first application of the Ising model to describe the global state of a firm.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 101 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 6, © Springer Science+Business Media, LLC 2012 102 6 Universal Features of Group Decision Making

6.1.1 The Model

A group of N agents is considered, with each one having attached to it a variable SQi where i D 1;2;:::;N. When the agent i is working at the maximum of its productivity, we take SQi D 1. In the opposite case, when it is not working at all, we put SQi D 0. Instead of the two values 0 and 1, we need to have ˙ to fit the Ising model. This is readily achieved by making the variable change Si D 2 SQi 1=2 . Now a maximum productivity by agent i is denoted by Si D 1 while a zero activity is defined by Si D1. Following the Ising model construction, we introduce a coupling Ji;j between two agents i and j . This coupling embodies their social exchanges. As in ferromagnetism, we take a positive coupling with Ji;j >0. The quantity Ji;j Si Sj is then a measurement of the discrepancy which exists between their individual states. This accounts for an eventual dissatisfaction generated by a difference in productivity. Otherwise, when they are synchronized, a negative dissatisfaction is obtained, which means that the satisfactions are alike. To account for the dissatisfaction created by the actual salary W with respect to an expected minimum fair salary expectation V , we introduce an external field denoted H D W V , which couples linearly with each agent to create a dissatisfaction HSi . Accordingly, when H>0, if the agent is working at maximum, it gets the maximum negative dissatisfaction H, i.e., a satisfaction while H<0, to reach the maximum negative dissatisfaction H, the agent must stop working. The first case is activated by having W>Vand the second by W

X XN H D Ji;j Si Sj H Si ; (6.1) .i;j / iD1 where (i, j) denotes all interacting pairs of agents.

6.1.2 The Operating Mechanism

We are now in a position to be able to postulate a principle for determining which state, working or not working, a given agent will select, given a distribution of the individual states of all the other agents. A reasonable postulate is to assume 6.1 The Strike Phenomenon 103

Fig. 6.1 Dissatisfaction function F versus M in the ordered phases. The cross x denotes the actual state of the system. It is stable since it is at the absolute minimum. Here H>0and the factory is at work

that an agent would like to minimize its dissatisfaction. A principle of minimum dissatisfaction is thus postulated to determine the eventual equilibrium global state of the system. It is analogous to the minimum energy principle which is activated at a temperature of absolute zero. In physics, the temperature T is a crucial ingredient that does not exist as such in social systems. However, the notion of a social permeability denoted 1=T might be a good counterpart. The next step is then to introduce a global dissatisfaction function F , the equivalent of the physical free energy. Formally, it can be calculated exactly, since it is an explicit function of H and T . However, in practice, for most of the cases, this can only be performed by numerical means. Approximate techniques exist to perform some of the analytical calculations. The most powerful one is called the mean field treatment since it neglects local fluctuations. Accordingly, we performed a mean field treatment of (6.1), which quite naturally reproduces the well-known phase diagram of the ferromagnetic Ising model in an external field. A key internal parameter to characterize the macroscopic state of the system is the order parameter XN M 1=N Si : (6.2) iD1 A nonzero M reveals an ordered phase while M D 0 denotes the disordered phase. Several ordered phases might exist. In the present case, two are present with M>0and M<0. However, they are symmetrical, i.e., they display the same broken symmetry. An important property relevant to the strike phenomena is the existence of the metastability phenomenon combined with the nucleation mechanism, which are obtained by reversing the sign of the external field H. This situation occurs in particular when the expected minimum fair salary V increases in the workers expectation while the current wages are unchanged. The phenomenon is illustrated in the series of Figs. 6.1–6.4. 104 6 Universal Features of Group Decision Making

Fig. 6.2 Dissatisfaction function F versus M in the ordered phases. The cross x denotes the actual state of the system. It is locally stable but could turn unstable since it is not at the absolute minimum. Here H<0and the factory is at work. The dashed arrow indicates the eventual jump into a strike state, which corresponds to the absolute minimum, driven by some external action such as the intervention of a handful of activists

Fig. 6.3 The limit of metastability has been reached. We still have H<0but its magnitude has increased. F has only one minimum

6.1.3 The Overlap with the Physical Model

The social model of strikes proposed here is identical to the Ising ferromagnet in an external ﬁeld. The treatment used to obtain the corresponding phase diagram is the classical mean ﬁeld theory. The totality of the associated rich properties are thus recovered and must be given a social interpretation. It is indeed a mapping from a physical reality to a social reality with all the limitations that such reasoning implies. 6.1 The Strike Phenomenon 105

Fig. 6.4 M versus H for K

6.1.4 The Novel Counterintuitive Social Highlights

If technically the model does not bring anything new, the transfer from statistical physics to the new field of social sciences sheds new light on it. It produces a rather large spectrum of novel insights that are rather counterintuitive and unexpected. Most salient are [4]: • From our individual two-state assumption for each agent, the two collective states of a firm, being working normally or on strike, appear to be symmetrically ordered phases. Accordingly, the worker’s involvement in an eventual strike will be identical to the prevailing amplitude of working just before the strike begins. • It appears from the hysteresis phenomenon, which is linked to metastability, that avoiding a strike is not equivalent to stopping a strike in terms of cost. Having a positive effective wage H>0guarantees a working state with a production value M>0. However, the wage situation may well-deteriorate even if the actual salary is kept unchanged. It might happen when the worker’s expectation of the minimum salary E increases while no increase in the current salary is given. Such a change induces a field modification from H>0to H<0. In this case, although H<0, the firm’s working state can persist with no strike occurring, creating the illusion that everything is alright while indeed the system has moved from a stable state to a metastable state. Only below the limit of metastability, i.e., a more negative value of H does the strike happen with certainty. • However, while within the metastable working state, any handful of activists, including external ones, with very little action, could suddenly precipitate the whole firm into striking. The same action would have no effect in the range H>0. This is the manifestation of the so-called nucleation phenomenon, which 106 6 Universal Features of Group Decision Making

takes place within the H range for which the striking phase corresponds to the absolute equilibrium and the working state is only a local equilibrium. • A remarkable feature is that our model provides us with an explanation of how and why it is often cheaper to avoid a strike by increasing W to reach a H>0 before the strike starts than to put a firm back to work once the strike has occurred. It is also the direct result of the existence of a hysteresis effect produced by the metastability. We can now have H>0and still have the workers staying on strike. This is the symmetrical situation of having them working when H<0. However, the major difference is that in the case of H<0, activists may well precipitate the firm into striking, that will not be the case when H>0. Then the minimum increase of the actual salary W to get the workers back to work is that which is necessary to get out of the metastable range as seen in Fig. 6.4. • In the case where the social permeability is weak enough to put the firm in the disordered state characterized by M D 0, the metastability vanishes. Then the current wage is instrumental in keeping the firm working or striking. As soon as H<0the agents strike and as soon as H>0, they work. The concern might be that M D 0 means an average individual production equal to zero. The workers are not much involved in their work with a low productivity.

6.1.5 Achievements of the Model

While the model provides a new view with a series of guidelines on how to manage wage policy and how to implement a strike, no speciﬁc predictions have been made to date concerning a precise situation.

6.2 How Do Groups Make Their Decisions?

The major contribution from the preceding model has been to initiate and to open up a new field of knowledge, taking over 30 years to attract a substantial part of the community of condensed matter physicists. These results in itself show the new insight that can be gained from sociophysics. Following the development of this model, I met in New York a leading social psychologist, Serge Moscovici. While discussing the possible applications of sociophysics, we decided to address the challenge of explaining the observed polarization and risk taken in a series of experiments conducted in psychology. This work led to the publication of a series of papers, which created a coherent and rather rich framework [5–9]. Indeed, it was found that groups that have no time limit constraint exhibit a trend towards an extreme decision instead of a balanced decision [10, 11]. As for the striking phenomenon, the Ising ferromagnetic model in an external field was perfectly adapted to the problem of a two-choice situation. However, while 6.2 How Do Groups Make Their Decisions? 107 taking again the same formal Hamiltonian given by (6.1), the parameters are given different meanings besides that of the coupling Ji;j , which keeps on measuring the strength of exchanges between two individuals, denoted i and j . After a series of very long in-depth discussions with Serge Moscovici, I concluded that using a modified version of a finite size of the so-called random field Ising ferromagnetic model in an external magnetic field at zero temperature would be perfectly adapted to the investigation of the social phenomenon of group decision making. Postulating a minimum of interpersonal conflicts, we obtained the psychosociological experimental fact that interactions produce a group polarization along a single choice, which is, however, found to be randomly selected in our approach. On this basis, a small external social pressure has a radical effect on the polarization. Individual bias related to personal backgrounds, cultural values, and past experiences can also be introduced via quenched local competing fields. These biases are shown to be instrumental in generating a larger spectrum of collective new choices beyond the initial ones, which in turn matches certain experimental findings. In particular, compromise is found to result from the existence of individual competing biases. Conflict is shown to weaken group polarization. The model yields new psychosociological insights about consensus and compromise in groups. To apprehend the numerous associated insights, we refer to the related papers for details [5–9, 12–15].

6.2.1 The Symmetrical Individual Versus the Symmetrical Group

In numerous cases, an individual has to make a choice between two given answers, which can be denoted by yes and no. In addition, a larger spectrum of answers can often be mapped at some approximate level into a two-answer case [16]. A variable ci D˙1 represents the choice of agents i with Yes ci DC1 and No ci D1. At this stage, the individual has no reason to favor either one of the two choices, so the yes and no are equally probable with a probability distribution function 1 p.c / ı.c 1/ ı.c 1/ ; i D 2 f i C i C g (6.3) where ı is the Kronecker function. For N individuals, each one making its choice independently, the aggregated collective choice of a group of N people

XN C D ci (6.4) iD1

p1 C 0 is thus zero on average, with fluctuations of the order of N .Theresult D creates a new qualitative choice which did not exist at the individual level. It can 108 6 Universal Features of Group Decision Making be understood as the perfect choice of compromise. Since the macroscopic quantity C is zero, the group aggregation process has no effect at the macrolevel, making the associated group symmetrical. It is a neutral group with no link among group members. The various choices of two people i and j are of two types. One type is where each makes an identical choice with ci D cj D˙1 and the other type is where they hold opposite choices with ci Dcj D˙1. The product ci cj discriminates these two situations with ci cj DC1 for agreement and ci cj D1 for conflict. The choice ˙1 is arbitrary, in accordance with both a symmetrical agent and a symmetrical group. This is the physical equivalent of time reversal invariance. The system is unchanged by reversing all individual spins. However, in a social system, although the individual has no a priori to select either one of the two choices, his or her actual choice will have a drastically different effect on the social implementation of the choice. Prior to the decision itself, although both choices are equally probable, both individuals could either argue for a long time or discuss and exchange information. On the other hand, they may well decide without any exchanges. A quantity is thus required to measure their choice of involvement. Calling J the exchange amplitude, the product Jci cj measures the amplitude of the pair involvement, either positive CJ , i.e., cooperation, or negative, meaning a conflict J . Restricting interaction to pairs, the overall group conflict is measured by the function X G J ci cj ; (6.5) where we have assumed a constant exchange J for all interacting pairs and represents all interacting pairs. G is the group internal conflict function.

6.2.2 The Random Symmetry Breaking Choice

For each one of the 2N configurations of the N individual choices, the internal conflict function G measures the corresponding conflict. It discriminates between various possible choices but does not induce a choice. To produce a group decision dynamics, we need to introduce a criterion to select the actual collective choice favored by the group. In the spirit of the physical minimum energy principle, and from psychological evidence, we postulate that “Each individual favors the choice which minimizes his or her own conflict.” Minimum conflict means maximum agreement. At this stage agents make choices but are not personally bound to them. Later, we will introduce the personal convictions of agents. If we proceed in a sequential form, selecting randomly one person who made a choice, yes or no, then we select all people interacting with it and they will make the same choice so as to minimize their own conflicts. By so doing, all people 6.2 How Do Groups Make Their Decisions? 109 interacting with them will again make the same choice so as to favor agreement and so on and so forth. If none of the agents are isolated and if there exist no isolated subgroups, the group ends up making the same initial choice that the first person selected. The net result of these successive choices is an extreme polarization of the collective choice with C D˙N . The sign, i.e., the polarization direction, is determined by the initial individual choice, which was made randomly. This polarization phenomenon holds whoever is chosen to be the initial person. Only the direction (yes or no) will change from one initial person to another. In real life situations, the above process starts simultaneously from several people. The dynamics of choice spreading is a rather complex phenomenon. Monte- Carlo simulations at zero-temperature of the Ising model show nontrivial behavior at all dimensions [17]. However, the group eventually succeeds in selecting only one choice, allowing everyone to minimize their own conflicts. We thus find that,

“Symmetrical groups polarize themselves towards an extreme choice. The direction of that choice is arbitrary. Each extreme is equally probable.” This shows that individual minimum conflict is identical to maximizing G.A positive conflict is an agreement. This polarization effect is identical to the physical spontaneous symmetry breaking phenomenon, which is instrumental in the making of collective phenomena in inert matter [18]. Individual local interactions make the group behave as a single super-person [19]. This super-person chooses between the two possible choices with equal probability in the same way as the isolated individual. Nevertheless, individuals within the group have lost their freedom of choice. Each individual must make a choice that is identical to that of the person with whom they interact. Individual freedom disappears in favor of the emergence of a group freedom. Perfect compromise has also disappeared. The group decision produces an effect on its surroundings that is somehow proportional to N since C D˙N . Without interactions, the individual choices ci D˙1 were overall self-neutralized macroscopically. Interactions produce strong individual correlations associated with the symmetry breaking. Results obtained from group decision making experiments conducted in social psychology can now be understood in a different way. The polarization effect was clearly evident in data reported in [16] but most theoretical explanations have been unconvincing in connecting choices at the individual level [20] and the group level [11]. We conclude from the model that the polarization effect arises quite naturally from first principles, i.e., from interactions among agents.

6.2.3 Anticipating the Group Choice

Anticipation is a very crucial feature in most social phenomena. To incorporate it, we rewrite G from (6.5)as 110 6 Universal Features of Group Decision Making ( ) I XN Xn G c c ; D 2 j.k/ i (6.6) iD1 kD1 where n is the number of people one individual interacts with. To keep the presentation simple this number is assumed to be equal for everyone. For small groups where everyone interacts with everyone n D N . Anticipating the emergence of a collective choice, each agent i thus tries to project its expectation of the overall ﬁnal group decision through the various choices cj of the agents with whom it discusses. Individual i extrapolates j ’s choice cj to the expected collective choice the group will eventually make without its own contribution. Accordingly individual i perceives j ’s choice as given by 1 cj ! .C ci /; (6.7) N 1 where C is the collective choice. Within this anticipation process, the function G is written 8 9 J XN

C XN G c ; a D N i (6.10) iD1

nJN where 2.N 1/ is a constant independent of the group choice. It is thus irrelevant to the collective choice. C is not yet the ﬁnal decision. Rather, it is the expected ﬁnal collective choice. Putting

XN Ga D Sa ci ; (6.11) iD1 where C S ; a N (6.12) is the equivalent of a group field which couples with each individual choice. The notion of field is very convenient to account for some external pressure towards 6.2 How Do Groups Make Their Decisions? 111 a definite choice. Within our convention of minimum conflict, the product Saci measures this influence. A positive field Sa >0favors a positive choice C1, while 1 is favored by a negative field Sa <0. The conflict amplitude is given by Sa. This expression is self-consistent since on the one hand, individual i wants to align itself along the virtual field Sa, and on the other, it contributes directly to this virtual field through its dependence on the collective choice C . Rewriting (6.11)as C 2 G a D N (6.13)

2 shows that maximizing Ga is identical to maximizing C . Its maximum value is reached at C 2 D N 2. This is an extreme polarization with either C DCN or C DN . Individual minimum conflict appears to be identical to the maximum of the group internal conflict function. In the above, the “group formation process by anticipation” is formally identical to the mean field theory of phase transitions. Whilst in physics it is an approximation, here it is exact, since it embodies the social mechanism of global anticipation.

6.2.4 Individuals Are Often Not Symmetrical

A pressure from the outside of the group can be applied to the group to break the symmetry between the two choices by favoring one speciﬁc choice. Individuals are also different and may well have internal biases towards one of the two choices while being against the other.

6.2.4.1 The Social Pressure

External pressure on group members is achieved by introducing a social field S, which differentiates the two possible choices. It turns the symmetrical individual into a social individual with p.ci D 1/ ¤ p.ci D1/. It is a similar effect to the anticipating field Sa. Each person’s conflict with S is measured by the product Sci . Agreement is associated with Sci >0, i.e., the choice is made along the field with S and ci having the same sign. In contrast, Sci <0represents a conflict between the individual and the social pressure, since S and ci have opposite signs. The measurement of surrounding group conflict is XN Gs Sci : (6.14) iD1

a Maximizing the sum G C Gs still results in an extreme polarization but now its direction is no longer random. The group choice is C DCN for S>0 and C DN when S<0. Even an extremely weak external pressure makes 112 6 Universal Features of Group Decision Making all the individuals align themselves along it. The group and the individual behave identically, both following the pressure induced by the external pressure. The symmetry breaking choice is no longer random. The super-person represented by the whole group is identical to the individual person. They are both aligned along the ﬁeld. This result is in contrast to the symmetrical state in which the individual loses its freedom of choice in favor of the group choice freedom.

6.2.4.2 The Social Representational State

Following the social literature we incorporate “individual social representations” [21] to account for individual differences in preferences about certain issues. A social representation varies in both direction and amplitude from one person to another. It depends upon cultural values, history, past experiences, ethics, and beliefs. It is attached to a person. The representational effect can be included within our formalism by introducing an individual local field. We call Si the internal social field attached to individual i. Its properties are similar to those of a social field S. The difference is that while the social field applies uniformly to each group member, an internal social field acts only on one person. The individual conflict with the internal social field is measured by the product Si ci . It is positive for a choice made along the representation (internal agreement with personal values) and negative otherwise (internal conflict with personal values). The measurement of group representation conflict is given by

XN Gr Si ci : (6.15) iD1

The knowledge of the distribution of individual representations is required to determine the group collective choice. The representation effect is enhanced in the isolated-person case where both exchange amplitude and social external field are zero. Then, the final decision results from every individual following his or her own representation. This gives XN S C i ; D S (6.16) iD1 j i j where the j:::j denotes the absolute value. Equation (6.16) illustrates the qualitative change driven by the existence of individual social representations. The actual value of C can now vary over the whole range of values N;:::;0;:::;CN . Compromise C D 0 is again a possible outcome. Individual representations are thus instrumental for making the model relevant to real situations in which collective choices are far richer than C D˙N . Prior to the group formation through the making of a collective choice, individuals have their own representations, which determine their a priori answers to 6.2 How Do Groups Make Their Decisions? 113 the initial question. All these representations result in either a yes or a no. Then, during the discussion, people start to interact through the yes and no distributions throughout the group. However, to reach a collective choice, due to the existence of opposite representations, people must construct new answers in addition to the initial yes and no. The spectrum of answers is thus enriched by the group building an answer driven by all individual representations. In the neutral state, groups do not produce new answers. We assume that once a final collective decision is reached, each group member C identifies with it, triggering its new individual choice to di D N ,whichmay differ from the initial ci . In the neutral state, di D˙1. Thus, in the process of building a new answer d, a new representation has been produced by the group. This new representation is integrated by each individual to yield the d choice. Group formation has qualitatively modified individual representations. If individuals resist adopting a representation that is in opposition to their own, via the group transformation, they eventually join a new common representation, which accounts for the overall balance of all the representations present. It articulates around a new answer which did not exist prior to the group forming. It is worth stressing our qualitative departure from statistical physics. This part is novel and does not apply to a physical system. Here we are not considering an average individual position, but a well-defined and fixed individual position. This position results from the group building an answer. In most cases, d is different from ˙1. We are thus transforming Ising variables ci D˙1 to continuous variables 1 C1 N di N . It is also a novel scheme, which does not exist in physics.

6.2.5 Life Is Not a Paradise

All the above effects add into an extended group internal conﬂict function G D Ga C Gs C Gr with

X XN XN G D J ci cj C S ci C Si ci ; (6.17) iD1 iD1

To maximize (6.17) is a more difficult task since competing effects are active. A given individual wants to minimize his or her own overall conflict, which results from three contributions. These are • Interacting group members: the individual wants to adopt the same final decision as preferred by the others. • External social field: the individual wants to comply to the external pressure from the surroundings. • Internal social field: the individual wants to comply to the internal pressure from its personal representations. 114 6 Universal Features of Group Decision Making

These three elements cannot necessarily be satisfied simultaneously. The individual wants to minimize its overall personal conflict, which in turn could result in simultaneous agreement with some of the above items and conflict with others. Writing (6.17) in the form XN t G D Si ci ; (6.18) iD1 where t Si D Sa C S C Si ; (6.19) is the resulting field applied to individual i in the group formation process, and makes clearer the eventual contradiction in which agent i can be caught. Maximum G and minimum individual conflicts are achieved when each individual follows its t t resulting field sign. If Si >0,thenSi D 1 and Si D1 for Si <0. t The case Si D 0 creates a doubt about agent i’s choice in a similar way as to the t isolated neutral case. To satisfy the Si sign implies satisfying simultaneously the signs of Sa, S,andSi . This possible contradictory competing effect is the signature of the psychological complexity involved in the decision making process. Each t person follows his or her resulting field Si , which in turn produces a collective choice C . Then this collective choice is individually integrated back by agent i with c d C i ! D N . Each agent ends up sharing individually the same final collective choice.

6.2.6 Some Emblematic Illustrations of the Model

To gain some feeling about the implication of (6.17)and(6.18), we consider four different speciﬁc cases. Each will illustrate some emblematic social situations.

6.2.6.1 Two Half/Half Balanced Opposite Social Representations

We consider an even group of N people in which half the agents share a positive social representation Si DCS0 with S0 >0and the other half have a negative social representation with the same amplitude Sj DS0. There is no external social ﬁeld, i.e., S D 0. The whole group has thus no net representation. The corresponding internal conﬂict function is N G C 2 S cC S c ; D C N C 2 0 i 0 j (6.20)

C where ci and cj are the people with respectively positive and negative representa- nIN tion. The constant 2.N 1/ has been introduced in (6.10). The collective choice 6.2 How Do Groups Make Their Decisions? 115

N C may be written as C D 2 .ci C cj /, which is the one that maximizes G.Since C only two different kinds of agents symbolized by ci and cj are involved, four choice configurations are possible: C (a) ci D cj DC1 with C D N and G.a/ D G.b/ D C N C (b) ci D cj D1 with C DN and G.a/ D G.b/ D C N C (c) ci Dcj DC1 with C D 0 and G.c/ D C NS0 C (d) ci Dcj D1 with C D 0 and G.d/ D NS0 Agreement occurs in (a), (b) and conflict in (c), (d). Clearly G.d/ < G.c/ reduces the choice to either (a) and (b) or (c). In the case >S0,wehave G.a/; .b/ > G.c/, indicating that the interaction strength that is proportional to nJ is stronger than S0. The group then polarizes with C D˙N . The direction of the extreme choice occurs at random. Half of the members are fully satisfied with both their social representation and that of their partners while the other half is in conflict with its own social representation. The “losing” subgroup has to build a new representation, which embodies some level of internal conflict. The “winning” part does not modify its initial representation. In this case, no new answer was built with ci ! d D˙1. On the other hand, strong representation, i.e.,

“Within a balanced social representation group, exchange favors a compromise. Weak exchange results in an extreme polarization along a random direction.”

6.2.6.2 Two Half/Half Unbalanced Opposite Social Representations

We consider the previous example but with unbalanced opposite social representations, the positive representation S0 being stronger than the negative one, denoted ˛S0 where 0<˛<1. The respective values of the internal conﬂict function change, to become, G.a/ N ˛ NS (a) D C C 2 0 G.b/ N ˛ NS (b) D C 2 0 G.c/ .1 ˛ /NS (c) D C 2 0 G.d/ .1 ˛ /NS (d) D 2 0 We have 0<˛<1) G.a/ < G.b/ and G.d/ < G.c/. However, in the case of >S0, the polarization direction is determined with C DCN . The balanced case ˛ D 1 makes the choice direction arbitrary, but now it is the strongest initial representation that wins. The discussion process within the forming group has made 116 6 Universal Features of Group Decision Making the weaker-biased people align themselves along the stronger ones with ci ! d D 1. To favor a compromise outcome, a decrease in exchanges among group members is required. For

“Within an unbalanced representation group, exchange favors extremism along the initially strongest social representation. Only a limitation of exchange may hamper extremism and produce a compromise.”

6.2.6.3 The Minority Effect

To have an equal number of people in two opposite social representation subgroups is not a common situation. Most of the time the group is divided between a majority of .N M/agents and a minority of M agents with M0, the majority holding an unequal negative representation Sj D˛S0. We choose for instance the case of a minority that is more motivated than the majority, i.e., 0<˛<1. C Denoting ci and cj the respective minority and majority agents, the collective C choice is given by C D Mci C .N M/cj . The internal conﬂict function is n o n 2 C G D JC C Mc .N M/˛cj S0: (6.21) 2.N 1/ i

The corresponding four choice configurations are: C (a) ci D cj DC1 with C D N and G.a/ D C N Cf.1 C ˛/M ˛N gS0 C (b) ci D cj D1 with C DN and G.b/ D C N f.1 C ˛/M ˛N gS0 cC c 1 C 2M N G.c/ .2M N/2 (c) i Dj DC with D and D C N C f.1 ˛/M C ˛N gS0 cC c 1 C 2M N G.d/ .2M N/2 (d) i D j D with D C and D C N f.1 ˛/M C ˛N gS0 The first two cases (a), (b) are minority–majority agreements while the other two (c), (d) are minority–majority splitting. A systematic analysis of the above expressions is complicated since now several parameters N , M , nJ , S0,and˛ are involved. Case (d) is never selected since indeed, nothing is satisfied in that case, neither the interactions, nor the representations. Two classes can be defined with respect to the respective effects of interactions over social representation. • Interactions over-rule social representations: the group is polarized along either one choice since G.a/ > G.c/ and G.b/ > G.c/. (1) When G.a/ > G.b/, the minority wins, turning the majority to its side. This happens for M<.N M/˛. The condition G.a/ > G.c/ is ensured by nJM > .N 1/˛S0. 6.2 How Do Groups Make Their Decisions? 117

(2) When M>.N M/˛,wehaveG.a/ < G.b/. The majority wins turning the minority to its side. The condition G.b/ > G.c/ is ensured by nJ.N M/ > .N 1/S0. It does not depend on ˛ since in both cases (b) and (c) the majority follows its own social representation ˛S0. • Social representations are over-ruling interactions: the group is balanced, i.e., G.a/ < G.c/ and G.b/ < G.c/. Condition G.a/ < G.c/ is ensured by nJM < .N 1/˛S0 and G.b/ < G.c/ by nJ.N M/ < .N 1/S0. A balanced collective choice reﬂecting the respective strength in numbers of each group is given by case (c).

6.2.7 The Leader Effect

In most groups, people are not all equal in status. The inequality can stem from either a strong character or an institutional position. Such situations can be taken into account by assigning a stronger individual field Si to the leading person i in the group. In other words, the leader case is a special case of a strong minority, which is reduced to one person. Now SI incorporates the individual character in addition to the person’s social representation. The above minority case equations can thus be used putting M D 1.Thevalue of ˛ determines the strength of the agent’s self-determination. A weak charismatic leader will have ˛ 0 and a strong charismatic leader will have ˛ 1.However, for more influential and authoritarian leaders, an external field would account for the pressure that the leader applies to all group members. Let us consider a leader denoted i D 1 with a positive social representation S1 D S0. The rest of the group has N 1 agents with an unequal negative social C representation Sj D˛S0. Denoting c0 and cj as the respective leader and C majority choices. The collective choice is given by C D c0 C .N 1/cj .The internal conflict function is n o n 2 C G D JC C c .N 1/˛cj S0: (6.22) 2.N 1/ 0 The corresponding four choice configurations are: C (a) ci D cj DC1 with C D N and G.a/ D C N Cf.1 C ˛/ ˛N gS0 C (b) ci D cj D1 with C DN and G.b/ D C N f.1 C ˛/ ˛N gS0 cC c 1 C 2 N G.c/ .2 N/2 .1 (c) i Dj DC with D and D C N Cf ˛/ C ˛N gS0 cC c 1 C 2 N G.d/ .2 N/2 .1 (d) i D j D with D C and D C N f ˛/ C ˛N gS0 The first two cases (a), (b) are leader-majority agreements while the other two (c), (d) are leader-majority conflicts. As always, case (d) is never selected since indeed nothing is satisfied in this case, neither the interactions nor the social representations. 118 6 Universal Features of Group Decision Making

• Interactions are over-ruling social representations: the group is polarized along either one of the choices since G.a/ > G.c/ and G.b/ > G.c/. (1) The leader wins, turning the group to its side if G.a/ > G.b/.Thisisthe 1 case if ˛ G.c/ is ensured by nJ > .N 1/˛S0. (2) The group wins turning the leader to its side when G.a/ < G.b/.This 1 happens when ˛ G.c/ is ensured by nJ > S0. The condition does not depend on ˛ since in both cases (b) and (c) the group follows its own representation ˛S0. • Social representations are over-ruling interactions: the group is balanced, i.e., G.a/ < G.c/ and G.b/ < G.c/. Condition G.a/ < G.c/ is ensured by nJ < .N 1/˛S0 and condition G.b/ < G.c/ by nJ < S0. A balanced collective choice reﬂecting the respective numerical strength of the leader and the group is given by case (c).

6.2.8 The Overlap with the Physical Model

The social model shares most of the statistical counterparts with the physical model except for two points. The first one is the consideration of the psychological effect of anticipation, which in turn makes exact a mean field treatment of a finite sized system as shown in Fig. 6.5. The second point is that while we start with Ising spins Si D˙1, once the equilibrium state is reached, we shift to continuous spin variables SQi with 1 SQi 1. In physics, M D is an average of flips from Si DC1 to Si D1 and vice versa. On the contrary, here we assume a new value of Si D M , in addition to the two initial values Si D˙1. Although the problem is solved using a mean field treatment, it has been given a frame which makes it exact. Moreover, the concept of anticipation has materialized via the existence of an order parameter, which is a priori unknown and enters the usual self-consistent mean field equation of state.

6.2.9 The Model’s Achievements

Several unexpected findings were obtained in this model, in particular with respect to group polarization, i.e., the emergence of an extreme choice, versus the establishment of a consensus at a moderate choice between the initial two extreme choices. The extremism is characterized by either one of the two group choices C D˙N ,whereN is the group size. A consensus will result in a group choice N C CN . The existence of individual biases represented by the local fields makes, in principle, any group choice between N and CN possible. We have built a conceptual methodology rather than a final complete theory. It has provided a series of unexplained experimental findings with a coherent References 119

Fig. 6.5 Point A represents the compromise choice C D 0 of a group of N agents without interaction among them .J D 0/. The vertical arrow to point B shows how beyond some increase in the setting of interactions J>0, the same group shifts its choice to a polarized choice .C ¤ 0/, and yet not an extreme one .C ¤˙N/ since D>0. The horizontal arrow to point C indicates that increasing the number of agents of the group within the same external conditions, in which J and D are unchanged, can turn back the collective choice to a consensus at C D 0 and unique explicative framework. It sheds surprising new light on the possible constitution of sensitive groups such as juries. The effect of a leader has also been studied. Our hypothesis is that group decision making obeys universal laws which are independent of the nature of the issue at stake. Our main conclusions are: • Exchanges among individuals do not aim at selecting an issue, but rather at aligning people along the same issue. The issue itself is random with respect to exchanges. • Exchanges among individuals do not favor compromise about an issue. On the contrary, they produce extremism with a polarization along one of the competing opinions. • Reducing individual exchanges favors compromise. • External social pressure is extremely efﬁcient in making the group select a given choice. • Individual social representations are a necessary ingredient for both weakening extremism and for opposing an external social pressure. These theoretical results must be set against the different data obtained from a large number of experimental studies which show that groups tend to polarize along an extreme position, reﬂecting the dominant pole of attitudes and not around an average position as a priori expected [11, 16].

References

1. R. K. Pathria, Statistical Mechanics, Elsevier Science and Technology (1996) 2. F. Reif, Fundamentals of Statistical and Thermal Physics, New edition,` Waveland Press, Inc. 3. S.-k. Ma, Statistical Mechanics, World Scientiﬁc (1985) 120 6 Universal Features of Group Decision Making

4. S. Galam, Y. Gefen and Y. Shapir, “Sociophysics: A mean behavior model for the process of strike”, Math. J. of Sociology 9, 1–13 (1982) 5. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: Consensus and attitude changes in groups”, Euro. J. of Social Psy. 21, 49–74 (1991) 6. S. Galam and S. Moscovici, “Compromise versus polarization in group decision making”, in Defense Decision Making, R. Avenhaus, H. Karkar and M. Rudnianski (Eds), Springer-Verlag, Berlin, 40–51 (1991) 7. S. Galam and S. Moscovici, “A theory of collective decision making in hierarchical and non- hierarchical groups”, Russian Psy. J. 13, 93–103 (1993) 8. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: II. Conformity and power”, Euro. J. of Social Psy., 24, 481–495 (1994) 9. S. Galam and S. Moscovici, “Towards a theory of collective phenomena: III. Conflicts and forms of power”, Euro. J. of Social Psy., 25, 217–229 (1995) 10. S. Galam and S. Moscovici, “Towards a theory of collective phenomena. I: Consensus and attitude change in groups”, European Journal of Social Psychology, 21, 49–74 (1991) 11. J. Davis , “ Group decision and social interactions: A theory of social decision scheme”, Psychological Review, 80, 97–125 (1973) 12. S. Galam, “When humans interact like atoms”, Understanding group behavior, Vol. I, Chap. 12, 293–312, Davis and Witte, Eds, Lawrence Erlbaum Ass., New Jersey (1996) 13. S. Galam, “Rational group decision making: a random field Ising model at T D 0”, Physica A, 238, 66–80 (1997) 14. S. Galam, “Universality of Group Decision Making”, Traffic and Granular Flow ’99 D. Helbing et al, Eds., Springer, Berlin (2000) 15. S. Galam and J. D. Zucker, “From Individual Choice to Group Decision Making”, Physica A 287, 644–659 (2000) 16. S. Moscovici and M. Zavalloni , “ The group as a polarizer of attitudes”, Journal of Personality and Social Psychology, 12, 125–135 (1969) 17. D. Stauffer , “Ising spinodal decomposition at T D 0 in one to five dimensions”, J. Phys. A 27, 5029 (1994) 18. Sh-k Ma, Modern Theory of Critical Phenomena, The Benjamin Inc.: Reading MA (1976) 19. J. C. Turner , “Rediscovering the Social Group”, Basil Blackwell: Oxford (1987) 20. E. Burnstein, and A. Vinokur , “Testing two classes of theories about group induced shifts in individual choices”, Journal of Experimental Social Psychology, 9, 123–137 (1973) 21. S. Moscovici, Social influence and social change, Academic Press: London (1976) Chapter 7 The Dictatorship Paradox of Democratic Bottom-up Voting

Although the question of the democratic legitimacy of leadership is not the main focus of this book, it is also included here to make the overview complete. Whatever is the political organization of a social, economic, or political group, hierarchies are present in most of the cases. Two main extreme hierarchies are found: authoritarian and democratic. In the first one, everything starts from the top and proceeds down towards the bottom while in the second case it is the reverse. However, it is often found that, despite the use of bottom-up majority rule voting, democratic hierarchies are de facto dictatorial in the sense that the leadership is maintained against the opposition from a bottom majority [1Ð13]. Surprising major political events, which have occurred in recent years were thus counter intuitive [14Ð19]. We start with a population having agents that hold either one of two political flags A and B. Each member has an opinion and respective proportions are p0 and .1p0/. Here we use political language, but the ideas may be applied to other social domains such as in management. Extracting at random some of the agents from the surrounding population, a hierarchy bottom is assembled. These bottom agents are then distributed randomly in groups of finite size r. This corresponds to the level (0) of the hierarchy. The first voting occurs, with everyone of these bottom groups electing a representative according to some well-defined voting rule Rr .p0/. Therefore, the group elects either an A with a probability

p1 D Rr .p0/ (7.1) or a B with probability .1p1/ asshowninFig.7.1. Within a democratic framework, this rule is a majority rule of the current composition of the group. The ensemble of elected representatives constitutes the ﬁrst level of the hierarchy. The number of elected representatives is a fraction 1=r of the number of bottom agents since r agents elect one representative. Once they have been elected, they form another series of ﬁnite sized groups, which in turn elect a higher level representative according to the same voting rule used to elect them. The process

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 121 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 7, © Springer Science+Business Media, LLC 2012 122 7 The Dictatorship Paradox of Democratic Bottom-up Voting

A B A A A single level hierarchy B A B B B B A B A B B A A B A A B A B B B A A B X: The A Population with N people both A and B President

X X X AAA A AAB (x3) Random selection of BBB B 3 agents ABB (x3)

Fig. 7.1 The simplest hierarchy has one single level with one voting group of three agents randomly selected from a population. The president uses a majority rule

Agents are selected randomly from the How it works population to form the ground level B A A A A B B B A A A A B A A B B B B B B A A A A B A

B BB A Representatives A A B A A First level

A B B B A B A Form new groups A A

B A A Second level

President A

Fig. 7.2 A more elaborate hierarchy with three voting levels still operates with groups of three people. A total of 40 agents is required

is repeated with pn D Rr .pn1/ until only one upper level is constituted by one single group, which eventually elects the hierarchy president. One typical hierarchy is shown in Fig. 7.2. 7.2 Incorporating the Inertia Effect of Being the Ruler 123

7.1 The Local Majority Rule Model

We consider groups of three agents randomly aggregated with a local majority rule [1Ð13]. Equation (7.1) yields the probability to have an A elected at level .nC1/ from a group at level n,

3 2 pnC1 P3.pn/ D pn C 3pn.1 pn/; (7.2) where pn is the proportion of elected A people at level n. Equation (7.2)hasthree ﬁxed points pd D 0, pc;3 D 1=2,andpt D 1. They correspond respectively to the disappearance of A, the threshold to deterministic full power, and the disappearance of B. While pc is unstable, pd and pt are stable. From p0 <1=2, repeating voting yields p0 >p1 >p2 < 1=2:::downwards pd D 0 while the ﬂow is backwards if p0 >1=2with p0

7.2 Incorporating the Inertia Effect of Being the Ruler

Majority rule is well defined for odd sized groups. However, it is not determined for even sized groups since local ties may occur such as in the case of a group size of 4 with 2 AÐ2 B configurations for which no majority prevails. Instead of discarding even sized groups, there is a certain interest in taking advantage of this deficiency. It is well admitted that to change a policy, a clear cut majority is required. When no majority exists, things stay as they are. This situation is a natural bias in favor of the status quo [1,2,4Ð7,9Ð12]. In most institutions, such a tie is avoided, for instance by giving one additional vote to the committee president. Such a scheme turns the voting function into being asymmetrical. Assuming the president is B, for an A to be elected at level n C 1 from level n the probability becomes 4 3 pnC1 P4.pn/ D pn C 4pn.1 pn/; (7.3) 124 7 The Dictatorship Paradox of Democratic Bottom-up Voting

Fig. 7.3 Given the same population with p0 D 0:40 for A and 1 p0 D 0:60 for B, two different three-agent voting group hierarchies are built. The hierarchy on the left hand side has six levels, which restores the democratic balance with p6 D 0:00. It requires a bottom of 729 agents with a total of 1,093 agents. On the right hand side, the hierarchy has only three levels, making the president an A with a probability of p3 D 0:20. Only 27 agents are necessary at the bottom for a total of 40 agents

while for B it is

4 3 2 2 1 P4.pn/ D pn C 4pn.1 pn/ C 2pn.1 pn/ ; (7.4) where the last term embodies the bias in favor of B. The associated stable fixed points are unchanged at (0) and (1). However, the unstable fixed point is drastically shifted from 1=2 to p 1 13 p C 0:77: c;4 D 6 (7.5) This sets the A threshold to the presidency at a much higher value than the expected 50%. Moreover, the process of self-elimination is accelerated as seen from the series p0 D 0:69, p1 D 0:63, p2 D 0:53, p3 D 0:36, p4 D 0:14, p5 D 0:01, and p6 D 0:00. A majority of 63% is suppressed democratically from the higher leadership levels within only five voting levels. 7.3 From Probabilistic to Deterministic Voting 125

An a priori reasonable bias in favor of the ruling party turns a majority rule democratic voting to an effective dictatorship outcome. To obtain the presidency, A must gain over 77% of the bottom support, a goal that is virtually out of reach in any democratic environment.

7.3 From Probabilistic to Deterministic Voting

Generalizing the above results to any size r, the voting function pnC1 D Pr .pn/ becomes lXDm rŠ l rl Pr .pn/ D pn.1 C pn/ ; (7.6) lŠ.r l/Š lDr where m D .r C 1/=2 for odd r and m D .r C 1/=2 for even r to account for the B favored bias. The two stable ﬁxed points pd D 0 and pt D 1 are independent of the group size r.Forr odd values, the unstable pc;r D 1=2 is also independent of r r the group size , for which there exists no bias. On the contrary, it variesp with for even values. It starts at pc;2 D 1 for r D 2, decreases to pc;4 D .1 C 13/=6 0:77 for r D 4, and then keeps on decreasing asymptotically towards 1=2 from the above [1, 2, 4Ð12]. Given a bottom support p0

pn pc;r .pn1 pc;r /r ; (7.8) which can be iterated to give

n pn pc;r .p0 pc;r /r : (7.9)

Taking the logarithm of both sides, the critical number of levels nc at which pn D is obtained as

ln.pc p0/ nc C n0; (7.10) ln r where n0 ln.pc;r /= ln r . Plugging in n0 D 1 while taking the integer part of the expression yields rather good estimates of nc in comparison to the exact estimates obtained numerically by iterations of (7.7). 126 7 The Dictatorship Paradox of Democratic Bottom-up Voting

The above expression is essential but does not allow us to elaborate a strategy since most organizations have a fixed structure, which cannot be modified at will before every new election, even though sometimes this happens. The number of hierarchical levels is thus fixed and constant.

7.4 The Magic Formula for Presidency

To take advantage of the above analysis, we need to reverse the question of ¥ How many levels are needed to obtain a tendency of self-elimination? to ¥Givenn levels, what is the necessary bottom support to get the presidency with certainty? The “power landscape” is not symmetrical for A and B. Here we address the dynamics of voting from the A perspective. To implement the reformulated operative question, we rewrite (7.9)as

n p0 D pc;r C .pn pc;r /r ; (7.11)

pn which exhibits two different critical thresholds. The ﬁrst one is d;r at which the A representatives vanish thanks to the democratic voting. With a certainty of the elected representatives at level n, the elected president is B with certainty. It is obtained by putting pn D 0 in (7.11) with

pn p .1 n/: d;r D c;r r (7.12) p 1 pn The value n D yields the second threshold f;r above in which the A obtains full and total power, including the presidential level. We get

pn pn n: f;r D d;r C r (7.13) p p pn < The existence of the two thresholds d;r and f;r produces a new region d;r p

n pn pn : r D f;r d;r (7.14) 7.5 A Simulation to Visualize the Dictatorship Effect of Democratic Voting 127

It shrinks as a power law of the number n of hierarchical levels. A small number of levels increases the threshold to a shift of presidency but simultaneously lowers the threshold for democratic disappearance. From (7.13), r D 4 yields D 1:64 and pc;4 0:77. Considering n D 3; 4; 5; 6; 7 pn level organizations, we get respectively for d;r the series 0.59, 0.66, 0.70, 0.73, pn and 0.74. In parallel, f;r equals 0.82, 0.80, 0.79, 0.78, and 0.78. The associated amplitude of the probabilistic region is 0.23, 0.14, 0.09, 0.05, and 0.04. These numbers clearly show the dictatorship character of the bottom-up voting process.

7.5 A Simulation to Visualize the Dictatorship Effect of Democratic Voting

To fully grasp the extent of the dictatorship phenomenon, we performed with Stephan Wonczak a large sized numerical simulation [8] with 16,384 bottom agents. Snapshots of typical situations are given in Figs. 7.4Ð7.7. The two competing A and B parties are represented respectively as white and black squares. In the case of a local tie, the bias is set in favor of the blacks. The bottom-up hierarchy operates with voting group sizes of 4 and has 7 voting levels in addition to the hierarchy bottom. The first three snapshots show a huge bottom white square majority. The A (white squares) are self-eliminated quite quickly. The percentages written on the lower right-hand part of the figure are for the white representatives at each level. The bottom is denoted (8) and the president is (1). The “Time” and “Generations” indicators should be discarded. The first Fig. 7.4 shows p0 D 52:17%, slightly over the expected 50% democratic threshold for taking over. However, three levels higher, no white squares appear. The bottom majority has self-evaporated. Figure 7.5 shows the same population with now a substantial increase in A (the white sites in the figures) support with a majority of p0 D 68:62%, rather more than the democratic balance of 50%. And yet, after four levels no more white (A) squares are found. The situation has worsened in Fig. 7.6 where the A (white sites in figures) support has climbed up to the huge value of p0 D 76:07%.Andyetsevenlevelshigher,a B (black) is elected with certainty, although its bottom support is as low as (black) 23.03%. Finally, Fig. 7.7 shows an additional very small 0.08% increase in A (white sites in the figures) support, putting the actual support at p0 D 77:05%, which in turn prompts an A to get elected (the white sites in the figures) as president. These simulations provide some insight about the often observed blindness of top leaderships towards the disastrous consequences of huge and drastic increases in dissatisfaction at the bottom level of their organization. For any given president, who could easily obtain direct information about the possible disagreements with its policy, it is impossible to recognize the real current state of affairs concerning the opposition, if he or she restricts his or her view of the reality of the situation to that of 128 7 The Dictatorship Paradox of Democratic Bottom-up Voting

Fig. 7.4 Even groups of four people can be used to build a bottom-up democratic eight level hierarchy. The white and black squares are denoted respectively the A and B members. At a tie, a B (black) is elected. The bottom level is denoted eight and the presidency one. At each level, 8,7,...,1,theproportions of A (white) are given. Here p0 D 52:17%, p1 D 34:69%, p2 D 11:91%, p3 D 0:78%, p4 D 0:00%, p5 D 0:00%, p6 D 0:00%, and p7 D 0:00%. The “Time” and “Generations” indicators should be discarded. The president is B (black) what is perceived by the immediate lower hierarchical level. As seen from Fig. 7.7, even when the opposition has reached a height of 68.62% support, the president still obtains 100% of totally satisﬁed votes from the two levels directly below him or her. Accordingly, he or she will arrive at a satisfying, albeit false, conclusion, so why should he or she make any change in policy? 7.6 From Two to Three Competing Parties 129

Fig. 7.5 Even groups of four people can be used to build a bottom-up democratic eight level hierarchy. The white and black squares are denoted respectively the A and B members. At a tie, a B (black) is elected. The bottom level is denoted eight and the presidency one. At each level, 8,7,...,1,theproportions of A (white) are given. Here the bottom A proportion has increased to p0 D 68:62% with then p1 D 63:23%, p2 D 52:25%, p3 D 33:59%, p4 D 12:50%, p5 D 0:00%, p6 D 0:00%, and p7 D 0:00%. The “Time” and “Generations” indicators should be discarded. The president is still B (black)

7.6 From Two to Three Competing Parties

The study of two competing parties to rule a bottom-up democratic hierarchy has revealed some rather surprising effects, in particular the dictatorship bias in favor of the ruling party. Often, a third party is present, making the competition for power 130 7 The Dictatorship Paradox of Democratic Bottom-up Voting

Fig. 7.6 Even groups of four people can be used to build a bottom-up democratic eight level hierarchy. The white and black squares are denoted respectively the A and B members. At a tie, a B (black) is elected. The bottom level is denoted eight and the presidency one. At each level, 8, 7, . . . , 1, the proportions of A (white) are given. Here the bottom A proportion has increased again to the huge value of p0 D 76:07% with then p1 D 75:68%, p2 D 75:20%, p3 D 73:83%, p4 D 73:44%, p5 D 62:50%, p6 D 50:00%, and p7 D 0:00%.The“Time”and “Generations” indicators should be discarded. The president is still B (black) more subtle. In most cases, this third party C is smaller than the two parties A and B. It is then the setting of alliances between the parties, which gives access to the presidency [3]. In going from two to three parties the tie occurs already within voting group sizes of three with the (A B C) conﬁguration, which is unsolved by majority rule. This is equivalent to the two party tie case with the (A A B B) conﬁguration. For this last case, the tie was resolved in favor of the ruling party, like giving an additional vote 7.6 From Two to Three Competing Parties 131

Fig. 7.7 Even groups of four people can be used to build a bottom-up democratic eight level hierarchy. The white and black squares are denoted respectively the A and B members. At a tie, a B (black) is elected. The bottom level is denoted eight and the presidency one. At each level, 8,7,...,1,theproportions of A (white) are given. Here the bottom A proportion has increased slightly from the huge value of p0 D 76:07%top0 D 77:05%. This slight increase of bottom A support raises the bottom support to just above the threshold and thus allows the A to take over the hierarchy with p1 D 77:25%, p2 D 76:46%, p3 D 78:52%, p4 D 75:00%, p5 D 81:25%, p6 D 75:00%, and p7 D 100:00%. The “Time” and “Generations” indicators should be discarded. The president is eventually A (white) to the committee president. The rationale was that since the ruling party had 50% of the votes, it is natural to require ﬁfty percent plus one vote from the opposition to take over the leadership. But here, the ruling party gets only one third of the votes, making the preceding rationale inapplicable. It is therefore through agreements between parties that the tie is resolved, for instance, with party A voting for party C at certain hierarchical levels and party C voting for party A at other levels. Usually 132 7 The Dictatorship Paradox of Democratic Bottom-up Voting

A will support C at low levels of the hierarchy and C will then help A to reach the presidency. A full chapter is devoted to a thorough investigation of the subtle case of three competing groups at the end of the book. We present here only a few illustrations to give a ﬂavor of the richness of the voting dynamics monitored by a large spectrum of settings for local alliances among parties, which can be varied from level to level. The voting functions for respectively A, B, and C in the most general case to go from level l up to level .l C 1/ can be written,

p P .p ;p / p3 3p 2 .1 p / 6˛p p .1 p p /; a;lC1 a;3 a;l b;l D a;l C a;l a;l C a;l b;l a;l b;l (7.15)

p P .p ;p / p3 3p 2 .1 p / 6ˇp p .1 p p /; b;lC1 b;3 a;l b;l D b;l C b;l b;l C a;l b;l a;l b;l (7.16)

pc;lC1 D .1 pa;lC1 pb;lC1/; (7.17) where ˛ and ˇ measure the average net results of the local alliances. From the study of two competing parties, we can conclude that any support higher than 50% at the bottom for a given party will guarantee the presidency provided that there exist enough hierarchical levels. Therefore, the goal of each party is to gain the maximum of the central triangle delimited by fpa;0 < 0:50; pb;0 < 0:50; pc;0 < 0:50g. This conclusion will become clear below from Figs. 7.9Ð7.11. The three extreme cases of alliances are set with .˛ D 1; ˇ D 0; D 0/, .˛ D 0; ˇ D 1; D 0/,and.˛ D 0; ˇ D 0; D 1/. Figure 7.8 shows the voting dynamics in each case for the four different bottom distributions fpa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18g, fpa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20g, fpa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43g,andfpa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53g.The flows correspond to a six-level hierarchy. The winner of the presidency is a direct function of the alliance set as seen from the figure (Figs. 7.8Ð7.11). We show a few other voting flows in Figs. 17.37Ð17.38. The respective set of alliances is .˛ D 0; ˇ D 0; D 1/, .˛ D 0:18; ˇ D 0:42; D 0:40/,and.˛ D 0:68; ˇ D 0; D 0:32/.

7.7 The Overlap with the Physical Model

The bottom-up democratic voting model has no direct counterpart in physics. Although we have two species A and B, our agents are not Ising-like variables. Each agent belongs to one party and does not change its afﬁliation. We are using a mixture of one-state variables instead of a homogeneous system of two-state variables. Nevertheless, our model borrows one major ingredient from statistical physics, which comes from the very scheme of real-space renormalization group techniques, which uses a local majority rule to deﬁne a superspin. But the analogy ends here since our implementation is performed in a totally different way than in physics where the superspin is a virtual entity, which allows us to perform some correlation 7.7 The Overlap with the Physical Model 133

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

B 1

0.8

0.6

0.4

0.2

A 0.2 0.4 0.6 0.8 1

Fig. 7.8 Above left: .˛ D 1; ˇ D 0; D 0/; above right: .˛ D 0; ˇ D 1; D 0/; Below: .˛ D 0; ˇ D 0; D 1/. In each case, the same four bottom supports are considered with fpa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18g, fpa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20g, fpa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43g, fpa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53g calculations. In our voting case, the local rule is operated to add a real new agent that is the representative of the group at the higher hierarchical level. This elected agent is not a substitute for the group. It is not a virtual superagent. The group and its elected representative are real agents, which are simultaneously present. All the hierarchical levels are real. In physics, renormalization group techniques are mathematical methods. They are used to extract certain physical quantities with a high accuracy, which characterize a given system. Applying a renormalization group scheme to a physical system does not modify the system. In our case, the system is built up step by step following the scheme principle. The hierarchy does exist. Accordingly, an n level hierarchy is different from an m level hierarchy. 134 7 The Dictatorship Paradox of Democratic Bottom-up Voting

Fig. 7.9 Two-dimensional voting ﬂow diagram for a three party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the tails of the arrows. Here the set of voting alliances is .˛ D 0; ˇ D 0; D 1/

Fig. 7.10 Two-dimensional voting ﬂow diagram for a three party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the tails of the arrows. Here the set of voting alliances is .˛ D 0:18; ˇ D 0:42; D 0:40/ 7.8 Achievements of the Model 135

Fig. 7.11 Two-dimensional voting ﬂow diagram for a three party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the tails of the arrows. Here the set of voting alliances is .˛ D 0:68; ˇ D 0; D 0:32/

7.8 Achievements of the Model

Novel counterintuitive social and political predictions are obtained from the bottom- up democratic hierarchical model. The results shed new light on some essential and surprising mechanisms of majority rule voting. It is very general and allows us to consider many different applications, such as the empirical difﬁculty in changing leaderships in well-established institutions. In particular, it exhibits paradoxical and unexpected explanations to a series of social features and historical events including an astonishing view about the sudden auto-collapse of Eastern European communist parties. While communist parties in power had seemed to be eternal, they collapsed virtually at once. Most explanations based the phenomena on some opportunistic change at the head of the various organizations involved, which were feasible due to the end of the Soviet army threat. However, the explanation could be of a different nature. Communist organizations are indeed based, at least in principle, on the concept of democratic centralism which is a tree-like hierarchy similar to our bottom-up model. Such a structure is very stable to changes at the bottom of the organization. To illustrate our viewpoint, imagine a communist party organized along our scheme with a voting group size of four. We saw that in that case, the threshold to take over the presidency for a challenging party is at 77%. In contrast, it is at only 23% for the ruling party. 136 7 The Dictatorship Paradox of Democratic Bottom-up Voting

In other words, the ruling party needs to preserve a support of just beyond 23% in order to be “democratically” reelected. Accordingly, the internal opposition to the orthodox communist leadership could have grown steadily over several decades without leading to any change at the top levels of the communist hierarchy. It could reach huge heights, as much as 76%, and still have no visible effect since only the top leadership makes the party policy, which is the one perceived from the outside. It thus creates an artificial feeling of a totally frozen situation, while indeed massive changes were occurring but confined to the bottom level and a few higher ones. Then, a little increase of additional opposition at the bottom of just one or two percent, which is negligible with respect to the previous seventy or so percent increase, will provoke a radical swing at the top of the hierarchy, creating the impression of an opportunistic move. Such a process is illustrated in the series of Figs. 7.4Ð7.7. From the outside, the several decades long increase in opposition was invisible. Indeed, it looked as though nothing was changing. And once the threshold was passed, the shift appeared to be instantaneous [16]. We do not pretend that what happened in Eastern communist parties obeyed the above simplistic mechanism, but that something analogous was also at work in the real process. We have a general explanation to show that what looked like a sudden and punctual decision of a top leadership could be indeed the result of a very long and solid phenomenon inside the communist parties. This explanation does not oppose the very many additional features that were instrumental in these massive collapses. However, it might well have singled out some trend within the internal mechanism of these organizations, which on the one hand, are extremely stable, and on the other, collapse at once. Our hierarchical model provides some new insight of such a rare historical event. In addition to the above explanation, using the hierarchical model, I was able to predict a political scenario, which could have happened in France, and which eventually did occur with respect to the extreme right party of the National Front. With the rise in popularity of this party, I enumerated the conditions for its success at a forthcoming election in a paper published in 1997 [14, 15]. The paper was perceived at the time as being too extravagant. When the scenario did happen in 2000 along the predicted lines, people were shocked and totally surprised. Such an occurrence of an event was perceived as not being possible, in particular since this party did not gain a significant increase of support from the electorate with respect to the previous election. Only one hundred thousand votes turned the balance in forces upside down. The extreme right party did win the first round against the socialist leader, producing a political earthquake in France. I denoted this effect as being the straw that not only breaks the camel’s back but also kills the camel! Fortunately, its leader eventually lost in the second final run thanks to a large Republican front against it [17Ð19]. The totally unexpected uprisings of 2010/2011 in Tunisia and Egypt (the case of Libya is different) could also be viewed from the same angle. In these cases, all the ruling institutions were taken by a total surprise with political situations being suddenly shifted from what was perceived as existing “forever”, to an incredible References 137 void. The implacable strength of those regimes suddenly appeared to be completely illusory. The result was that they collapsed within a few weeks under the pressure of sustained street demonstrations. But indeed, in each case this was the result of a very long and unnoticeable process, which by chance crossed a tipping point, making the “invisible iceberg” of many years brutally visible and solid.

References

1. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, J. of Math. Psychology 30, 426Ð434 (1986) 2. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943Ð951 (1990) 3. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191Ð200 (1991) 4. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organi- sations”, Models of self-organization in complex systems (Moses) Akademie-Verlag, Berlin V. 64, 53Ð59 (1991) 5. S. Galam, “Paradoxes de la regle majoritaire dans les systemes hierarchiques”,« Revue de Bibliologie, 38,62D-68 (1993) 6. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132Ð139 (1999) 7. S. Galam, “Real space renormalization group and totalitarian paradox of majority rule voting”, Physica A 285, 66Ð76 (2000) 8. S. Galam and S. Wonczak, “Dictatorship from Majority Rule Voting”, Eur. Phys. J. B 18, 183Ð 186(2000) 9. S. Galam, “Democratic Voting in Hierarchical Structures”, Application of Simulation to Social Sciences, G. Ballot and G. Weisbush, Eds. Hermes, Paris, 171Ð180 (2000) 10. S. Galam, “Building a Dictatorship from Majority Rule Voting”, ECAI 2000 Modelling Artiﬁcial Societies, C. Jonker et al, Eds., Humboldt U. Press (ISSN: 0863-0957), 23Ð26 (2001) 11. S. Galam,“How to Become a Dictator”, Scaling and disordered systems. International Work- shop and Collection of Articles Honoring Professor Antonio Coniglio on the Occasion of his 60th Birthday. F. Family. M. Daoud. H.J. Herrmann and H.E. Stanley, Eds., World Scientiﬁc, 243Ð249 (2002) 12. S. Galam, “Dictatorship effect of the majority rule voting in hierarchical systems”, Self- Organisation and Evolution of Social Systems, Chap. 8, Cambridge University Press, C. Hemelrijk (Ed.) (2005) 13. S. Galam, “Stability of leadership in bottom-up hierarchical organizations”, Journal of Social Complexity 2 62Ð75 (2006) 14. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 15. S. Galam, “Crier, mais pourquoi”, Liberation,« Vendredi 17 Avril, 6 (1998) 16. S. Galam, “Le vote majoritaire est-il totalitaire?”, Pour La Science, Hors serie,« Les Mathematiques« Sociales, 90Ð94 July (1999) 17. S. Galam, “Citation in front page of the Figaro in an editorial from Jean dOrmesson”, Le Figaro, Mardi 4 Juin, 1 (2002) 18. S. Galam, “Risque de raz-de-maree« FN, Entretien, France Soir, La Une et 3, Mercredi 5 Juin (2002) 19. S. Galam, “Le FN au microscope”, Le Minotaure 6, 88Ð91, Avril (2004) Chapter 8 The Dynamics of Spontaneous Coalition–Fragmentation Versus Global Coalitions

In the study of group decision making, I introduce the Ising ferromagnetic model, which assumes that interactions amongst agents tend to align themselves along the same opinion. It is inspired from the existence of ferromagnetism in inert matter. However, nature is rather rich and usually covers the whole spectrum of symmetries. Therefore, anti-ferromagnetism, i.e., a material in which interactions force the coupled spins to be anti-parallel. Accordingly, it was natural to explore this feature and to introduce within the social models couplings amongst agents, which push them apart in regards to their respective choices. Indeed, the ﬁrst paper to take advantage of this option was from political scientists [1]. They considered countries instead of individuals, which was more appealing. Unfortunately, they got confused about different spin glass models, making their paper misleading [2]. On this basis, I then undertook the study of the dynamics of spontaneous coalition forming versus fragmentation. I introduced a scheme to build up coalitions by the free will of the agents [2Ð6]. The model applies equally to a group of ﬁrms, people, or any social bodies.

8.1 The Two-Country Problem

A world which was reduced to one single country could appear as being a beautiful perspective in the light of all the past wars which have been driven by the numerous fights amongst different countries. To try to reach an understanding of these past and still present wars and conflicts, we start to describe a world consisting first of two countries and then of three before extending the approach to N countries. Given two countries i and j , they might either cooperate or be in conflict (see Fig. 8.1). In the first case, it would be “paradise” and in the second case “hell”. Having two situations, we introduce a two-state variable Si D˙1.The states C1 and 1 are symmetrical. Four possible configurations are then possible with fSi D˙1; Si D˙1g. The countries are cooperating when Si D Sj D˙1 and in conflict when Si DSj D˙1.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 139 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 8, © Springer Science+Business Media, LLC 2012 140 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions

Defining the problem

Heaven: One single country Life is beautiful Adding a country

Hell or Paradise?

Compete Collaborate

Fig. 8.1 The two-country option: to compete or to collaborate, that is the question

On this basis, to evaluate the cost of the two countries either cooperating or being in conflict, we note that they have been developing a propensity to either cooperate or to be in conflict during the many years of their mutual existence. This propensity depends on many different ingredients, including geography, resources, religion, language, historical events, and other characteristics. We make the hypothesis that the net result of this common life history ends up with a quantitative bilateral bond, which can be represented by a quantity Gi;j .Itis positive .Gi;j >0/if the tendency is to cooperate and negative .Gi;j <0/if the tendency is towards conflict. An absence of links is also possible with Gi;j D 0. The Gi;j is a quenched variable since it is the balance of accumulation over a long period of time, which is not given to being modified at once by free will. It can be modified only over a long stretch of historical time. The Gi;j >0coupling does not depend on i and j separately. It is a common bond between the couple .i; j /.We also assume symmetrical coupling with Gi;j D Gj;i . The cost of the two countries i and j attitudes is measured by the quantity

Hi;j DGi;j Si Sj ; (8.1) which equals Gi;j for cooperation and Gi;j for conflict. At this stage, we need a criterion to determine each country’s choice for a given Gi;j . We make the assumption that each country wants to minimize its cost. This implies that a negative cost is beneficial for the country. Therefore, Gi;j >0favors cooperation with Si D Sj while Gi;j <0pushes towards conflict, i.e., Si DSj . It is worth underlining that for a pair of countries, which have a negative coupling, to be in conflict is a stable configuration. Conflict does not mean active war (see Figs. 8.2Ð8.4). 8.1 The Two-Country Problem 141

Fig. 8.2 The two-country S =± option: quantifying the Si =±1 j 1 problem with Ising variables

+1, +1 +1, -1 -1, -1 -1, +1

Collaborate Compete

What’s the cost ?

Fig. 8.3 Gi;j which results G : Intensity of exchange > 0 or < 0 from the past history between ij the two countries is given and there exists no way of changing its value on a short The cost of both countries’ attitude time scale. It is a bond pair exchange, with which both countries have to deal. It determines the common cost G SS for the two countries’ ij i j behavior Si =−Sj =±1⇒−Gij

Si = Sj =±1⇒+Gij

Fig. 8.4 Setting a minimum We need a criterion cost as a criterion for each country’s better choice

Each country acts in order to minimize its cost with respect to countries it has an interaction with

Gij > 0 ⇒ Si = Sj =±1

Gij < 0 ⇒ Si =−Sj =±1 142 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions

8.2 The Unstable Three-Country Problem

A two-country problem finds its equilibrium rather easily. Indeed it is the past common attitudes which determine the present ones. A past of conflict creates present conflict while a past of cooperation produces present cooperation. But most of the time a country has more that one neighbor. Let us investigate the case of three neighboring countries 1, 2, and 3. As for the two-country case, we have a set of bilateral exchanges G1;2;G1;3; and G2;3. In the case where G1;2;G1;3; and G2;3 are all positive, our criterion of conflict minimization makes the three countries cooperate in a stable spontaneously formed alliance with S1 D S2 D S3 D˙1. However, in the case where one of the couplings, say G2;3, is negative the situation is turned into an unstable situation with a perpetual cycle of an ongoing formation of alliances followed by fragmentation followed by new alliances and so on. To illustrate such infernal dynamics, let us take G1;2 D G1;3 DG2;3 D G (see Fig. 8.5). The situation is shown in Fig. 8.6. If countries 1 and 3 decide to cooperate in order to minimize their mutual conflict .G1;3 >0/,wehaveS1 D S3 D˙1.In thefigureitisC1. Then, country 2 has to decide which attitude to adopt. Since G1;2 >0, it wants to cooperate with country 1 but simultaneously G2;3 <0,so it wants to be in conflict with country 3. The simultaneously applied criterion of minimum conflict does not provide a single choice. With both being equivalent, let us take S2 D1. Two alliances have thus been spontaneously formed, one with the two countries 1 and 3 with S1 D S3 DC1 and one alliance reduced to the country 2 with S2 D1. The respective costs for countries 1, 2, and 3 are .H1;3 C H1;2 DG C G D 0/, .H2;1 C H2;3 DCG G D 0/,and.H3;1 C H3;2 DG G D2G/.Only country 3 is at its absolute minimum conflict with 2G. Both other countries have their conflicts equal to 0.

For a pair of countries, life is simple

But with a third partner things get already complicated

For instance Gij = G ji

Fig. 8.5 From the simple two-country problem to the G = G = G = G > unstable three-country set 12 23 31 0 8.2 The Unstable Three-Country Problem 143

1 +

G > 0 G > 0

23 + + 2 - -G < 0

What to do ? Choose + or - at random

Cost: 0

2 coalitions: one with 1 and 3, -+ the other with 2

Cost: 0 Cost: -2G

Fig. 8.6 The unstable three-country alliances breaking dynamics

Each of them can be tempted to shift alliance so as to try to decrease their conflicts down to the absolute value of 2G. For instance, country 2 is at maximum with country 1 but at a minimum with country 3. It can thus shift from S2 D1 to S2 DC1. By so doing, its conflict becomes .H2;1 C H2;3 DG C G D 0/,which is globally unchanged. Now it is at a minimum with country 1 but at a maximum with country 3. It is still not at minimum conflict. In addition, the country 2 shift has changed the level of conflict of countries 1 and 3. Country 1 is at its minimum .H1;3 C H1;2 DG G D2G/ while country 3 has raised its minimum conflict from 2G to .H3;1 C H3;2 DG C G D 0/. The possible shifts are from either one of the countries 2 and 3 at no cost, with neither gain nor loss. The cycle is endless, as shown in Fig. 8.7. The cycle is not even predictable since two countries might be tempted to shift alliance simultaneously. When two countries have an interest in shifting alliance either to lower their current cost or because it is at zero cost, it is not possible to know in advance which one will actually move. It is not possible either to determine when. Figure 8.8 illustrates the complexity of the situation showing from one given alliance set four possible alliance breaks. Although our three-country case could look artificial, in the real world, indeed, several such cases do exist with the famous three cases of fIsrael, Iran, Iraqg, fFrance, England, Germanyg,andfSerbia, Bosnia, Croatiag (see Fig. 8.9). The situation appears rather inextricable and yet we consider only three countries. Enlarging the number of connected countries makes the dynamics more subtle and totally unpredictable. Many international situations involve many countries 144 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions

One illustration of endless dynamics

0 +2G + +

- + + + +2G +2G -2G 0

0 + 0 -

... - + - + -2G 0 0 -2G

Fig. 8.7 One particular path of the alliances breaking dynamics for three countries

Cost: 0

Cost: -2G -+Cost: 0

1 2 3 Cost: -2G Cost: 0 Cost: 2G

+ - -

- - - + - -

Cost: 0 Cost: 0 Cost: 0 Cost: -2G Cost: 2G Cost:2G

Fig. 8.8 The alliances breaking dynamics for three countries is not predictable since many different paths are possible from a given set of alliances 8.2 The Unstable Three-Country Problem 145

Some real life examples of endless coalition forming

Israel Serbia

Iran Iraq Bosnia Croatia

France

England Germany

Fig. 8.9 Three real life sets of unstable three-country sets

-G(-3G) 1 -2G(-2G) Another

2 illustration 3 -G(-3G) with 5 actors

4 -2G(-4G)

5 0(-2G)

All bonds < 0 => competition

Fig. 8.10 Countries 1 and 4 are allied against countries 2, 3, and 5. Only country 2 has an absolute minimum conflict at 2G while all the others are at an amplitude of C2G above their absolute values as indicated in the figure distributed in various subsets of bond couplings. We now present one specific illustration with five countries as shown in the series of Figs. 8.10Ð8.16. The first figure considers a situation with countries 1 and 4 against countries 2, 3, and 5. Within such a set, only country 2 has an absolute minimum conflict while all the others are at an amplitude of C2G above their absolute values as indicated in Fig. 8.10. This creates a large spectrum of possible spontaneous individual shifts. 146 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions

-G(-3G) 1 -2G(-2G) 5 has shifted alliance 2 3 -3G(-3G)

4 0(-4G)

5 0(-2G)

All bonds < 0 => competition

Fig. 8.11 From the alliance set of Fig. 8.10, country 5 joined the 1Ð4 alliance leaving the 2Ð3 alliance at zero cost

-3G(-3G) 1 0(-2G) 4 has shifted alliance 2 3 -G(-3G)

4 0(-4G)

5 -2G(-2G)

All bonds < 0 => competition

Fig. 8.12 From the alliance set of Fig. 8.11, country 4 joined the 2Ð3 alliance leaving the 1Ð5 alliance at zero cost

Figure 8.11 shows a move from country 5 at zero cost to join the 1Ð4 alliance, leaving the 2Ð3 alliance. This move makes country 3 happy which is now at its minimum cost with 3G, but country 4 is unhappy since its cost has jumped from 2G to zero. Therefore, country 4 shifts alliance, which puts its cost back to 2G. The move makes countries 5 and 1 happy, with their costs reaching their minimum at respectively 2G and 3G as indicated in Fig. 8.12. Only country 2 is unhappy with a cost at zero instead of 2G. 8.2 The Unstable Three-Country Problem 147

-G(-3G) 1 2G(-2G) 2 and 4 have shifted alliance 2 3 -3G(-3G)

4 2G(-4G)

5 0(-2G)

All bonds < 0 => competition

Fig. 8.13 From the alliance set of Fig. 8.12, countries 2 and 4 have shifted alliance simultaneously at zero cost to form a large alliance with 1 and 5. Country 3 is alone against everyone, which is its best optimal positioning at a cost of 3G

-3G(-3G) 1 0(-2G) 2, 4 and 5 have shifted alliance 2 3 G(-3G)

4 2G(-4G)

5 2G(-2G)

All bonds < 0 => competition

Fig. 8.14 From the alliance set of Fig. 8.13, a multiple shift has involved countries 2, 4, and 5 leaving country 1 isolated and happy as seen from the various country costs

After these successive moves, the alliance set remains unstable. In Fig. 8.13,the two countries 2 and 4 shift simultaneously at zero cost to form a large alliance with 1 and 5. Country 3 is alone against everyone, which is its best optimal positioning at a cost of 3G. It is worth reminding that all bond exchanges are negative, making country 3’s position the most favorable one. At the same time within the 148 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions

G(-3G) 1 -2G(-2G) 3, 4 and 5 have shifted alliance 2 3 3G(-3G)

4 2G(-4G)

5 2G(-2G)

All bonds < 0 => competition

Fig. 8.15 From the alliance set of Fig. 8.14, countries 3, 4, and 5 with positive cost have shifted at once to produce the new conﬁguration with country 2 alone against the 1Ð3Ð4Ð5 alliance. But country 3 is at a higher cost of C3G

-G(-3G) 1 -2G(-2G) 3 has shifted alliance 2 3 -3G(-3G)

4 0(-4G)

5 0(-2G)

All bonds < 0 => competition

Fig. 8.16 From the alliance set of Fig. 8.15, country 3, which had its highest possible cost at C3G, has shifted to reach its most favorable cost at 3G set, countries 2 and 4 are at a great disadvantage, with heavy costs at C2G.This hints at a next shift from them, which is shown in Fig. 8.14, where country 5 has also shifted at no cost. The new conﬁguration is that everyone is against country 1. The main loser from the previous multiple shifts is country 3, which now has a cost of C1G. 8.3 Superposing Current Pair Bonds to Historical Ones 149

-G -2G(-4G) -2G(-2G) 1 Almost same 2 3 -G(-3G) configuration: one additional link 4 -2G(-4G)

0 5 -G(-3G)

It is stable ! All bonds < 0 => competition

Fig. 8.17 The addition of one negative bond between countries 1 and 5 reverses the whole situation from being unstable to stable, with country 5 shifting alliance

This “chaotic situation” (not to be taken in the mathematical sense of chaos) keeps on forever, with the next step involving the three countries with positive cost shifting at once to produce the new conﬁguration shown in Fig. 8.15. Now, country 2 is alone against the 1Ð3Ð4Ð5 alliance. But country 3 being at a higher cost of C3G, it shifts alliance, yielding the set given in Fig. 8.16. By so doing it turns immediately its heavy cost of C3G to the very favorable value of 3G. Nevertheless, if some regions of the world such as those in Fig. 8.9 are known to be unstable, others are quite stable. Indeed it depends directly on the set of bonds which couples each pair of countries in a given subset. The case of ﬁve countries shown in the series of Figs. 8.10Ð8.16 was found to be unstable. However, adding one negative bond for instance between countries 1 and 5 can reverse the whole situation by making it stable, as shown in Fig. 8.17. Here, no country gains an advantage from breaking the current alliance with countries 1 and 4 against countries 2, 3, and 5.

8.3 Superposing Current Pair Bonds to Historical Ones

The remark at the end of the last subsection about the addition of a single pair bond to stabilize an infernal situation of continually changing alliances is of course significant and prompts us to ask the question, “why not just apply this recipe in the real world?”. The answer is twofold. Firstly, we have emphasized the fact that 150 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions these pair bonds are built over long historical periods and thus are not given to modification. Secondly, it opens the way to circumvent this intrinsic difficulty by creating temporal bonds superimposed on the untouchable ones. The central idea is to neutralize some of the bonds, which are at the core of the instability by creating super bonds. However, the characteristics of these super bonds is to be transient. They will be active as long as they are being activated by the countries involved. As soon as they are left to themselves, they will fade away, bringing back the command to the historical bonds. We now implement this idea by building our coalition model. While the above analysis is grounded on each country’s choice with respect to its neighbors, we now introduce the process by which a supra alliance is first set by one or several countries, with other countries eventually joining it. This option is now a witting political choice. We start from a group of N countries and the existence of two supra alliances denoted A and B. Each country is free to join either one of two coalitions under the condition to be accepted since now the a priori existence of these supra alliances presupposes central commands which define the respective policies. We keep the variable Si D˙1 attached to each country with i D 1;2;:::;N.WetakeSi DC1 if country i joins coalition A and Si D1 if it joins coalition B. These supra alliances always have a principal founding country like, among others, America, Russia, Egypt, or Iran. On this basis, each neighboring country has also developed a “natural” trend to join or to repulse one of the two coalitions. To account for this natural trend we introduce a new variable i to be attached to each country. It is i DC1 if country i would rather join A and i D1 if it would rather join B. No natural a priori is also possible and denoted by i D 0.These variables are quenched as the Gi;j . Once countries belong to one of the supra alliances, new cooperating exchanges are activated among the members of each alliance as well as competition between members of different alliances. Measuring the exchanges between a pair of countries .i; j / by a coupling Ji;j >0, we define an effective coupling Ji;j i j , which can be either positive or negative depending on whether countries i and j are cooperating (belonging to the same alliance) or competing (belonging to opposite alliances). An additional propensity Ji;j i j Si Sj is thus created in addition to the historical Gi;j . While the i are not given to change, the coupling Ji;j >0is directly monitored by the will of both the countries involved and the alliance central command. It is essential to strongly underline the fundamental nature of this coupling, which can exist only through active exchanges. Any halt in the solid exchanges makes the coupling evaporate. Taking into account both kinds of exchanges, the countries’ choices are now determined by an extended cost

Hi;j D .Gi;j C i j Ji;j /Si Sj ; (8.2) between countries i and j instead of from (8.1). In addition to the spontaneous trend of each country, we also need to incorporate the possible direct beneﬁts or inconveniences, which result from economic and 8.4 From Binary to a Multiple Coalitions 151 military pressure attached to a given alignment. A variable ˇi D˙1 is introduced to account for this effect. It is ˇi DC1 infavorofA,ˇi D1 for B, and ˇi D 0 for no direct pressure. The amplitude of this economic and military interest or handicap is measured by a local positive ﬁeld hi , which also accounts for the country’s size and importance. The cost of actual country choices becomes

Hi;j D .Gi;j C i j Ji;j /Si Sj ˇi hi Si : (8.3)

We are now in a position to add all costs, which result from each pair coupling to obtain the total system cost as

XN ˚ XN H D Gi;j C i j Jij Si Sj ˇi hi Si ; (8.4) .i>j / i where N is the number of countries and .i > j / means all distinct pairs to avoid double counting [2Ð4, 6].

8.4 From Binary to a Multiple Coalitions

Up to now we have considered only two alliances but sometimes, several different alliances may exist simultaneously. Indeed, in effective terms, these multiple alliances often lead to extra alliances among them, which ends up with two competing groups. Then, it is each alliance which has to be treated as the individual actor creating a super alliance level. It is thus of interest to extend the analysis to multiple alliances. On this basis, we extended with Florian the previous bimodal approach to that of a multimodal one [5]. A case of three competing alliances can be found in the software domain where Windows, Mac OS, and LinuxnUnix are used as competing computer operating systems. To generalize the previous binary study to a multiple one is readily implemented thanks to the so-called Potts variables, which are a straightforward generalization of the Ising variables. We take Si D 1; 2; 3; : : :; q instead of Si D˙1,where q is the number of possible alliances. To arrive at the most general case, we put q D N accounting for the extreme possibility in which each country makes its own single alliance, i.e., in effect no alliance set. Before, the number of alliances was set at two and thus was an external parameter of the problem. Now the number q of actual coalitions is an internal degree of freedom, which can vary from 1 to N . It will be through the solving of the dynamics that the number of coalitions will be determined. It is no longer known a priori. The possibility of neutrality can be included by assigning one additional value to the possible N discrete values. To preserve the Ising-like counting of being or not in the same alliance using the product Si Sj D˙1, we substitute the function ı.Si ;Sj / for the product Si Sj , 152 8 The Dynamics of Spontaneous CoalitionÐFragmentation Versus Global Coalitions where ı.x; y/ is the Kronecker delta function with ı.x;x/ D 1 and ı.x; y/ D 0 for x ¤ y. Accordingly, the cost produced by pairs of historical exchanges writes XN Hij D Gij ı.Si ;Sj /: (8.5) i>j

This value is equal to Gij when Si and Sj belong to the same coalition and zero otherwise. For the bimodal case, we have ˙Gij and for the multimodal case it is Gij or 0. Once the model ingredients are deﬁned, the key issue becomes the determination of these propensities Gij . There exists no validated recipe up to date, only suggestions for their formulation. Studying the case of ex-Yugoslavia, we suggested the following expression X8 Gij D qikqjlwkl; (8.6) k;l where qik represents the percentage of ethnic group k in entity i and wkl represents the pairwise propensity between ethnic groups k and l [5]. For k D l, wkk DC1.Fork ¤ l,thewkls are computed as the sum of two terms. One stands for religion and the other for language: wkl D !religion.k; l/ C !language.k; l/.See[2Ð6] for more details.

8.5 The Overlap with the Physical Model

Several features are directly taken from the physics of a special family of alloys called spin glasses. The subject has been an intense source of study for several decades in both condensed matter physics and statistical physics notwithstanding the huge development of simulation capacities due to the very long relaxation times involved [7]. The presence of quenched disorder has also contributed to the simultaneous richness and difficulty of solving the problem. In our model, we use two categories of spin glass models, one with random bonds and one with random site bonds. The major difference between them is the existence of “frustration effects” in the first one and its absence in the second one. The definition of frustration is given in the example with a three-country system asshowninFigs.8.5 and 8.6. The presence of frustration produces an infinite degeneracy of the physical ground state, which in turn creates memory effects as well as pinning of domain walls. These two features make numerical simulations extremely difficult and uncertain. In addition, the equivalence of a quenched random field has also been used. Although the above ingredients are taken from physics, the novelty of my model is to have considered a superposition of a random bond spin glass, a random site spin glass, namely, the so-called Mattis model [8], and a random field system [9Ð11], leading to the unique Hamiltonian given by (8.4), which has not been considered in physics before. References 153

8.6 Achievements of the Model

The stability of the cold war period and the instabilities of Eastern Europe, which followed the auto-dissolution of the Warsaw pact, have been given a new surprising explanation. Several counterintuitive hints on how to determine some speciﬁc international policies have been elaborated, in particular with respect to enlarging the European Union as well as to extending the NATO coalition to incorporate the former Warsaw pact countries. For a detailed presentation, we refer to [2Ð6]. The stability of China has also been discussed along these lines. The question of evaluating quantitatively the propensities has been addressed in the case of the former ex-Yugoslavia with a signiﬁcant feasibility [5], but more work using real data needs to be performed. Several interesting results were found with respect to the Second World War and the fragmentation of ex-Yugoslavia. The case of Kosovo was also given some unusual explanations.

References

1. R. Axelrod and D. S. Bennett, “A landscape theory of aggregation”, British J. Political Sciences 23, 211Ð233 (1993) 2. S. Galam, “Comment on A landscape theory of aggregation”, British J. Political Sciences 28, 411Ð412 (1998) 3. S. Galam, “Fragmentation versus stability in bimodal coalitions”, Physica A 230, 174Ð188 (1996) 4. S. Galam, “Spontaneous coalitions forming: a model from spin glass”, arXiv:cond- mat/9901022, I. Kondor and J. Kertesz, Eds., Budapest (1999) 5. R. Florian and S. Galam, “Optimizing Conflicts in the Formation of Strategic Alliances”, Eur. Phys.J.B16, 189 (2000) 6. S. Galam, “Spontaneous Coalition Forming. Why Some Are Stable?”, Lecture Notes in Computer Science. 5th International Conference on Cellular Automata for Research and Industry, Springer-Verlag, Heidelberg, 1Ð9 vol. 2493 (2002) 7. Binder and A.E Young, “Spin glasses: experimental facts, theoretical concepts, and open questions”, Rev. Modem Physics 58, 801Ð976 (1986) 8. D.C. Mattis, “Solvable spin systems with random interactions”, Phys. Lett. A 56 421Ð422 (1976) 9. S. Galam and A. Aharony, A new multicritical point in anisotropic magnets. I. Ferromagnets in a random longitudinal field, Journal of Physics C: Solid State 13 1065Ð1081 (1980) 10. S. Galam and A. Aharony, A new multicritical point in anisotropic magnets. II. Ferromagnets in a random skew field, Journal of Physics C: Solid State 14 3603Ð3619 (1981) 11. S. Galam, A new multicritical point in anisotropic magnets. III. Ferromagnets in both a random and a uniform longitudinal field, Journal of Physics C: Solid State 15 529Ð545 (1982) Chapter 9 Terrorism and the Percolation of Passive Supporters

Dealing with terrorism is extremely delicate. Too much sorrow, passion, hate, and death are at stake. But that should not prevent us from trying to model the issue to try to single out its bare mechanisms and to eventually be able to curb its destructive capacity. The use of equations combined with what could look a priori like a simplistic viewpoint, but in fact is inspired from the physics of disorder, may indeed provide a new insight, which could eventually open unexpected paths to help in reducing terrorism. Any improvement is urged and should be welcomed. But modeling does not mean discovering the one and only “truth”. The problem is multi-fold, difﬁcult, complex, emotional, and operated by archaic passions. This is precisely why the use of “nonhuman” frames and concepts could turn out to be quite productive in providing us with a different paradigm for encompassing the phenomenon, which may lead to useful solid steps in curbing terrorism. September 11, 2001 marked a cornerstone for terrorism, in terms of its capacity for destruction and its range of action. The attack on the US has made terrorism a permanent threat worldwide. Before this dramatic attack, terrorism had been conﬁned to local geographic areas. Emblematic cases are found in Europe with Corsica, Northern Ireland, and Euskadi. And then countries that were very far from the core of traditional terrorism became aware that they also had turned into potential targets. The threat has been validated by the renewal of attacks in Madrid and London. While the clear source of the problem is the terrorist groups themselves, all priorities have been combined to try and neutralize them. However, from an empirical observation on the ground it seems that it is impossible to eradicate terrorism. On this basis, I have suggested that it is of importance to look at terrorism as being a duality. On the one hand, there is the terrorist organization itself on which all focus is concentrated, but on the other stands the issue of the social permeability to terrorist movements. And this last point is the key issue for reducing terrorist activities. This is my main hypothesis. Of course, a terrorist network itself is required for practicing terrorism, but in order to be able to act, a terrorist must move so as to reach the target and needs

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 155 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 9, © Springer Science+Business Media, LLC 2012 156 9 Terrorism and the Percolation of Passive Supporters

Remembering Mao Zedong

Fig. 9.1 The famous Mao Zedong statement about guerilla warfare to find a path within the social space. Accordingly, the terrorist range of potential destruction is determined by the actual social permeability of its members. Without a terrorist, there exists no terrorism, but without an open social space for the terrorist movement, there exists no terrorist action. How does it work? As Mao Zedong stated during the Chinese revolution, it is essential for a guerrilla to win the support of the people, who are like the ocean in which they must swim like fish (Fig. 9.1). For a terrorist, this means that he or she must find the water in which they can swim. The “water” can be reduced to a series of “droplets” formed by individual passive supporters. Passive supporters are not connected to terrorist activity, but they would not oppose a terrorist moving in the case where he could do so. The supporters share the goal of the terrorism, at least in part and are sympathetic to it. They have an individual and personal view which does not need to be explicitly expressed (Fig. 9.2). For most of the passive supporters, they are never confronted with any terrorist movement. They have a silent and dormant attitude which is almost never activated. It is worth stressing that many reasons such as fear, tolerance, indifference, or profit can be the motivation for becoming a passive supporter as opposed to merely sharing a global goal. Indeed, I am presenting a model for any kind of underground activity (Fig. 9.3).

9.1 The Geometry of Terrorism

We can then ask the question of what is the connection between passive supporters and the movement of terrorists? From a physicist’s viewpoint, it is a geometrical problem with a random distribution of active sites, consisting of the 9.1 The Geometry of Terrorism 157

Among people, you, me, others, some...

Are not involved with terrorism Are against killing of civilians (?) But support the goal of terrorism

THEY ARE PASSIVE SUPPORTERS

Fig. 9.2 Anyone could be a passive supporter of a terrorist cause and indeed some people are

Fig. 9.3 Passive support Instead of passive results from either a deliberate position coming support the from sharing a common cause motivation can be: or goal, or a threat, or fear Fear Tolerance Indifference

Profit Like with the Mafia,gangs, illicit practices, illegal activities... passive supporters. On this basis, I have applied percolation theory [1] to address the problem and to determine the range of social space which is open to a terrorist group’s movement and thus to its potential power for destruction [2, 4, 5, 10]. Passive supporters are people, who are sympathetic to the terrorist cause, but without any active involvement. They are essentially dormant and do not need to identify themselves. They are unknown. Only their proportion can be roughly estimated using polls and in some cases through elections when associated political groups, often called the political window, are running. The very fact that the passive supporters are unknown makes the percolation theory particularly well adapted to the problem since it deals with a random distribution of active sites. Accordingly, we attribute some space to each person in society, with this person being able to 158 9 Terrorism and the Percolation of Passive Supporters

Each individual on Open earth occupies some local space I Open Open

Open The agent can let the space either open or close to crossing by a terrorist

Closed

Closed I Closed

Closed

Fig. 9.4 Each agent controls a local piece of space, at least, by eye. The agent can thus either allow a free crossing of the space by a terrorist or it can close it off

Fig. 9.5 A portion of the I I I I I earth’s surface: each small square is occupied by one individual I. All spaces are I I I I I open

I I I I I

either allow a free passage to a terrorist or report to the police about them passing through. It is thus equivalent to giving to the agent the choice to open or to close its local area. See Fig. 9.4. A whole area can thus be divided into local sets which are opened or closed to a terrorist’s movement as shown in Fig. 9.5. For a given territory, a terrorist who wants to commit a certain action must be able to move through it from a departure point to the arrival point where the target is located. He or she has to ﬁnd a series of adjacent open local spaces as shown in Fig. 9.6. Open spaces can exist and yet not be accessible since they must be adjacent to one another via a nearest neighbor path. A series of clusters of open spaces may well be distributed over the whole of the territory involved (Fig. 9.7). Usually most of the clustered open spaces are disconnected and thus are not accessible from the terrorist’s base. They are thus only virtually open to a 9.1 The Geometry of Terrorism 159

I I I I I I I I I I I I

I I I I I I I I I I I I A I I I I I I I I I I I I

I I I I I I I I I I I I

I I I I I I I I I I I I D I I I I I I I I I I I I

I I I I I I I I I I I I

D: departure A: arrival

Fig. 9.6 Each individual space is open to different types of motion. From a departure case to reach an arrival case, several different paths may be taken. The one that is used is unpredictable

V O S

A O S

T B

V O S V O S

Fig. 9.7 A space with a random distribution of passive supporters. TB designates the terrorist’s base. The associated active open space (AOS) is shown. In addition there exist six Virtual Open Spaces (VOSs), which are not accessible to the terrorists terrorist’s movements. We deﬁne them as virtual open spaces (VOSs). Only one of these open spaces is actively open to terrorist action, the one which is connected to the terrorist base. There, the threat is real. We call it the active open space (AOS). Figure 9.7 illustrates the situation. 160 9 Terrorism and the Percolation of Passive Supporters

Newly AOS

AOS

VOS VOS

Fig. 9.8 From the conﬁguration of Fig. 9.7, one agent has turned into a passive supporter. The net effect is to attach one former VOP to the AOS which is thus enlarged as shown

At this stage, it is necessary to underline the fact that to be a passive supporter is a dynamical state. An agent can turn to a passive supporter at any time and simultaneously a passive supporter can become an opponent, for instance after a killing of civilians. But these individual shifts do not occur everyday. Although the associated space conﬁguration is volatile, it is certainly stable over some medium period of time. However, a few shifts can have a drastic effect on the extension of the OAS. Figure 9.8 shows an example of one agent becoming a passive supporter, which in turn makes one of the VOS coalesce with the AOS, which at once then becomes much larger. The reverse situation may occur as well, as show in Fig. 9.9. In this case, a passive supporter quits its status, so disconnecting a piece of the AOS, which becomes a VOP.

9.2 Local Versus Global Terrorism: A Uniﬁed Frame

Reviewing a series of terrorist groups, it appears that passive supporters are found signiﬁcantly only within a limited geographical area around the terrorist core. Most terrorist causes are connected to a geographical area, usually with an independence claim, though with a few exceptions. Far away from this region, most people do not even know what the problem is about. The Northern Ireland, Basque, Corsica, 9.2 Local Versus Global Terrorism: A Uniﬁed Frame 161

A O S

T B

V O S V O S

Newly One individual is no longer a passive supporter V O S disconnecting part of the active open space

Fig. 9.9 From the configuration of Fig. 9.8, one agent has stopped being a passive supporter. The net effect is to disconnect one piece of the AOS, which is thus reduced in size, with the creation of anewVOS and Kurd cases belong to the first group while the Red Brigades in Italy belong to the second more rare category. The terrorist activity is thus usually confined within a closed geographical area as schematically shown in Fig. 9.10. What could be framed as “classical terrorism” is basically a local terrorism. Another illustration is shown in Fig. 9.11. This indicates that three different areas can be naturally defined. The first one, area A, includes the territory in which the terrorists are free to act; they are at home. The second one, area B, incorporates the territory, which in principle could be attached to area A, but would require an increase in the number of passive supporters. The last one, area C, is the rest of the world, in which a priori the terrorist group cannot be operative on a solid and regular basis. In a physical system, the occurrence or not of percolation is evaluated for a specific sample, but here what matters is to determine the minimum size for which an observation concludes the existence of a percolation effect. While in physics, this effect would be looked upon as being an artifact of a finite size, here it is the relevant point to evaluate. We are thus reversing the physical approach. Such a change of viewpoint is driven by the very fact that passive supporters are distributed with different densities within each of the regions A, B, and C. 162 9 Terrorism and the Percolation of Passive Supporters

No terrorism out Classical V O S terrorism

No terrorism out No terrorism out

Geographically localized A O S

Fig. 9.10 Classical terrorism is local. Outside a given geographical area, it does not exist. Within the area are the AOS and a few VOS

Natural Terrorism Frontier to has been Terrorism always localized … up to Sept 11 Area C

T B

A O S

Area A The distribution of VOS covers V O S only some geographical area Area B

Fig. 9.11 Classical terrorism deﬁnes spontaneously three different areas. Area A determines the territory in which terrorists are at home. Area B incorporates the territory, which could be attached to area A given an increase in the number of passive supporters. Area C is the rest of the world, where a priori the terrorist group cannot be operative on a regular basis

For the moment, the value of the percolation threshold pc of the social system is unknown. However, we can state that in area A, p>pc where p is the density of passive supporters, in area B p

The whole world is one unique AOS and the terrorist threat is real everywhere. We are considering long term movement of terrorists, not just single and unique acts as manifested for instance with the Japanese Red Army attack at Ben Gurion airport on May 30, 1972. We are now in a position to claim that terrorism obeys a universal mechanism. The only difference between one type of terrorism and another is the value of their percolating range, i.e., the geographical extension to which their respective AOS extends. But of course this effect makes a signiﬁcant difference for the corresponding number of people threatened by a given terrorist activity. The value of this range of extension of the AOS is independent of the terrorist net itself. It depends only on the support the terrorist group gets. Within our percolation frame, destroying a terrorist network does not only suppress the success of terrorist acts, which is of course of major importance. But destruction of a terrorist cell or even of the terrorist network does not eradicate the associated terrorism since the attached social permeability is unaffected by these actions. On the contrary, it could generate an increase in the number of passive supporters. Then, as soon as a new terrorist group is formed, it can strike immediately, taking advantage of the existent AOS and VAS. This conclusion leads us to state that military action against terrorists cannot destroy the threat of terrorism. It can only reduce it temporally. The situation sounds rather hopeless.

9.3 What Is to Be Done?

The reduction of the density of passive supporters below the percolation threshold becomes the major strategic goal of an efficient fight against international terrorism in order to reduce it back to some local geographical area. However, even a reduction of a few percent of the passive supporter density would require neutralizing millions of people, either physically or ideologically, making both options morally unacceptable and totally unpractical within reasonable ethical action. Moreover, the passive supporters being unknown it would be necessary to neutralize a huge number of people opposed to terrorism to be able to reach them. In other words, to neutralize x passive supporters, one needs to neutralize randomly 100 agents whose 100 x are opposed to terrorism, which is totally absurd, as well as being totally unacceptable. At this stage, the conclusion appears to be very pessimistic, with no apparent solution to reduce the current world level of terrorism threats. The lack of a solution comes from the fact that pc being fixed by the ground topology and the social structure of society, which are also not given to modification, to complete the condition ppc, in which long range terrorism is defeated, requires a reduction in the density p of the passive supporters. But as stated above, it is simultaneously impossible, unacceptable, and inefficient to neutralize millions of people with the destruction of a good part of the planet, although it could be done by mad leaders using the existing weapons of mass destruction. 164 9 Terrorism and the Percolation of Passive Supporters

On this basis, after quoting Mao Zedong, we can now invoke Vladimir Ilitch Oulianov, otherwise known as Lenin, with his famous “What is to be done?” book. From the above conclusion of there being no solution, the new idea was to consider reversing the usual physical scheme so as to suppress a percolation. It was then important to remember the fact that we are not dealing with a physical system but with a social system and what is given to change is different for the two systems. Instead of modifying the value of the density of passive supporters, as in a physical system, with Alain Mauger we argued that it is the value of the percolation threshold itself that should be modified [6]. Such a solution would leave the sites themselves untouched, i.e., avoiding all the ethical and moral obstacles involved in reducing the number of passive supporters. The immediate question is then how to do it? Before finding out how to modify a percolation threshold, we must understand how the percolation threshold of a geometrical network is calculated. It appears to be a very difficult mathematical task. Up to date only a very few types of lattice networks have allowed an exact calculation of their respective percolation threshold. For all other lattices, numerical simulations are required. It then appears that the percolation threshold depends primarily on two independent parameters, which are the connectivity of the network q, i.e., the number of nearest neighbors a site has, and the dimensions of the space d. For instance, a square lattice has q D 4 with d D 2 and yields pc D 0:59. Its cubic extension at d D 3 has q D 6 with pc D 0:31. Going to four dimensions, the hypercube has q D 8 and pc D 0:20. Increasing either the dimension and/or the connectivity, produces a reduction in the value of the percolation threshold since more numerous paths are created to connect one site to another. For a social application of percolation, the connectivity may be of the order of 5–20 for a dimension a priori equal to 2, the surface of the earth, or 3 by adding the vertical extension of buildings. The associated percolation threshold is unknown but could be evaluated in principle using Monte-Carlo simulations. Fortunately, a few years ago, we discovered with Alain Mauger a universal formula for all percolation thresholds with an explicit dependence on q and d [7]. It is written

b pc D aŒ.d 1/.q 1/ ; (9.1) where a D 1:2868 and b D 0:6160. Most known thresholds are reproduced to an excellent accuracy. The formula is shown in three dimensions in Fig. 9.12 as a function of q and d in the range q D 5;:::; 20and d D 2;:::; 10. Its intersection with the plane pc D 0:10 is shown. It is seen that the percolation threshold is a decreasing function of both q and d. To grasp this dependence more quantitatively, the Galam–Mauger formula is shown in Fig. 9.13 for ﬁxed dimensions as a function of connectivity. Dimensions d D 2;4;10 are illustrated. It is seen that the threshold values drop with the connectivity. We are thus in a position to calculate the percolation threshold of a social system with the above rough estimates of q D 20 and d D 3. The result is pc D 0:14.It is worth noting that pc D 0:36 at q D 5 and d D 3 but it shrinks to pc D 0:054 9.4 The Various Flags of a Terrorist Group 165

Fig. 9.12 Representation of the Galam–Mauger universal formula for percolation thresholds pc D aŒ.d 1/.q 1/b ,whered is the dimension, q the connectivity, a D 1:2868 and b D 0:6160 as a function of q and d in the range q D 4;:::; 22and d D 2;:::; 13. The intersection with the plane pc D 0:10 is shown. Above the plane, terrorism is localized geographically while below, it extends to the whole world at q D 20 and d D 10.Atd D 2,wehavepc D 0:55 at q D 5 and pc D 0:21 at q D 20. These results indicate that to percolate, a terrorist group must gain a substantial proportion of passive support if in nonurban areas and strong support in urban areas. This result is coherent with the approximate data available concerning European terrorist groups. However, it is inconsistent with international terrorism for which qualitative estimates mentioned in newspapers put the level of support to its cause at around 10%. These ﬁgures indicate that a world percolation is taking place although we have p>pc. Something sounds wrong, either the theory or the calculation of pc.

9.4 The Various Flags of a Terrorist Group

From the above alternatives, I would prefer not to give up my model but then a novel concept is required. If it is the calculation of pc that is wrong, it implies that it is the evaluation of either the coordination number q or the dimension d, or both, which 166 9 Terrorism and the Percolation of Passive Supporters

Percolation threshold

0.5 Fixed connectivity Dimension d=2 0.4

0.3

d=4 0.2

0.1

d=10 Connectivity 10 15 20 25 30

Fig. 9.13 Representation of the universal Galam–Mauger formula for fixed dimensions as a function of connectivity. The variation of pc is plotted as a function of q at dimensions d D 2; 4; 10. It is seen that the threshold values drop with the connectivity are wrong. However, the coordination number q is the result of cultural and social evolution. The figure is fixed by a social structure which is not given to immediate change. The only option left is then the dimension. A priori, the dimension is also fixed, since we live in a two- or three-dimensional space. At this stage, the instrumental breakthrough was to realize that we were using a static geometric model to describe a situation with combined static aspects and some dynamics of the agents. Indeed the connections among agents are different depending on the agent’s localization. In the social sphere of life, we preserve our links with individuals or groups even when we are not actually together. Therefore, it makes sense to add these different links. The way to achieve the integration of the various connections among agents is to postulate that while we are moving in three-dimensional space, we are actually living in a social space whose dimensions are higher. It can be shown that adding virtual dimensions onto a two-dimensional plane produces additional real bonds at a larger distance on the plane [6]. This explains why this process decreases the percolation threshold. For instance, at d D 2 we have pc D 0:59 for nearest neighbor connections (q D 4), pc D 0:41 for nearest neighbor plus next nearest neighbor connections (q D 4C4 D 8)andpc D 0:29 for nearest neighbor plus next nearest neighbor connections (q D 4 C 4 C 4 D 12)[8]. The strategic focus then becomes on how to add or to suppress a virtual social dimension. We call each of these social dimensions a flag since it allows agents to identify or not with it. Accordingly, as soon as a terrorist group is formed, it waves 9.5 The Overlap with the Physical Model 167 the flag of its main claim, usually the independence or the autonomy of a particular territory. The immediate state repression against the newly formed terrorist group creates another flag. People can be against state repression if it is felt as being excessive even though they may not identify with the independence flag themselves. With these minimal ingredients, the effective dimension is at d D 4 or d D 5.The associated percolation thresholds then become realistic with respect to the known support for European movements. But this is not the case for international terrorism which can count on only a few percent of passive supporters throughout the world. It thus becomes necessary to consider the very specific nature of international terrorism, which in fact waves a series of independent flags, which cover several different world issues at the same time. These different flags can include the independence of several different territories, as well as religions, ethnicities, history, super powers, and even social and economic causes. These factors can all add up to a social dimension of roughly 10. Taking d D 8 yields already pc D 0:06 for q D 20. Now, only 6% of support is enough to yield world percolation. It is the first time in history that one type of terrorism embodies so many different flags. It is therefore on the ground of social paradigms that the fight to curb terrorism might be efficient. The security challenge is not military but social in the neutralization of these additional social dimensions. A new strategic scheme to suppress the passive supporter percolation without touching the passive supporters themselves is thus envisioned as a viable solution. Reducing the dimension of its social space, i.e., dealing with the various independent flags displayed by the terrorist group [5, 7, 9] p>p pp p could turn a situation with c into c where c c keeping unchanged. In the first case, terrorism occurs at a world scale, while in the second, it is geographically localized. The model strategy applies to a large spectrum of clandestine activities including guerilla warfare as well as tax evasion, corruption, illegal gambling, illegal prostitution, and black markets [4].

9.5 The Overlap with the Physical Model

The unique but essential overlap with physics is the use of the theory of percolation. But then the approach has been very different. In physics, given a system, we calculate its percolation threshold and measure the density of active sites to determine if percolation is active or not. The active sites are the ingredients which can be modiﬁed. Here we are tackling the problem with a totally different approach. First, we determine the size of the system within which percolation may occur. Then for a given density of active sites, we modify the dimensions of the social space in order either to produce percolation, from the terrorist’s side, or to suppress percolation, from the institutional side. Up until now, these dynamics were left free to the terrorist organization. 168 9 Terrorism and the Percolation of Passive Supporters

9.6 Achievements of the Model

We have shown why military action alone cannot eradicate terrorist activities. Moreover, it has been highlighted how direct repression can be counterproductive. It was also found that all types of terrorism and underground activities belong to the same frame and obey the same rules. The model opens the way to a renewal of the means of curbing terrorism, taking over the ﬁght to construct or deconstruct the social space in which the range of terrorist activities is determined. We have obtained a counterintuitive approach to implement an efﬁcient policy against global terrorism.

References

1. D. Stauffer and A. Aharony, “Introduction to percolation theory” Taylor and Francis, London (1994) 2. S. Galam, “The September 11 attack: A percolation of individual passive support”, Eur. Phys. J. B 26 Rapid Note, 269–272 (2002) 3. S. Galam, “La detection´ des reseaux´ terroristes”, Strategie´ et decision:´ La crise du 11 septembre, General Loup Francart et Isabelle Dufour, Economica, Paris (2002) 4. S. Galam, “Global physics: from percolation to terrorism,: guerilla warfare and clandestine activities”, Physica A 330, 139–149 (2003) 5. S. Galam, “Terrorisme et percolation”, Pour La Science 306, 90–93, Avril (2003) 6. S. Galam and A. Mauger, “On reducing terrorism power: a hint from physics”, Physica A 323, 695–704 (2003) 7. S. Galam and A. Mauger, “Universal formula for percolation thresholds”, Phys. Rev. E 53, 2177–2181, (1996) 8. K. Malarz and S. Galam, “Square-lattice site percolation at increasing ranges of neighbor bonds”, Phys. Rev. E 71, 016125-1-4 (2005) 9. S. Galam, “Global terrorism versus social permeability to underground activities”, in Econo- physics and Sociophysics: Trends and Perspectives, B. K. Chakrabarti, A. Chakraborti, A. Chatterjee (Eds.), Chap. 14, Wiley-VCH Verlag GmbH & Co. KGaA: Weinheim (2006) 10. S. Galam and A. Mauger, “On reducing terrorism power: a hint from physics”, Physica A 323, 695–704 (2003) Chapter 10 The Modeling of Opinion Dynamics

10.1 An Overview

Among the numerous topics of social sciences covered by sociophysics, the study of opinion dynamics has become a mainstream of research [1–23]. It is a critical subject since public opinion has become a central issue in modern societies, making the understanding of its underlining mechanisms a major challenge [24–26]. Any progress could have significant effects on the way of tackling sensitive decisions to which each society as well as the world at large are confronted. Our approach in tackling opinion phenomena relies on a few simple assumptions, which in turn provide a series of astonishing and powerful results [1, 3, 4, 27]. In particular it allows us to discover that the dynamics of opinion formation follows some flow, whose direction appears to be determined by the existence of thresholds in the initial public support for the competing issues. Most models yield such threshold dynamics. Indeed, it has been shown that they all belong to one single probabilistic sequential scheme [28]. It is worth noticing that in 2005, for the first time, a highly improbable political vote outcome was predicted using a model of sociophysics [3, 27]. Moreover, the prediction was made several months ahead of the actual vote against all the predictions of the polls and analyses made at the time [29–31]. The model deals with the dynamics of spreading of a minority opinion in public debates using a two state variable system. It applies to a large spectrum of issues including national votes such as the recent French vote, behavior changes such as smoking versus nonsmoking, support or opposition to a military action such as the war in Iraq, rumors such as the French hoax about September 11 [32], and reform proposals [33–37]. The model uses two state variables to study the forming of a public opinion from a public debate. Agents are floaters who discuss in small groups using a one- person-one-argument principle. At each cycle of the discussion, they update their individual opinion according to a local majority rule. The associated dynamics is driven by repeated local opinion updates operated throughout. Local ties may occur in even-sized groups. They are solved in favor of either of the opinions according

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 169 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 10, © Springer Science+Business Media, LLC 2012 170 10 The Modeling of Opinion Dynamics to the common beliefs of the agents. The resulting forming of opinions is found to follow threshold dynamics with a separator ac;r which determines the direction of an opinion flow toward either one of two attractors aA and aB at which opinion A and opinion B are respectively the winning majority. When all agents are floaters, the two attractors are single opinionated with aA D 1 and aB D 0 where only one opinion survives the public debate in the full population. For an opinion A initial support at >ac;r , there exists a number n of successive updates which makes the flow to reach the A winning attractor with atC1 atC2 > >atCm aB where the number m is different from n.Theyare both integers which can be calculated. Taking small values, they diverge at ac;r . For odd-sized update groups, ac;r D 1=2. Even sizes allow the inclusion of doubt when there is an opinion tie in a group. In the case of a doubt, the collective belief is evoked to produce a local bias in favor of either one of the opinions. Such bias may shift ac;r anywhere between 0 and 1 depending on the distribution of both the population collective belief and the local updates in group sizes. When ac;r 1=2, the associated dynamics gives rise to the occurrence of minority opinion spreading. Focusing on the no tie case for which ac;r D 1=2, we study the effect of including heterogeneous agents such as contrarians [38] and inflexibles [39] in addition to floaters. It is found that these have a drastic effect on the opinion dynamics. Contrarians are agents who deliberately oppose the local majority by shifting to the other opinion, whatever is the majority opinion [38, 40–43]. At very low densities, they create a stable coexistence between a majority and a minority with aA ¤ 1 and aB ¤ 0. The threshold is unchanged at 1=2. However, beyond some critical values, they turn the dynamics into being thresholdless [38]. One unique attractor aA D aB D ac;r D 1=2 drives the dynamics. Whatever the initial conditions are, the public debate brings the collective opinion to exactly fifty–fifty. This surprising mechanism of threshold erasing was used to explain the famous 2000 Bush — Gore presidential election in the United States. It was then predicted that fifty–fifty elections were about to occur again and often in voting democracies and it did happen several times since, like in Germany, Italy, and Mexico [44]. The majority level at which contrarians operate can be global instead of local using results from polls. It gives rise to chaotic behavior around 50% [40]. Inflexible agents are agents who never shift their opinion during small group discussions. They were found to produce similar effects as contrarians but with the novelty of asymmetry since the densities of inflexibles for each opinion are usually not equal. In particular, considering one-sided inflexibles makes the associated opinion certain to gain the support of the whole population. Even if an opinion is supported by only a very low density of inflexibles against a huge majority in favor of the other opinion, the debate will reverse the ratio so that eventually the whole population will be aligned along with the inflexibles. Accordingly, the expected democratic character of a free public debate may turn into a dictatorial machine to propagate the opinion of a tiny minority against the initial opinion of the overwhelming majority. It may shed a new and counterintuitive light on the social aspects of the global warming phenomenon [45]. 10.2 Why Is Public Opinion Often Conservative? 171

10.2 Why Is Public Opinion Often Conservative?

All over the world and more speciﬁcally in democratic countries, public opinion seems to be rather conservative in the face of issues open to public debate such as with a reform proposal, a referendum, a behavioral change such as stopping smoking, or a rumor propagation. Refusal seems to be the rule rather than acceptance of change (Fig. 10.1). Notwithstanding this observation, many individuals want things to be changed. They want change, yet they refuse most proposals of reforms, just like myself (Fig. 10.2). I have also experienced this paradox several times. Once, a few years ago, I was desperately hoping for some reform in the French university system.

All over the world and more specifically in democratic countries public opinion seems to be rather conservative while facing issues opened to a public debate like with a reform proposal, a referendum, a behavioral change (stop smoking), a propagating rumor

Something rather strange seems to always happen REFUSAL

Fig. 10.1 Democratic public debate often yields refusal of a possible change

THE PROBLEM

MY SEL F

I WANT THINGS TO CHANGE

I AM AGAINST THE REFORM

Fig. 10.2 I want things to change. I am offered some reform. I am against it. I am not the only one 172 10 The Modeling of Opinion Dynamics

Fig. 10.3 The problem THE SOLUTION sounds as though it is psychological in nature. Is the solution from a shrink? No, it CALL A SHRINK ? is a sociophysical problem

CALL A PHYSICIST !

(It’s much cheaper)

Fig. 10.4 I do hold an opinion but a collective choice has to be made. And I want to both validate my opinion and inﬂuence others to add weight to the ﬁnal collective choice

I hold an opinion, how to proceed?

Then, when a reform proposal was suggested which I found rather good, a few days later I was demonstrating against it! What happened? Indeed, it is what happens to all of us as part of a social group. Start with a case where changes are known to be desperately needed. An initial majority of people support the reform project against a very small minority. Then a wide debate is undertaken so as not to impose the reform on the people by force. As a result the project is rejected by a huge majority of people. Can this be understood? Should we call a shrink? (Fig. 10.3). Like everyone else, I myself hold an opinion with respect to any given public issue concerning a political choice (Fig. 10.4). How do I proceed in getting involved in the public debate? Yet to validate my opinion, I also want to argue with others again and again (Fig. 10.5). To model the problem, I consider two separate mechanisms for opinion forming asshowninFig.10.6. One mechanism is external to the agent. It acts directly on it via the media, leaders, and education to drive its personal choice. The second is internal and results from interactions among individuals. In the case of two 10.3 The Local Majority Model and the Existence of Biases 173

Fig. 10.5 I get active in the So indeed, what is ongoing public debate. I want the problem? to argue, again and again, just like everyone else Once more, MYSELF

I have an opinion, I want to argue, again and again

But, I am not the only one

Fig. 10.6 An individual We consider two separate opinion is ﬁrst shaped by mechanisms in the external mechanisms. Then it opinion forming evolves through an internal mechanism of discussions among agents One One external internal

Acts directly on Results from single individuals, interactions medias, leaders, among education individuals competing opinions A and B for a group of N agents we associate a variable ai D˙1 to each agent i to identify its actual choice. To analyze the dynamics of opinion forming I split the two mechanisms into two; external and internal. The external one is activated ﬁrst to determine the initial values at and .1 at / of respective opinions at a time t. They can be estimated using polls. They are the direct result of each agent’s information and beliefs. Given these initial conditions, the external mechanism is turned off and then the internal mechanism is activated to study the internal dynamics of opinion.

10.3 The Local Majority Model and the Existence of Biases

To study the internal dynamics driven by discussions among agents we need to elaborate a scheme to monitor individual changes of mind. But that is a rather complicated and complex psychosociological process, which is not yet understood. 174 10 The Modeling of Opinion Dynamics

Fig. 10.7 Individual change Open debate While discussing an issue, of opinion is mimicked by a means local people may change their physicist-like scheme discussion mind: dynamical volatility

HOW?

Very complicated and complex psycho-sociological process

We make it simple and fair: one person one vote A local group with a local majority rule polarization

Local Discussions

B A AA B BB AA AA B A ABB A BB BB BB AAB A A A B AA

A A AA A BB AA AA B B AAA A Polarization BB process BB BB BBB B B A A AA

Fig. 10.8 Small groups of agents are formed where a local majority rule is applied to produce local polarization with every agent in the group sharing the same opinion

Accordingly, we mimic these unknown cognitive processes by a simple and fair rule in the same way that physicists are skilled in dealing with complex phenomena. We give each agent one single argument, in favor of an opinion and against the other (Fig. 10.7). We then constitute random small groups of r people in discussion and we assume that locally all agents adopt the opinion which has won the majority of the arguments as shown in Fig. 10.8. Minority agents eventually update their own opinion. Group size r may vary with r D 1;2;:::;L. However, using a local majority rule does not operate in the case of a tie in an even-sized group. Then, a common belief “inertia principle” is applied to select 10.3 The Local Majority Model and the Existence of Biases 175 either of the opinions A or B with respective probabilities k and .1 k/,where k accounts for the collective bias produced by the common beliefs of the group members. A k noninteger value accounts for the fact that real societies are divided into disconnected subgroups, which may share different collective beliefs with not everyone discussing with every one else [27]. For instance, in the case of a reform proposal, most cultures share the wisdom that states that in the case of doubt, it is better to keep the situation unchanged. This situation is a tilt towards the status quo, which would put k D 0 if opinion A corresponds to a “yes” to the reform. But for other issues, one social subgroup may have a common belief yielding k D 1 while another one has an opposite common belief with k D 0, resulting on average in an effective bias 0ac;r , one update gives atC1 >at . Repeating the update moves the support closer to the attractor aA D 1. Indeed, there exists a number n to reach it with the series at

A population of 33 people with 22 in favor of the Day “1” morning reform and 11 against it

People on their own

Fig. 10.9 A population of 33 people with 22 in favor of the reform and 11 against it

The same people at lunch

22 in favor and 11 against Day “1” lunch

Six persons

Three persons

Five persons Two persons They are discussing One person

Two persons

Four persons One person

Four persons One person Three persons

One person

Fig. 10.10 The same people at lunch, 22 in favor and 11 against 10.3 The Local Majority Model and the Existence of Biases 177

Luch is over

20 in favor and 13 against Day “1” end of lunch

Six persons

Three persons

Five persons Two persons One persons

Two persons

Four persons One person

Four persons One person Three person

One person

Fig. 10.11 Lunch is over, with 20 in favor and 13 against

Day “1” afternoon

People on their own

Fig. 10.12 The same population of 33 people with now 20 in favor of the reform and 13 against it 178 10 The Modeling of Opinion Dynamics

Dinner time

20 in favor and 13 against Day “1” dinner

Six persons

Three persons

Five persons Two persons One person They are discussing Two persons

Four persons One person

Four persons One person Three persons Usually group compositions One person are different

Fig. 10.13 Dinner time, 20 in favor and 13 against

Dinner is over

14 in favor and 19 against Day “1” end of dinner

Six persons

Three persons

Five persons Two persons One person

Two persons

Four persons One person

Four persons One person Three persons

One person

Fig. 10.14 Dinner is over with 14 in favor and 19 against 10.3 The Local Majority Model and the Existence of Biases 179

One day later Day “2” morning

People on their own

Fig. 10.15 The same population of 33 people with now only 14 in favor of the reform and a majority of 19 against it

Lunch time

14 in favor and 19 against Day “2” lunch

SIX PERSONS

THREE PERSONS

FIVE PERSONS TWO PERSONS

ONE PERSONS

They are discussing TWO PERSONS

FOUR PERSONS ONE PERSONS

FOUR PERSONS ONE PERSONS THREE PERSONS

ONE PERSONS Usually group compositions are not too different

Fig. 10.16 Lunch time with 14 in favor and 19 against 180 10 The Modeling of Opinion Dynamics

Lunch is over

0in favor and 33 against Day “2” end of lunch

SIX PERSONS

THREE PERSONS

FIVE PERSONS TWO PERSONS

ONE PERSON

TWO PERSONS

FOUR PERSONS ONE PERSON

THREE PERSONS

ONE PERSON

Fig. 10.17 Lunch is over with 0 in favor and 33 against

A population of 33 persons with an unanimity against the reform Day “2” afternoon

The person in charge of the reform is dismissed

Fig. 10.18 The initial population of 33 people which started with 22 in favor of the reform and 11 against it ends up rejecting unanimously the reform 10.3 The Local Majority Model and the Existence of Biases 181

Fig. 10.19 The opinion ﬂow The opinion for the A opinion with flow a ;r D 1=2. It is democratic c diagram since the initial aggregated majority becomes the collective majority 0 1 1/2 Total Total disappearance Spreading

Democratic dynamics

Fig. 10.20 The opinion ﬂow The opinion for the A opinion with flow a ;r >1=2. It is totalitarian c diagram Bias in favor of since the initial aggregated the status quo minority becomes the collective majority 01 1/2 Total Total disappearance Spreading

Totalitarian dynamics

Fig. 10.21 The opinion ﬂow The opinion for the A opinion with flow a ;r <1=2.Thisisnovelty c diagram Bias in favor of driven since the initial the novelty aggregated minority becomes the collective majority. It can be equally assimilated to 01 totalitarian dynamics, which seems indeed to follow 1/2 minority spreading dynamics Total Total disappearance Spreading

Novelty dynamics 182 10 The Modeling of Opinion Dynamics

The opinion has totally disappeared. From at atC1 >atC2 > >atCm D amC1 D aB D where opinion A has now disappeared. In the case of odd-sized groups, ac;r D 1=2. But for even sizes, the possibility of local doubt at a tie breaks the symmetry between opinion A and B according to the value of k, which is determined by the distribution of collective beliefs in the population. The threshold value ac depends on r and k [27]. Figures 10.19–10.21 illustrate the three cases with ac;r D 1=2 (top), ac;r <1=2(middle), and ac;r >1=2 (bottom). For odd sizes and for even size with k D 1=2 we have ac;r D 1=2. Solving the problem exactly in the case of a group size of r D 4 yields for the threshold p 6k 1 13 36k C 36k2 a D ; c;4 6.2k 1/ (10.3) which yields ac;4 D 0:23; 0:77; 1=2 for respectively k D 1; 0; 1=2. This shows explicitly how the existence of doubt combined with a collective belief, which favors an opinion, here A, can turn the public debate into a machinery to propagate a minority opinion when k D 1. To win the majority, the initial A support must satisfy at >0:23. Increasing the group size r dampens this effect by making ac;r closer asymptotically to the value 1=2. For instance, taking k D 1 and considering the series of group sizes r D 2; 4; 6; 8; 10; 12; 14; 16; 18; 20 yields respectively the critical values ac;r D 0; 0:23; 0:35; 0:39; 0:42; 0:44; 0:45; 0:47; 0:46; 0:47 as shown in Fig. 10.22. Nevertheless, it is worth emphasizing that people most often discuss in small groups whose size never exceeds a few individuals. Larger groups usually split into smaller ones. Extreme cases with respectively k D 0, k D 1=2 and k D 1 are emblematic of numerous social situations as shown in Fig. 10.23. For instance, in the case of a reform proposal, the tie is broken along the common belief of the group in order to preserve the status quo. If you are not sure that a reform will improve the situation, which is already accepted a priori, it is much safer to preserve the status quo. This yields ac;4 D 0:77. It means that in order to win a public debate a support for a reform must start with individual support of over 80%, which is almost impossible in a democratic society. At the other extreme where k D 0, it is the tip toward novelty with ac;4 D 0:23. Here it is enough to convince 25% of the people to adopt a new product so as to have the other people adopt it by the simple effect of imitation. For a political vote, in case of a doubt, agents will just preserve their current choice making k D 1=2, which ensures a democratic balance of the debate. In addition, several situations can receive a new surprising reevaluation using (10.3). For instance, we can compare the apparently identical situations of implementing a ban on smoking in America and in France. In both countries, the same information is available and yet it has been achieved more “naturally” in America than in France where only the reinforcement of several successive repressive laws was able to change the situation. The explanation comes from the acceptance of the differences in the notions of individual freedom in both countries (Fig. 10.24). 10.3 The Local Majority Model and the Existence of Biases 183

a c,r 0.5

0.4

0.3

0.2

0.1

r 10 20 30 40 50

Fig. 10.22 The threshold ac;r as a function of even sizes r D 2; 4; 6; 8; 10; 12; 14; 16; 18; 20; 30; 50 for k D 1

However with For instance, in respect to case of a reform innovation like proposal, the tie is computers or cars, the tie is broken broken along the But in the case of a along the novel common belief of presidential vote, the product: the group: status quo means ⇓ that each voter keeps ⇓ its opinion Preserves the status quo ⇓ If B corresponds to If B corresponds to k=1/2 novelty acceptance ⇓ ⇓ ⇓ k=0 p = 0.50 k=1 c ⇓ ⇓ pc = 0.23

pc = 0.77

Fig. 10.23 The extreme cases of a bias always in favor of either one opinion or of none 184 10 The Modeling of Opinion Dynamics

Fig. 10.24 The extreme In case of cultural cases of a bias always in the attitudes, favor of either one opinion like freedom, k may be 0 in one culture and 1 in another A = non-smokers

75 % of non- 25 % of nonsmokers in smokers in France are not America are enough to forbid enough to smoking forbid smoking everywhere and everywhere indeed do accept smoking everywhere

While in America, individual freedom for an individual means not to be bothered by others’ actions, in France it is the right of the individual to do what he or she wants despite bothering others. These two philosophical approaches yield in the case of a tie within a group to respectively k D 0 in France and k D 1 in America if opinion A is to stop smoking and B is to keep on smoking. Accordingly, having 75% of nonsmokers in France is not enough to forbid smoking everywhere and indeed means accepting smoking everywhere. In contrast, having 25% of nonsmokers in America is enough to forbid smoking everywhere.

10.4 The Appearance of Nonthreshold Dynamics

From Fig. 10.22, it appears that while the threshold value decreases with smaller group sizes, it reaches zero at r D 2, i.e., for pairwise discussions. This implies a rather strong effect in the opinion dynamics bias driven by discussions within couples. Indeed, many exchanges occur by pairs. It is noticeable that for pairwise groups the threshold ac;2 D 0, i.e., any A opinion is certain to invade the whole of the population given k D 1. This is the ﬁrst appearance of thresholdless dynamics. Any initial support for opinions A and B ends up with the whole population sharing opinion A. Unless we start at at D ac;2 D 0 with the whole population holding opinion B from the start, which in turn means no dynamics at all. Studying the r D 2 case more generally, (10.2) writes: 2 atC1 D at C 2kat .1 at /: (10.4) 10.5 Mixing the Group Sizes 185

Fig. 10.25 The variation of at 1 a a tC1 as a function of t from 1 (10.4). It includes the series k D 1 (above external line), k D 0:90 (upper Only opinion A inside), k D 0:80, k D 0:70, 0.8 k D 0:60,andk D 1=2 (the diagonal straight line). Any initial A support at ¤ 0 is Only opinion B 0.6 expanded through the public debate towards total invasion at aA D 1. The opposite dynamics associated to an 0.4 initial A support at D 0:30 is shown for k D 0:20

0.2

at 0.2 0.4 0.6 0.8 1

It yields two opposite regimes, both being thresholdless as illustrated in Figs. 10.25 and 10.26. In the range 0 k<1=2, the separator is ac;2 D 1 with the unique attractor aB D 0. Any initial condition, besides at D 1 leads to atCn D 0.In contrast, for 1=2 < k 1, we have the separator ac;2 D 1 with the unique attractor aA D 1. Any initial condition, besides at D 0 leads to atCn D 1. The case k D 1=2 produces an invariant dynamics with atC1 D at . Agents do change their opinions individually but on average, the global supports do not. It is worth stressing that such an effect has been corroborated from an analysis of data from the 2004 American presidential election to analyze the marriage gap, i.e., the difference in voting for Bush and Kerry between married and unmarried people. It appears to be possible to interpret this marriage gap in terms of our model with a positive value of k and Bush denoted by opinion A [21].

10.5 Mixing the Group Sizes

However, in real life people do not always discuss in successive groups of the same size. Therefore, to make our model more realistic we consider the distribution given by pr the probability of having a local group of size r with the constraint,

XL pr D 1; (10.5) rD1 186 10 The Modeling of Opinion Dynamics

Fig. 10.26 The variation of at 1 a a tC1 as a function of t from 1 (10.4). It includes the series k D 0 (below external line), k D 0:10 (below Only opinion A inside), k D 0:20, k D 0:30, 0.8 k D 0:40,andk D 1=2 (the diagonal straight line). Any initial A support at ¤ 0 is Only opinion B shrunk through the public 0.6 debate towards total disappearance at aB D 0.The opposite dynamics associated 0.4 to an initial A support at D 0:30 is shown for k D 0:20 (lower part) 0.2

at 0.2 0.4 0.6 0.8 1 where r D 1;2;:::;L stands for the respective sizes 1;:::;L with L being the larger group [3]. Accordingly, the general update equation writes: 8 9 XL < Xr = j r r r rj r 2 2 atC1 D pr Cj at .1 at / C kV .r/C r at .1 at / ; (10.6) : 2 ; rD1 r j DNŒ2 C1 C r rŠ N r C 1 r C 1 V.r/ where j .rj/ŠjŠ, 2 Integer Part of 2 and N r N r1 V.r/ D 1 r V.r/ D 0 r 2 2 .Itgives for even and for odd. The occurrence of local ties in even-sized groups produce from (10.7)an asymmetry in the polarization dynamics toward either one of the two competing opinions with a threshold ac;, which may be very unfair for one of the two opinions. We illustrate the process by choosing for the size distribution the set p1 D 0:10; p2 D 0:25; p3 D 0:10; p4 D 0:30; p5 D 0:10; p6 D 0:15, which yield ac; D 0:214; 1=2; 0:786 for respectively k D 1; 1=2; 0. Figures 10.27 and 10.28 illustrate the above set of pr for the three different cases k D 1; 1=2; 0 for the same initial minority support at D 0:30. When the collective beliefs favor opinion A, its initial support at D 0:30 is seen to increase dramatically, driven by the public debate. Only four updates are enough to make opinion A the majority. For a neutral tie effect, it will decrease toward zero while for a bias in favor of opinion B, it falls off very quickly to zero support as seen in Figs. 10.27–10.29. 10.5 Mixing the Group Sizes 187

Fig. 10.27 The variation of at 1 the general update from 1 (10.7) for the set of size distributions p1 D 0:10; p2 D 0:25; p3 D 0:10; Only opinion A p4 D 0:30; p5 D 0:10; 0.8 p6 D 0:15 and an initial A support at D 0:30.Here k D 1 which yields Only opinion B 0.6 ac; D 0:214.The A minority wins the majority rather quickly Separator 0.4

0.2

at 0.2 0.4 0.6 0.8 1

Fig. 10.28 The variation of at 1 the general update from 1 (10.7) for the set of size distributions p1 D 0:10; p2 D 0:25; p3 D 0:10; Only opinion A p4 D 0:30; p5 D 0:10; 0.8 p6 D 0:15 and an initial A support at D 0:30.Here the tie is neutral with Only opinion B 0.6 k D 1=2 which yields ac; D 1=2. The A support decreases towards zero Separator 0.4

0.2

at 0.2 0.4 0.6 0.8 1 188 10 The Modeling of Opinion Dynamics

Fig. 10.29 The variation of at 1 the general update from 1 (10.7) for the set of size distributions p1 D 0:10; p2 D 0:25; p3 D Only opinion A 0:10; p4 D 0:30; p5 D 0.8 0:10; p6 D 0:15 and an initial A support at D 0:30.Here the tie breaking is in favor of Only opinion B the B opinion with k D 0 and 0.6 ac; D 0:786.TheA support disappears fast Separator 0.4

0.2

at 0.2 0.4 0.6 0.8 1

10.6 Heterogeneous Agents and the Contrarian Effect

Up to now, we have considered an identical behavior for all agents in making their opinion. Each one gives its own opinion in a discussion group, and eventually all shift along the same opinion, the one which wins the majority of the arguments. In the case of a local tie, it is the collective belief bias that sets up the choice. However, it is clear that in real-life situations, agents may exhibit different types of behavior in their cognitive process of choosing their opinion. Among others, we introduced the contrarian behavior to account for agents who determined their choice, not with respect to some convincing criterion but just to oppose what the others think [38, 40]. Accordingly, a contrarian is someone who deliberately decides to oppose the prevailing choice of the majority around it whatever that choice is. In the case of update groups of odd sizes, (10.1) is changed to ! Xr r a D .1 2c/ am.1 a /rm C c; tC1 m t t (10.7) rC1 mD 2 where c is the density of contrarians among the floater population. It shows that the associated dynamics is changed since at D 0 and at D 1 yielding respectively atC1 D c and atC1 D 1 c, are no longer fixed points of the equation atC1 D at . However, ac;r D 1=2 is unchanged as a fixed point. 10.7 Thresholdless Driven Coexistence 189

Fig. 10.30 The two aA,B contrarian attractors ac 1 A;B from (10.8) as a function of contrarian density c.At c D 1=6 ac ac both A and B merge with the separator 0.8 ac;3 D 1=2 to turn the dynamics thresholdless. In the range 1=6 c 1=2,the dynamics leads 0.6 systematically to a ﬁfty–ﬁfty coexistence between opinions AandB 0.4

0.2

c 0.1 0.2 0.3 0.4 0.5

Solving the fixed point equation for r D 3 yields p 1 2c ˙ 1 8c C 12c2 ac D ; A;B 2.1 2c/ (10.8) whichareshowninFig.10.30. It is seen that at low densities of contrarians, increasing c makes both attractors move symmetrically towards the threshold a D 1=2 ac <1 ac >0 c;3 with A and B . They are found to all merge with ac D ac D a D 1=2 c D 1=6 0 c 1= A B c;3 at exactly . In the region the opinion dynamics yields a stable coexistence between an A majority (B) and a B minority (A). Figures 10.31 and 10.32 show the two cases of c D 0:05 < 1=6 and c D 0:35 > 1=6. The evolution of at D 0:30 is shown in both cases. The first one leads to a B majority and an A minority while the second one produces a fifty/fifty support.

10.7 Thresholdless Driven Coexistence

From c D 1=6 and beyond in the range 1=6 c 1=2, the dynamics becomes thresholdless. Instead of having a ﬂow which puts ahead one of the two competing opinions, it now reduces any initial difference to zero to establish a perfect equality between the two competing opinions A and B as seen in Fig. 10.30. 190 10 The Modeling of Opinion Dynamics

Fig. 10.31 The evolution of at 1 a a tC1 as a function of t for 1 c D 0:05 < 1=6.The evolution of at D 0:30 is A majority shown. The dynamics leads to a B majority and an A 0.8 minority B majority 0.6

0.4

Separator

0.2

at 0.2 0.4 0.6 0.8 1

Fig. 10.32 The evolution of at 1 atC1 as a function of at for 1 c D 0:35 > 1=6.The evolution of at D 0:30 is shown. Contrary to the case of Fig. 10.31 the dynamics 0.8 now leads a ﬁfty/ﬁfty coexistence 0.6

0.4

Separator

0.2

at 0.2 0.4 0.6 0.8 1

ac Driven by the local discussions and the contrarian effect the two attractors A ac and B have disappeared to the profit of one unique attractor located at the former separator ac;3 D 1=2. Once the equilibrium is reached, agents keep on updating their opinion due to the contrarians who never settle on a fixed opinion. However, the net effect of these individual opinion changes is perfectly self-balanced. 10.8 The One-Sided Inflexible Effect and the Global Warming Issue 191

Fig. 10.33 The alternating at 1 opinion dynamics for 1 c>1=2with c D 0:70 for which the dynamics is thresholdless with a convergence towards the 0.8 attractor ﬁfty/ﬁfty. The evolution from at D 0:30 is shown for six successive updates in both cases 0.6

0.4

0.2

Single attractor Fifty fifty at 0.2 0.4 0.6 0.8 1

Any election to be held in that state yields very narrow results which are by nature disputable due to incompressible counting errors which score less than the expected statistical ﬂuctuations. For larger values of the contrarian density with c>1=2, the dynamics becomes alternating. In the range 1=2 c 5=6, the majority is shifted from one opinion to the other at each new update, but with the difference in amplitude being shrunk until it reaches zero at the same attractor 1=2. Beyond, with 5=6 < c 1, the dynamics is still alternating, but now 1=2 is again a separator with two alternating attractors ac <1 ac >0 a D a A and B given by the ﬁxed point equation tC1 t . Figures 10.33 and 10.34 exhibit the two cases c D 0:70 and c D 0:90 for an initial A support at D 0:30. The contrarian behavior was also extended to the level of the majority at the collective choice given by polls [40]. The effect is similar to the previous one but now a chaotic behavior is obtained at around 50%.

10.8 The One-Sided Inﬂexible Effect and the Global Warming Issue

Another specific feature of human character is the inflexible attitude. An inflexible agent sticks to its opinion whatever arguments are given to it, and never shifts opinion [39]. The effect on the dynamics is similar to the contrarian effect with the introduction of an asymmetry between the two opinions A and B, depending on the ratio in the respective proportions of inflexibles. In particular, the separator is no longer located at 1=2 and the two attractors are no longer symmetrical [39]. 192 10 The Modeling of Opinion Dynamics

Fig. 10.34 The alternating at 1 opinion dynamics for 1 c>1=2for the case c D 0:90 where there exists A majority attractor alternating attractors. The evolution from at D 0:30 is 0.8 shown for six successive updates in both cases

0.6

0.4

0.2

Separator

at 0.2 0.4 0.6 0.8 1

While it seems natural to have inflexible agents on both sides of any social issue at stake, there are some specific cases that do not obey this logic. In particular there exist certain issues in which some agents are convinced that they have “scientific proof” to justify their particular opinion, as against only doubts that prevail on the other side, such as the question of human-caused global warming [45]. In this problem, some agents believe that scientific proof has been obtained to link global warming to the man-made production of carbon dioxide. On the other side, other agents, much less numerous, claim that there is no scientific proof of human culpability. But at the same time they have neither a proof of another cause nor the proof that man is not guilty. In between these two groups, the majority of agents are floaters. To model the public debate associated with such a situation we consider a proportion q of inflexible agents in favor of opinion A with other agents being floaters that obey a local majority rule. We also assume that initially the majority of floaters support opinion B. For r D 3 the corresponding update becomes, instead of (10.1), 3 2 2 atC1 D at C 3at .1 at / C q.1 at / ; (10.9) where the last term accounts for the configurations where two opinion B holders discuss with one inflexible A agent, yielding no change in the respective opinions by the local update. It is worth underlining that at q by definition of the inflexibles. To study the effect on the dynamics, we solved the new fixed point equation atC1 D at , which yields two attractors aA D 1 and p q 1 a D 1 C q 1 6q C q2 ; (10.10) B 4 10.9 The Thresholdless Case 193

Fig. 10.35 The evolution of at 1 atC1 as a function of at from 1 (10.9)forp q D 0:05 < .3 2 2/ 0:172.The evolution from at D 0:30 is Only opinion A shown to lead to a very large 0.8 B majority with a tiny A minority A inflexibles 0.6

0.4

Separator

0.2

at 0.2 0.4 0.6 0.8 1 with a separator located at p q 1 a D 1 C q C 1 6q C q2 : (10.11) c;3 4

aq aq Both B and c;3 are defined The first effect of the one-sided inflexibles is to prevent the total disappearance of opinion A even if all the floaters support opinion B. An initial situation with a 2 3 small inflexible minority at D q strengthens the A supportp to atC1 D q C q q after one update. However, in the range 0 q .32 2/ 0:172, any A support satisfying at < 0:172 leads to the B opinion eventually winning the public debate. On the contrary, any initial support which satisfies at > 0:172 ends up with a total victory of the A opinion even with 80% of initial support for opinion B. The process is illustrated in Fig. 10.35 with q D 0:05 and at D 0:30.

10.9 The Thresholdless Case

The surprising additional effect of including one-sided inﬂexible agents is that with q aq aq increasing the inﬂexible densityp, both B and c;3 move toward one another to q D 3 2 2 0:172 aq D aq 0:293 eventually merge at with B c;3 ,andthen disappear. One example is given in Fig. 10.36 with q D 0:20 and at D 0:30. This means that in the range q > 0:172, any B support, even huge, systematically loses the public debate against an ultrasmall A minority. The various regimes are 194 10 The Modeling of Opinion Dynamics

Fig. 10.36 The evolution of at 1 atC1 as a function of at from 1 (10.9)forp q D 0:20 > .3 2 2/ 0:172.The evolution from at D 0:30 is Only opinion A shown to produce a huge 0.8 support in favor of opinion A, which eventually gains the A inflexibles full support of the population 0.6

0.4

0.2

at 0.2 0.4 0.6 0.8 1

Fig. 10.37 The three fixed Fixed points points from (10.9)asa 1 function of the inflexible density q. The attractor aA D 1 is independent of q aq 0.8 but both B (lower curve) and aq c;3 (middle curve) get closer and closer with increasing q to merge andp disappear at 0.6 q D .3 2 2/ 0:172 aq D aq 0:293 with B c;3 (small empty circle). The 0.4 arrows show the direction of the opinion flow driven by the public debate. The gray area is the incompressible 0.2 inflexible proportion of the A opinion given by q q 0.2 0.4 0.6 0.8 1 shown in Fig. 10.37. The gray area represents the incompressible A inflexible domain. The A attractor aA D 1 is independent and stays unchanged. The above results may shed light on the mechanisms by which the public debate about the human culpability with respect to global warming has gained such an increasing support all over the world. But this is not a formal proof, it is only another way of looking at the associated dynamics. 10.10 Extending the Competition to Three Opinions 195

10.10 Extending the Competition to Three Opinions

While many issues end up between a binary choice, yes or no, in favor or against, accept or reject, some others involve a third option, which makes the dynamics much more subtle and highly nonevident. As for the even-sized groups where a tie break can occur, an identical tie is obtained for odd sized groups with three competing opinions A, B, and C for a configuration (A B C). The majority rule does not operate. Moreover, the tie cannot be resolved by invoking common beliefs and social representations. For two competing opinions with fifty/fifty support it makes sense, for instance, to preserve the status quo. But for three opinions it is more difficult since each opinion has one third of the arguments, and an agreement among the agents is necessary to resolve the tie. This might correspond to the alliance set among the various parties during an election. As soon as two parties make an alliance to join their votes, the tie is resolved in favor of one of their respective representatives. The voting rules are shown in Fig. 10.38 for the most general case. For each configuration of three agents with a local majority of 3 or 2, the agents still adopt the opinion which obtained the majority of the arguments. Accordingly 2 A, 2 B, 2 C adopt respectively opinions A, B, C whatever is the third opinion. All possible cases of local alliances can be considered by taking a probability ˛ to have (A B C) ! (A A A), a probability ˇ to have (A B C) ! (B B B) and a probability to have (A B C) ! (C C C) with the condition ˛ C ˇ C D 1. Without loss of generality, we select ˛ and ˇ to be two independent parameters with D 1 ˛ ˇ. Denoting atC1;btC1;ctC1 the respective proportions in support of A, B, and C at level .l C 1/, we have at the lower level l just below, the proportions at ;bt ;ct of A, B, and C representatives. These proportions obey the constraint at C bt C ct D 1

Fig. 10.38 The rules of opinion updates for groups of size 3 196 10 The Modeling of Opinion Dynamics since we assume that every agent has an opinion. We thus choose at and bt as the two independent variables with ct D 1 at bt . The associated update functions for respectively A, B, and C are

3 2 atC1 A3.at ;bt / D at C 3at .1 at / C 6˛at bt .1 at bt /; (10.12)

3 2 btC1 B3.at ;bt / D bt C 3bt .1 bt / C 6ˇat bt .1 at bt /; (10.13)

ctC1 D .1 atC1 btC1/: (10.14)

I first considered the problem in the early 1990s [46] restricting the investigation to the extreme case of D 1. In this case, two leading opinions A and B will lose the public debate if both have an initial support of 50%. Here, a tiny C minority will eventually spread over the entire population. More than two decades later, I investigated with Stephan Gelke and Luca Peliti the entire flow diagram of opinion dynamics for three competing opinions A, B, and C in the general case .˛; ˇ/ with updated local groups of size 3 [47]. The associated flow diagram is two dimensional and exhibits a very rich variety of highly nonlinear behavior. Several fixed points are involved. The model was also extended to consider a distribution of local groups whose sizes may vary from r D 1 up to r D L where L is some integer that is usually smaller than 10 [3]. For a detailed analysis, I refer the reader to the original paper [47].

10.11 The Reshufﬂing Effect and Rare Event Nucleation

Before closing the discussion on opinion dynamics modeling, it is worth com- menting on the use of iterated local updates with probabilities in the analytical treatment of the problem. It implies a reshuffling of agents between two successive updates. Many physicists have looked at this scheme as a mean field treatment due to the random reshuffling. But it appears that it is not the case since the local update integrates local fluctuations as has been proven by Monte-Carlo studies of the reshuffling scheme for a two dimensional ferromagnetic Ising system [48]. This reshuffling effect was earlier investigated using a cellular automata simulation [1]. It has allowed the discovery of the occurrence of very rare events, which can, under some specific conditions, nucleate and invade the whole system [2, 49]. An application to cancer tumor growth was made performing numerical simulations with Radomski [50]. Several opinion dynamics models have been proposed, each one grounded in a specific local update. However, I was able to demonstrate that most of them are identical, using a unifying frame for local updates and calculating the associated phase diagram [28]. 10.12 The Overlap with Physical Systems and Other Sociophysics Models 197

10.12 The Overlap with Physical Systems and Other Sociophysics Models

It is worth stressing that here, when an update takes place, it is applied to the same and full population of agents. Agents do shift their opinion eventually. It is different from our voting model where a group of agents elect one representative as shown in Fig. 10.39. Starting with nine agents (A A B B A B A A A) distributed as (A A B), (B A B), (A A A) yields the same nine agents but with different opinions with (A A A B B B A A A) after one update. In contrast to the voting model, the same configuration (A A B), (B A B), (A A A) produces the addition of a group of three (A B A) above it. These are different from the first nine. In the first case, the agents are the same but their respective opinions may have changed. In the second case, the agents do not shift in their opinions and new agents are added within a hierarchical structure. Performing one additional update in the voting model adds a second level with one agent (A), the president, while in the opinion model the update first requires a reshuffling of the nine agents and then a distribution in the three groups of three agents each. In our case, two kinds of configurations (A A B), (B A B), (A A A) or (A A B), (B A B), (A A A) are possible. Then the update is implemented leading respectively to (A A A), (B B B), (A A A), and (A A A), (A A A), (A A A). Some agents have again modified their opinion. At this stage, no more updates can be performed in the voting model, unless the initial assembly of nine agents is increased to 27 agents. In the opinion model, another update can be performed from the configuration (A A A), (B B B), (A A A). The above analysis demonstrates that sometimes the same mathematical equation, here a local majority rule, can create two very different realities. Although the mathematical equations between our two models of voting and opinion are identical, the content, the meaning, the implementation, and the result are totally different. The update scheme is taken from real space renormalization group transformation. In physics, this is a mathematical “trick” to extract some long range correlations while keeping the physical system unchanged. Here, it corresponds to either a real change of the system or to the building of a hierarchical structure.

AAA-BBB-AAA AAA-AAA-AAA

Fig. 10.39 The same local A AAB-BAB-AAA AAB-BAA-BAA majority rule yields very AAA-BBB-AAA different social applications. A-B-A In the voting case, a AAB-BAB-AAA AAB-BAB-AAA representative is elected and no individual opinion is Voting Opinion changed. In contrast, in opinion dynamics, agents do AAB-BAB-AAA Physics change their opinion AAB-BAB-AAA 198 10 The Modeling of Opinion Dynamics

10.13 Achievements of the Model

The model has proved itself in yielding heuristic power. In 2005, for the first time a highly improbable political vote outcome was predicted using it. The prediction concerned the prediction of the “No” victory to the French referendum about the project for a European constitution. Moreover, the prediction was made several months ahead of the actual vote in opposition to all the polls and analyses made at the time. Already in earlier publications of the model, the conclusions about this prediction were already pointed out, as seen in this excerpt of a paper published in 2002 [3]: To give some real life illustrations of our model, we can cite events related to the European Union which all came as a surprise. From the beginning of its construction there has never been a large public debate in most of the countries involved. The whole process came through government decisions and most people always seemed to agree on this mode of construction. At the same time, European opponents have been systematically urging for public debates. Such a demand sounds like it is absurd, knowing that a majority of people are in favor of the European Union. But anyhow, most European governments have been reluctant to hold a referendum on the issue. At odds with this attitude, several years ago, French president Mitterand decided to run a referendum over accepting the Maastricht agreement [11]. While a large success of the Yes vote was taken for granted, in reality it only just made it a bit beyond the required fifty percent. The more people discussed, the less support there was for the proposal. It is even possible to conjecture that an additional two weeks extension of the public debate would have made the “No” vote win! And again two years later I wrote [4]: Applying our results to the European Union leads to the conclusion that it would be rather misleading to initiate large public debates in most of the countries involved. Indeed, even starting from a huge initial majority of people in favor of the European Union, an open and free debate would lead to the creation of a huge majority hostile to the European Union. This provides a strong ground to legitimize the on-going reluctance of most European governments to hold a referendum on associated issues. At the end of 2004 in France, Jacques Chirac decided to hold a referendum over adopting the project of the European constitution. I then applied my model and the conclusion was that indeed the “No” vote would win [29–31]. At that moment, a huge majority of people were in favor of the “Yes” vote. Almost all political leaders were in favor of the “Yes” vote, as well as all the media, and the majority of the population. France could not say “No” to Europe; that would have been totally absurd. I myself could not believe in the model’s prediction. On May 29th, 2005, the“No”wonthevotewith55%. It was the first time that a political vote outcome was predicted using a model from sociophysics. Moreover, it was a highly improbable event. In addition, the prediction was made several months ahead of the actual vote, in opposition to all the polls and predictions made at the time. 10.14 In the Meantime 199

Fig. 10.40 A minority An opinion, which is spreading phenomena occurs at odds with a shared for most public debates common belief, must dealing with a key issue start from individual ! related to a major change support of more than 80 % of the whole population!

To survive a public and open debate!

It is important to stress that it was not a heads or tails result, nor even a possible random outcome with a low probability, but it was indeed a zero probability event. But, of course, this does not mean that the model is exact. It only validates the model and the approach in a way that strongly suggests that sociophysics may become a real predictive tool in the future. It illustrates the difficulty for an opinion in favor of major change to survive a public debate (Fig. 10.40). Our model has also led to the prediction of an increase of occurrence of voting at fifty–fifty in democratic countries, which has occurred several times in a row [44].

10.14 In the Meantime

Sociophysics emphasizes the biases which may be involved in the holding of a public debate to decide about some major social issue. Within our model, we saw how a free public debate produces quite naturally a dictatorial machine to propagate the opinion of a tiny minority against what could have been the initial opinion of an overwhelming majority. While rationality is applied using a majority rule, the possible occurrence of local doubts opens the way to having the collective beliefs making the choice. In the case of threshold dynamics these collective belief biases can shift the threshold from 50% to any side at values from 10% to 90%. The model applies to a large spectrum of social, economic, and political phenomena. In particular, it applies to the propagation effects such as that of fear and the spreading of rumors. It was used to explain the French hoax about September 11 [32]. Moreover, in 2005 for the first time, the outcome of a highly improbable political vote was predicted several months ahead of the actual vote [3, 51]. The victory of the “No” vote to the French referendum on the European constitution was confirmed by the actual vote. It is worth stressing that earlier polls and all analyses were predicting a victory of the “Yes” vote. The heuristic power of sociophysics was 200 10 The Modeling of Opinion Dynamics clearly demonstrated as being feasible. Nevertheless, one result is not sufficient to make a definite conclusion. Clearly, more studies are needed. We have given a possible explanation of the growing public support for the thesis of human guilt concerning the current global warming. It has been shown why the associated opinion dynamics will make the opinion of human guilt certain in becoming unanimous despite there being no scientific proof of its correctness. In particular, the model shows how powerful the driving force is of individual self- confidence to get a very small minority to successfully reverse a huge opposite majority opinion. We have presented a simple model which is able to reproduce some of the complexity of the social reality. It suggests that the direction of the inherent polarization effect in the formation of a public opinion driven by a democratic debate is biased from the existence of common beliefs within a population. Homogeneous versus heterogeneous situations were shown to result in different qualitative outcomes.

References

1. S. Galam, B. Chopard, A. Masselot and M. Droz, “Competing Species Dynamics”, Eur. Phys. J. B 4, 529–531 (1998) 2. B. Chopard, M. Droz and S. Galam, “An Evolution Theory in Finite Size Systems”, Eur. Phys. J. B 16, Rapid Note, 575–578 (2000) 3. S. Galam, “Minority Opinion Spreading in Random Geometry”, Eur. Phys. J. B 25 Rapid Note, 403–406 (2002) 4. S. Galam, “The dynamics of minority opinion in democratic debate” Physica A 336, 56–62 (2004) 5. S. Pajot and S. Galam, “Coexistence of Opposite Global Social Feelings: The Case of Percolation Driven Insecurity ”, Int. J. Mod. Phys. C 13 1375–1385 (2002) 6. S. Solomon, G. Weisbuch, L. de Arcangelis, N. Jan and D. Stauffer, “Social percolation models”, Physica A 277/1–2 (2000) 239–247 7. G. Deffuant, D. Neau, F. Amblard and G. Weisbuch, “Mixing beliefs among interacting agents” Advances in Complex Systems 3 (2000) 87–98 8. K. Sznajd-Weron and J. Sznajd,“Opinion evolution in closed community”, Int. J. Mod. Phys. C 11 (2000) 1157–1165 9. R. Hegselmann and U. Krausse, “Opinion Dynamics and Bounded Condence Models, Analysis and Simulation”, J. Artif. Soc. Social Sim 5 (2002) no. 3 10. F. Slanina and H. Lavicka, “Analytical results for the Sznajd model of opinion formation,” Eur. Phys.J.B35 (2003) 279–288 11. M. Mobilia and S. Redner, “Majority versus minority dynamics: Phase transition in an interacting two-state spin system”, Phys. Rev. E 68 (2003) 046106 (1–11) 12. L. Behera and F. Schweitzer, “On Spatial Consensus Formation: Is the Sznajd Model Different from a Voter Model?”, Int. J. Mod. Phys. C 14 (2003) 1331–1354 13. C. J. Tessone, R. Toral, P. Amengual, H.S. Wio, and M. San Miguel, “Neighborhood models of minority opinion spreading”, Eur. Phys. J. B 39 (2004) 535 14. M. C. Gonzalez, A. O. Sousa and H. J. Herrmann,“Opinion formation on a deterministic pseudo-fractal network”, Int. J. Mod. Phys. C 15 No. 1 (2004) 45–57 15. J. J. Schneider and C. Hirtreiter, “The Impact of election results on the member numbers of the large parties in Bavaria and Germany, Int. J. Mod. Phys. C 16 (2005) 1165–1215 References 201

16. R. Lambiotte and M. Ausloos, “Coexistence of opposite opinions in a network with communities”, eprint (2007) arxiv:physics/0703266 17. M. Ausloos and F. Petroni , “Statistical dynamics of religions and adherents”, Euro. Phys. Lett. 77 (2007) 38002 (1–4) 18. S. Fortunato and C. Castellano, “Scaling and Universality in Proportional Elections”, Phys. Rev. Lett. 99 (2007) 138701 19. C. Castellano, S. Fortunato and V. Loreto, “Statistical physics of social dynamics”, Rev. Mod. Phys. 81, 591–646 (2009) 20. P. Contucci and S. Ghirlanda, “Modeling Society with Statistical Mechanics: an Application to Cultural Contact and Immigration”, Quality and Quantity 41 (2007) 569–578 21. Kulakowski and Nawojczyk, “The Galam Model of Minority Opinion Spreading and the Marriage Gap”, eprint (2007) arXiv:physics/0703268 22. R. Lambiotte, J. Saramaki, V. D. Blondel, “Dynamics of latent voters”, arXiv:0811.1464v1 [physics.soc-ph] 23. A. C. R. Martins, “Continuous opinions and discrete Actions in opinion dynamics problems”, Int. J. Mod. Phys. C 19, 617–624 (2008) 24. A. C. R. Martins, “Mobility and Social Network Effects on Extremist Opinions”, Phys. Rev. E 78, 036104 (2008) 25. R. Vicente, A. C. R. Martins and N. Caticha, “Opinion Dynamics of Learning Agents: Does Seeking Consensus Lead to Disagreement?”, J. Stat. Mec. P03015 (2009) 26. A. C. R. Martins, C. B. Pereira and R. Vicente, “An Opinion Dynamics Model for the Diffusion of Innovations”, arXiv:0809.5114v3 [physics.soc-ph] 27. S. Galam, “Heterogeneous beliefs, segregation, and extremism in the making of public opinions”, Phys. Rev. E 71, 046123-1-5 (2005) 28. S. Galam, “Local dynamics vs. social mechanisms: A unifying frame”, Europhys. Lett. 70, 705–711 (2005) 29. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Interview par P. Lehir, Le Monde, Samedi 26 Fevrier,´ 23 (2005) 30. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Le Monde, Lundi 11 Avril, 15 (2005), Reproduced in the international weekly selection of Le Monde 2005 31. S. Galam, “Les mathematiques´ s’invitent dans le debat´ europeen”,´ Le Monde, Lundi 11 Avril, 15 (2005), Reproduced in “TA NEA”, Greek daily newspaper March 3 (2005) 32. S. Galam, “Modeling Rumors: The No Plane Pentagon French Hoax Case”, Physica A 320, 571–580 (2003) 33. S. Galam, “Les reformes´ sont-elles impossibles?”, Le Monde, Mardi 28 Mars, 18–19 (2000) 34. S. Galam, “Quand les reformes´ democratiques´ butent sur les lois physiques”, Entretien, Sciences et Vie 1017, 92–97, Juin (2002) 35. S. Galam, “Notre avenir en equations”,´ Entretien, Le Minotaure, 554–559 2, Juin (2003) 36. S. Galam, “Il faut bien plus qu’une majorite”,Tribune,D´ ecisio´ N 7 (www.decisio.info), Janvier (2004) 37. S. Galam, “L’opinion se modelise”,´ Le 8 d’Entrecom, 5–6 2, Mai (2004) 38. S. Galam, “Contrarian deterministic effect: the hung elections scenario”, Physica A 333, 453–460 (2004) 39. S. Galam and F. Jacobs, “The role of inﬂexible minorities in the breaking of democratic opinion dynamics”, Physica A 381 366–376 (2007) 40. C. Borghesi and S. Galam, “Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect”, Phys. Rev. E 73 066118 (1–9) (2006) 41. D. Stauffer and S. A. Sa Martins, “Simulation of Galam’s contrarian opinions on percolative lattices”, Physica A 334 (2004) 558–565 42. M. S. de la Lama, J. M. Lopez´ and H. S. Wio, “Spontaneous emergence of contrarian-like behaviour in an opinion spreading model”, Europhys. Lett. 72, (2005) 851–857 43. H. S. Wio, M. S. de la Lama and J. M. Lopeza,´ “Contrarian-like behavior and system size stochastic resonance in an opinion spreading model”, Physica A 371 (2006) 108–111 202 10 The Modeling of Opinion Dynamics

44. S. Galam, “From 2000 Bush–Gore to 2006 Italian elections: voting at fifty–fifty and the contrarian effect”, Quality and Quantity Journal 41 579–589 (2007) 45. S. Galam, “Les scientifiques ont perdu le Nord”, (2008) Plon, Paris 46. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191–200 (1991) 47. S. Gekle, L. Peliti, and S. Galam, “Opinion dynamics in a three-choice system”, Eur. Phys. J. B 45, 569–575 (2005) 48. A. O. Sousa, K. Malarz, and S. Galam, “Reshuffling Spins with Short Range Interactions: When sociophysics produces physical results”, Int. J. Mod. Phys. C 16 1507–1517 (2005) 49. S. Galam, B. Chopard and M. Droz, “Killer geometries in competing species dynamics”, Physica A 314, 256–263 (2002) 50. S. Galam and J. P. Radomski, “Cancerous tumor: the high frequency of a rare event”, Phys. Rev. E 63, 51907–51911 (2001) 51. S. Galam, “Pourquoi des elections´ si serrees´ ?”, Le Monde, Mercredi 20 Septembre, 22 (2006) Chapter 11 By Way of Caution

This overview of my salient models of sociophysics exhibits the potential power of sociophysics with respect to a very large spectrum of social, political, economical, and strategic applications as seen in Fig. 11.1. However, it is of crucial importance to keep in mind that we are only using models to mimic part of the reality. These models are only an approximation of that reality and are not the reality itself (Fig. 11.2). To forget the difference may lead to some misunderstandings and misleading conclusions of what should be done in the face of reality, with a misuse of the approach. The limits of the approach must always be discussed before making any prediction. At this stage, the collaboration with researchers from the social sciences could be valuable for making predictions on precise social events. Last, but not least, I would like to assess once more that dealing with the shaking of the paradigms of human behavior, we should always keep in mind that in case humans can be similar in some aspects with atoms, making an error modeling inert matter could have different consequences than with human beings as illustrated in thedrawingshowninFig.11.3. Therefore, a great deal of caution is and will be always required before trying to apply a model of sociophysics.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 203 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 11, © Springer Science+Business Media, LLC 2012 204 11 By Way of Caution

The model is… right !

$$$

Fig. 11.1 Happiness with the successful setting of a great model of human behavior

To conclude,

Remember it is only a model of v reality NOT Reality …

Fig. 11.2 A model may well be quite efﬁcient in predicting some aspect of the reality but it is not the reality itself 11 By Way of Caution 205

Oops, I forgot a minus… $$$ somewhere

Fig. 11.3 The model was indeed right . . . up to a wrong sign Part III Democratic Voting in Bottom-Up Hierarchical Structures: From Advantages and Setbacks to Dictatorship Paradoxes Chapter 12 Highlights of the Part

In order not to discourage any impatient readers, we outline here the main steps of our investigation together with some of the major ﬁndings [1Ð19]. It has to be read “alamani` ere”` of a research physical paper. In other words, not many details are given. Explanations are limited and no backward motivation is given. It should be an incentive to plunge into the heart of the work contained in the following chapters.

12.1 Dictatorships Can Be Democratic

The paradoxical use of democratic bottom-up voting is demonstrated using concepts from real space renormalization group theory developed in statistical physics to tackle the very difﬁcult problem of critical phenomena and phase transitions. It is shown how dictatorship effects are driven “naturally” from the use of local majority rules. This democratic twist is produced by the existence of critical thresholds in the voting ﬂow monitored by repeating votes. Accordingly, within a democratic hierarchy a huge bottom majority can be self-eliminated while climbing up the hierarchical levels through democratic voting. It is even possible to make the election of a president belonging to a given party certain by using democratic voting. For two competing parties, a critical threshold, i.e., a separator, delimits the basins of attraction, each one being associated to the leadership of a party. Above its critical threshold, a party gains more and more representation while iterating voting, until eventually reaching full power at the top of the hierarchy. The threshold for reaching power with certainty varies from 23%upto77% of the bottom support. New light, which is rather surprising, is shed in particular on the last century’s Eastern European collapse of communism. Unexpected and counterintuitive insights are obtained with respect to a large spectrum of social and political phenomena.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 209 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 12, © Springer Science+Business Media, LLC 2012 210 12 Highlights of the Part

12.2 What It Is About

A bottom-up hierarchy is built using local majority rules to climb from one level to the next. The frame maybe a political group, a ﬁrm, a society, or any human organization. Starting from the bottom, local groups of agents are constituted by random selection from the population concerned. These local groups constitute the level 0 of the hierarchy. We focus on a two party A and B competition whose respective support within the population are p0 and 1 p0. Each of the agents supports a party. Once formed, each bottom group elects a representative, either A or B using a local majority rule within itself. These elected people constitute the ﬁrst hierarchical level of the hierarchy denoted level 1. The same local group voting process is repeated at level 1 with the elected agents fromlevel0tobuildalevel2byelecting new representatives and so on. The president is elected at the last level of the hierarchy designated as the top level.

12.3 The Wonderful World of Democracy

To start, we consider a voting group size of 3. It the simplest case with three agents randomly selected from the population. Several groups are formed and each one elects a representative using a local majority rule. A group with 3A or 2A and 1B elects a A and otherwise a B. The associated probability to have an A elected is thus,

3 2 p1 P3.p0/ D p0 C 3p0.1 p0/; (12.1) where P3.pn/ denotes the voting function, here a simple majority rule. At level 1, the same group forming and voting is repeated now using the elected agents. New agents are elected and constitute level 2. The process can be repeated again and again, provided that enough bottom groups are formed since a given level has a total number of agents that equals one third of the total number of agents at the level just below it. Generalizing the voting probability to have an A elected at level (n+1) from level n we get,

3 2 pnC1 P3.pn/ D pn C 3pn.1 pn/; (12.2) where pn is the proportion of A elected people at level n. P .p / D p p D 0 p D 1 p D 1 Solving 3 n n three ﬁxed points B , c;3 2 and A are obtained. At pB D 0, no A is elected with the dictatorship situation and only B is democratically elected. At pA D 1, it is the opposite situation, with only A being elected. The middle ﬁxed point pc;3 is unstable while both pB and pA are stable. p < 1 p D 0 p > 1 p D 1 Starting from 0 2 leads to B and 0 2 drives to A .Thevalueof pc;3 determines the critical threshold to have repeating votes to drive a party toward either full power at pA or total disappearance at pB. 12.4 Not Ruling Is Bad for You 211

For instance, starting from a support of p0 D 0:45 at the bottom we get successively p1 D 0:42, p2 D 0:39, p3 D 0:34, p4 D 0:26, p5 D 0:17, p6 D 0:08 down to p7 D 0:02 before p8 D 0:00. Within eight levels, 45% of the population is found to be self-eliminated. However, the overall process preserves the democratic character of a global majority rule voting within seven levels since it is the bottom majority that gets the presidency. We conclude that majority rule voting produces the self-elimination of any A 1 proportion as long as the bottom support is less than 50%, i.e., p0 < 2 .This democratic self-elimination process requires a sufﬁcient number of voting levels to be completed. The situation is symmetrical with respect to A and B. The threshold to full power is the same at 50% for both of them. To determine the number of levels required to ensure a full leadership by the bottom majority party is thus instrumental. To make sense, this number must be small enough, since most organizations have only a few levels, usually less than 10. The scheme is wonderful since it simultaneously distributes local power to the minority party and ensures the organization presidency to the majority party. The whole thing is at a very low cost in terms of manpower since only a small part of the population needs to be involved.

12.4 Not Ruling Is Bad for You

From the above, we found that to obtain a top leadership requires having more than 50% of the support at the bottom, which is a fair constraint. However, in real life situations, taking over the presidency for a challenging party appears to be much more difficult than to reach the support of the bottom majority. Indeed, frequent political changes are not perceived as being good for the efficient functioning of an organization. Accordingly, a tip is often given to the ruling party to reinforce the organization’s stability, by breaking slightly the symmetry between the status of the two parties, which are respectively in power and challenging the current power. Such a procedure is implemented through various mechanisms, like attributing one additional vote to the president or by allowing the president to designate some of the committee members. These internal features of institutional structures, which aim to stabilize the edge of the top leadership sounds wise and democratic. However, while they are supposed to give a little tip in favor of the current rulers only in the rare cases of balance with the opposition party, they turn out to produce in practice a strong bias in favor of the current rulers. For instance, we will see how the reasonable statement “to make a change, you need a majority” becomes an incredible advantage in favor of the current ruling party. The associated subtle breaking of the democratic balance is nicely illustrated using even sized voting groups. Let us take the simplest case with four agents per group and assume that B is the ruling party. To get an A elected from one voting group, the challenging party needs to obtain either 4A or 3A and 1B. The tie case is 2 AÐ2 B votes for a B. Going from voting 212 12 Highlights of the Part groups with three to four agents, the salient new feature is the appearance of tie configurations 2 AÐ2 B for which there exists no majority. In most social situations, no majority implies de facto, no change. Therefore, a tie is a bias in favor of the rulers. It makes the voting function asymmetric. The probability of getting an A elected at level n C 1 becomes

4 3 pnC1 P4.pn/ D pn C 4pn.1 pn/: (12.3)

In contrast, the probability of having a B elected is

4 3 2 2 1 P4.pn/ D pn C 4pn.1 pn/ C 2pn.1 pn/ ; (12.4) where the last term embodies the bias in favor of B. The associated stable ﬁxed points are still pB D 0 and pA D 1. However, the unstable one is drastically shifted to p 1 C 13 p D 0:77 c;4 6 (12.5) for the A. Simultaneously, the B threshold to stay in power is down to about 23%, making both situations very different. To take over the presidency, A needs to get more than 77% of the support at the bottom while to stick to power, B only needs to keep their support above 23%. In addition to the asymmetry, the bias makes the number of levels to democratic self-elimination smaller than in the previous case of a size of three. Starting again from p0 D 0:45 leads to p1 D 0:24, p2 D 0:05 and p3 D 0:00. Instead of eight levels, three are enough to make the A disappear. For instance, starting far above 50% with p0 D 0:70 the associated voting dynamics yields p1 D 0:66, p2 D 0:57, p3 D 0:42, p4 D 0:20, p5 D 0:03,andp6 D 0:00. Within only six levels, 70%ofa population is thus self-eliminated democratically. Using an a priori reasonable bias in favor of B has turned a majority rule democratic voting system into a dictatorship outcome. To get to power, A must pass above 77% of overall support which is almost out of reach in any normal democratic system with two competing parties.

12.5 Big Is Better

Many organizations have voting group sizes of larger than 3 and 4. Extending the above cases to any size r leads only to more complicated equations but the main features remain unchanged. For an r-sized cell, the voting function pnC1 D Pr .pn/ becomes, lXDm rŠ P .p / D pl .1 p /rl ; r n lŠ.r l/Š n n (12.6) lDr 12.6 What Matters 213

rC1 r where m D 2 for odd r,andm D 2 C 1 for even r thus accounting for the bias in favor of B. The two stable fixed points pB D 0 and pA D 1 are invariant with size 1 change. For odd sizes, the unstable fixed point is also unchanged with pc;r D 2 .For even sizes, the asymmetry between the threshold values for respectively rulers and nonrulers weakens with increasing sizes but stays non-zero.p 1C 13 For the A party, the threshold to power is pc;4 D 6 for size 4. It decreases p D 1 !1 1 asymptotically toward c;r 2 for r . But it always stays larger than 2 , still making it hard for the opponents to pass over the barrier, since in democratic countries the difference between two candidates is usually of only a few percent. In parallel, increasing the sizes of the voting groups reduces the number of levels necessary to get to the stable fixed points.

12.6 What Matters

Given a bottom support p0 we calculate pn, the corresponding value for A support after n voting levels. Expanding the voting function pn D Pr .pn1/ around the unstable ﬁxed point pc;r yields

pn pc;r C .pn1 pc;r/r ; (12.7)

dPr .pn/ r jp Pr .pc;r/ D pc;r where dpn c;r with . Rewriting the last equation as,

pn pc;r .pn1 pc;r/r ; (12.8) we can then iterate the process to get,

n pn pc;r .p0 pc;r/r ; (12.9) from which we get, n pn pc;r C .p0 pc;r/r : (12.10)

C Two different critical numbers of levels nc;r and nc;r can be obtained. The ﬁrst one corresponds to pn D 0 and the second one to p C D 1. Putting pn D pn D c;r nc;r c;r 0 in the above equation gives,

1 pc;r nc;r ln ; (12.11) ln r pc;r p0 which is deﬁned only for p0

C 1 1 pc;r nc;r ln ; (12.12) ln r p0 pc;r which is now deﬁned only for p0 >pc;r since pc;r <1. Only above pc;r can the A proportion increase to one. The above expansions turn out to be rather good estimates even in the vicinity of the two stable ﬁxed points 0 and 1 if taking the integer part and adding 1.

12.7 The Strategic Key

Organizations are set with a ﬁxed number of levels, which cannot be modiﬁed. On this basis, to be practical, the question of “How many levels are needed to self-eliminate a party?” must be turned into “Given n levels, what is the minimum bottom support to take over the presidency with certainty?”. Or alternatively, for the ruling party, “Given n levels, what is the critical overall support above which the ruling party is automatically reelected?”, which means the critical support above which it is necessary to worry about the current ruling policy even if it is unpopular. n To implement this operative question, we rewrite pn pc;r C .p0 pc;r/r as

n p0 D pc;r C .pn pc;r/r : (12.13)

This equation yields two critical thresholds. The ﬁrst one is the disappearance pn threshold r;B which gives the value of support under which the A disappears with certainty at the top level of the n-level hierarchy. It is obtained by plugging pn D 0 into the above equation with

pn D p .1 n/: r;B c;r r (12.14) p D 1 pn In parallel, n gives the second threshold r;A above which the A obtains full and total power with pn D pn C n : r;A l;r r (12.15) pn

n seen from the equations, this democratic region shrinks as a power law r of the number n of hierarchical levels. A small number of levels raises the threshold to a value giving a total reversal of power but simultaneously lowers the threshold for nonexistence. The above formulae are approximate but give the right quantitative behavior with pn n C 1 pn n C 2 r;B ﬁtting to the value and r;A to thep one. Let us illustrate the case r D 4 D 1:64 p D 1C 13 n D 3; 4; 5; 6; 7 of where and c;4 6 . Considering level pn 0:59 0:66 0:70 0:73 0:74 pn organizations, r;B equals respectively , , , , . In parallel, r;A equals 0:82, 0:80, 0:79, 0:78 and 0:78. These series emphasize dramatically the dictatorship character of the voting process.

12.8 Visualizing the Democratic-Driven Dictatorship Twist

Some snapshots of a large scale numerical simulation are shown in Figs. 12.1Ð12.4 to illustrate the impressive effect of the phenomena. The A and B parties are represented respectively in green and red squares. The bias is in favor of the red ones, i.e., a tie of 2 red 2 green votes for a red. A structure with eight levels is shown. The mechanism of massive democratic self-elimination is clearly seen. The written percentages are for the green representation at each level. ¥ Figure 12.1: Groups of four people in an eight level hierarchy. Parties A and B are denoted respectively by green and red squares. The bias is in favor of the red squares, i.e., a tie of 2Ð2 votes for a red square. The written percentages are for the green representation at each level. The bottom green support is 52:12%. After three levels, all the green squares disappear. The “Time” and “Generations” indicators should be discarded. ¥ Figure 12.2: The same as Fig. 12.1 with a bottom green support of 68:62%. ¥ Figure 12.3: The same as Fig. 12.1 with a bottom green support of 72:01% bottom A (green) support, far more than 50%. After ﬁve levels no more green squares are found. ¥ Figure 12.4: The same as Fig. 12.1 with a bottom green support of 76:85% bottom A (green) support. Finally the A (green) gets the presidency.

12.9 The Key Conﬁgurations to Inﬁltrate a Party

Our approach is based on iterated votes whose associated dynamics is monitored by the existence of critical thresholds. The full process is described in terms of probabilities with the fundamental assumption of a random selection of the bottom agents to constitute the voting groups. Nevertheless, we will discover the existence of very rare unexpected events, which is not a real problem in itself. But the real 216 12 Highlights of the Part

Fig. 12.1 A population of 65,536 agents with 52:12% bottom A (green) support. After three levels, no more green squares appear 12.9 The Key Conﬁgurations to Inﬁltrate a Party 217

Fig. 12.2 A population of 65,536 agents with 72:01% bottom A (green) support, far more than 50%. After ﬁve levels no more green squares are found 218 12 Highlights of the Part

Fig. 12.3 A population of 65,536 agents with 76:03% bottom B (green) support. Now the seven levels are needed to get a red square (B) elected. The bottom B support (red) is only of 23:97% 12.9 The Key Conﬁgurations to Inﬁltrate a Party 219

Fig. 12.4 A population of 65,536 agents with 76:85% bottom A (green) support. Finally, the A (green) gets the presidency 220 12 Highlights of the Part fascinating discovery will be the possibility of making certain a very rare event by using either the coring of the hierarchy or lobbying, which will appear as a means of frightening to break down the democratic functioning of an organization for the beneﬁt of an ultrasmall minority.

12.10 Life Is More Risky with Three Competing Parties

Many social situations involve more than two competing parties. Consider the case of three parties A, B, and C with voting groups of size 3. Before, with two competing parties, majority rule was always applicable in this case. But now the A B C configuration is unsolved since no majority prevails, which is similar to the previous 2 AÐ2 B configuration. In the three party situation, the bias is set to favor the ruling party. For multiparty competition, the bias is usually resolved by agreements between parties. In most cases, the two largest parties, say A and B, are hostile to each other, while the smallest one, C, could compromise with either one of them. Along this line, the A B C configuration elects a C with a coalition with either A or B. In addition, 2A or 2B are still necessary to elect respectively an A or a B. Otherwise, a C is elected. Therefore, the elective functions for A and B are the same as in the two party model. This means that the critical threshold for achieving full power by A and B is still 50%. In other words, for bottom support to both A and B of less than 50%the C gets full power. The required number of levels is unchanged. For instance, take 39% for both A and B. The C is thus left with 22%. We have the series for respectively A, B, and C: 34%, 34%and32% at the first level; 26%, 26% and 48% at the second level; 17%, 17%and66% at the third level; 8%, 8%and84% at the forth level; 2%, 2%and96% at the fifth level; and 0%, 0%and100%atthe sixth level giving total power to the C minority within only six levels. It is possible to generalize the present approach to as many groups as wanted. The analysis becomes much more involved but the mean features of voting flowing toward a fixed point are preserved. Moreover, power will go even more rarely to the largest groups, as seen with the above A B C case.

12.11 Eastern European Communist Collapse Was Not Sudden

As an application of the model, we comment on an explanation to last century’s generalized autocollapse of eastern European communist parties, which would apprehend it not as a sudden event but as the result of a very long and slow process. While communist parties seemed eternal, once they collapsed many explanations were based on an opportunistic change within the various organizations. For the References 221 non-Russian countries, it was the end of the Soviet army threat. Maybe part of the explanation could indeed be related to our hierarchical model. Communist organizations are based, at least in principle, on the concept of democratic centralism, which is nothing else than a tree-like hierarchy. Suppose then that the critical threshold to power was of the order of 80%, as in our previous example. We could then consider that the internal opposition to the orthodox leadership grew a lot and massively over several decades to eventually reach and pass the critical threshold. Once the threshold was passed, the internal opposition was suddenly everywhere. The immediate collapse of eastern European communist parties could have been the result of indeed a very long and solid phenomenon within the communist parties themselves. Such an explanation is not opposed to additional constraints but emphasizes some trend of the internal mechanism within these organizations. At this stage, it is of importance to stress that modeling social and political phenomena is not aimed at stating an absolute truth but instead at singling out some basic trends within a very complex situation. All models can be questioned, and can be changed eventually.

References

1. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, J. of Math. Psychology 30, 426-434 (1986) 2. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943-951 (1990) 3. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191-200 (1991) 4. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organi- sations”, Models of self-organization in complex systems (Moses) Akademie-Verlag, Berlin V.64, 53-59 (1991) 5. S. Galam, “Paradoxes de la rgle majoritaire dans les systmes hierarchiques”,« Revue de Bibliologie, 38, 62-68 (1993) 6. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132-139 (1999) 7. S. Galam, “Real space renormalization group and totalitarian paradox of majority rule voting”, Physica A 285, 66-76 (2000) 8. S. Galam and S. Wonczak, “Dictatorship from Majority Rule Voting”, Eur. Phys. J. B 18, 183-186 (2000) 9. S. Galam, “Democratic Voting in Hierarchical Structures”, Application of Simulation to Social Sciences, G. Ballot and G. Weisbush, Eds. Hermes, Paris, 171-180 (2000) 10. S. Galam, “Building a Dictatorship from Majority Rule Voting”, ECAI 2000 Modelling Artiﬁcial Societies, C. Jonker et al, Eds., Humboldt U. Press (ISSN: 0863-0957), 23-26 (2001) 11. S. Galam,“How to Become a Dictator”, Scaling and disordered systems. International Work- shop and Collection of Articles Honoring Professor Antonio Coniglio on the Occasion of his 60th Birthday. F. Family. M. Daoud. H.J. Herrmann and H.E. Stanley, Eds., World Scientiﬁc, 243-249 (2002) 12. S. Galam, “Dictatorship effect of the majority rule voting in hierarchical systems”, Self-Organisation and Evolution of Social Systems, Chap. 8, Cambridge University Press, C. Hemelrijk (Ed.) (2005) 13. S. Galam, “Stability of leadership in bottom-up hierarchical organizations”, Journal of Social Complexity 2 62-75 (2006) 222 12 Highlights of the Part

14. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 15. S. Galam, “Crier, mais pourquoi”, Liberation,« Vendredi 17 Avril, 6 (1998) 16. S. Galam, “Le vote majoritaire est-il totalitaire ?”, Pour La Science, Hors serie,« Les Mathematiques« Sociales, 90-94 July (1999) 17. S. Galam, “Citation in front page of the Figaro in an editorial from Jean dOrmesson”, Le Figaro, Mardi 4 Juin, 1 (2002) 18. S. Galam, “Risque de raz-de-maree« FN, Entretien, France Soir, La Une et 3, Mercredi 5 Juin (2002) 19. S. Galam, “Le FN au microscope”, Le Minotaure 6, 88-91, Avril (2004) Chapter 13 Basic Mechanisms for the Perfect Democratic Structure

The democratic majority rule voting in democratic bottom-up hierarchical structures is shown to produce under certain conditions dictatorship effects by keeping a minority in power against a huge increase of the opposition. No coercion is necessary, only democratic voting. This result sheds new light in particular on the internal strength of former communist organizations. It also provides a scheme to assess the conditions by which a minority extremist group can quickly get into power by democratic means and without any intermediate step. In turn, it also deﬁnes the conditions by which to avoid such a danger.

13.1 Starting from a Naive View of Former Communist Organizations

For the ﬁrst illustration of our approach, we consider the former communist organizations in order to make the following observation. During the many long years of their existence, they seemed totally frozen in their leadership. Despite very different internal and external situations, all of them, apart from Hungary and Czechoslovakia, remained unchanged, keeping the same leaders in power for decades and showing a total lack of internal democracy. While in communist countries, this lack of democracy was obvious, due to the strong use of political police and physical repression, within democratic countries the absence of organized physical threats could temper such an analysis. But yet the same frozenness did prevail among western communist organizations, with the exception of the Italian communist party. Nevertheless, during the period of the cold war, whilst non-communists were obviously convinced of the dictatorship character of communist organizations, communist party members kept on dismissing their claim of the existence of an internal dictatorship. On the contrary, communist party members usually claimed that the

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 223 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 13, © Springer Science+Business Media, LLC 2012 224 13 Basic Mechanisms for the Perfect Democratic Structure

Fig. 13.1 During the cold From outside: war, non-communists claimed that the communist party was It is a dictatorship a dictatorship, in opposition to the statements of the communist members who were saying that it was democratic. Who was right? Who was wrong? Maybe both From inside: statements are right It is democratic simultaneously Communist party ?

party was functioning democratically, with open and free internal discussions, but within the logic of the so-called democratic centralism. Facing this question of the democratic functioning of communist parties together with the contradictory claims, any fair observation would conclude that either one of the sides is lying or is misinformed and of course that only one truth exists. Communist parties were either dictatorships or democratic. Observations from the outside led to the first conclusion, at odds with the internal view of the militants (Fig. 13.1). The above description may not be exact; however, what matters at this stage is that it raises an interesting paradox. To help give shape to it, we take a naive a priori view of human beings in which firstly, people do not lie, and secondly, they tell the truth about what they feel. On this basis, we consider that both “communists” and “anti-communists” are simultaneously right about their respective claims of democracy and dictatorship. Such an observation drives us to the contradictory possibility of having a democratic functioning of a democratic organization that eventually leads to a dictatorship. At least, such an explanation is a plausible way of formulating the contradictory observation. The associated question is to find out if in principle a dictatorship can arise spontaneously and democratically from and against the will of the majority. And if the answer is yes, we then need to find out if such a dictatorship can be stable against a growing majority of the people, while yet preserving a democratic mode of functioning, which means having periodic free elections. Following our general framework defined in the first chapter, we have gone through the first two steps: Step 1: “Choose a particular phenomenon : bottom up democratic voting in hierarchical structures”. Step 2: “Single out one salient paradoxical feature : democratic voting in hierarchical structures may lead to a stable dictatorship”. 13.2 Setting Up the Simplest Form of a Voting Process 225

13.2 Setting Up the Simplest Form of a Voting Process

Once the above two ﬁrst steps of our general scheme have been completed, we can proceed with the third one which is: Step 3: “Deﬁne quantitatively the phenomenon in its simplest form”. To complete such a step, we forget about communist organizations and real complicated politics and we take a step back so as to take a much simpler virtual frame. We create a model which, although very crude, can yet illustrate the phenomenon at stake. This phenomenon is being able to have a voting system which is simultaneously democratic and dictatorial. We thus start considering an assembly of N agents, each one supporting either one of two competing political policies A or B. Then the assembly has to elect a president using a democratic voting scheme where each agent casts one single vote according to its supporting politics. Of course, in most voting situations, the vote itself is preceded by a debate, which in turn may produce some dynamics in the respective support for politics A and B. It involves the effects of debates, charismatic people, advertising, and many other features of individual opinion changes. This very important and complex issue will be studied in a forthcoming chapter. Here we start from a population in which each agent has made up its mind and sticks to its opinion in favor of either A or B before voting. When it comes to actually voting, it casts the corresponding ballot. On this basis, we give some initial support distribution, p0 and 1 p0 for respectively policy A and B. Various voting organizations will be studied and compared. In particular, we will investigate hierarchical organizations in which voting is used locally at various levels to go up the hierarchy from the bottom to the designation of the president at the top level. It implies that voters designate representatives and that the president is not directly elected by all the agents. Local votes designate a representative at the immediately higher hierarchical level. We will study the result of the voting dynamics depending on the structural organization of the voting process. But before evaluating the effect of a voting bottom-up hierarchical structure on the designation of the president, we start from the direct voting from all the agents. The situation corresponds to the one vote designation of the president. In this case, since we are using the democratic criterion of a majority to designate the winner, 1 1 the result is deterministic. The president is A if p0 > 2 and B when p0 < 2 .It embodies the basic axiom of democratic voting where the president is elected by the current majority. At this level, we are not considering the existence of nonvoting agents but a turnout of 100%. We can represent the outcome probability p1 using a step function (see Figs. 13.2 and 13.3) 1 if p0 >1=2 p1 D Pall.p0/ (13.1) 0 if p0 <1=2: 226 13 Basic Mechanisms for the Perfect Democratic Structure

Fig. 13.2 The whole p 1 population voting step 1 function (13.1). The result is deterministic, either “yes” or Separator “no” depending on the 50 % density p0 of A supporters. 0.8 p > 1 When 0 2 , an A is elected p < 1 against a B for 0 2 B elected 0.6

0.4 A elected

0.2

p 0 0.2 0.4 0.6 0.8 1

Fig. 13.3 The result for one A The whole vote in favor of from a group voting group which includes the whole population with a density p0 of A supporters and .1 p0/ for B.Theresult Deterministic is deterministic, either “yes” outcome: Yes or No or “no” depending on p0

The unique unknown is the support for A

Knowing the support for A gives the outcome

But such a deterministic procedure suffers from several setbacks as soon as it is applied to an organization and not to a whole country. The first major setback is that it requires every agent to vote, which can be difficult to achieve in large organizations. The second one, more problematic, is that it somehow makes the organization fragile, in the sense that the minority opinion is not taken into consideration at all. The last but not least deficiency of a single, direct, all agents voting is the absence of any intermediate levels between the president and the assembly of agents, which often can amount to more than several thousand people. In addition, these people are often scattered in several locations which can encompass the whole planet. With the ongoing globalization of the economy, such a picture is becoming the norm. 13.3 The Single Random Small Group Voting Scheme 227

13.3 The Single Random Small Group Voting Scheme

To account for the above difficulties, most organizations use hierarchical structures where local committees elect representatives. Before going onto the investigation of such structures, we first focus on the discrepancy between small group voting and all agents voting. We thus start considering the simplest case where only three agents are randomly chosen from the whole population to make up a committee, which in turn elects the president (Fig. 13.4). We then have to calculate the associated value of p1 which is a step function in the case of all agents voting (see (13.1)). For a group of three agents to elect an A, it is necessary to have in the group either f3A; 0Bg or f2A; 1Bg. Knowing the densities p0 and .1 p0/ for support of respectively the policies of A and B within the full population, the associated 3 2 probabilities are evaluated to be p0 for the first configuration and 3p0.1 p0/ for the second one which can be achieved from three different possibilities. Adding them yields the corresponding probability

3 2 p1 P3.p0/ D p0 C 3p0.1 p0/; (13.2) where P3.pn/ denotes the voting function, here a simple majority rule applied to a group of three agents. It is no longer a step function as shown in Fig. 13.5. While (13.1) always produces a deterministic voting outcome in favor of the current majority, using (13.2) turns the process probabilistic for the whole range 0

A B A A B A B B B B A B A B B A A B A A B A B B B A A B

X: the A Population with N people President Fig. 13.4 Three agents are picked randomly from the X X X population of N agents AAA A supporting either opinion A AAB (x3) or opinion B. The randomly Random selection of selected group of three agents BBB B 3 agents elects the president ABB (x3) 228 13 Basic Mechanisms for the Perfect Democratic Structure

Fig. 13.5 The result for one p 1 A vote is in favor of from a 1 group of three agents picked randomly from a population Separator with a density p0 of A 50 % supporters and .1 p0/ for 0.8 B. The result is now p < 1 probabilistic. For 0 2 ,itis p

1 p >p 0 2 yields 1 0 B victory Deterministic

A victory 0.4

0.2

p 0 0.2 0.4 0.6 0.8 1

Fig. 13.6 The result for one A The random vote in favor of from a three agent group of three agents picked group voting at random from a population scheme with a density p0 of A supporters and .1 p0/ for Probabilistic outcome: B. The result is now maybe Yes, maybe No probabilistic

Knowing the support for A in the whole population does not give the outcome

But a value of p0 D 30% yields p1 D 22% which becomes a likely event. It means that a majority of 70% has only a 78% chance of winning the election. The situation gets much worse for the current majority for the minority A support in the vicinity of 50%. There is an advantage of 5%, which is huge in a democratic voting system with a majority of 55%, that results in only 58% to obtain the presidency. The closer to p0 D 50%, the more the voting outcome is similar to the ﬂipping of a slightly imperfect coin. For instance, a majority of 52% wins only with 53%, in contrast to all agent voting where it wins at 100%. In other words, the more the two opinions have similar support, the more the outcome depends strongly on the selection of the three voters. At the other extreme, when the difference is very large, such as at 90%and10%, the majority wins at 97% making the result almost independent of the choices of the three voting agents (Fig. 13.7). 13.4 Fluctuations, Group Sizes, and Democratic Balance 229

Fig. 13.7 The result for one p 1 A vote in favor of from a 1 group of r agents picked at random from a population Separateur with a density p0 of A 50 % supporters and .1 p0/ for 0.8 B. Five different cases are shown for r D 5; 15; 35; 115; 1115.The 0.6 Deterministic result is now probabilistic for Bvictory Deterministic certain ranges of p0 and Avictory deterministic elsewhere. As seen from the ﬁgure, 0.4 increasing the value of r r=5 reduces the range of the r=15 r=35 values of p0 which yields a 0.2 r=115 probabilistic outcome r=1115

p 0 0.2 0.4 0.6 0.8 1

13.4 Fluctuations, Group Sizes, and Democratic Balance

Increasing the difference between the two supports thus reduces the fluctuation drive to undermine the expected democratic balance with the current majority winning the election. Or from another viewpoint, we notice that small differences in support of both opinions is highly enhanced by considering a group of only three agents. It produces an overamplification of the statistical fluctuations associated to the random selection of three agents. On this basis, it can be hinted that there may exist a path for damping such a fluctuation effect, in particular, around 50%. Clearly, making the whole population vote suppresses any fluctuation while they are huge for a group of three agents. It thus shows that we need to study in more detail the effect of the group size on the amplitude of the voting fluctuations. To achieve this study, the natural frame is to generalize the one voting group scheme from three agents to a number r of agents still randomly selected from the population. In the case of a group of r agents electing an A president, it is necessary to achieve in the group one of the following configurations frA; 0Bg, f.r1/A; 1Bg, .r 2/A; B . rC1 /A; r1 /B f g... , f 2 2 g. Adding the various associated probabilities to obtain such configurations yields the probability of p1 D Pr .p0/ to have an A elected by a group of r agents randomly selected from the population. The result r turns out to be the truncated binomial expansion of f1 C .1 p0/g with ! Xr r p P .p / pm.1 p /rm; 1 D r 0 m 0 0 (13.3) rC1 mD 2 230 13 Basic Mechanisms for the Perfect Democratic Structure

Table 13.1 The result of the single, random group voting scheme is shown for respective sizes of r D 3; 5; 35; 115; 1115,andr D all and a wide variety of values p0 from 0:100 up to 0:900 p0 (Total p1; p1; p1; p1; p1; p1; support) r D 3 r D 5 r D 35 r D 115 r D 1;115 r D all 0.100 0.028 0.009 0.000 0.000 0.000 0 0.200 0.104 0.058 0.000 0.000 0.000 0 0.300 0.216 0.163 0.006 0.000 0.000 0 0.400 0.352 0.317 0.114 0.015 0.000 0 0.450 0.425 0.407 0.275 0.141 0.000 0 0.470 0.455 0.444 0.360 0.259 0.022 0 0.490 0.485 0.481 0.453 0.415 0.252 0 0.500 0.500 0.500 0.500 0.500 0.500 0.500 0.510 0.515 0.519 0.547 0.585 0.745 1 0.530 0.545 0.556 0.640 0.741 0.978 1 0.550 0.575 0.593 0.743 0.859 1.000 1 0.600 0.648 0.683 0.886 0.985 1.000 1 0.700 0.784 0.837 0.994 1.000 1.000 1 0.800 0.896 0.942 1.000 1.000 1.000 1 0.900 0.972 0.991 1.000 1.000 1.000 1

r rŠ where m mŠ.rm/Š is a binomial coefficient. We can now estimate the size effect on the discrepancy between the outcome of an election using Pr versus Pall which always yields a deterministic result. Some cases of different values of r are shown in Fig. 13.7 using (13.3). For instance, for p0 D 30%wegetp1 D 16% to win for a group of five agents against p1 D 22% for the former case of three agents. It falls down to p1 D 1% for a group size r D 35 and reaches p1 D 0:00% already at a size of r D 39.The notation 0:00 means that it is not exactly zero but zero to within a precision of two digits. In other words, for a population of any size where two competing opinions score supports of respectively 30%and70% a randomly selected group of only 39 agents yields the same outcome of electing a candidate from the majority. It thus appears that the single random group voting system can substitute itself for the whole population voting system provided its size is so fitted to recover a deterministic result according to the rule that an increased support in a minority opinion can be damped by increasing the sample voting group size. Table 13.1 shows that for any value of p0 0:400 there exists a reasonably small value of r which guarantees the recovery of the voting outcome of whole population voting. Such an equivalence holds even for p0 D 40% where the probability to win the election at a size of r D 35 is p1 D 11% while it goes down to p1 D 1%atasize of r D 115 and eventually to p1 D 0:00%forasizeofr D 167. Moreover, to reach p1 D 0:000 asizeofr D 1;115 is required. Yet, to gather 1;115 agents is certainly a much easier task than to require a few hundred thousand agents or more to actually go and vote. 13.5 Limits of the Single Group Voting Scheme 231

Fig. 13.8 Increasing the size Increasing the of the random voting group size of the r D 3 from produces a range random selected p For some range of A of values of 0 where the agent group support: probabilistic outcome becomes voting scheme deterministic as with the outcome -> maybe Yes, whole population voting maybe No group. Elsewhere, the outcome is still probabilistic For other range of A Knowing the support support: deterministic for A in the whole outcome -> Yes or No population can yield the outcome for some range of values

If using single, small size group voting appears to preserve the whole voting procedure as long as p0 <0:47or p0 >0:53, the situation becomes more tricky in the vicinity of 50% as seen from Table 13.1 in the range of 0:470 < p0 < 0:530, much larger sizes r are required to recover the probability of p1 D 0:00.For instance, with p0 D 49%, the probability to win the election is p1 D 25%atasize of r D 1;115. It reaches p1 D 0:00% only from a size of r D 16;151. On this basis, the feasibility of the single random group voting scheme becomes questionable when the respective support comes closer to one another near the democratic threshold of 50%. Nevertheless, while dealing with very large popu- lations of the order of millions, a group consisting of tens of thousands of agents is still much smaller and easier to achieve (Fig. 13.8).

13.5 Limits of the Single Group Voting Scheme

However, should the single random group voting scheme be put into effect, it would face two practical problems. The ﬁrst one is that the actual value of p0 is not known, although it may be estimated using polls within certain error bars. If p0 could be determined exactly at the time of the actual voting, then it could be directly used to designate the winner. The existence of incompressible error bars imposes the performance of the measurement from a real voting process. Moreover, many election outcomes with totally unexpected results with respect to the poll predictions have proven that one cannot give up the actual voting. In addition, the required corresponding size of r depends on the p0 value. And since p0 varies over a timescale that could be short, the determination of the right size guarantees that the democratic balance is subject to errors. But the second practical problem is much more serious. It deals with the random character of the selection of the r agents who make up the few members of the voting group. Any correlation in the choice could shift the outcome in favor of either one of 232 13 Basic Mechanisms for the Perfect Democratic Structure the opinions, making the operating scheme null and void in particular with respect to its capability to reproduce the same result as with the all agents voting scheme. On this basis, if such a procedure of electing a president was to be adopted in the cases where it was feasible, the technical selection of the group voting members ought to be perfectly random. But to validate the totally random character of a selection of r agents from a much larger population is far from being easy to achieve. Even using a computer, the production of random numbers requires complicated algorithms. Up to date, the best series obtained are not totally random. What this means is that a correlation exists on a horizon of some length intervals although these can be rather long. Using them could provide an adequate frame to practically achieve the single random group voting scheme. In place of selecting randomly a number r of agents from the whole population, it could be easier to determine the current value of p0. Extended polls could make this possible. Once p0 is known, the associated value of r can be estimated. To dampen the poll error bars, a larger value of r can be chosen. Once a p0 and a r are in hand, the probability p1 can be evaluated from (13.3). The last step then reduces to the use of a random number generator to determine the opinion elected. The technique is called the Monte Carlo scheme and is widely used in numerical simulations. It is extremely powerful and yields very accurate results when applied, among others, to physical problems. It consists in choosing a random number between 0 and 1 using a random number generator. Its value is then compared to p1. In the case of p1 it is a B member that is designated. At this stage it appears that single random group voting may be a useful, efﬁcient and much lighter procedure for designating a president. It operates well provided that the difference between the two competing opinions is large enough. However, the democratic feasibility of the scheme becomes doubtful when both competing opinions get closer to the 50% marks for the respective supports. In this case, the single random group voting scheme produces a drastic softening of the democratic majority rule principle. It gives too much power to the minority with almost the same probability of having its candidate elected as the one from the majority at the expense of the legitimate right for this very majority to win the presidency. On this basis, noticing in todays democracies that the score differences between running candidates is most often less than a few percent invalidates the single random group voting prospect. It is unfortunate for our theoretical prospect but it is a fact.

13.6 Including Even-Sized Voting Groups

Up to now, to avoid the problem of not having a majority at a tie in an even sized group we have restricted the size of voting groups to odd values. This is without a loss of generality as is demonstrated now by also including the possibility of having even sized voting groups. 13.6 Including Even-Sized Voting Groups 233

In such a case of an even-sized voting group, the procedure is simply, in the case 1 of a tie, to elect either an A or a B with the probability of 2 . The symmetry between A and B is thus conserved and the corresponding results appear to be identical as for odd sizes. This can be proven mathematically. For instance, considering a group of size r D 4 we would have 4 3 2 2 p1 D P4.p0/ D p C 3p .1 p/ C 3p .1 p/ ; (13.4) where the last term accounts for ties with f2A; 2Bg. There exists six various tie configurations, giving three of them in favor of A and three in favor of B. Expanding 3 2 3 (13.4) yields p1 D2p C 3p which turns out to be equal to P3.p0/ D p C 3p 2.1 p/. We can repeat the demonstration for a size of r D 6 for which 6 5 4 2 3 3 p1 D P6.p0/ D p C 6p .1 p/ C 15p .1 p/ C 10p .1 p/ (13.5) with now 20 tie configurations, half of them contributing to the election of an A as 5 4 3 seen in the last term. Expanding (13.5) yields p1 D 6p 15p C 10p which turns 5 4 3 2 out to be equal to P5.p0/ D p C 5p .1 p/ C 10p .1 p/ . We have thus shown that even-sized groups of six and four agents are identical to odd-sized groups of respectively five and three agents when the tie contributions for even sizes are equally associated to each opinion. Generalizing the voting function Pr from (13.3) to include any size r of both odd and even values yields the formula ! ! r X n h io r r m rm r r r r P .p / p .1 p / kı N p 2 .1 p / 2 ; r 0 m 0 0 C 2 2 r 0 0 r 2 mDNŒ2 (13.6) r rŠ k 0 k 1 NŒx x where m mŠ.rm/Š, is a real with , Integer part of and ıfxg is the Kronecker function, i.e., ıfxgD1 if x D 0 and ıfxgD0 if x ¤ 0. The presence of k in the second part of (17.1) allows the distribution of the vote in any proportion from an even-sized group to a tie between both political parties A and B. Accordingly, in a local group of any even size at a tie, there exists a probability k of electing an A representative, and a probability .1 k/ of electing a B representative. At this stage of the presentation, no advantage is given to either one of the two competing political parties and in turn we restrict the tie case to an 1 equal contribution to both A and B by taking k D 2 . r r NŒr 1 From (17.1), any odd-sized value of yields f 2 2 gD 2 making the Kronecker function always equal to zero, which in turn cancels the last term and thus recovering (13.3). Indeed, for odd sizes there exists no tie. On the contrary, any even r r r r r 2 r size makes f NŒ gD0 with the last term contributing as k r p0 .1 p0/ 2 . 2 2 2 In a more readable manner, (13.6) can simply be written in two different expressions as ! Xr r P .p / pm.1 p /rm r 0 D m 0 0 (13.7) rC1 mD 2 234 13 Basic Mechanisms for the Perfect Democratic Structure for odd sizes of r,and ! ! r X r r m rm r r P .p / p .1 p / k p 2 .1 p / 2 ; r 0 m 0 0 C r 0 0 (13.8) r 2 mD 2 C1 when r is even. The above identity between groups of respective sizes of 3, 4, 5, and 6 can now be extended to any pair of odd size r D 2w 1 and the associated even size r D 2w where w is an integer. Using (17.1), it can be demonstrated that P2w1.p0/ D P2w.p0/ for any integer w. This means for instance that it is the sizes r D 3 and r D 4 that are identical and not r D 4 and r D 5 under the k 1 constraint D 2 . Nevertheless, it is worth mentioning that even sizes will be studied on their own later since they allow very interesting developments when the symmetry between A and B is broken in a local group at a tie favoring one opinion at the expense of the other. In this case, we lose the identity P2w1.p0/ D P2w.p0/.

13.7 Setting Up the Perfect Democratic Structure

If the above analysis has led to the discarding of the single random group voting scheme, it nonetheless opens the way to a novel vision of organizing the perfect democratic structure. Indeed, we found that to consider a voting group of three randomly selected agents yields a probability of p1 p0 if p0 > 2 . This shows that the process either shrinks or inﬂates the initial value p0 of the A supporters. Since to produce a democratic scheme we aim at a deterministic outcome so as to guarantee the presidency to the current majority, the variation from p0 to p1 hints at extending the single three agent voting scheme to constitute many such groups. However, several such groups voting simultaneously do not result in the designation of one president but in a series of elected local representatives as shown in Fig. 13.9. All these elected representatives constitute a new population of agents where the difference in proportions of support for respectively A and B opinions has now been shifted from .p0;1p0/ within the initial population to .p1;1p1/. The associated amplitude of the difference in respective support for A and B turns out to be always increased from any value d0 j p0 .1 p0/ jDj 2p0 1 j to d1 j 2p1 1 j with 1 d . d 3 3d /; 1 D 2 0 C 0 (13.9) making d1 >d0 whatever is p0.SeeFig.13.10. Following this observation, looking at Fig. 13.10 suggests repeating the above process to constitute a new population of higher representatives where once again the difference in respective support will be increased. We thus repeat the previous 13.7 Setting Up the Perfect Democratic Structure 235

Fig. 13.9 Many single voting groups of three agents B A A A are constituted. Each one A B B B elects a representative. All A A A A B A A elected representatives B B constitute a new population. B B B B A The pyramid is shown from A A the bottom down to the top A B A

B B B A A A B AA New population of elected representatives

Fig. 13.10 The amplitude of d 1 the difference between the 1 respective support for both opinions using single voting groups of three agents. Any initial difference d0 is 0.8 ampliﬁed to d1 by the local votes. The value d0 D 0 is a Perfect ﬁxed point but any small 0.6 departure from it is Equality One opinion automatically increased by the simultaneous voting of all has disappeared local groups. After several 0.4 iterations of local voting, the collective vote eventually reaches 100% for one opinion 0.2

d 0 0.2 0.4 0.6 0.8 1

process, now using the elected agents. Accordingly, new groups of three agents are formed at random and then each one elects a new higher ranking representative. A second subpopulation is constituted where respective support for A and B is now .p2;1 p2/ obtained from .p1;1 p1/ instead of .p0;1 p0/ in (13.2). The associated new difference in support becomes from (13.9), 1 d . d 3 3d / 2 D 2 1 C 1 (13.10) with d2 >d1 >d0. 236 13 Basic Mechanisms for the Perfect Democratic Structure

Ground people

B A A A A B BB A A A A B A A BB BB BB A A A A B A

B BB A Representatives A A B A A First level

A B B B A B Form new groups A A A

B AA Second level

President A

Fig. 13.11 A democratic bottom up hierarchy with nine groups of three agents at the ground level totaling 27 agents. The ﬁrst level has three groups of three agents giving nine elected representatives. The second level has three agents. Adding all the agents involved and the elected president yields 27 C 9 C 3 C 1 D 40 agents. The result may be probabilistic depending on the value p0. Here, we have 15A (in red)and12B (in blue) at the bottom level. At the ﬁrst level, it has 5 A versus 4 B. The second level has 2A and 1B that elects a president A. The pyramid is shown from the bottom down to the top

In the case where we decided to stop the process of iteration with the above two levels of elected representatives, in order to have the president elected from the second level, the number of corresponding agents is restricted to three. In turn, this implies having only nine agents underneath at the first level distributed within three groups. These nine agents have to be elected from nine groups of three agents each. The corresponding 27 agents are ground agents who have been randomly selected from the population as shown in Fig. 13.11. By so doing, we have built a simple democratic hierarchy with two levels in between the population and the president as seen in Figs. 13.12 and 13.13.Atthe ground level, where the first nine groups of three agents are formed, the probability of finding an A-supporter is p0. At the first hierarchical level, the probability of having a representative who supports A among the nine agents is p1. It becomes p2 at the second level to eventually reach p3 for the probability of having an A president elected. However, for many ranges of values of p0, p3 is neither zero nor one. Depending on the initial value p0 it may well be at a value between zero and one, making the presidential election probabilistic. For instance, starting from an initial p0 D 0:40 we get successively p1 D 0:35, p2 D 0:28 and p3 D 0:20 as seen in Fig. 13.11.But an initial p0 D 0:20 would give successively p1 D 0:10, p2 D 0:03 and p3 D 0:00. 13.7 Setting Up the Perfect Democratic Structure 237

In other words A pyramid to power

President p3 X

X XX p2

XXX XXX XXX p1

XXX XXX XXX XXX XXX XXX XXX XXX XXX p0

Ground people 27+9+3+1=40 persons

Fig. 13.12 The same two level democratic bottom up hierarchy as in Fig. 13.11 with 15A (in red) and 12B (in blue) at the bottom level. But the distribution at the ﬁrst level is slightly different with now 6A versus 3B. However, at the second level the same conﬁguration with 2A and 1B is obtained. The elected president is, therefore, again A

President: Level 3

Level 2

Level 1

Bottom: Level 0

Fig. 13.13 Another schematic view of the precedent democratic bottom up hierarchy of Fig. 13.12 with now a slightly different distribution of agents at the bottom: 14A (in red)and13B (in blue). Higher up it is identical in numbers but not in geographical positions. The result may still be probabilistic depending on the value of p0 238 13 Basic Mechanisms for the Perfect Democratic Structure

An illustration President A p6 = 0.00

p5 = 0.03

p4 = 0.10

p3 = 0.20

p2 = 0.28

p1 = 0.35 p0 = 0.40

1 President 3 The democratic balance is restored 9 with 6 levels 27 81 and 1093 persons 243 The ground 729

Small structure

Fig. 13.14 A democratic bottom up hierarchy with 243 groups of three agents at the ground level totaling 729 agents. The ﬁrst level is built out of 81 groups of three agents giving 243 elected agents. The second level has 27 groups of three agents giving 81 elected representatives. The third level has thus nine groups with a total of 27 agents. The fourth level shrinks to three groups of three below the ﬁfth level of one single group of three elected agents. Adding all of the agents involved and the elected president yields 729 C 243 C 81 C 27 C 9 C 3 C 1 D 1;093 agents. The result may turn probabilistic depending on the value p0

So although the hierarchy is indeed democratic in the sense that all votes are bottom up and obey a majority rule, it often violates the democratic balance since, for instance a minority of 40% may well win the presidency at a rate of one fifth with p3 D 0:20. On the contrary, an initial minority of 20% is certain to lose the presidential election. The democratic voting machine appears to be democratic for a certain range of respective support for A and B and somehow dictatorial for another range. Dictatorial here means giving the presidency to the minority at the expense of the existing majority. However, the above figures hint at a way of reestablishing the democratic balance. Increasing the number of levels will mechanically produce a further increase, level by level in the amplitude difference in the respective support so as to eventually reach the democratic limit of one, at which point the president is elected in a deterministic manner. Deterministic means here that the knowledge of the respective bottom supports for A and B allows the exact prediction of the winning policy, but only at the level of the president. In terms of the amplitude in the difference of the support, it means that increasing the number of hierarchical levels eventually produces an amplitude of one as shown in Fig. 13.10. For instance, going back to the series p0 D 0:40, p1 D 0:35, p2 D 0:28,andp3 D 0:20, we find that repeating the voting process yields p4 D 0:10, p5 D 0:03,andp6 D 0:00 as illustrated in Fig. 13.14. Thus, building 13.7 Setting Up the Perfect Democratic Structure 239

p 1 p 1 1 1

Separator Separator 50 % 50 % 0.8 0.8

Deterministic Deterministic 0.6 0.6 B victory Deterministic B victory Deterministic

A victory A victory 0.4 0.4

0.2 0.2

p 0 p 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 13.15 The variation of the density in elected representatives from the bottom of the hierarchy up through various levels. In the left part of the figure, an initial proportion of p0 D 0:40 is shown to shrink to respectively p1 D 0:35 at the first level of the hierarchy, p2 D 0:28 at the second level, and p3 D 0:20 at the third one. The parallel evolution for the current majority is also shown with the series .1p0/ D 0:60, .1p1/ D 0:65, .1p2/ D 0:72 and .1p3/ D 0:80.Theright part of the figure shows the same situation as above but now with two additional levels in the hierarchy which ensure the democratic balance with on the one hand p4 D 0:10, p5 D 0:03 and p6 D 0:00, and on the other, .1 p4/ D 0:90, .1 p5/ D 0:97 and .1 p6/ D 1:00. It is seen that an initial proportion p0 >0:40would not achieve the democratic balance within the existence of only five levels between the bottom and the president

ﬁve successive hierarchical levels between the bottom and the top guarantees the democratic balance at the top with a president belonging to the majority, together with a distribution of local powers to the minority, which has a support of 40% within the population as shown in Fig. 13.15. The associated amplitudes in the difference in respective support are respectively d0 D 0:20, d1 D 0:30, d2 D 0:44 and d3 D 0:60, d4 D 0:80, d5 D 0:94,and d6 D 1:00. However, within ﬁve hierarchical levels an initial value p0 >0:40 would again yield a probabilistic outcome for the presidential election at the top level with p0 ¤ 0:00. The building of our democratic bottom-up hierarchy is thus found to systematically amplify the initial majority at the bottom step by step while climbing up each new higher level within the hierarchy. Therefore, given any initial value of p0 at the bottom, there always exists a series of successive n bottom-up levels i D 1; :::; n characterized at each level by a proportion pi D Pr .pi1/ of A elected representatives. The associated series p0, p1..., 1 1 pn obeys either p0 >p1 > ::: > pn if p0 < 2 or p0 2 . To support this statement, Table 13.3 shows the bottom-up evolution of an initial p0 D 0:49 for a series of group sizes of r D 3; 5; 7; 9; 11; 13; 15. This reveals that the successive proportions of A elected representatives always decreases quite rapidly to reach zero within only a small number nc;r of levels. More precisely, we have nc;r D 11; 8; 6; 5; 5; 5 for respectively r D 3; 5; 7; 9; 11; 13; 15. 240 13 Basic Mechanisms for the Perfect Democratic Structure

This data hints at the fact that there always exists some ﬁnite number nc;r such p 0 p a< 1 p 1 a> 1 that nc;r D for 0 D 2 . Similarly, by symmetry, when 0 D 2 ,i.e., 1 p 1 at the same distance from 2 ,weget nc;r D . Indeed, on one of the extreme sides of the smallest voting group of three agents, we found that nc;r is rather a small number. On the other extreme side, using the larger possible group r D all,wehave nc;r D 1. On this basis, noticing that by nature the value of nc;r decreases with an increasing size in r, we infer that nc;r is always rather a small number, which in turn makes our scheme of building such a bottom-up democratic hierarchy a realistic hypothesis.

13.8 The Dynamics Driven by Repeated Democratic Voting

To go from the numerical evidence of the existence of nc;r to the proof itself, we study the properties of the dynamics of (13.6). The first step is to single out the existence of eventual fixed points p of the voting function from the equations Pr .p / D p . k 1 Within the symmetric frame D 2 ,(13.6) reduces to (13.3) as noticed earlier. 1 This yields the three unique solutions p D 0; 2 ;1. We denote them respectively pB;pc;pA. The first one pB D 0 means that if the probability to have an A elected is zero, the voting process will not produce spontaneously elected A representatives, keeping the probability p equal to zero. At the other extreme, at pA D 1 only A 1 agents are elected at any vote. In between, at pc D 2 the probability of having an A elected is equal to the probability of having a B elected. Any vote will preserve this equality. At the two fixed points pB D 0 and pA D 1, any election is a deterministic event with a hundred percent prediction of the outcome. Of course, for a group of size r, the voting function Pr .p0/ is a polynomial of degree r which has r fixed points. However, pB;pc;pA are the only ones which satisfy the “physical constraint” 0 p 1. The next step to determine the properties of the dynamics of (13.6)istoevaluate the respective stabilities of the above three fixed points pB;pc;pA. This is achieved using the standard technique of Taylor expansions around the respective fixed points. Indeed, expanding the voting function pn D Pr .pn1/ given by (13.6) around the unstable fixed point p and using Pr .p / D p yields at the first order,

pn p C .pn1 p /r ; (13.11)

dPr .pn1/ r .p / p where dpn1 j . Rewriting it as

.pn p / .pn1 p /r (13.12) 13.8 The Dynamics Driven by Repeated Democratic Voting 241

shows that if r <1, the voting process decreases the distance .pn1 p / from the ﬁxed point p since r <1H) .pn p /<.pn1 p /. This occurs whenever pn1

p. Therefore, a vote from one level up to the following one drives the proportions of elected A agents closer to the fixed point value p. Accordingly, the corresponding fixed point is stable, i.e., if the proportion p deviates from p, the voting dynamics brings it back to it. A stable fixed point attracts the voting dynamics. This is called an attractor of the dynamics. In the opposite case, when r >1, the voting process increases the distance .pn1 p / from the fixed point p since r >1H) .pn p />.pn1 p /. Whenever pn1

p, the dynamics puts the proportions of the elected A agents further from the fixed point value p. Any deviation from the fixed point p is enhanced by the voting dynamics. It is an unstable fixed point. However, the moving away proceeds in the same direction as the initial deviation from the unstable fixed point. As such it is a separator of the dynamics since it delimits two different and well separated regions within the dynamics. Accordingly, a fixed point p characterized by r <1is an attractor of the dynamics while a fixed point p characterized by r >1is a separator of the dynamics. To apply the above criteria to the fixed points of (13.3), we calculate explicitly r .p / with respectively p D pB;pc;pA. For the two extreme values pB D 0 and pA D 1, we found r .pB/ D r .pA/ D 0 making both fixed points stable. At the 1 middle value fixed point pc D 2 ,weget ! 1 Xr r r .p / D .2m r// ; c 2r1 m (13.13) rC1 mD 2 which can be reduced to ! r r 1 r .p / D : c 2r1 r1 (13.14) 2

.p />1 r 3 .p / 3 Equation (13.14q ) yields qr c for any size from D with 3 c D 2 to .p / 2 p r 2r r 1 r c ! 2 for as seen in Fig. 13.16 and Table 13.2. .r 3 / Using the above results r .pB/ D r .pA/ D 0 and r .pc/>1, the ﬂow diagram of the dynamics driven by repeated voting can be constructed as shown in Fig. 13.17. It exhibits two attractors located at respectively pB D 0 and pA D 1, the separator 1 1 being located at .pc/ D 2 . An initial support p0 > 2 leads toward an election at 100 p < 1 % provided enough voting levels are present. When 0 2 , it leads toward an election at 0% for the A politics. Before closing this section, it is worth noticing that some limit should be added to (13.11) to ensure that pn, which is a probability, stays positive from pn1.To 242 13 Basic Mechanisms for the Perfect Democratic Structure

lambdar,c lambdar,c 2.75 25

2.5 20 2.25

2 15

1.75 10

1.5

5 1.25

r r 4 6 8 10 200 400 600 800 1000

.p / r .p / D 3 Fig. 13.16 The variation of r c as a function of the size . It starts from 3 c 2 and is greater than 1 for any value of r, always making the associated ﬁxedq point unstable. The thick line on the left side is (13.14) with the limits for large values of r being 2 p r above the line and .r 2 / q 3 2r .p / r below. The right side shows r c forlargervaluesof

Table 13.2 The variation of r .pc/ as a function of the local group size with r D 3; 5; 7; 9; 11; 13; 15; 101; 1001 p D 1 . It is always larger than 1, making the ﬁxed point c 2 unstable Groupsize r D 3rD 5rD 7rD 9rD 11 r D 13 r D 15 r D 101 r D 1;001

r .pc/ 1.5 1.87 2.19 2.46 2.71 2.93 3.14 8.04 25.25

Separator: 1/2

0 1

B elected A elected

p > 1 Fig. 13.17 The density of elected representatives flow diagram. An initial support 0 2 leads 100 p < 1 0 to an election at % while 0 2 leads to an election at %. The respective ranges of attraction p D 0 p D 1 .p / D 1 of the two attractors located at B and A are delimited by the separator c 2 avoid a negative probability, we define the range for pn1, which ensures pn 0 by solving (13.11)inpn1 with pn D 0.Weget, 1 pn1 pc 1 ; (13.15) r which thus defines the validity of (13.11). This condition is a function of r as shown in Fig. 13.18 where the gray area defines the domain of the allowed pn1. 13.9 Some Comments About Zero 243

Fig. 13.18 The limit of p n-1 validity of the Taylor 1 p D 1 expansion around c;r 2 as a function of r.Theupper dark area ﬁts to the condition on pn1 to ensure pn 0 0.8

0.6

0.4

0.2

r 20 40 60 80 100

13.9 Some Comments About Zero

At this stage, it is worth stressing that when quoting pnc;r D 0 and pnc;r D 1, zero and one are not to be taken as the exact integer values but as approximate values, deﬁned with a certain number of digits, here in most of our calculations, to within two digits. A numerical precision of two digits means that there exists an extremely small possibility of having the elected president belonging to the minority. For instance, the above illustration with p0 D 0:40 yields p6 D 0:00222. The notation 0:00 means a possible counterdemocratic outcome with a chance of 2 out of 1,000 votes. To move from a two digit precision to a three digit precision requires increasing the hierarchy by adding only one subsequent level. Two additional levels 10 lead to a precision of p8 D 8:75 10 . Using (13.3), Table 13.3 gives the evolution of the probability of having an elected representative climb the hierarchy level by level for a representative whose associated politics has a support of p0 D 0:49 at the bottom level. A series of increasing voting group sizes with r D 3; 5; 7; 9; 11; 13; 15 are considered. Data are given within a two digit precision to account for the small initial difference in support d0 D 0:02. Table 13.4 gives the same results as Table 13.3 still using (17.2) but with a four digit precision. It starts from an initial value of p0 D 0:4995 at the bottom level, which allows accounting for a very small difference in the initial supports of d0 D 0:001. The total numbers of respective levels to reach the democratic balance are in fact not too far apart from the numbers from Table 13.4 since for such a value p0 D 0:4995, a two digit precision would lead to the conclusion of an equality with p0 D 0:50. Most of the discrepancies come from the fact that the increase in the 244 13 Basic Mechanisms for the Perfect Democratic Structure

Table 13.3 The evolution of the probability of having a representative elected climbing the hierarchy level by level for a representative whose associated politics has a support of p0 D 0:49 at the bottom level nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 0 0.49 0.49 0.49 0.49 0.49 0.49 0.49 1 0.48 0.48 0.47 0.48 0.47 0.46 0.48 2 0.47 0.46 0.44 0.45 0.43 0.39 0.45 3 0.46 0.42 0.38 0.39 0.31 0.21 0.35 4 0.43 0.36 0.25 0.25 0.09 0.01 0.11 5 0.40 0.25 0.08 0.05 0.00 0.00 0.00 6 0.36 0.10 0.00 0.00 0.00 7 0.29 0.01 8 0.21 0.00 90.11 10 0.03 11 0.00 The hierarchy is built in order to ensure the election of the president from the other politics which scores the majority of 0:51 at the bottom. Depending on the local group sizes r D 3; 5; 7; 9; 11; 13; 15 less levels are necessary to guarantee the democratic balance. Data are obtained using (13.3) initial difference while climbing up the hierarchy driven by the voting process is very slow, in the vicinity of 50%. This detail will appear explicitly in the following section. 13.9 Some Comments About Zero 245

Table 13.4 The evolution of the probability of having a representative elected climbing the hierarchy level by level for a representative whose associated politics has a support of p0 D 0:995 at the bottom level nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 0 0.4995 0.4995 0.4995 0.4995 0.4995 0.4995 0.4995 1 0.4992 0.4990 0.4989 0.4987 0.4986 0.4985 0.4984 2 0.4988 0.4982 0.4976 0.4969 0.4963 0.4957 0.4950 3 0.4983 0.4967 0.4947 0.4925 0.4900 0.4873 0.4844 4 0.4974 0.4938 0.4885 0.4816 0.4731 0.4630 0.4513 5 0.4962 0.4884 0.4749 0.4549 0.4277 0.3928 0.3505 6 0.4943 0.4782 0.4453 0.3903 0.3109 0.2127 0.1141 7 0.4914 0.4593 0.3819 0.2467 0.0912 0.0099 0.0000 8 0.4871 0.4240 0.2526 0.0464 0.0001 0.0000 9 0.4807 0.3598 0.0759 0.0000 0.0000 10 0.4711 0.2506 0.0009 11 0.4568 0.1041 0.0000 12 0.4354 0.0096 13 0.4036 0.0000 14 0.3572 15 0.2917 16 0.2056 17 0.1095 18 0.0333 19 0.0032 20 0.0000 The hierarchy is built in order to ensure the election of the president from the other politics which scores the majority of 0:5005 at the bottom. Depending on the local group sizes r D 3; 5; 7; 9; 11; 13; 15 less levels are necessary to guarantee the democratic balance. Data are obtained using (13.3) Chapter 14 Going to Applications

14.1 The Practical Scheme

Another meaningful representation of the evolution of the respective proportions in voting for A and B agents is to draw the proportions pn as a function of n as shown in Fig. 14.1 for various sizes of voting groups as illustrated in the figure. Starting from a certain initial value po,theseriesp1, p2, ..., pn is plotted for several values of r. Increasing r reduces the number of levels necessary to reach the democratic balance within a fixed precision. It is seen once more how a proportion of support that is lower than the democratic threshold of 50% shrinks quite quickly while climbing up the hierarchy. Moreover, small voting groups with a size of 3; 5; 7 require between five and ten levels to leverage to almost zero 48% of support at the bottom. In contrast, a larger group of size 19 achieves the goal within only four levels, the fourth being the president. It is worth stressing that using larger voting groups does not make a real difference in terms of the number of levels. As seen from the Fig. 14.1, increasing the group size from r D 19 to respectively r D 49; 99; 199 yields very close curves. These results are consistent with the data shown in Table 13.1. To formalize the above finding and to ensure the building of our perfect 1 democratic bottom-up hierarchical structure we now proceed, given p0 < 2 ,to analytically calculate the critical number of levels nc;r,atwhichpnc;r D with being a very small number. The value of determines the level of confidence of the prediction to totally satisfy the democratic balance, i.e., having an elected president who belongs to the majority at the bottom. As is seen from Tables 13.3 and 13.4, the smaller is , the larger is the corresponding pyramid since more levels are involved. However, if dealing with some is necessary while doing numerical calculations since the proportions are real numbers, when performing analytical calculations we can consider the integer value of zero. In other words, starting from a certain case 1 p0 < 2 for a pyramid using local voting groups of size r, we can formally evaluate n p 0 the number c;r of hierarchical levels at which nc;r D . The upper index means

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 247 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 14, © Springer Science+Business Media, LLC 2012 248 14 Going to Applications

Probability p0

0.4 3

0.3 7

0.2 49

0.1 199

Levels 0 2 4 6 8 10

Fig. 14.1 The evolution of the probability to be elected at the various levels of the bottom-up democratic hierarchy starting from a support of p0 D 0:48 at the bottom for a series of group sizes r D 3; 5; 7; 19; 49; 99; 199.Thedashed lines represent the larger groups r D 49; 99; 199 and show very little difference between them. In all the cases, less than ten levels are enough to recover the democratic balance with a zero probability to have the president elected for the minority even if it scores 48%

that p0

n .pn pc;r/ .p0 pc;r/r : (14.1)

n n p 0 At D c;r, nc;r D turning (14.1)to

nc;r pc;r r ; (14.2) pc;r p0 from which, taking the Logarithm on both side, we obtain, 1 1 n : c;r ln p0 (14.3) r 1 ln pc;r

1 When p0 >pc;r D 2 , from symmetry we get 1 1 nC ; c;r ln p0 (14.4) r 1 ln pc;r 14.1 The Practical Scheme 249

e Table 14.1 Exact numerical estimates nc;r for nc;r using (17.2) and the values obtained from (14.5) for the bottom value p0 D 0:49 as a function of the series of voting group sizes of r D 3; 5; 7; 9; 11; 13; 15

p0 D 0:49 r D 3rD 5rD 7rD 9rD 11 r D 13 r D 15 e nc;r 12 8666 5 5 nc;r 9:65 6.22 4.99 4.34 3.93 3.64 3.42 nNc;r 11 8665 5 5 Values from (14.6)arealsogiven

C where we used p C D 1 instead of pn D 0. The upper index means that p0 > nc;r c;r pc;r. Equations (16.8) and (16.7) combine into the single equation

1 1 n ; c;r ln p0 (14.5) r 1 ln j pc;r j which exhibits a logarithmic singularity at the unstable fixed point separator pc;r. Such a singularity shows up the fact that when at the hierarchy bottom p0 is very close to pc;r, many voting levels are necessary to increase the initial small difference in respective supports for A and B as was already noticed in the numerical illustrations given earlier. Table 14.1 compares the exact numerical estimates for nc;r obtained by successive iterations of (17.2) to the values obtained from (14.5) in the case p0 D 0:49 as a function of the series of voting group sizes of r D 3; 5; 7; 9; 11; 13; 15. From Table 14.1 two remarks arise. Firstly, the number of levels being by nature an integer, (14.5) should be modified accordingly. Secondly, it can be noticed that to have the value of nc;r given by (14.5) to match exactly, besides the value of r D 3, e to the exact estimates nc;r, it is sufficient to take the integer part of (14.5)andtoadd C2 to the result, leading to the effective expression " # 1 1 n N 2: Nc;r ln p0 C (14.6) r 1 ln j pc;r j

It is worth emphasizing that from the C2 addition, one C1 is the consequence of having to take the integer part from nc;r while the second C1 contribution is an ad hoc fit by hand. Figure 14.2 illustrates the dependence of nNc;r and nc;r for the two voting group sizes r D 3 and r D 15 as a function of 0 p0 1. The stair functions correspond 1 to nNc;r. Both curves show a singularity at the unstable fixed point pc;r D 2 separator, which means that in the very vicinity of 50% many voting levels are required to significantly enlarge the difference in respective supports while climbing up the hierarchy. Much fewer stairs appear for the larger size of r D 15 in agreement with the limiting case of r D all, which yields nc;r D 1. 250 14 Going to Applications

a a n c ,nc n c ,nc 20

15 6

10 4

5 2

p p 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0

Fig. 14.2 The variation of nNc;r and nc;r for the two voting group sizes r D 3 (left part)andr D 15 (right part) as a function of 0 p0 1. The stair functions correspond to nNc;r. Both curves have p D 1 a logarithmic singularity at the unstable ﬁxed point c;r 2 separator. Much fewer stairs appear for the larger size r D 15

14.2 The Physicist’s Corner: Trying to Be a Little More Mathematical (To Be Omitted by Lovers of Simplicity)

Physicists are used to making the kind of ad hoc “arrangement” we did above to pass from nc;r to nNc;r in order to obtain a simple formula with a minimum working effort, which yields rather satisfactory results. In physics, at a later point in time, a mathematical physicist will then eventually grasp the problem and derive a more rigorous formulation. This sort of event does not raise much excitement among the large majority of physicists since, as was emphasized in the first chapter, they don’t care too much about the rigorous aspects of the mathematics. What they do care for is the practical power obtained by playing around with the mathematics. Nevertheless, to illustrate our statement and for a kind of formal fun, we now present an illustration of the way in which a researcher may go about trying to derive a more accurate analytical calculation of nc;r. As will appear at the end of this section, our mathematical tribulations will prove to be useless, but they constitute a very good initiation to the way in which research works. But the reader who dislikes splitting hairs would find it boring to go through so many formulae and figures for nothing. I advise her or him to just move on to the next section, this one having nothing to do with sociophysics in itself. At this point of our analysis, it appears that the above Taylor expansion around 1 the unstable fixed point pc;r D 2 separator approximates the curve pn D Pr .pn1/ given by (17.2), by the straight line pn D p C.pn1p /r obtained from (13.12). It is a line which is by construction a tangent to the curve at the fixed point pc;r as 14.2 The Physicist’s Corner: Trying to Be a Little More Mathematical 251

p n p n 1 1

Separateur Separateur 50 % 50 % 0.8 0.8

Deterministic Deterministic 0.6 0.6 B election B election

0.4 0.4

0.2 0.2

p p 0.2 0.4 0.6 0.8 1 n-1 0.2 0.4 0.6 0.8 1 n-1

Fig. 14.3 The curve pn D Pr .pn1/ from (17.2). The left part corresponds to the case r D 3 with its two graphical approximations. The straight line given by (13.11) below the curve comes from p D 1 the Taylor expansion of (17.2) around the separator c;r 2 . The parabolic curve given by (14.9) above the curve results from the expansion around the attractor pB D 0 The right part shows the same curve pn D Pr .pn1/ from (17.2) with r D 13. The discrepancy from the parabolic curve is larger than for r D 3 shown in Fig. 14.3 for the case r D 3. We can see that while it is quite close to the curve in the range 0:25 p0 pc;r, it starts to deviate from it when p0 <0:25. The deviation already starts for p0 <0:35in the case of r D 13. Accordingly, another Taylor expansion should be performed on the other side around the stable ﬁxed point pB D 0 attractor. Then, coupling it to the previous one around pc;r would optimize the process. However, as has been noticed above, r .pB D 0/ D 0, which thus requires the expansion to be started at a higher degree. But the following higher order derivatives also vanish at pB D 0 since (13.3) is a polynomial of degree r to the power of pn1. Therefore, the ﬁrst nonzero derivative at pB D 0 is the derivative at the order of the lowest degree of the polynomial, rC1 p v r which turns out to be v D 2 .Derivingv times the term n1 where v from (13.3) yields

.v/ ˇ d Pr .pn/ˇ .v/.p / ˇ r B .v/ d pn pB D vŠ; (14.7) which in turn allows us to construct the ﬁrst nonzero term of the Taylor expansion 1 .v/ v pn p C .pn1 p / ; (14.8) B vŠ r B 252 14 Going to Applications which reduces, substituting (14.7)andpB D 0,to

v pn pn1; (14.9) which is shown in Fig. 14.3 for the case of r D 3. It is interesting to note that we could have reached (14.9) directly by noticing that (13.3) is a polynomial of degree r to the power of pn1. This implies that around p 0 pv n1 D the leading term is the one which has the lowest degree, here being n1. r Its coefﬁcient being v we recover (14.9). However, the same property hints at the fact that this leading term will become insufﬁcient to mimic the curve behavior in the vicinity of 0 with an increasing higher degree of r since more and more terms are neglected as shown in the lower part of Fig. 14.3 with r D 13 as compared to the upper part with r D 3. Iterating (14.9) once leads to v v pn pn2

1Cv v2 D pn2 (14.10) and .n 2/ additional successive iterations lead to

1CvCv2C:::Cvn1 vn pn p0 n 1v n 1 v D v p0 ; (14.11) 1 2 ::: n1 1vn where we used C v C v C C v D 1v . Before proceeding, pointing out some of the restrictions in the use of (14.9)isin order since pB D 0 was found to be an attractor of the voting process. The condition pn

1 pn1 <1v : (14.12)

1 1 1 p < r 3 1v As seen in Fig. 14.4 it yields n1 3 for D with the limit ! 4 for r !1. We now calculate the number of levels ncc to reach pB D 0 at the top level from p0 at the bottom. However, from (14.11) it can be seen that we cannot put pn D 0 to vn extract some meaningful information since it would yield p0 D 0. Accordingly, we choose some to determine the level of conﬁdence of the “deterministic outcome” of the election at the top level as was already discussed earlier while we investigate some numerical data and discus the exact meaning of zero within our framework.

Putting pnc;r D in (14.11) leads to ! 1 1 1v n ln : cc D ln 1 (14.13) ln v 1v p0 14.2 The Physicist’s Corner: Trying to Be a Little More Mathematical 253

Fig. 14.4 The limit of p n-1 validity of the Taylor expansion around pB D 0 as a function of r.Thedark area 0.3 ﬁts to the condition 1 pn1 <1v 0.25

0.2

0.15

0.1

0.05

r 100 200 300 400 500

e Table 14.2 Exact numerical estimates nc;r for nc;r using (13.3) and the values obtained from (14.13) for the bottom value p0 D 0:25 as a function of the series of voting group sizes of r D 3; 5; 7; 9; 11; 13; 19

p0 D 0:25 r D 3rD 5rD 7rD 9rD 11 r D 13 r D 19 e nc;r 42222 2 1 ncc 3.94 2.66 2.21 1.98 1.83 1.74 1.56

Table 14.2 compares the results given by (14.13) and the exact estimates obtained from successive iterations of (17.2) for the bottom value p0 D 0:25 as a function of the series of group voting sizes r D 3; 5; 7; 9; 11; 13; 19. The results are reasonably good, but here we do not revisit ncc to improve them 1 as we did for the expansion around 2 by deriving nNc;r in (14.6). To use ncc in place of nNc;r would mean to apply it for initial p0 values around 0:40, which is larger than 1 1 its upper limit of validity between 3 and 4 given by (14.12). Therefore, we need to p < 1 p 0 combine (14.9) and (13.11) to successive iterations from 0 2 down to nc;r D via an intermediate value pi . Equation (13.11) is used ﬁrst down to some value pi , and then (14.9) is used from pi down to pnc;r D 0. To optimize the combination we determine the value pi which minimizes the distance between the two curves, which are located respectively on each side of the curve pn D Pr .pn1/ as seen in Fig. 14.3. This difference can be written

v fp C .pi p /r gpi ; (14.14) 254 14 Going to Applications

Fig. 14.5 The limit of 1 validity of the Taylor p D 1 expansion around c;r 2 as a function of r with pn1 belonging to the upper dark 0.8 area.Thelower gray part p D 1 below n1 4 deﬁnes the domain where the second Taylor expansion around 0.6 pB D 0 should be used. The white part in between is where there exists a difﬁculty 0.4

0.2

5 10 15 20 25 30

which yields upon derivation with respect to pi the optimum value

1 r v1 pi D v 1 ; D 4 (14.15) where (13.14) was used to substitute for r. The optimum is thus a constant that is independent of r. Somehow the result of (14.15) is intuitive since it splits the 1 0 pn1 2 exactly in the middle. p 1 However, the value i D 4 raises a difficulty. We saw in the determination of 1 pn1 p .1 / the range of validity of (13.11) that it must fulfill the condition c r given by (13.18). The associated domain of validity for pn1 is shown in Fig. 13.18. 1 Adding the constraint to use (13.11) down to pn1 D 4 creates a whole domain where indeed (13.11) is not valid as shown in the white part of Fig. 14.5, while indeed it already appears from r D 7. The alternative for overcoming this difficulty is to shift from (13.11) to (14.9)at 1 1 pn1 > p .1 / a value of 4 , i.e., in the vicinity of c r given by (13.18). However, as seen from the lower part of Fig. 14.3 for the case r D 13 there exists some range of values of pn1 where neither (14.9) nor (13.11) is valid. This region is obtained by adding the respective domain of validity of both Taylor expansions shown in Figs. 13.18 and 14.4 as shown in Fig. 14.6 in the middle white part. 14.2 The Physicist’s Corner: Trying to Be a Little More Mathematical 255

Fig. 14.6 The limit of p n-1 validity of the Taylor 1 p D 1 expansion around c;r 2 as a function of r with pn1 belonging to the upper dark area.Thelower gray part 0.8 p D 1 below n1 4 deﬁnes the domain where the second Taylor expansion around 0.6 pB D 0 should be used. The white part in between is where there exists a difﬁculty 0.4

0.2

r 10 15 20 25 30

At this stage, we come to the negative conclusion that combining an additional Taylor expansion around pB D 0, although it sounds like a good idea, is just not doable in practice. We can now close this pedagogical section by stressing that there exist other techniques besides Taylor expansions for approximating functions. It means that our initial goal in deriving a more exact formula for the determination of nc;r could eventually be fulﬁlled at some mathematical price. But such an achievement would yield very little return and would make the reading of the formula heavy and fuzzy. This is particularly true since we have been able to derive a rather powerful formula with (14.6). Moreover, since we can evaluate nc;r exactly by direct iterations of Pr .pn/ knowing that it is always a small number, as was demonstrated earlier, one could ask from the beginning, why try to obtain an approximate analytical formula? The answer lies in the fact that having such analytical formulae as (14.5)or(14.6) makes it possible to grasp the fundamental trends underlying the whole process such as the singularity in the number of levels at pc;r seen in Fig. 14.2. It also leads to insight on how to use the results for practical applications to real social hierarchies as is shown in the next section. It is worth keeping in mind that contrary to what has been assumed so far, there exist many situations where the respective position of A and B politics is not symmetrical within the voting process. Later in our development the current symmetry will be broken, in particular to account for the advantage of being already in power with respect to the challenging political party. This inertia will 256 14 Going to Applications be embedded in voting groups of even sizes when a tie occurs at a local vote. As 1 1 a consequence, the current hypothesis k D 2 will no longer be valid and k ¤ 2 1 will drive pc;r ¤ 2 for even sizes. This will lead to surprising and counterintuitive p 1 results. However, c;r D 2 is always satisﬁed for odd-sized voting groups since odd sizes are independent of k. These developments are given in the next chapter on how to build the perfect democratic dictatorship.

14.3 The Magic Formula

In the previous section, we were able to calculate the number nc;r of bottom-up hierarchical levels, which are required, given some support p0 for the political party A, to make sure the president ultimately elected at the top belongs to the political party enjoying the majority support at the bottom. The result is embedded in (14.6). It shows that the number of levels nc;r is a function of the current p0. Such a feature prompts the practical remark that most social and political organizations do not change their structure at every election or decisional event. They are set once, and then do not change for some period of time, that can be more or less long. The number of hierarchical levels is thus ﬁxed and constant. On the other hand, the support for the respective competing political parties A and B is a function of time, and may vary quite substantially from one election to another. This very fact jeopardizes the use of (14.6) with respect to any application to real situations. Therefore, to make our analysis useful, the above conclusion leads us to invert the initial question

“How many levels are needed to eliminate the minority tendency?” into the new question

“Given n levels, what is the minimum overall support at the bottom if any, for a speciﬁc political party to obtain full power at the top with certainty?”

For the case where the whole population votes at once and with a single vote the answer is 50% as expected in majority rule democratic voting. But with small group voting and bottom-up hierarchical voting the answer is expected to be different. To implement this new operative question it is enough to invert (14.1)toget

n p0 D pc;r C .pn pc;r/r : (14.16)

p 0 pn Then, putting n D yields the higher limit r;B below in which any vote produces a victory of the A party at the top level. It deﬁnes a lower magic threshold below which A gets a zero probability to win the presidency with 14.3 The Magic Formula 257

pn p .1 n/ r;B D c;r r 1 .1 n/; D 2 r (14.17)

p 1 p 1 since here c;r D 2 . In parallel, putting n D into (14.16) gives a second pn threshold, the higher magic threshold r;A, above which the A gets the top of the hierarchy with a total certainty where

pn p .1 n/ n r;A D c;r r C r 1 .1 n/: D 2 C r (14.18)

1 n Equations (14.17)and(14.18) exhibit a symmetrical shift ˙ 2 r with respect 50 1 n to %. The threshold for certain failure has been decreased by 2 r while the 1 n threshold for certain victory has been increased by C 2 r . This symmetrical shift 50 pn pn 1 around % creates a new domain with r;B C r;A D . In the previous approach, given p0, we evaluated the number of levels required to 1 reach either one of the two attractors, pB D 0 in the case where p0 < 2 ,andpA D 1 p > 1 in the case where 0 2 . This constitutes a scheme to recover the deterministic outcome of the single shot whole population voting. Here, we also get deterministic p pn p pn votes but only when either 0 r;B or 0 r;A. Political party B loses in the ﬁrst case with pn D 0, and wins in the second one with pn D 1. pn

n pn pn ; r r;A r;B n D r : (14.19)

r 1 1= For a ﬁxed value of voting group size , it yields its maximum value r Dq r at n 2r n D 1 with r ! 0 for n 1 since always r >1. In addition, r ! for r 1 n . 2r /n=2 0 r 1 n H) r ! ! for with any value . The probabilistic domain implies a region of coexistence in which, because the results of the election process are only probabilistic, they may trigger some unexpected shift in the elected leadership at the top of the hierarchy. Within this region, no political party is sure of winning, making alternating leadership a reality. The majority may lose and the minority may win, the respective probability being, however, larger for the political party which has a majority at the bottom. 258 14 Going to Applications

Fig. 14.7 The phase n n p r,A ,p r,B diagrams associated to 1 (14.17)and(14.18)asa function of the number of the bottom-up hierarchical levels. The case for r D 3 is shown 0.8 in the upper part of the ﬁgure while the lower part shows Deterministic A victory the case for r D 15. In both the upper white parts,any 0.6 initial p0 yields at the top of Probabilistic A-B victory the hierarchy a deterministic victory for the political party 0.4 A. In both the lower white p parts, any initial 0 yields at Deterministic B victory the top of the hierarchy a deterministic failure for the 0.2 political party A with a victory for B. In the shaded areas in between, the result at the top is probabilistic. A n 2 4 6 8 10 12 drastic shrinkage of the probabilistic area is seen pn ,pn r D 3 r,A r,B when going from 1 (upper part)tor D 15 (lower part)

0.8

Deterministic A victory

0.6

Probabilistic A-B victory

0.4

Deterministic B victory

0.2

n 2 4 6 8 10 12

n This coexistence region shrinks as a power law r whose exponent is the number n of hierarchical levels. Therefore, having a small number of hierarchical levels increases the threshold to a certain reversal of power since the current minority will need to reach more than 50% in order to win the presidency for certain. But, simultaneously, it lowers the threshold from which the minority, still being in the minority, starts to have a nonzero chance of winning against the current majority as illustrated in Fig. 14.7. 14.3 The Magic Formula 259

pn Table 14.3 The value r;B given by (14.17) for the series of local group sizes r D 3; 5; 7; 9; 11; 13; 15 as a function of a number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 1 0.17 0.23 0.27 0.30 0.31 0.33 0.34 2 0.28 0.36 0.40 0.42 0.43 0.44 0.45 3 0.35 0.42 0.45 0.47 0.48 0.48 0.48 4 0.40 0.46 0.48 0.49 0.49 0.49 0.49 5 0.43 0.48 0.49 0.49 0.50 0.50 0.50 6 0.46 0.49 0.49 0.50 0.50 0.50 0.50 7 0.47 0.49 0.50 0.50 0.50 0.50 0.50 8 0.48 0.50 0.50 0.50 0.50 0.50 0.50 9 0.49 0.50 0.50 0.50 0.50 0.50 0.50 10 0.49 0.50 0.50 0.50 0.50 0.50 0.50 .r; n/ p

At this stage, (14.17)and(14.18) are approximate formulae which result from a 1 Taylor expansion in the vicinity of the unstable ﬁxed point pc;r D 2 . We thus have to check their accuracy as we did above with nc;r givenby(14.5). In this case, we have to eventually modify nc;r to nNc;r given (14.6) to get a more accurate formula with respect to its predictions. Table 14.3 gives the estimates for the lower magic threshold below which, any support for A at the bottom of the hierarchy yields a zero chance of having one of pn its representatives elected at the top level. The data are obtained from r;B given by (14.17) for the series of local group sizes r D 3; 5; 7; 9; 11; 13; 15 as a function of the number of bottom-up hierarchical levels with n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. .r; n/ p pn top presidential level. However, as soon as 0 r;B A gets a nonzero chance of p pn winning the presidency. The associated probability increases from zero at 0 D r;B 50 p 1 50 100 p pn to %at 0 D 2 . Then it keeps increasing from to %at 0 D r;A (see Fig. 14.7). We discuss in this section a comment on the meaning of zero and what is implied in choosing the precision of two digits in the calculations, which is used most of the time. To illustrate this effect, Table 14.4 gives the same estimates as in Table 14.3 but including four digits. It is seen that almost all the results of a lower magic threshold at 0:50 have been split into different lower values. A ﬁve digit precision would split the remaining six values of 0:50. We proceed by coming back to a two digit precision. By symmetry, we build Table 14.5 for the values of the higher magic threshold pn 100 pn 1 pn r;A given by (14.18)fora % A victory using r;A D r;B with results from Table 14.3. To check the accuracy of the data given in Table 14.3, we insert each one of P .pn / the lower magic thresholds into the associated voting function r r;B ,whichwe iterate until we reach at some j level a value pj 0:004 since we are considering 260 14 Going to Applications

pn Table 14.4 The value r;B givenby(14.17)asinTable14.3 but now using a four digit precision. It splits almost all values of 0:50 to different lower values A ﬁve digit precision would split the remaining six 0:50 nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 1 0.1667 0.2333 0.2714 0.2968 0.3153 0.3295 0.3409 2 0.2778 0.3578 0.3955 0.4174 0.4318 0.4419 0.4493 3 0.3518 0.4241 0.4522 0.4664 0.4748 0.4802 0.4839 4 0.4012 0.4595 0.4782 0.4864 0.4907 0.4932 0.4949 5 0.4342 0.4784 0.4900 0.4944 0.4966 0.4977 0.4984 6 0.4561 0.4885 0.4954 0.4977 0.4988 0.4992 0.4995 7 0.4707 0.4939 0.4979 0.4991 0.4995 0.4997 0.4998 8 0.4805 0.4967 0.4990 0.4996 0.4998 0.4999 0.5000 9 0.4870 0.4982 0.4996 0.4998 0.4999 0.5000 0.5000 10 0.4913 0.4991 0.4998 0.4999 0.5000 0.5000 0.5000

pn Table 14.5 The value r;A givenby(14.18) for the series of local group sizes of r D 3; 5; 7; 9; 11; 13; 15 as a function of the number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 1 0.83 0.77 0.73 0.70 0.69 0.67 0.66 2 0.72 0.64 0.60 0.58 0.57 0.56 0.55 3 0.65 0.58 0.55 0.53 0.52 0.52 0.52 4 0.60 0.54 0.52 0.51 0.51 0.51 0.51 5 0.57 0.54 0.51 0.51 0.50 0.50 0.50 6 0.54 0.51 0.51 0.50 0.50 0.50 0.50 7 0.53 0.51 0.50 0.50 0.50 0.50 0.50 8 0.52 0.50 0.50 0.50 0.50 0.50 0.50 9 0.51 0.50 0.50 0.50 0.50 0.50 0.50 10 0.51 0.50 0.50 0.50 0.50 0.50 0.50 .r; n/ p >pn For a given couple of values ,any 0 r;A ends up with a victory for A at the top presidential level a two digit precision. We then compare j to n. For instance, Table 14.6 gives the various corresponding series obtained for the case of r D 3 from which the associated values of j are extracted. This shows a systematic discrepancy of C2in pn the number of levels. It means that the value r;B from Table 14.3 becomes accurate if it is associated to a hierarchy with n C 2 levels instead of n. However, our “physicist like arrangement” of noticing that adding a systematic C2 levels makes (14.17) accurate was based on the results using only the voting local group size of r D 3. To check the robustness of such a finding we now repeat the same evaluation as above in the case of local voting groups of a larger size, e.g., r D 15. The results are given in Table 14.7. Using a larger-sized voting group drastically reduces the coexistence area as seen in the Fig. 14.7 and confirmed by Table 14.7. The unstable fixed point separator is obtained already with five hierarchical levels, which means that the coexistence area has totally disappeared. Using a higher digit precision would slightly enlarge it. 14.3 The Magic Formula 261

p D pn n D 1; 2; :::; 10 Table 14.6 The variation of 0 r;B as a function of given by (14.17) while climbing up a hierarchy using local voting groups of a size of r D 3

p0 D pn p p p p p p p p p p p p nj 3;B 1 2 3 4 5 6 7 8 9 10 11 12 1 0.17 3 0.07 0.02 0.00 2 0.28 4 0.19 0.09 0.02 0.00 3 0.35 5 0.28 0.20 0.10 0.03 0.00 4 0.40 6 0.35 0.29 0.20 0.10 0.03 0.00 5 0.43 7 0.40 0.35 0.29 0.20 0.10 0.03 0.00 6 0.45 8 0.43 0.40 0.35 0.29 0.20 0.10 0.03 0.00 7 0.47 9 0.46 0.43 0.40 0.35 0.29 0.20 0.11 0.03 0.00 8 0.48 10 0.47 0.46 0.43 0.40 0.35 0.29 0.20 0.11 0.03 0.00 9 0.49 11 0.48 0.47 0.46 0.43 0.40 0.35 0.29 0.20 0.10 0.03 0.00 10 0.49 12 0.49 0.48 0.47 0.46 0.43 0.40 0.35 0.29 0.20 0.10 0.03 0.00 P .pn / The successive values are obtained using the voting function r r;B , which is iterated until we reach at some j level a value of pj 0:004 since we are considering a two digit precision. The number j is compared to the corresponding number n, which has been used to evaluate the lower pn p magic threshold r;B taken as 0

Table 14.7 The variation of p D pn p p p p p p D pn n 0 15;B j 1 2 3 4 5 0 r;B as a function of n D 1; 2; :::; 10 given by 1 0.34 2 0.10 0.00 (14.17) while climbing up a 2 0.45 3 0.34 0.10 0.00 hierarchy using local voting 3 0.48 4 0.45 0.34 0.10 0.00 group sizes of r D 15 4 0.49 5 0.48 0.45 0.34 0.10 0.00 5 0.50 0.50 0.50 0.50 0.50 0.50 The successive values are obtained using the voting function P .pn / j r r;B , which is iterated until we reach at some level a value of pj 0:004 since we are considering a two digit precision. The number j is compared to the corresponding number n, which has pn p been used to evaluate the lower magic threshold r;B taken as 0

With respect to the accuracy of (14.17), the results in Table 14.7 show a systematic discrepancy of C1 between the prediction value given by n and the exact value j . The discrepancy was C2 for the smaller size r D 3. pn We can thus conclude that to associate the value r;B for the lower magic threshold given by (14.17) to a bottom-up voting hierarchy with n C 2 levels yields a correct prediction. Indeed, to eventually have one missing level can lead to the violation of the democratic balance while one superﬂuous level will only reinforce the democratic balance. On this basis, we modify (14.17)and(14.18)toget 1 pNn D .1 nC2/ r;B 2 r (14.20) and 1 pNn D .1 C nC2/; r;A 2 r (14.21) 262 14 Going to Applications

n n Fig. 14.8 The same phase p r,A ,p r,B diagram as in Fig. 14.7 but 1 now using improved (14.20) and (14.21) instead of (14.17) and (14.18). The case for r D 3 is shown in the upper 0.8 part of the ﬁgure while the Deterministic A victory lower part shows the case for r D 15.Forn D 1 and n D 2 the result is always 0.6 probabilistic Probabilistic A-B victory

0.4

Deterministic B victory 0.2

n 2 4 6 8 10 12

n n p r,A ,p r,B 1

0.8

Deterministic A victory

0.6

Probabilistic A-B victory

0.4

Deterministic B victory

0.2

n 2 4 6 8 10 12

pn pn where Nr;B and Nr;A are the lower and higher magic thresholds which characterize a bottom-up voting hierarchy with n levels. These thresholds provide the correct qualitative behavior and rather precise quantitative estimates. Their evolution as a function of the number of levels n for the two ﬁxed sizes r D 3 and r D 15 are shown in Fig. 14.8. The probabilistic area has been enlarged. It now appears that for n D 1 and n D 2 the result is always probabilistic as seen directly from (14.20)and(14.21), which p2 0 p2 1 yields respectively Nr;B D and Nr;A D . However, it should be stressed that while 14.4 What It Means in Terms of Global Size 263 the persistent probabilistic character of the result for one and two level hierarchies makes sense for a small size of r, this is not true for large voting group sizes as seen in Table 13.1. This restriction comes from the fact that all formulae in this section are derived from a Taylor expansion around pc;r, which was shown to be valid only in some range of values of p0. And indeed, it is far from pc;r that a single large voting group can yield a deterministic outcome with just one vote. We have now identiﬁed the various phases of the dynamics of getting to power by democratic bottom-up voting via local groups of size r within a pyramidal hierarchy with n levels above the bottom one. For two competing parties A and B, if p0 is the support for A at the bottom level we have respectively 8 ˆ B p pn < if 0 Nr;B B A pn

14.4 What It Means in Terms of Global Size

Up to now we have been somehow advocating the advantageous features of using a bottom-up democratic hierarchy with local voting groups instead of one direct single vote of all the agents concerned. However, to complete the case we need to N n n evaluate the number of agents G;r , which are required to build an -level given hierarchy using groups of size r. Indeed, if this global number of required agents is not much smaller than the total size of the corresponding population, building a hierarchy becomes useless, since this requires turning to a more complex and demanding structure than that necessary just for a single global vote. Since each group of size r elects one representative, the president being elected at the last level n, while going down the hierarchy, at each level down the number of representatives is multiplied by r. We thus have 1 at level n, r at level .n 1/, r2 at level .n 2/ ..., and rn at level 0, the bottom. Adding all agents yields the global size of the hierarchy

N n 1 r r2 ::: rn G;r D C C C C rnC1 1 D (14.23) r 1

n nl with Nl;r D r agents at each level l where l D 0; 1:::; n. 264 14 Going to Applications

N n r D 3; 5; Table 14.8 Values of G;r for the series of local group sizes of 7; 9; 11; 13; 15 as a function of the number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 3rD 5rD 7rD 9rD 11 r D 13 r D 15 1 4 6 8 10 12 14 16 2 13 31 57 91 133 183 241 3 40 156 400 820 1,464 2,380 3,616 4 121 781 2,801 7,381 16,105 30,941 54,241 5 364 3,906 19,608 66,430 177,156 402,234 813,616 6 1,093 19,531 137,257 597,871 2106 5106 107 7 3,280 97,656 900,800 5106 2107 7107 2108 8 9,841 488,281 7106 5107 2108 9108 3109 9 29,524 2106 5107 4108 3109 1010 41010 10 88,573 5107 3108 4109 31010 1011 61011 For a given couple of values .r; n/ the total number of agents involved is evaluated. Numbers larger that 106 are rounded off

N n r Table 14.8 gives a series of values for G;r for various values of size and number of levels n, and before proceeding further in our investigation for the perfect democratic scheme, it is of practical importance to discuss these numbers. Indeed, if building a specific .r; n/ hierarchy requires millions of agents, the whole project becomes a questionable task. Up to now, we have discussed numbers of levels lower or equal to 10 with voting group sizes of up to 15, i.e., small realistic numbers. However, even within this small range of values, Table 14.8 shows that half of the cases have numbers of the order of or larger than 106 making a more detailed analysis necessary. N n Along the same lines, Fig. 14.9 shows the evolution of G;r for the fixed voting group sizes r D 3; 5; 7; 9; 11; 13 as a function of the number of levels n. Figure 14.10 shows the same but for various fixed numbers of levels n as a function of the voting group size r. The global size explosion occurs very quickly for rather small values of both r and n.FromFig.14.9, we can conclude that only for the smaller voting group size r D 3, a hierarchy can be built with up to ten levels, but for r D 5 this number has to be reduced to eight, to seven for r D 7,tosixforr D 9 and to five for larger groups. These findings are fully consistent with the numbers given in Table 14.8 and impose setting a reasonable limit to building an efficient bottom-up hierarchy in terms of voting group size and number of levels.

14.5 The Physicist’s Corner: To Make It Simpler

To go deeper into the understanding of the dependence of the global size of a .r; n/ hierarchy on both r and n we open our physicist’s corner in order to gain to a better grasp of (14.23). 14.5 The Physicist’s Corner: To Make It Simpler 265

n N G,r n N G,r

2·109 4·106

9 r=13 3·106 r=7 1.5·10

9 2·106 1·10

8 1·106 r=3 5·10

r=9 n n 2 6 8 10 2 4 6 8 10

N n Fig. 14.9 The evolution of G;r for a series of ﬁxed voting group sizes as a function of the number of levels n.Ontheleft side r D 3; 5; 7 and r D 9; 11; 13 on the right side. The divergence occurs ﬁrst for the larger values of r

n N G,r

n N G,r 6 4000 2·10

n=4

3000 1.5·106

2000 1·106 n=7

1000 500000 n=5

n=2

r r 4 5 6 7 8 9 10 4 5 6 7 8 9 10

N n Fig. 14.10 The evolution of G;r for a series of ﬁxed voting group sizes as a function of the number of levels n.Ontheleft side r D 3; 5; 7 and on the right side r D 9; 11; 13. The divergence occurs ﬁrst for the larger values of r

N n Looking at it, a mathematician would be pleased with it since G;r is expressed in a compact exact formula with a ratio of two simple terms. But a physicist will not know what to extract from it in terms of a “physical meaning.” To allow such an extraction, the physicist will immediately notice that r and n being integers larger than one, one could consider that rnC1 1, and therefore neglects the 1 to keep 266 14 Going to Applications

Fig. 14.11 Evolution of a 8 linear dependence versus a logarithmic one. The second 7 one is much slower

6 y=n

1 z = Ln r

n,r 0 2 4 6 8 10 12 14 only the power law rnC1. Applying the same approximation to the denominator, i.e., r 1 r and combining both, yields the result

NN n rn; G;r (14.24) which is much more readily interpretable than (14.23) since both variables are separated and appear only once. Since the global size explodes rather quickly as a function of both r and n, as seen from Table 14.8, it is better to examine the variation NN n n r of its logarithm with ln G;r D ln . In this case, only the size dependence is logarithmic, while it is linear with the number of levels. A smoother sensitivity is thus expected for varying sizes of r in contrast to a higher one with the number of hierarchical levels n, as shown in Fig. 14.11. From the form of the logarithm, taking the exponential recovers the initial NN n n ln r expression of the global size written as G;r e , which emphasizes ﬁrstly, why the global size increases so quickly with both r and n, and secondly, how it is more sensitive to n than to r due to the Logarithm. But before using the simple form given by (14.24), we notice that substituting NN n N n .r; n/ G;r for G;r assumes de facto that the total number of agents involved in a N n rn bottom-up hierarchy is mainly concentrated in the bottom level 0, since G;r n n and N0;r D r . The question is therefore “Does-it make sense?” To answer this basic question prompts the opening of another parenthesis about the way physicists play with formulae and numbers. Indeed, an a priori look at the sum 1 C r C r2 C :::: C rn from (14.23) explains the counter intuitive aspect of stating that the entire series 1 C r C r2 C :::: C rn1 can be neglected with respect to 14.5 The Physicist’s Corner: To Make It Simpler 267 thesingletermrn. In particular, since already the last term of the series is just rn1, i.e., rn divided by r the group size, which is a small number. This remark leads to the conclusion that (14.24) is not a valid assumption. Yes, but maybe it is somehow valid, in particular to help grasp the trend N n associated with the evolution of G;r as a function of the two independent variables r and n. Let us go further in our investigation by rewriting (14.23)as 1 1 N n D rn 1 (14.25) G;r rnC1 1 1 r

1 1 and expanding 1 in powers of r to get 1 r 1 1 1 1 N n rn 1 1 C C C C ::: ; G;r rnC1 r r2 r3 (14.26) which is equal to 1 1 1 1 1 N n rn 1 C C C C ::: ::: : G;r r r2 r3 rnC1 rnC2 (14.27)

Canceling the appropriate terms leads to 1 1 1 1 N n D rn 1 C C C C ::: C ; (14.28) G;r r r2 r3 rn which turns out to be an exact result as can be seen by noting ﬁrst that

nC1 1 1 1 1 1 1 1 C C C C ::: C D r r r2 r3 rn 1 r 1 1 rnC1 1 D : (14.29) rn r 1 Then, multiplying by the prefactor rn recovers exactly (14.23). N n rn From (14.28) the level of approximation in taking G;r is precisely quan- tiﬁed. Table 14.9 gives each contribution from the various terms 1=r; 1=r2; :::; 1=r 5 for the series of voting group sizes of r D 3; 5; 7; 9; 11; 13; 15. It appears that only the ﬁrst correction 1=r is substantial. All the others contribute very little. N n rn r 3; 5; 7; 9 Using G;r is thus quite accurate except for D . In any case, the approximation 1 NN n rn 1 C G;r r (14.30) 268 14 Going to Applications

Table 14.9 Values of the corrections 1=r; 1=r 2; :::; 1=r 5 in (14.9) for the series of local group sizes r D 3; 5; 7; 9; 11; 13; 15 Correction r D 3rD 5rD 7rD 9rD 11 r D 13 r D 15 1=r 0.33 0.20 0.14 0.11 0.09 0.08 0.07 1=r 2 0.11 0.04 0.02 0.01 0.01 0.01 0.00 1=r 3 0.04 0.01 0.00 0.00 0.00 0.00 0.00 1=r 4 0.01 0.00 0.00 0.00 0.00 0.00 0.00 1=r 5 0.00 0.00 0.00 0.00 0.00 0.00 0.00 The few substantial corrections are limited to 1=r and 1=r 2 only for r D 3

is always excellent. It shows that in a .r; n/ bottom-up hierarchy most agents are N 0 rn N 1 rn1 concentrated on the two lower levels with respectively G;r D and G;r D . For large voting groups (r>9), most agents are at the bottom level.

14.6 Putting a Limit on the Global Size

Having a better feeling of how the global size varies, it is now useful for any practical applications to put in a global constraint in terms of the number of agents involved. To do this, we can simply set the maximum global size allowed for building a .r; n/ bottom-up democratic hierarchy. A useful way is to decide on some upper value as a power form as 10a where a is an integer. N n <10a On this basis, the inequality G;r determines the relationship between the number of levels n and the size r to satisfy the size constraint. Using (14.23)and taking the Logarithm on both sides yield ar;n

1 1 rnC1 ar;n ln : (14.31) ln 10 1 r

Figure 14.12 shows a series of curves ar;n as a function of n for a ﬁxed size r. The curves are linear. Areas which satisfy ar;n

Fig. 14.12 Aseriesof ar,n curves ar;n as a function of n 12 for a ﬁxed size r with r D 3; 5; 7; 9; 11; 13; 15.The curves are linear in n. Areas 10 r=15 which satisfy ar;n

r=3 a=3 n 2 4 6 8 10

Fig. 14.13 A similar series ar,n of curves ar;n as in Fig. 14.12 8 but now as a function of r for a ﬁxed value of n with n=8 7 a=5 n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. The curves are no longer linear. The same areas for 6 ar;n

2 a=3 1 n=2 r 4 6 8 10 12 14 270 14 Going to Applications

Fig. 14.14 Evolution of nM n M as a function of the size r for agivena with a D 3; 4; 5. 10 The allowed values of n must obey n

a=5

a=3 r 5 10 15 20

from which the inequality ar;n

a ln 10 1 n D M ln r r ln r a ln 10 (14.33) ln r is the maximum number of levels allowed to keep to the size constraint. The second term has been neglected since the larger contribution coming from r D 3 is only 0.30. It goes down to 0.04 for r D 11 and ! 0 for r 1. Figure 14.14 shows the evolution of nM as a function of the size r for a given a. On the other hand, given a ﬁxed number of levels n and still keeping the 10a upper limit, the associated constraint on the size is r

a rM 10 n : (14.34)

Figure 14.15 shows the corresponding range allowed in the size of the voting group. N n To sum up the limit in the global size G;r ,Fig.14.16 shows the three- dimensional behavior as a function of both r and n. The sudden and huge increase 14.6 Putting a Limit on the Global Size 271

r M r M 100 2500

80 2000

60 1500 a=5 a=5

40 1000

500 a=3 20

a=3

n n 2 4 6 8 10 2 4 6 8 10

Fig. 14.15 Evolution of rM as a function of the number of levels n for a given a with a D 3; 4; 5. The allowed values of r must obey r

r 10 4 8 6 8 n 6 10 4 2

1.5·107

n N G,r 1·107

5·106

N n r n Fig. 14.16 The three-dimensional behavior of the global size G;r as a function of both and . The sudden increase is very sharp and occurs for a very precise range of values of r and n occurs for a very precise range of values of r and n. Therefore, any practical building of a hierarchy must be conﬁned within these ranges, with a size r and a number of levels n, which are outside of the region of divergence in order to make it a reasonable task. Chapter 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

15.1 Phase Transitions and Critical Phenomena

From our current level of investigation of the use of majority rule voting in bottom-up hierarchical structures, we are now in a position to touch on one of the most fascinating features of the “life” of inert matter; the phase transition of a given system from one global organization into another. An example in matter would be going from a liquid to a gas or from a conductor to a super-conductor. The mechanisms at work in such a metamorphosis are universal, which means that they obey the same restricted set of laws whatever the physical nature of the matter undergoing the phase transition. The most salient signature of a phase transition is the divergence of the so-called correlation length. The correlation length is the length beyond which a physical microscopic change of one degree of freedom has no effect at all on the other degrees of freedom. In particular, it defines the extent of a correlation between two fluctuations. It is therefore sufficient to take a sample of the size of the correlation length to study all the properties of a given material. In most cases, the correlation length is of a finite size and increases slightly with increasing temperature. However, it diverges in the vicinity of a phase transition, following a power law of the distance to the critical point with a negative exponent. At the critical point, it is infinite, making all scales of all fluctuations correlated. It is a “total mess” and was intractable for centuries. The tool to handle this huge theoretical difficulty was discovered only in the early nineteen seventies with the so-called “renormalization group techniques.” Our scheme to build a bottom-up hierarchical structure embodies exactly the main features of the phenomenon of phase transition. Given a proportion p0 of agents in favor of the A policy, we have determined what is the minimum size of the bottom level of the hierarchy to ensure the recovery of the all-voting, single shot democratic result. The aim is to get an A president with certainty at the top when 1 p0 > 2 , and a B president otherwise [1–19]. It determines the perfect sampling of the whole population. To add more agents has no effect.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 273 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 15, © Springer Science+Business Media, LLC 2012 274 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

Indeed, we saw that when p0 is of the order of 0:45 or lower, the corresponding bottom size is rather small and it turns the scheme into a very proﬁtable substitute to the all-voting scheme. In contrast, when p0 is larger, the required hierarchy goes out of reasonable proportions and diverges in size in the close vicinity of 0:50.These features demonstrate that the bottom size of the hierarchy is the equivalent of a correlation length. The same result holds on the other side of 0:50. To exploit this equivalence, we postulate a power law behavior for N as a function of the distance of p0 from below the critical point value pc;r D 0:50 writing,

N 0 .p p /vr ; G;r c;r 0 (15.1) where vr is a positive number. In addition, since we saw that the main contributions to the global size of the hierarchy come from the bottom and the ﬁrst levels, we can N 1 extend the power law form given by (15.1)to G;r with,

N 1 .p p /vr ; G;r c;r 1 (15.2) which leads to, vr 1 vr .pc;r p0/ D r .pc;r p1/ ; (15.3) N 1 r1N 0 where we used the intrinsic equality G;r D G;r . We can then evoke (13.12) to get, vr 1 r D r ; (15.4)

dPr .pn/ r jp where dpn c;r is given by (13.14). Taking the logarithm on both sides yields the value of the critical exponent,

ln r vr D ; (15.5) ln r which appears to be a function of the group size single variable r. This is shown in ln 3 Fig. 15.1. It varies from v3 D 3 D 2:71 to the large size limit ln 2 1 2 2 vNr 2 1 ln C ; (15.6) ln r 3r which reduces to, 1 2 vNr 2 1 ln ; (15.7) ln r for r1. It depends only on the voting group size r and is always larger than two, as seen in Fig. 15.1. N 0 .p p /vr From (15.7) the minimum global size G;r c;r 0 of the “correlated” bottom level to build a bottom-up democratic hierarchy is readily calculable. While 15.1 Phase Transitions and Critical Phenomena 275

Fig. 15.1 The variation of vr the exponent vr given by 2.7 (15.1) as a function of the voting group size r 2.6

2.5

2.4

2.3

2.2

2.1

r 50 100 150 200 250 300

N0 G,r N0 20000 G,r 1·106

17500 p0=0.45

800000 p0=0.498 15000 p0=0.48

p0=0.497 12500 p0=0.46 600000 10000

7500 400000 p0=0.49

p0=0.495 5000 p0=0.47 200000 2500

r r 20 40 60 80 100 20 40 60 80 100

N 0 D .p p /vr Fig. 15.2 The minimum global size G;r c;r 0 of the “correlated” bottom level to build a bottom-up democratic hierarchy as a function of r for a series of initial support p0.Theleft side shows the cases for p0 D 0:45; 0:46; 0:47; 0:48,andtheright side cases that are closer to 0.50 with p0 D 0:49; 0:495; 0:497; 0:498 it is not too large for any range of values of p0 outside the vicinity of pc;r for any r, it explodes very close to it as shown in Fig. 15.2. Therefore, the whole hierarchical scheme loses its practical interest very near to 50%, as appears from Fig. 15.3 where the number of agents reaches massive values. It is worth stressing that the same applies for p0 above pc;r writing p0 pc;r instead of pc;r p0. Taking the absolute value j pc;r p0 j makes the formulae valid for the whole range 0 p0 1. 276 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

0 N G,r 0 1000 N G,r 5·108

800 4·108

600 3·108

400 2·108

200 1·108

p p 0.2 0.4 0.6 0.8 1 0 0.49 0.5 0.51 0.52 0

N 0 Dj p p jvr Fig. 15.3 The minimum global size G;r c;r 0 of the “correlated” bottom level to build a bottom-up democratic hierarchy as a function of the initial support p0 for r D 3 (inside curves)andr D 15.Theleft part shows data up to 1,000 agents while the right part goes up to 5 108

15.2 Revisiting the Practical Scheme

We are now in a position to evaluate quite naturally the number of levels nd N 0 associated to a bottom size G;r , knowing by construction, that the corresponding hierarchy ends up with an elected president at a level nd, who belongs to the opinion that is the majority of the whole population. 0 n 0 N r d N p p vr Combining both expressions G;r D and G;r Dj c;r 0 j leads to

ln j pc;r p0 j nd D : (15.8) ln r

This number nd represents, in terms of physics, the minimum number of levels required to integrate out all “local ﬂuctuations,” which could occur in the agent distribution at the bottom level. By so doing it makes the “ground state” of the system emerge, step by step, while climbing up the hierarchy level by level. It is a process that enables the current majority to be discovered, which may be unknown from the start. It means that at level nd the elected president belongs to the global majority with a total certainty. Indeed, we have reached the same expression nc;r introduced earlier in (14.5) by a different way. However, comparing (15.8) to (14.5) from the practical scheme section, a discrepancy is found with

ln pc;r nd nc;r D : (15.9) ln r 15.3 Revisiting the Magic Formula 277

e Table 15.1 Exact numerical estimates nc;r for nc;r using (17.2) and the values obtained from (14.5) for the same bottom value p0 D 0:49

p0 D 0:49 r D 3rD 5rD 7rD 9rD 11 r D 13 r D 15

nc;r 9.65 6.22 4.99 4.34 3.93 3.64 3.42 nNc;r 118665 5 5 e nc;r 128666 5 5 nd 11.36 7.33 5.88 5.11 4.62 4.28 4.02 nNd 128665 5 5

p 1 >1 n >n Since c;r D 2 and r , we conclude that d c;r, which implies that (15.8) produces larger values for the critical number of levels than (14.5). To evaluate an actual number of levels, we have to take the integer part of (15.8)andaddC1 to get ln j pc;r p0 j nNd N C 1: (15.10) ln r

Table 15.1 reproduces Table 14.1 including values given by (15.10)foraseriesof local group sizes r. It appears that all exact estimates are now recovered with nNd, including the case r D 3, and without the ad hoc additional C1 (therefore, C2 in the formula of (14.6)) is used to improve nNc;r,making(15.10) a rather accurate and solid result.

15.3 Revisiting the Magic Formula

The above ﬁnding of a more accurate expression for the number of critical levels using the power law assumption (15.1) also prompts us to revisit the magic formula expression and its associated probabilistic region. In the magic formula section, we had to invert (14.1) to determine the magic thresholds for p0. They were obtained by putting in respectively p0 D 0 and p0 D 1. The two thresholds delimit the strategical phase diagram to respectively be certain to reach the presidency and to lose it also with certainty, for a given bottom-up .r; n/ democratic hierarchy. Here, we get them quite directly from (15.8)whichis identical to

nd j pc;r p0 jD r : (15.11)

Relaxing nd to any value n directly yields via p0 the two magic thresholds. The lower magic threshold being smaller than pc;r,weget,

pn p n r;B D c;r r (15.12) 278 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human and pnC p n r;A D c;r C r (15.13) for the higher magic threshold, which is greater than pc;r. It is worth noticing that, although similar, these two expressions differ from the earlier ones obtained in (14.17) and (14.18). The extent of the associated probabilistic area is now,

n pnC pn r r;A r;B n D 2r (15.14)

n instead of r previously. The probabilistic area is now twice the previous one. Such an increase goes along the ad hoc “arrangement” made at the end of the magic formula section to improve the two formulae, making them fit the exact estimates. There, we shifted n to n 2. It was also noticed that while the 2 correction was adequate for small values of r (Table 14.6), a 1 correction would have been sufficient for a larger value of group size such as r D 15 (Table 14.7). Figure 15.4 incorporates the various formulae obtained for both magic thresholds. The inside curves, previously shown in Fig. 14.7 are given by (14.17) and (14.18). The associated enclosed probabilistic area is found sandwiched between the curves from (15.12)and(15.13), which were found to yield excellent results. They are denoted as being “almost exact” in the figure. The corresponding enclosed probabilistic area is effectively doubled, in agreement with (15.14). The dots show the exact estimates of the magic thresholds. The values are given in Table 15.2. While 17 levels are required to shrink the probabilistic area to a width of 2103 for voting groups of size r D 3, it reduces to six levels only for groups of r 15 p 1 pe 1 pe size D . Due to the symmetry with respect to c;r D 2 we have r;B D r;A. In addition, the upper outside curve represents (14.21) where a 2 correction is performed on the exponent of (14.18). At r D 3,itismuchcloserto(15.13), with almost exactly the same results as with (14.21), thus justifying the 2 correction, which is seen to overestimate the magic thresholds by a small amount. On the contrary, a 1 correction can be misleading since it underestimates the threshold, as seen by the fact that the corresponding curve is located between (14.17) and (15.12). However, at r D 15 the situation is reversed, with the 1 correction becoming more appropriate, as seen with the much larger distance between (14.21) and (15.13) than between (15.12) and the corresponding 1 correction curve. The above discussion demonstrates the net advantage of the present correlation length approach, which is found to yield the best approximation, as seen from Fig. 15.4. It is a nice illustration of the “physicist’s magic” in postulating the power law behavior for the hierarchy with the two lowest level numbers of agents. 15.4 Rare Dictatorial Events Versus Antidemocratic Ones 279

n n Fig. 15.4 Comparison of the p r,A ,p r,B higher and lower magic 1 thresholds given by respectively (15.12), (15.13), Almost exact (14.20), (14.21), (14.18) and (14.16) with a 1 correction 0.8 in the exponent instead of the 2. Equations (15.12)and 2 correction (15.13) are denoted to be “almost exact” since they 0.6 yield excellent estimates. While the 2 correction is r D 3 good at (top ﬁgure)a 0.4 1 correction is better at r D 15 (bottom ﬁgure). Dots 1 correction are the exact estimates of the magic thresholds 0.2

Almost exact

n 2 4 6 8 10 12

n n p r,A ,p r,B 1

Almost exact

0.8

2 correction

0.6

0.4

1 correction

0.2

Almost exact

n 2 4 6 8 10 12

15.4 Rare Dictatorial Events Versus Antidemocratic Ones

The democratic bottom-up hierarchical leadership is organized and operates via repeated majority rule voting. The two main features are on the one hand, that the presidency goes to the global majority, i.e., the opinion which holds more than 50% within the whole population, and on the other, that some local power is given 280 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

Table 15.2 Exact estimates of the magic thresholds for both sizes r D 3 and r D 15.Dataare given to within a precision of 103. The probabilistic area shrinks to a width of 2 103 within 17 levels. Only six levels are required for groups of size r D 15. Due to the symmetry with respect p D 1 pe D 1 pe to c;r 2 we have r;B r;A Magic n D 1nD 2nD 3nD 4nD 5nD 6nD 7nD 8nD 9nD 10 n D 17 pe rD3;A 0.952 0.853 0.760 0.682 0.623 0.583 0.556 0.537 0.525 0.517 0.501 pe rD3;B 0.048 0.147 0.240 0.318 0.377 0.417 0.444 0.463 0.475 0.483 0.499 pe rD15;A 0.774 0.594 0.531 0.510 0.503 0.501 pe rD15;B 0.226 0.406 0.469 0.490 0.497 0.499

to the minority at various levels of the hierarchy, more at lower levels than at higher, depending on the current minority strength. In addition, the “global cost” is drastically reduced in terms of setting up the corresponding elections. However, while the 50% threshold to get the presidency for sure is the outcome of a random distribution of agents at the bottom, it also requires a number of levels that is equal or greater to the critical number nc;r. This condition ensures that all local ﬂuctuation effects, which may occur from the initial distribution of agents within the bottom voting groups, are averaged out. In the case where the number n is smaller than nc;r, we saw that the presidential outcome becomes a probabilistic issue. Indeed, organizations are set up to survive several general elections, making the number n of levels a ﬁxed variable. In contrast, the respective supports p0 and .1 p0/ for each opinion are varying variables. The associated democratic voting phase diagram was found to exhibit three phases separated by the two magic pn pnC thresholds r;B and r;A given respectively by (15.12)and(15.12). In the two p pnC regions 0 r;B and 0 r;A , the outcome of the voting at the presidential level n is expected with certainty. p

Fig. 15.5 A winning bottom distribution of a minority number of A-agents, 4 against 5 B-agents for a r D 3; n D 2 hierarchy. It is an antidemocratic bottom conﬁguration

discriminating between two kinds of bottom configurations, both producing the dictatorial effect of having a minority president at the top of the hierarchy. Some bottom configurations exhibit an overall local majority that is opposed to the global majority. Once they occur, the bottom up democratic voting yields the presidency to the associated opinion, which is in majority at the hierarchy bottom, although it is in minority within the whole population. The dictatorial switch is driven by a fluctuation in the selection of the agents that constitute the bottom level. But then the bottom-up process is democratic since it preserves that majority. The probability of such configurations can be calculated and reduced by building additional levels as was seen in the previous chapter. At odds with the previous analysis, some bottom configurations, while preserving the minority status of the global minority opinion, actually thwart the democratic bottom up voting process by producing a bottom minority president. We call these winning minority bottom configurations antidemocratic. They are rarer than the dictatorial ones. An illustration of such antidemocratic bottom configurations is shown in Fig. 15.5 for a .r D 3; n D 2/ hierarchy. The 4 A-agents present at the bottom against a majority of 5 B-agents obey a distribution of the type .2;2;0/among the three bottom voting groups of size 3 each. There exist 27 different antidemocratic rearrangements of this minority winning configuration. The associated total probability

4=5 4 5 P3;2 .p0/ D 27p0.1 p0/ (15.15) is shown in Fig. 15.6 as a function of p0. It reaches its maximum value 0.0557 at 4 p0 D 9 . This ratio of 4 against 5 is the unique one which can reverse the bottom overall minority status by giving it the presidency. Four agents is the minimum number to achieve this process. Three are not enough and ﬁve are already the majority. These ﬁgures are a function of both the size r of the voting group and the number n of actual levels. 282 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

4 5 P 3,2

0.05

0.04

0.03

0.02

0.01

p0 0.2 0.4 0.6 0.8 1

4=5 Fig. 15.6 The probability P3;2 .p0/ of bottom antidemocratic rearrangements for a .r D 3; n D 2/ hierarchy with the 4-A agents against 5 B-agents (15.15). The maximum value is 0.0557 p D 4 at 0 9

15.4.1 Another Viewpoint

Using the previous finding, we can indeed tackle the bottom-up voting problem from a different point of view. Instead of studying the iteration process we consider directly the bottom of the hierarchy and determine all the configurations which lead to an A victory. Along this line, we enumerate all A winning configurations. In addition to the above antidemocratic ones, we have the series of democratic configurations, which are, in terms of the number of A agents in each one of the three voting groups, of the type .2;2;1/, .3;2;0/ for a ratio of 5 to 4, .3;3;0/, .2;2;2/, .3;2;1/for a ratio of 6 to 3, .3;3;1/, .3;2;2/for a ratio of 7 to 2, .3;3;2/ for a ratio of 8 to 1, .3;3;3/and for a ratio of 9 to 0 and all associated permutations. All these winning configurations add up to,

4 5 5 4 6 3 P3;2.p0/ D 27p0.1 p0/ C 99p0.1 p0/ C 84p0.1 p0/ 7 2 8 1 9 C36p0.1 p0/ C 9p0.1 p0/ C p0: (15.16)

As expected from the coherent requirement of both approaches, (15.16) turns out to embrace at once the initial voting process that is repeated twice to go from the bottom level to the ﬁrst one, and then from the ﬁrst one to the second one for the 15.4 Rare Dictatorial Events Versus Antidemocratic Ones 283

Fig. 15.7 Both probabilities P3,2 ,P3 P .p / P .p / 3 0 and 3;2 0 as a 1 function of p0 with the ﬁrst one being contained within the second one. Two iterations on the inside curve 0.8 are identical to one iteration on the external curve 0.6

0.4

0.2

p 0.2 0.4 0.6 0.8 1 0 presidency. Here, we immediately have the probability for the president from the probability at the bottom. Indeed, it follows from the equality P3fP3 .p0/gDP3;2.p0/. Figure 15.7 shows both probabilities P3.p0/ and P3;2.p0/ as a function of p0 with the ﬁrst one being contained within the second one. But if it is possible to solve the problem both ways for a very small hierarchy of the type (r D 3; n D 2/, it goes rapidly out of hand for larger hierarchies. To tackle the problem by considering the peculiar bottom distributions is therefore a more complicated and complex way to solve the problem than by using our level- by-level repeated voting scheme. However, this study leads quite naturally to the discovery of the possibility of “strategic nesting,” which is a very crucial ingredient for evaluating the social and political validity of the voting democratic procedures, as is demonstrated in the following section.

15.4.2 The Physicist’s Corner

Using the above findings makes it possible to shed light on the effect of fragmentation of a single group into smaller subgroups. We saw that for a hierarchy .r D 3; n D 2/, iterating P3.p0/ twice as required to fulfill our bottom up hierarchical election of a president is identical to calculating directly from the bottom all the winning configurations as given by P3;2.p0/ in (15.16). In parallel, looking at all the winning configurations from a single voting group of size 3 3 D 9 we get

9 8 7 2 6 3 5 4 P9.p0/ D p0 C 9p0.1 p0/ C 36p0.1 p0/ C 84p0.1 p0/ C 126p0.1 p0/ : (15.17) 284 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

Comparing it to (15.16) shows that all coefficients associated to the configurations .9A; 0B/; .8A; 1B/; .7A; 2B/; and.6A; 3B/ are identical. The differences arise from both the coefficient of configuration .5A; 4B/, which is 126 in (15.17)as against 99 in (15.16), and the additional term in (15.16), which accounts for a contribution from the minority configuration .4A; 5B/ with a coefficient of 27. 4 5 This last term 27p0.1 p0/ is exactly the one that is associated to the antidemocratic configurations found above. By symmetry between the two opinions, similar configurations also exist in favor of the B opinion with the possibility of a winning B minority of four agents. This is why, on the one hand the coefficient of 126 associated to the configuration .4A; 5B/ in (15.17) is reduced to 12627 D 99 in (15.16), and on the other, there exists a symmetrical contribution in favor of A in (15.16) with the coefficient 27 at the expense of B from the configurations .4A; 5B/. It thus appears that the fragmentation of the group of nine agents into three separate groups of three agents each, has created the possibility of the above reversal of the majority rule with a bottom minority winner but restricted to the case of the ratios of 4 to 5 and 5 to 4. The numbers and values of these ratios are a function of both r and n.

15.5 From Rare Antidemocratic Events to the Radical Efﬁciency of Geometric Nesting

Up to now, we have considered a population that is composed of well-behaved agents who follow faithfully the fair rules of the democratic game set up to build a bottom-up hierarchy. Nevertheless, the discovery of the existence of antidemocratic bottom conﬁgurations opens the way to the common practice of human beings of optimizing the rules for their own proﬁt. This means that it is possible to bypass the rules of a given system by using the rules themselves, while having in mind a good understanding of the “geometry” of the system. Then, the strategy is to achieve geometric nesting, which in our case would be by acting on the distribution of agents within the bottom voting groups. This could be achieved in particular by optimization and correlations.

15.5.1 Monitoring the Rare Antidemocratic Bottom Conﬁgurations

The main concept is to turn the rare antidemocratic bottom configurations into a controlled deterministic event. Indeed, by directly monitoring the distribution of the opinion amongst the members within the bottom voting groups, a rather small group is enough to infiltrate the pyramidal organization and obtain the top leadership by democratic means, without changing the rules. To achieve such a pacific antidemocratic takeover, the group needs to figure out the right overall geometry to maximize its voting power. 15.5 From Rare Antidemocratic Events to the Radical Efficiency of Geometric Nesting 285

A A B

A A BBAA BBB

AAB AAB BBB AAB BBB AAB BBB BBB BBB

Fig. 15.8 The optimized distribution of 8 A-agents against 19 B-agents within the bottom voting groups of a .r D 3; n D 3/ hierarchy, which in turn ensures the A certain democratic winning of the presidency

rC1 This task would require for the nesting group to have firstly, no more than 2 of its agents in a single voting group of an odd size of r. Secondly, the group should rC1 rC1 “occupy” at this 2 rate only a number of 2 adjacent groups. It can then ignore all other groups with a zero presence. Thus, it will get the minimum number of representatives at the upper level, who in turn will guarantee the election of the second level members, and so on and so forth. An illustration of such a nesting distribution is shown in Fig. 15.8 for the case of a .r D 3; n D 2/ hierarchy. There, a minority of 8 A-agents, who are strategically nested, are sufficient to hold up the presidency against the majority of 19 B-agents. It is worth stressing that this antidemocratic bottom configuration can in principle always occur by chance under a random selection of the voting agents. However, it is with a very low probability, as appears from its calculation. The first contribution comes from the 8 19 probability of having 8 A-agents 8 and 19 B-agents and is equal to p0.1 p0/ . The second contribution arises from the number of distinct rearrangements of this minority winning configuration, which scores .3 3 3/2 3 D 2187.Thefinal result is 8=19 8 19 P3;3 D 2187p0.1 p0/ ; (15.18) whose variation as a function of p0 is shown in Fig. 15.9. Its peak reaches the 8 maximum value of 0.00016 at p0 D 27 . This proves how little chance there is of the occurrence of any such family of antidemocratic bottom configurations. It is therefore the collective global monitoring of a given nesting group which may transform an extremely rare event into a certain one. The 8 versus 19 antidemocratic bottom configuration corresponds to the minimum minority size to reach the presidency. There exists many more antidemocratic bottom configurations covering the range from 8/19 up to 13/14, which all yield a democratic victory to the minority, provided their distributions are antidemocratic. 286 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

8 19 p 3,3

0.00015

0.000125

0.0001

0.000075

0.00005

0.000025

p0 0.2 0.4 0.6 0.8 1

8=19 Fig. 15.9 The probability P3;3 .p0/ of bottom antidemocratic rearrangements for a .r D 3; n D 3/ hierarchy with the 8-A agents against 19 B-agents (15.15). The maximum value is 0.00016 p D 8 at 0 27

15.5.2 When the Radical Efﬁciency Turns Nasty

From revisiting the magic formula section, we discovered that, given an existing .r; n/ hierarchy, a phase diagram to the presidency can be constructed as a function of the support p0 for one of the two competing opinions. There exist three different phases. Two are deterministic, which means that in each one of them, the outcome of the elected president can be cast with certainty in favor of one of the two opinions, independently of the variation of p0 within the associated phase. It thus allows the estimation of how much a decrease in support of a particular tendency, which is in power, can eventually be accommodated without losing new elections. In between these two phases stands a probabilistic phase where the election outcome cannot be predicted more precisely than in terms of respective chances. The range of this uncertain phase is a function of the two independent variables r and n, the voting group size and the number of hierarchical levels. It shrinks with an increase in either one or both of them. The complete phase diagram is shown in Fig. 15.10 with the two magic thresholds given by (15.12)and(15.13). For ﬁxed values of the pair .r; n/, the power phase diagram reduces to one dimension as a function of the unique variable p0. In the case of the .r D 3; n D 3/ p3 1 . 3 /3 0:20 p3 hierarchy considered above, we have 3;B D 2 2 and 3;A D 1 . 3 /3 0:80 2 C 2 . This is shown in Fig. 15.11. To win the presidency with certainty, the A opinion must attract more than eighty percent of support from the total population. But as we have discovered, using 15.5 From Rare Antidemocratic Events to the Radical Efﬁciency of Geometric Nesting 287

r 5 8 10 10 15 n 6 4 2

0.8

0.6 n n p r,A ,p r,B 0.4

0.2

Fig. 15.10 The phase diagram for winning the presidential election in a .r; n/ hierarchy as a function of the A support p0. The probabilistic phase is delimited by the two magic thresholds respectively (15.12)and(15.13)

B-president A-president

Probabilistic president p 0

0 0. 20 0. 80 1

Fig. 15.11 The phase diagram for winning the presidential election in a .r; n/ hierarchy as a function of the A support p0. The probabilistic phase is delimited by the two magic thresholds respectively (15.12)and(15.13). Here it corresponds to a .r D 3; n D 3/ hierarchy for which p3 D 1 . 3 /3 0:20 p3 D 1 C . 3 /3 0:80 3;B 2 2 and 3;A 2 2 a nesting strategy makes it possible to get rid of these global constraints. While antidemocratic bottom configurations are by nature a rare event and should not destabilize the voting system, in contrast, a nesting strategy could do so as it is the tool for making certain a very rare event. But a group performing a nesting strategy by taking over an antidemocratic bottom configuration can indeed hold a fair and democratic system in hostage. It turns into a totally nasty practice when it is not seen by the other parties. Therefore, in such a situation, in principle, a zero density proportion of a given opinion may find itself holding up the presidency of a democratic structure, provided the “gang” knows and understands the required geometry. To calculate the minimum numbers of nesting agents to hold up a democratic bottom-up voting hierarchy we come back to the implementation of the optimizing r rC1 task of a majority rule vote. For a group of size , it is sufficient to have 2 agents. Any additional agents are superfluous with respect to winning the vote. Then the 288 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

r +1 r 1 2 2

A … A B … B

AAA…ABB…B … AAA…ABB…B

r +1 r 1 r +1 r 1 2 2 2 2 BBBBBB…B … BBBBBB…B r r +1 r 1 r 2 2

Fig. 15.12 An antidemocratic bottom configuration for a n D 2 hierarchy with voting groups of size r featuring the case of the minimum number of A-agents to win the presidency goal is to win the vote at the first hierarchical level where the elected representatives elect a higher level representative. The same goal applies there of just having a rC1 number of 2 representatives. rC1 To reach it requires winning the votes of 2 distinct bottom groups from the larger ensemble of r groups, which elect the r first level representatives, who in turn vote for the second level representative. All other groups can be totally ignored since it is of no use to have any agent in them to win the presidency. The corresponding scheme is shown in Fig. 15.12. The associated number of A-agents present at the rC1 rC1 rC1 2 bottom is thus . 2 /. 2 / D . 2 / . This configuration can be modified by performing all permutations within the groups and within the group lc;r;n of groups, and yet achieving the same performance of electing an A president. To add one hierarchical level requires reproducing it r times in the structure of Fig. 15.12. Extending the process to an n level hierarchy thus requires at the hierarchy bottom a minimum number of A-agents equal to, r 1 n l C c;r;n D 2 (15.19) 15.5 From Rare Antidemocratic Events to the Radical Efficiency of Geometric Nesting 289

Fig. 15.13 The evolution of rc,r,n c;r;n for three ﬁxed values of 1 the voting size groups with r D 3; 7; 35 as a function of the number of levels n 0.8

0.6

0.4 r 7

r 3

0.2

r 35 n 0 2 4 6 8 10

N 0 rn with respect to a total number of G;r D agents. The associated proportion is 1 1 n c;r;n D 1 C 2n r 1 n 1 C ; 2n r (15.20) which goes to zero rather quickly as seen from Figs. 15.13 and 15.14.Thefirst one shows the evolution of c;r;n for a fixed value of the voting size group r as a function of the number of levels n. In the second one, it is n which is fixed and r which varies. The tremendous advantage in terms of manpower appears explicitly from both figures. Already at r D 7, the dependence on r vanishes with respect to the main contribution 2n. For a fixed value of n, the evolution does not change much above a level of 5. Of course, our examples sound totally exaggerated, and they are. But the purpose is not to describe a precise real situation, but on the contrary, to illustrate the possibilities, which do exist in real organizations, of completely diverting the democratic dynamics at the expense of the majority of the people, and to the advantage of an infinitesimally small group playing the structure against the democratic system. At the same time, it also shows up new possibilities for thwarting these “geometric coups.” A particular example would be by using a random redistribution of people from their different committees. 290 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

Fig. 15.14 The evolution of rc,r,n c;r;n for three ﬁxed values of 1 the number of levels with n D 3; 5; 10 as a function of the voting size groups r 0.8 n 1

0.6

n 3 n 7

0.4

0.2

r 0 5 10 15 20

15.6 More About Hierarchies

Before moving to the next major step in our analysis of the various effects of bottom-up local majority rule voting we mention several additional aspects of a hierarchical system, which improve the realistic part of our model. Having stressed the mechanisms by which the hierarchy operates, it now becomes possible and desirable to relax some of the constraints so as to move it closer to some real-life situations.

15.6.1 Randomness Is Sufﬁcient at the Bottom

In the definition of the rules of building up the hierarchy, the elected representatives need to be reshuffled at each level with a random distribution within the new voting groups. Such a procedure is redundant since once we have a random distribution at the bottom, the evaluation of the probabilities at higher levels follows the repeated iteration of the same formula p1 D Pr .p0/ from (13.3) with pn D Pr .pn1/. Indeed, the fact that p0 is a probability automatically gives the status of probability to all subsequent pn. In other words, without reshuffling at level n, the probability of finding an A-agent is still pn. In parallel, on average, the density of elected A-agent representatives at level n is also pn. 15.6 More About Hierarchies 291

Nevertheless, it should be emphasized that if reshufﬂing elected representatives at higher levels is not required to ensure the validity of the equations, it is a very powerful tool for neutralizing possible tentatives of geometric nesting. A random redistribution of agents at higher levels can thus increase substantially the preserving of the democratic character of the whole process, while involving smaller numbers of agents.

15.6.2 Geography and Multisize Combination of Voting Groups

Throughout the current presentation, we have considered bottom-up democratic hierarchies in which all voting groups, whatever their level position, have the same unique size r. However, real hierarchies involve voting groups of different sizes, which may vary from very small values like r D 3 up to very large ones with r D 103 or much higher. These various sizes are usually shaped by the geometry of the existing hierarchy, which determines the constitution of the voting groups. And most often the hierarchy structure is based on the geographical extension of the organization. Accordingly, the pyramidal structuring follows some archetype, which articulates along a series of guidelines to account for both the geographic spreading of agents and the associated heterogeneity of their respective support for the two competing opinions. In its simplest form, the various level steps may be summarized as being connected to area committees, localities, districts, states, and national representation before the president. Usually, the bottom groups are of much larger sizes than higher level groups. One emblematic illustration is the two level American voting system for the President where the bottom groups are the various American states, each having millions of voters, while the ﬁrst level is the Electoral College composed of a few hundred presidential Electors, and the second is the President. • The committees:

The bottom level is constituted through various committees spread throughout the entire territory. Each committee reﬂects the local tendencies of the people from the area surrounding it. Moreover, it is not connected to other committees. Their number being measured by the integer L0, we denote each one of them by the index j with j D 1; 2; :::; L0. Different tendencies could then be expected in different parts of the national territory, making the value of former p0 a heterogeneous variable, which may well vary from one area to another. It thus depends on the speciﬁc committee j j with a value p0 for the associated A support. In addition, these committees can j have different sizes of r0 . 292 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

• The localities:

Depending on geographical, economical, and historical reasons, each locality, associated to an index i, is formed of elected representatives from a certain i number r1 of committees. Often they are geographical neighbors. Their number is denoted by L1 with i D 1; 2; :::; L1. The probability of having committee j sending an A-representative to a locality i is given by the function

i j;k j j p1 D P0 .p0 /; (15.21)

which extends and generalizes the former p1 D Pr .p0/ from (13.3). For a i committee i of size r1, the average proportion of A-agents becomes

ri X1 1 i j;k pi D p ; (15.22) 1 ri 1 1 kD1

where a different speciﬁc committee j is associated to each value of k.

• The districts:

Localities in turn constitute districts which cover larger parts of the territory. In a similar manner to that for the locality formation, each locality elects a representative to be a member of the district to which it belongs.

• The states:

As for the building of our previous hierarchies, the preceding process is repeated to constitute the state levels. Each district sends its elected representative to the state to which it belongs.

• The national representation:

Each state then elects its representative to be a member of the last voting group, which is the national representation.

• The presidency:

The last step is obtained from the national representation, which elects the president. 15.6 More About Hierarchies 293

r j Level n with Ln groups of various sizes n

L 2 xx…A…xxx n 1 2 … i … Ln 1 r 1, 2, ……...k, …………rn r rn n n i

j i j k j j j j,r j p , = R p ,1 p ,2 p n 1 k = r n n 1( n 1, n 1,... n 1 ) with 1,2,..., n

j xx…A…xxx L 1 2 j n 1 L 1 2 … 1, 2, ……...l, …………r … r n 1 r r n 1 n n 1 n 1 1

j l j p , with l = r n 1 1,2,..., n 1 L r j Level (n-1) with n 1 groups of various sizes n 1

i j;k Fig. 15.15 The probability pn that a group j from level .n 1/ sends an A-representative at position k in the group i at level n.Atlevel.n 1/,theyareLn1 different voting groups k of respective sizes rn1 with k D 1; 2; :::; Ln1. At a higher level nLn different voting groups k are constituted with elected representatives from level .n 1/. Their respective sizes are rn with k D 1; 2; :::; Ln

The series of the above steps of the detailed building up of the hierarchy can be formalized quantitatively in a general form as j i j;k j j;1 j;2 j;r0 pn D Pn1 pn1;pn1; :::pn1 ; (15.23)

i j;k which determines the probability pn that a group j from level .n 1/ sends an A-representative at position k in the group i at level n. Now, within a voting group each position has its own probability of having an elected A representative, that depends on the composition of the associated voting group. The process is depicted in Fig. 15.15. The above case with area committees, localities, districts, states, national representation, and president corresponds to n D 5. With (15.23) and Fig. 15.15 a general algorithm is now constituted. This equation makes it possible to solve any hierarchical bottom-up structure by evaluating the associated parameters, which are the respective values of n, the number of levels, k Ln, the number of different voting groups at level n, rn , the respective sizes of the 294 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

j j j;1 j;2 j;r0 voting groups k D 1; 2; :::; Ln, and the various functions Pn1.pn1;pn1; :::pn1 /. Once the above dimensions have been evaluated, the associated phase diagram to 1 power can be built. Since it was found that 2 is the unique unstable ﬁxed point for all voting groups of any size r, this implies that it remains the same for any combination of sizes.

15.6.3 A Digression of About a Fifty Percent Score: What Is the Meaning of a Majority?

Throughout our study, we have encountered several times various difficulties, which all arise when the competing opinions score very similar overall supports in the vicinity of 50%. The deep origin of these qualitative and quantitative “difficulties,” which emerge at around 50%, is rooted in the “physical nature” of what is at stake at 50%. It was touched upon in the section on phase transitions and critical phenomena in Chap. 15. Namely, it is the critical threshold for a so-called phase transition of second order which occurs in our hierarchical system. The associated critical phenomenon is at the heart of a huge part of all collective changes which take place in nature. Nevertheless, we could instead question the validity of the democratic dogma of all power to the majority, precisely when the voting conditions correspond to very close voting. The fact of giving 100% of the power to the opinion which won 50% C one ballot of the votes could shed a doubt on the legitimacy of such a procedure. The principle behind the democratic paradigm is to avoid making a collective decision to depend only on a few people. Near 50%, even if millions of agents vote, the net result eventually depends on only the choice of a few thousand agents. Even if, contrary to authoritarian systems, these very people are unknown, the fact that a collective decision is linked to a few number of people is not very satisfactory. But here the difficulty could be circumvented. Due to such a lower limit of the only one additional single vote syndrome, we could revisit the democratic dogma of majority to nuance it a little bit. Indeed, on the one hand it sounds fair to give full power to a clear majority, let us say, of at least 53%. But on the other, it could sound illegitimate to do so when the difference between the current majority and minority is of the order of only a few percent. In that case, using our one random group voting scheme could provide an alternative to the designation of a president. It opens the way to building a radically new way of apprehending the foundations of democracy by introducing a probabilistic paradigm. Instead of giving so much power as a consequence of such a small difference in the number of ballots, the single random group voting scheme considers these ballots as being meaningless and substitutes them for a random outcome. In other words, in the range of 0:47 < p0 <0:53, voting would be monitored using a process that was made random on purpose. Within such a hypothetical frame, the References 295 current weak majority would be no longer certain to win the election. It only has a probability of doing so, but with a probability that is slightly higher than the one for the strong minority. In addition, we will discover later that in some conditions, which are becoming more and more common in western-like democracies, the dynamics of public opinion can paradoxically lead to a stable equilibrium state characterized by a perfect equality between the two competing opinions. In this case, the election outcomes are very close with less than one percent difference between the two leading candidates. Such a narrow margin creates a strong temptation to contest the results, which in turn, may produce the basis of social instabilities. To invent a new distribution of power in these very close elections is thus a vital goal in order to preserve the legitimacy of the general framework of majority rule voting, and in turn democracy itself.

References

1. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, J. of Math. Psychology 30, 426-434 (1986) 2. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943-951 (1990) 3. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191-200 (1991) 4. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organi- sations”, Models of self-organization in complex systems (Moses) Akademie-Verlag, Berlin V. 64, 53-59 (1991) 5. S. Galam, “Paradoxes de la rgle majoritaire dans les systmes hierarchiques”,´ Revue de Bibliologie, 38, 62-68 (1993) 6. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132-139 (1999) 7. S. Galam, “Real space renormalization group and totalitarian paradox of majority rule voting”, Physica A 285, 66-76 (2000) 8. S. Galam and S. Wonczak, “Dictatorship from Majority Rule Voting”, Eur. Phys. J. B 18, 183-186 (2000) 9. S. Galam, “Democratic Voting in Hierarchical Structures”, Application of Simulation to Social Sciences, G. Ballot and G. Weisbush, Eds. Hermes, Paris, 171-180 (2000) 10. S. Galam, “Building a Dictatorship from Majority Rule Voting”, ECAI 2000 Modelling Artiﬁcial Societies, C. Jonker et al, Eds., Humboldt U. Press (ISSN: 0863-0957), 23-26 (2001) 11. S. Galam,“How to Become a Dictator”, Scaling and disordered systems. International Work- shop and Collection of Articles Honoring Professor Antonio Coniglio on the Occasion of his 60th Birthday. F. Family. M. Daoud. H.J. Herrmann and H.E. Stanley, Eds., World Scientiﬁc, 243-249 (2002) 12. S. Galam, “Dictatorship effect of the majority rule voting in hierarchical systems”, Self- Organisation and Evolution of Social Systems, Chap. 8, Cambridge University Press, C. Hemelrijk (Ed.) (2005) 13. S. Galam, “Stability of leadership in bottom-up hierarchical organizations”, Journal of Social Complexity 2 62-75 (2006) 14. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 15. S. Galam, “Crier, mais pourquoi”, Liberation,´ Vendredi 17 Avril, 6 (1998) 296 15 Touching on a Fundamental Aspect of Nature, Both Physical and Human

16. S. Galam, “Le vote majoritaire est-il totalitaire?”, Pour La Science, Hors serie,´ Les Mathematiques´ Sociales, 90-94 July (1999) 17. S. Galam, “Citation in front page of the Figaro in an editorial from Jean dOrmesson”, Le Figaro, Mardi 4 Juin, 1 (2002) 18. S. Galam, “Risque de raz-de-maree´ FN”, Entretien, France Soir, La Une et 3, Mercredi 5 Juin (2002) 19. S. Galam, “Le FN au microscope”, Le Minotaure 6, 88-91, Avril (2004) Part IV The Risky Business of Alliances in Bottom-Up Democratic Voting with Three-Choice Competition Chapter 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

While we have shown up several democratic advantages in the use of bottom-up voting hierarchies, we can now undertake the deconstruction of the “beautiful world of democracy” to account for some very speciﬁc and inherent features of human behavior, and in particular, spontaneous wise “cheating.” We have already encountered the solid human tendency of cheating while discovering the natural existence of dangerous bottom conﬁgurations, which in turn opens the way to geometrical nesting. However, we will now realize that more fallacious cheating does not come from the motivated trend of a few people, but indeed from wise beliefs, which eventually emerge from human communities. Among these beliefs stands the sound belief that while facing a leadership, in the case of a tie one should preserve the current leadership. In others words, in order to turn down a given leadership you must get at least half the votes plus one. This cornerstone of the political setting is demonstrated to create a rather unbalanced bias in favor of the current ruling party. Indeed it makes any democratic shift of power almost impossible, turning the democratic bottom-up voting system into a democratic dictatorship [1–13].

16.1 The Inertial Effect of Being in Power

As seen in the previous chapters, within our bottom-up democratic hierarchical scheme, the requirement for a challenging party to exceed the threshold of 50% of full support so as to topple the leadership, is a fair constraint, which meets the mere notion of a democratic choice. In parallel, the inherent difﬁculties of implementing a given policy in practice from the top of any organization are expected most often to produce substantial dissatisfaction at the bottom of the organization at least in the ﬁrst stages of the implementation. However, it is of central importance to allow the completion of

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 297 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 16, © Springer Science+Business Media, LLC 2012 298 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures policies, which could be at first unpopular, but necessary and beneficial in the long term. It would be counterproductive to provoke regular and frequent shifts of top leaderships as a result of any decision or orientation. In other words, fluctuations in the mood of the overall community of agents must not drive automatically and immediately, shifts in the leadership. No organization would survive frequent drastic changes at the top. This is why, among other things, for a minimum of efficiency, designations of each level of the executives are not made every week or month, but usually on a time scale of one to several years. Nevertheless, there exists a qualitative gap between avoiding frequent shifts at the top of the leadership and preventing a new mature majority at the bottom to take over power. Unfortunately, or fortunately, depending on one’s interests, in real life, shifts of leadership seem to occur less frequently than expected. From an empirical observation, to stick to power seems easier than to get into power, whatever the general conditions. Such an observation leads to the statement that a clear, although not always well defined, asymmetry appears quite naturally in a large spectrum of social, economic, and political organizations between the ruling party and the challenging one. All kinds of tips are provided de facto and even from the law, so as to strengthen the stability of the current leadership. This is true for most human activities. Indeed, to be in charge gives substantial additional power, which breaks the symmetry in running for power, between the two positions of being respectively in power and in the opposition. And such a fact is usually well accepted by everyone since everyone understands that institutions need stability. However, it often appears that the purpose of avoiding erratic and frequent changes of leaders has evolved into an efficient system, whose main purpose becomes to oppose any political change. Mechanisms initially aimed to avoid unjustified volatile instabilities are often found to have created huge barriers so as to prevent any change. This is so even when it is strongly required and requested by a majority of the organizational members. These kick-backs are in part the result of structural manipulations such as “playing with the rules,” as encountered earlier for instance with the discovery of the natural existence of dangerous bottom configurations, which in turn opens the way to the possibility of geometrical nesting and its dictatorship counterpart. But other ways of fiddling with the system are unexpected and unseen from the normal functioning of the hierarchy. They are of a crucial importance since they are hidden, and thus it is impossible to act on them. This fiddling with the system is much more fallacious than the “playing with the rules” principle, since they are not the fact of a few motivated unfair people, but embedded in the democratic functioning of the system. They are based on a consensus shared by most people. On this basis, after having shown up several democratic advantages in the use of bottom-up voting hierarchies, we can now undertake the deconstruction of this “beautiful world of democracy” we have built, to account for some specific and inherent features of human organizations. Significant advantages exist by the mere fact that leading an organization produces many side effects in favor of the ruling party, which in turn can be 16.1 The Inertial Effect of Being in Power 299 apprehended from the well-observed fact that it is easier to stay in power than to gain power against the current leadership. We can thus postulate an empirical existence of some principle of inertia of the status quo at the top level of organizations. This inertia of power is the outcome of complicated and fuzzy mechanisms, which combining together, create it. Here, we tackle this effect in the way that physicists do, by implementing the simplest frame that is able to reproduce a similar effect of inertia within our scheme. Indeed our simple modeling of bottom-up voting systems provides a straightforward illustration of these complicated mechanisms that induce stability, which take place in real organizations. To make our point clear, we revisit the case of voting in even-sized groups. Before, voting to a tie in an even-sized group was attached to a probabilistic 1 outcome of 2 to elect either an A or a B. Such a choice was shown to make the voting functions obey P2w1.p0/ D P2w.p0/ for any integer w. The associated threshold for power was 50%. 1 Now we simply break the probability 2 tie symmetry to favor the current leadership. We then state that at a tie in an even-sized group, a member of the current leadership is elected. From now on, for the sake of simplicity, and without the loss of generality, we assume that B is in power, and that A is the challenging party. Such a rule makes P2w1.p0/ ¤ P2w.p0/. At a tie, a B is elected. In the following lines, we study the effects of such a tie break on the voting dynamics, while climbing up the hierarchy. To obtain a deterministic outcome at the top is still the goal of the construction. It is worth noting that a counterpart of our tie bias assumption exists in some real organizations where the tendency towards the status quo is materialized by granting one additional vote to the president of the committee. Requesting that in order to change a current leadership, a majority of at least one ballot is required, sounds more than a reasonable rule. Accordingly, within our simple scheme, a tie means no majority, which in turn means no change in the leadership. This arrangement does not appear to be so much of a significant advantage to the leading party since intuitively the probability of voting at a tie for the two competing candidates sounds a priori to be a little likely event. And indeed it is for large-sized groups. However, it is not the case for small-sized groups, as shown in (Fig. 16.1). By symmetry, the tie probability is expected to be symmetrical in p and .1 p/, r r r 2 from its evaluation, which can be written r p0 .1 p0/ 2 for any even value 2 1 of r. This equation shows that its maximum value always occurs at p D 2 .This maximum reaches its highest value 0:375 at r D 4. It then decreases for an increasing size of r and goes to zero for very large sizes with 0:120 at r D 44 and 0:038 at r D 444 as seen in (Fig. 16.1). Unfortunately, in the following, such a wise principle of requesting one more ballot to change a leadership is shown to break down the democratic balance by a dramatic proportion. What looks as being quite a reasonable rule at the level of one committee can become an incredible manipulation at a collective democratic aggregated level. Understanding the quantitative effect of the democratic twist 300 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.1 The probability of proba having a tie as a function of the proportion p in a group of respective sizes of 0.35 r D 4; 44; 444.The maximum value occurs at 0.3 p D 1 0:375 r D 4 2 with for . It decreases to 0:120 for 0.25 r D 44 and 0:038 for r D 444 and goes to zero for very large sizes 0.2 r=44

0.15 r=4

0.1 r=444 0.05

p 0.2 0.4 0.6 0.8 1 driven by such a wise arrangement opens the way to creating an amazingly efﬁcient tool for democratically imposing by voting a dictatorship in the hands of a minority. The subject of this chapter is to single out the mechanisms involved in this astonishing democratic making of a dictatorship. We revisit the previous steps (Chaps. 13–15) which are aimed successfully at building the perfect democracy, to demonstrate how the “reasonable” tie breaking vote turns the enterprise into a formidable machinery for producing dictatorships.

16.2 The Dramatic Effect of Tie Break Voting in the Single Random Group Voting Scheme

We start out by revisiting the previous investigation by extending the initial simplest case, where only three agents are randomly chosen from the population (Fig. 13.4), to four agents still randomly chosen from the population (Fig. 16.2). As before, a local majority rule is applied to elect the group representative with conﬁgurations f4A; 0Bg; f3A; 1Bg electing an A and f0A; 4Bg; f1A; 3Bg aB. Nevertheless, the additional tie conﬁguration f2A; 2Bg elects a B to embed what we have called “the inertia of power.” It breaks quite naturally the symmetry between the two competing parties depending on their respective status, B being in power and A in opposition. The probability for an A to be elected by a voting group of size 4 is thus 4 2 p1 P4.p0/ D p0 C 4p0.1 p0/; (16.1) 16.2 The Dramatic Effect of Tie Break Voting in the Single... 301

Fig. 16.2 Four agents are A B AA picked randomly from the B A BB B population of N agents B A B A B B supporting either opinion A AAB AAB A B or opinion B. The randomly BBAAB selected group of four agents The elects the president according A Population with N persons President to (16.1)

XXXX AAAA A AAAB (x4) Random selection of BBBB 4 agents ABBB (x4) B AABB (x6)

Fig. 16.3 The result for one p 1 vote in favor of A from a 1 group of four agents picked at random from a population Separator with a density p0 of A 77 % supporters and .1 p0/ for 0.8 B. The result is still probabilistic but now p1 p0 for p0 >0:77 B victory

Deterministic 0.4 A victory

0.2

p 0 0.2 0.4 0.6 0.8 1

instead of (13.4) where P4.p0/ denotes the new voting function for a group of four 2 2 agents. Earlier, without the tie breaking, we had the additional term 3p0.1 p0 /, which was then shown to reduce the expression to that of P3.p0/. In contrast, the probability that B would win the election is given by,

4 3 2 2 .1 p1/ D .1 p0/ C 4p0.1 p0/ C 6p0.1 p0/ ; (16.2) where the last term accounts for the tie conﬁguration. The voting function P4.p0/ is no longer a symmetrical function as for P3.p0/ shown in Fig. 13.5. The difference is striking and signiﬁcant as seen from Fig. 16.3. The part of the curve below the diagonal is now much larger. It indicates that the separator is no longer located at 50% but much higher above it. 302 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

In terms of political consequences, this result implies that the challenging opposition party needs to gain more than 50% of support to obtain the presidency. In contrast, the ruling party needs less than 50% of support to stay in power. For a voting group size of 3, an A minority of 10% gets a probability of p1 D 0:03 to win the election. The value shrinks to p1 D 0:00 for a voting group size of 4 with tie breaking. A value of p0 D 30% yielded p1 D 0:22 against a value now of 0:08. Before, an A majority of 70% had a higher probability of winning the election with 78%. Now its probability is lower at 65%. This last example demonstrates that 50% is no longer the frontier to get from the general support, either an improved or a decreased probability of winning the election in a single voting group. To be more precise, in the evaluation of the change induced in the new dynamics by (16.1), we determine the values of its fixed points by solving the equation P4.p/ D p. The former two fixed points pB D 0 and pA D 1 are unchanged. 1 However, the in-between fixed point pc D 2 has shifted to 1 p p .1 13/ 0:77: c;4 D 6 C (16.3) To determine their respective stabilities, we calculate as before the stability dPr .pn1/ r .p / p eigenvalue D dpn1 j , which comes from (16.1)

2 4 D 12p .1 p/: (16.4)

It gives zero for both pB D 0 and pA D 1 making them stable since it is less than one. As a consequence, by coherence of the dynamics, pc;4 must be unstable. It is indeed, since the associated value 4 1:64 is larger than one. The separator, for the A opinion voting dynamics, has thus jumped from the democratic threshold of 50% to the rather antidemocratic value of 77%forA. Simultaneously, for B, it has shrunk to 23%, making both situations drastically different from the perspective to remain in or to conquer power. The democratic balance has been broken into a highly undemocratic frame.

16.3 Varying the Voting Group Size

We now extend the single voting group scheme from a size of four agents to any even number r of agents, still randomly selected from the population. The voting designates an A agent for any one of the following conﬁgurations: frA; 0Bg, .r 1/A; 1B .r 2/A; B . r 1/A; . r 1/B f g, f g... , f 2 C 2 g. Adding their respective probabilities yields the probability p1 D Pr .p0/ for a group size of r to elect an A. r This is the asymmetric truncated binomial expansion of f1 C .1 p0/g with ! Xr r m rm p1 Pr .p0/ D p0 .1 p0/ ; (16.5) r m mD 2 C1 16.3 Varying the Voting Group Size 303

Table 16.1 Values of the Group Attractor Separator pc;r for Attractor various ﬁxed points for group size r B elected A threshold to win A elected sizes r D 4; 6; 8; 10; 12; 14; 16; 18; 20; 30; 40; 50; 70; and 100 2 0 1p None 1C 13 40 6 0.767 1 60 0.653 1 80 0.604 1 10 0 0.579 1 12 0 0.563 1 14 0 0.552 1 16 0 0.544 1 18 0 0.538 1 20 0 0.534 1 30 0 0.521 1 40 0 0.515 1 50 0 0.512 1 70 0 0.508 1 100 0 0.506 1

r rŠ k 0 where m mŠ.rm/Š is a binomial coefﬁcient. It is identical to (17.1) with D . k 1 A nonzero value of , that is larger than D 2 , makes it possible to soften the inertia principle. This effect is studied later at the end of the chapter. From (16.1) it is found that the two attractors pB D 0 and pA D 1 are independent of the group of size r. Nevertheless, the separator pc;r is a function of r as seen in Table 16.1. It reaches its largest value pc;2 D 1 for size r D 2.It then decreases with increasing r, with respectively 0:767 at r D 4, 0:512 at r D 50, 0:508 at r D 70 and reaches 0:506 at r D 100. For larger sizes, it follows the p 1 asymptotic limit c;r ! 2 . Holding a very small advantage in the vicinity of 50% may turn out to be decisive for winning an election process, in particular when the total opinion of the population is split into two almost equal parts. In contrast, any odd-sized voting p 1 group yields c;odd D 2 , the perfect democratic balance. Using (17.3), the associated functions Pr are illustrated for a series of values of r in Fig. 16.4. As for odd sizes, we can estimate, in the case of a tie breaking scheme, the discrepancy in the outcomes from an election, using on the one hand Pr and on the other, Pall , which always yields a deterministic outcome. Table 16.2 contains a series of values of p1, the probability of having an A elected from one single random voting group size of r, given a total bottom support p0.The values p0 vary from 0:000 up to 1:000 by successive increments of 0:100. For each one of these value, the associated p1 is calculated for group sizes of respectively r D 4; 6; 36; 116; 1116 and r D all. As expected, the discrepancy produced by using the single random group voting scheme depends on the group size r, as for the odd case. It is also found again, that while a probabilistic outcome is obtained for small sizes of r, a deterministic result is reached beyond some value of r. This value depends on p0. The deterministic 304 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.4 The result for one p 1 vote in favor of A from a 1 group of r agents picked at random from a population Separator p with a density 0 of A 77 % supporters and .1 p0/ for B. 0.8 Five different cases are shown 55 % for r D 6; 16; 36; 116; 1116. The result is still probabilistic Deterministic p 0.6 for some range of 0 and B victory Deterministic deterministic elsewhere. But A victory now the separator is seen to start at 0:77 at r D 4 and then 0.4 decreases toward 0:50 at r D 1; 116. Increasing the r=16 value of r reduces the r=6 difference with odd-sized 0.2 r=36 group voting. r=116 r=1116 p 0 0.2 0.4 0.6 0.8 1

Table 16.2 Given the total bottom support p0, the table gives the probability p1 to have an A elected from one single random voting group size of r Group size r 4 6 36 116 1;116 all Total support p0 p1 p1 p1 p1 p1 p1 0.000 0.000 0.000 0.000 0.000 0.000 0 0.100 0.004 0.001 0.000 0.000 0.000 0 0.200 0.027 0.017 0.000 0.000 0.000 0 0.300 0.084 0.070 0.003 0.000 0.000 0 0.400 0.179 0.179 0.083 0.011 0.000 0 1 0.500 0.312 0.344 0.434 0.463 0.488 2 0.600 0.475 0.544 0.854 0.982 1.000 1 0.700 0.652 0.744 0.991 1.000 1.000 1 0.800 0.819 0.901 1.000 1.000 1.000 1 0.900 0.948 0.984 1.000 1.000 1.000 1 1.000 1.000 1.000 1.000 1.000 1.000 1

The values of p0 vary from 0:000 up to 1:000 by successive increments of 0:100,usingfor each value the series of group sizes of respectively r D 4; 6; 36; 116; 1116 and r D all outcome obtained for a large enough size is still obtained by a monotonic decrease 1 1 (p0 < 2 ), or increase (p0 2 ), from the initial probabilistic value p1 corresponding to the case of r D 4. The ﬁrst qualitative difference with the odd case in Table 16.2 is for p0 D 0:600 and p0 D 0:700, p1

Differences induced by the tie breaking can be seen for values of p0 > 0:500. 1 1 For instance, p0 D 0:60 > 2 yields p1 D 0:475 < 2 with r D 4 against p1 D 1 1 0:648 > 2 with r D 3.Atr D 6, the balance is restored with p1 D 0:544 > 2 , while r D 5 yields the higher value of p1 D 0:683.However,atr D 116 and r D 115, the outcomes are almost equal with respectively p1 D 0:982 and p1 D 0:985. Both cases reach p1 D 0:000 at a size of r D 1;115. Yet, to gather 1,115 agents is certainly a much easier and less costly task than to have a few hundred thousand agents or more to actually go voting. 1 Similar results are obtained below 2 . An initial value of p0 D 30% yields p1 D 16% for a group of ﬁve agents, against p1 D 22% for three agents. Now we get p1 D 8%andp1 D 7% for respectively r D 4 and r D 6. It reaches p1 D 0:00% already at a size of r D 32 instead of r D 39 for the odd case. Therefore, given a population of any size, with two competing parties scoring respectively 30 and 70% of support, a randomly selected group of only 32 agents yields the same outcome, having the whole population voting. This equivalence holds true within the error bar as discussed earlier, here to within two digits. Increasing it requires a larger size to reach zero. We could infer the above similarities since increasing the voting group size reduces the tie breaking effect, and in turn reduces the discrepancy between odd and even sizes. Accordingly, we can conclude that using a unique and small (with respect to the total population) sized group, voting appears to preserve the all agent voting procedure.

16.4 From the Perfect Democratic Structure to the Perfect Democratic Dictatorship: The Simplest Case

In the odd-sized case, we saw that while using the single random small voting scheme, an effective construction to restore the democratic outcome at the top leadership, i.e., to ensure the presidency to the majority at the bottom, is to build a bottom-up voting hierarchy using the property p1 2 , which makes a very big difference, from a democratic perspective. Indeed it turns the whole democratic machinery of repeated bottom- up voting into a robust scheme to establish a dictatorship democratically as seen below. We use the word dictatorship in the sense of having a minority leading the organization. The paradox here is that while in a usual dictatorship the minority leads against the will of the majority by using force, here it does so according to the will of the majority by just making agents vote. This strange feature produces a very strong stability at the top of the leadership. 306 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.5 Many single voting groups of four agents are B A AA constituted. Each one elects a AA A B BB representative. In the case of AA AA B A AA AA a local tie, a B is elected. All BB elected representatives BB BB AA constitute a new population AA AA A B AA

B B B A A A B AA New population of elected representatives

We now demonstrate this statement, ﬁrst in the simplest case of voting groups of a size of 4, for which pc;rD4 0:77. We keep to the same goal as for the odd case, i.e., to reach a deterministic outcome of the elected president. It is again the property p1

p > 1 p > 1 where the C corresponds to 0 2 and the to 0 2 . Both curves are combined in Fig. 16.6. Contrary to what was found in the odd case, the initial amplitude d0 is not always increased. Four different regimes are found, of which three occur when d0 0:61. It corresponds to 0:23 < d0 < 054. 2. However, for 0:50 < p0 <0:61the majority–minority status is reversed at once 1 with p1 < 2 whatever p0. This situation corresponds to 0

d 1 1

0.8

0.6

0.4

0.2

d 0 0.2 0.4 0.6 0.8 1

Fig. 16.6 The amplitude of the difference in the respective support for both opinions using single voting groups of four agents. Four different regimes are obtained. When 0 d0 < 0:535, d1 0 p < 1 d >d Then, when 0 with 0 2 the amplitude always satisﬁes 1 0 (upper dashed curve). After some iterations of local voting, the collective vote eventually reaches 100% for one opinion. It is B when d0 < 0:535, i.e., p0 <0:77and A otherwise

1 status in the third regime. Indeed d0 D 0:23 yields d1 D 0,i.e.,p1 D 2 with p0 D 0:61, which shows that any lower value results in a minority support. In other words, an A majority of 61:4% shrinks within one voting cycle to 3 50%. Then from 50%, i.e., d D 0, it declines to d1 D 8 0:37. Accordingly, (16.6) produces a self-elimination of a majority to eventually elect a president belonging to the minority provided p0 2 with p0 d1 >d2 > ::: > dm where m corresponds to the ﬁrst level at which dm 0:23.There,the 308 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.7 A single level B hierarchy with local voting groups of a size of 4. It requires 16 agents chosen at random at the bottom for a AB AB total of 21 agents including the president. In the ﬁgure, 11A and 5B are present, yielding a tie at the upper level and eventually a B president AAAA B AA B AAA B B AA B

bottom A majority becomes the minority. Simultaneously, the bottom minority B gains the majority status. Then, the amplitude starts to inﬂate, dmC1

A B B A

AABA ABAB BB AB AAAB

A A B A A A B B B A A B A A A B A A B A A B A B A A A B A A A B B A A B A B A A A B B A A A B A A B B A A A A A B B A A B A A A

Fig. 16.8 A two level hierarchy with local voting groups of a size of 4. There are 4 16 D 64 agents at the bottom of which 42 support the A opinion, giving to it a large majority. The B scores only 22, one more than half the number of A. At the ﬁrst level, 16 agents are elected, with 9A and 7B. The A preserves its majority although it is a bit weaker. The repeated election process yields a tie at the second level where one single group is present with 2A and 2B. The president is a B

Ground people AA AA AA A B AA AA BB B A AA BB BB AA AA B A A B AA AA A B AA BB A B A B AA AA AA AA AA BB BB B A AA AA

B B A A Representatives B B AA A B A A A B A A First level

A B A B B B B B A A A A Form new groups A A A A

BBA A Second level

President B

Fig. 16.9 The same two-level democratic bottom-up hierarchy as in Fig. 16.8 with a larger population of 44A agents. The B at 20 gets less than half the A support. At the ﬁrst level 10A and 6B are elected. A tie is obtained at the second level, which yields a B president. The hierarchy is drawn with the bottom above and the president below 310 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

President: Level 3

Level 2

Level 1

Bottom: Level 0

Fig. 16.10 Another schematic view of the precedent democratic bottom-up hierarchy with an even larger support for A with 45 agents. The B are down to 19. At the ﬁrst level, A is still in majority with ten elected representatives and only six for B. But a tie is still obtained at the second level, leadingthentoaBpresident

Thus, building five successive hierarchical levels between the bottom and the top guarantees first, a reversing of the democratic balance after three levels, and second, the keeping in power of the current minority within two additional levels. Five levels are thus sufficient to democratically keep power for the minority of only 30%atthe bottom, so implementing a democratic hierarchical bottom-up dictatorship. The dynamics is now rather different from the one obtained in the case of odd- sized groups where the same five level hierarchy restores the democratic balance with a bottom minority of p0 D 0:40. There we found the series p0 D 0:40, p1 D 0:35, p2 D 0:28 and p3 D 0:20, p4 D 0:10, p5 D 0:03 and p6 D 0:00.However, more agents (5,461) are required in the case of r D 4 than for r D 3 (1,093). The associated amplitudes in the difference in the respective supports are respectively for the shrinking part d0 D 0:50, d1 D 0:30, d2 D 0:14 and d3 D 0:16, d4 D 0:60, d5 D 0:94, d6 D 1:00 for the inflating one. Contrary to the odd case, climbing up each new hierarchy higher level does not systematically amplify the initial majority at the bottom, but quite the opposite, it may well shrink it. Nevertheless, contrary to a brutal dictatorship using coercion to enforce its power, the democratic dictatorship allows in principle a power shift at the bottom. However, more than a huge majority is required at the bottom before it may happen. For instance, if the A opinion scores a p0 D 0:80 support at the bottom, it gets the leadership, as seen from the series p0 D 0:80, p1 D 0:82, p2 D 0:85, p3 D 0:89, 16.4 From the Perfect Democratic Structure to the Perfect... 311

Fig. 16.11 A democratic bottom-up hierarchy with 1; 024 groups of four agents at the ground level, totaling 4;096 agents. The ﬁrst level is built out of 256 groups of four agents giving 1; 024 elected agents. The second level has 64 groups of four agents giving 256 elected representatives. The third level has thus 16 groups with a total of 64 agents. The fourth level shrinks to four groups of four below the ﬁfth level of one single group of four elected agents. Adding all the agents involved and the elected president yields 4;096 C 1;024 C 256 C 64 C 16 C 4 C 1 D 5;461 agents. Here we have the series p0 D 0:70, i.e., a huge bottom support in favor of A, p1 D 0:65, p2 D 0:57 and p3 D 0:42. The ratio of majority/minority has been reversed. Going further up yields p4 D 0:20, p5 D 0:03 and p6 D 0:00. The result may turn probabilistic at the top, depending on the value of p0

p4 D 0:93, p5 D 0:98 and p6 D 1:00. In this case, which corresponds to p0 >pc;r, climbing up the hierarchy systematically ampliﬁes the initial majority at the bottom with d0 D 0:60, d1 D 0:64, d2 D 0:70 and d3 D 0:79, d4 D 0:86, d5 D 0:96 and d6 D 1:00. From the above illustration, we expect, as for the odd case, the existence of some

ﬁnite number nc of levels, such that pnc D 0 for any value p0 D apc;r we reach pmc D 1 for a different number of levels mc ¤ nc . Notice that 2pc;r a D 1 a p 1 only when c;r D 2 . Therefore, given any initial value of p0 at the bottom, there always exists a series of successive nc bottom-up levels, characterized at each level i D 1; :::; nc ,bya proportion pi D Pr .pi1/ of A elected representatives. The associated series p0, p1..., pnc obeys either p0 >p1 > ::: > pnc D 0 if p0 pc;r. 312 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

To sum up the above results, when voting groups are of a size of 4, to get to power, A must exceed 77% of the overall support, which is virtually out of reach in any normal democratic situation, where two parties compete for power. On the contrary, to stick to power, B only needs to keep its overall support above 23%. It is not a minor difference and leads to the following conclusions. (a) These results can be seen as the signature of a dictatorship effect. Majority rule voting has become a machinery to produce a dictatorship with the efﬁcient self- democratic elimination of a huge majority. (b) On the other hand, they could be interpreted as being a possibility given to a current leadership of conducting an unpopular policy, which is nevertheless necessary, and thus preserving its leadership position against an eventual fall of support during the implementation of the policy. Afterwards, the same leadership can recover a majority support at the bottom, once the positive effects of its unpopular policy have materialized. Whatever view is shown up, it is of a basic importance to be aware of both, as well as the existence of thresholds to power, and the fact that they are of different values with a drastic asymmetry in favor of the current leadership. Therefore, to know precisely what are their respective values is of primary strategic importance. In addition, for a group size of 4, the tie driven bias makes the number of levels recover a quasi deterministic outcome even smaller than in the case of a voting group size of 3. But now the deterministic outcome can be opposite to the deterministic result of the whole population voting. For instance, from p0 D 0:45 we obtain the series p1 D 0:24, p2 D 0:05 and p3 D 0:00. Three levels are now enough to make the A disappear from the elected representatives instead of eight levels. To emphasize the above democratic self-elimination effect, let us start from a support of p0 D 0:70 infavorofA,which is much higher than 50%. Climbing up the hierarchy, the associated voting dynamics yields for the various densities of representatives p1 D 0:66, p2 D 0:57, p3 D 0:42, p4 D 0:20, p5 D 0:03 and p6 D 0:00. Amazingly, 70% of a population is thus democratically self-eliminated within six voting levels. The current majority becomes a minority within only three levels.

16.5 The Dynamics Driven by Repeated Democratic Voting

We now proceed to evaluate, given an initial support p0

Fig. 16.12 The density of elected representatives ﬂow diagram. An initial support p >p > 1 0 c;r 2 leads to an election at 100% for A while p < 1 0 2 leadstoanelectionat 0%. The respective ranges of attraction of the two attractors located at pB D 0 and pA D 1 are not equal and delimited by the separator p > 1 c;r 2

1 1 by the separator pc;r > 2 with pc;rD4 0:77. An initial support p0 >pc;r > 2 100 p < 1 0 leadstoanelectionat % for A while 0 2 leads to an election at %. We thus start from C 1 1 pc;r nc;r ln (16.7) ln r p0 pc;r and 1 pc;r nc;r ln ; (16.8) ln r pc;r p0

dPr .pn1/ where we used p C D 1, pn D 0 and r .p / D jp . The upper index nc;r c;r dpn1 C means that respectively p0 >pc;r and p0

16.5.1 The Physicist’s Corner

The various values of pc;r for r D 2 to r D 100 are given in Table 16.1.The associated variation as a function of the voting group size is shown in Fig. 16.13. p 1 It also includes the odd-sized cases where all points sit on the straight line c;r D 2 . 1 For even sizes, pc;r starts from a value of 1 and then goes asymptotically towards 2 . The various values can be ﬁtted by 0:432 pNc;r 0:398 C ; (16.9) logŒr which is shown in the upper part of Fig. 16.13. 314 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

p c,r

0.9

0.8

0.7

0.6

0.5

r 10 20 30 40 50

p c,r 1

0.9

0.8

0.7

0.6

0.5

r 10 20 30 40 50

Fig. 16.13 The variation of pc;r as a function of the voting group size r is shown for both odd pN D 1 (triangles)andeven(large dots) sizes. For the first case, it fits perfectly the straight line c;r 2 . 1 For even sizes, it starts from a value of 1 and then goes asymptotically toward 2 .Intheupper part, the fitted curve pN shown is obtained by using (16.9)andthelower part uses (16.10)

Looking at the ﬁtted curve from (16.9), it appears that it yields good results for the data used, i.e., for r D 2; :::; 50. Unfortunately the limit when r !1is wrong p 0:398 1 since then Nc;r ! ¤ 2 . While it is possible to impose the right limit to the ﬁtting function, the price to pay, if keeping the one over log dependence, is not worth it since the accuracy in the region of interest for r 100 would be worse. Otherwise a more complicated formula should be used, losing part of the simple“feeling” about the behavior of pc;r as a function of the size of r. For instance, we could use 1:402 0:149 pNc;r 0:512 C ; (16.10) r logŒr 16.5 The Dynamics Driven by Repeated Democratic Voting 315

na ,n a c c n c ,nc 35 17.5 30 15

25 12.5

20 10

15 7.5

10 5

5 2.5

p p0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

Fig. 16.14 The variation of nNc;r (step function)andnc;r (continuous curve) for the two voting group sizes r D 4 (left part)andr D 10 (right part) as a function of 0 p0 1.Both curves exhibit a logarithmic singularity at the unstable ﬁxed point separator pc;r D 0:77 (left)and pc;r D 0:58 (right). Less stairs are required for the larger size r D 10 which is shown in the lower part of Fig. 16.13. The large size limit is much better 1 with pNc;r ! 0:512 but we have one additional term in r . It is often more fruitful to deal with simple expressions, even if the precision is not excellent. We could add more terms so as to improve the accuracy, but what for? We now examine the values of the numbers of levels given by (16.7)and(16.8) by taking, as for the odd case, the integer part and adding C2 with C; C; nNc;r N nc;r C 2: (16.11)

C; C; Figure 16.14 illustrates the dependence of nNc;r (step function) and nc;r (continuous curve) for the two voting group sizes r D 4 and r D 10 as a function of 0 p0 1. Both curves show a singularity at the unstable ﬁxed-point separator, pc;rD4 D 0:77 (left, r D 4)andpc;rD10 D 0:58 (right, r D 10). In their immediate vicinity, many voting levels are required to signiﬁcantly enlarge the difference in the respective supports while climbing up the hierarchy. For identical p0, less stairs are required for the larger size of r D 10, but with a lower pc;r. Table 16.3 compares the exact numerical estimates for nc;r obtained by successive iterations of (16.5) to the values obtained from (16.11) for both cases p0 D 0:49 (used in the similar table for odd sizes) and p0 D 0:59 as a function of the series of voting group sizes of r D 4; 6; 8; 10; 12; 14; 16. The accuracy is rather good, as seen from the table. To appreciate the changes induced by the tie effect, data for the odd case are also exhibited for p0 D 0:49. The speeding up of the dynamics appears from examining the table with much less levels for the even case than for the similar odd one. Indeed, for the even case, three levels are enough to self-eliminate democratically a minority 316 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

e Table 16.3 Exact numerical estimates nc;r using (16.5) and the equivalent values obtained from (16.7)or(16.8) for the bottom value p0 D 0:49, as a function of the series of voting group sizes r D 4; 6; 8; 10; 12; 14; 16 p0 D 0:49 r D 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 e nc;r 3333 3 3 3 nc;r 2.05 1.99 1.98 1.98 1.98 1.97 1.95 nNc;r 4333 3 3 3 p0 D 0:49 r D 3rD 5rD 7rD 9rD 11 r D 13 r D 15 e nc 128666 5 5 nc 9.65 6.22 4.99 4.34 3.93 3.64 3.42 nNc 118665 5 5 p0 D 0:59 r D 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 e nc;r 4565 4 3 3 nc;r 2.95 3.35 4.45 nNc;r 456 C nc;r 3.81 2.68 2.22 1.96 C nNc;r 5443 Similar results for the odd cases r D 3; 5; 7; 9; 11; 13; 15 are reproduced. The case of p0 D 0:59 is also investigated. There, depending on the value of r, the initial value p0 D 0:59 is below pc;r for r D 4; 6; 8 and above for r D 10; 12; 14; 16 of 49%. The process of reaching a deterministic outcome being always quicker than e in the odd case, with nc being always smaller, it makes it lighter in terms of the agents involved and the associated infrastructures for the building up of a bottom- up democratic hierarchical dictatorship. However, while a majority of 59%isself- eliminated for r D 4;5;8,itwinsforr 10. For the odd case it has been possible to evaluate the instrumental quantities for the general case of voting groups of size r, due to the fact that the separator r p 1 is independent of with c;r D 2 . This is not the case for even sizes due to the mathematical difﬁculty of solving analytically a polynomial of a degree larger than 3. Nevertheless, the problem can be solved numerically for each value of r.

16.6 From the Magic to the Machiavelli Formula

In the odd case, we were able, given a bottom-up democratic hierarchical structure, to determine three different situations for each opinion, according to the outcome of the voting process at the top level of the hierarchy. Given an opinion, in the ﬁrst situation it is certain to be self-eliminated before reaching the last level. In the second situation, the self-elimination becomes probabilistic with a nonzero chance to reach the top level with the other opinion being then self-eliminated. Finally, in the last situation the top leadership is reached with certainty for that opinion. 16.6 From the Magic to the Machiavelli Formula 317

The same scheme applies for the even case, and the associated lower and upper magic thresholds are still given by

pn p .1 n/ r;B D c;r r (16.12) and pn p .1 n/ n r;A D c;r r C r (16.13) p pn p 1 with 0 r;B ! n D and 0 r;A ! n D . But now the magic thresholds 1 are no longer symmetrical with respect to pc;r since pc;r ¤ 2 . It is therefore more appropriate to rewrite them as

pn p np r;B D c;r r c;r (16.14) and pn p n.1 p / r;A D c;r C r c;r (16.15) n to show up the asymmetry around pc;r and to emphasize the inequality r .1 p /<np p > 1 pn

2 to one. The r .p 1 / extension of both regions are a function of as the distance c;r 2 . The phase diagram is shown in Fig. 16.15. From Fig. 16.15, the probabilistic area is seen to shrink quite quickly with an increase in either the number of levels n or the size r, in a similar manner to the odd case. However, here the area is asymmetric. Its extent is still given by

n pn pn r r;A r;B n D r : (16.16)

As can be apprehended intuitively, it exhibits a maximum line as a function of r 1 at n D 1 with r D 1=r . The maximum of that line is located at r D 4 and monotonically decreases towards zero for an increase in r.Table16.4 contains values of r for the series of group sizes of r D 4; 6; 8; 10; 12; 14; 16. pn Table 16.5 gives the estimates for the lower magic threshold r;B given by (16.12) for the series of local group sizes of r D 4; 6; 8; 10; 12; 14; 16 as a function of the number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. For a set of .r; n/ p pn as soon as 0 r;B, A gets a nonzero chance of winning the presidency. The outcome is probabilistic. 318 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.15 The phase n n p r,A ,p r,B diagram associated to (16.12) 1 and (16.13) as a function of the number of the bottom-up Deterministic A victory hierarchical levels. The case of r D 4 is shown in the 0.8 upper part of the ﬁgure while Probabilistic A B victory the lower part shows the case for r D 16. In both of the upper white parts, any initial 0.6 p0 yields at the top of the hierarchy a deterministic Deterministic B victory victory for the political party 0.4 A. In both of the lower white parts, any initial p0 yields at the top of the hierarchy a deterministic failure for the 0.2 political party A with a victory for B. In the shaded areas in between, the result at the top is probabilistic. A n 2 4 6 8 10 12 drastic shrinkage of the probabilistic area is seen pn ,pn r D 3 r,A r,B when going from 1 (upper part)tor D 15 (lower part). The asymmetry of the probabilistic area can be explicitly seen 0.8

Deterministic A victory

0.6 Probabilistic A B victory

0.4

Deterministic B victory

0.2

n 2 4 6 8 10 12

Table 16.4 The variation of r .pc / as a function of the local group size with r D 4; 6; 8; 10; 12; 14; 16. It is always larger than 1, making the ﬁxed-point pc;r unstable Group size r D 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 r .pc;r/ 1.64 2.01 2.31 2.58 2.81 3.03 3.24 16.6 From the Magic to the Machiavelli Formula 319

pn r D 4; 6; 8; 10; Table 16.5 The value r;B given by (16.12) for the series of local group sizes 12; 14; 16 as a function of a number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 1 0.30 0.33 0.34 0.35 0.36 0.37 0.38 2 0.48 0.49 0.49 0.49 0.49 0.49 0.49 3 0.59 0.57 0.56 0.55 0.54 0.53 0.53 4 0.66 0.61 0.58 0.56 0.55 0.54 0.54 5 0.70 0.63 0.59 0.57 0.56 0.55 0.54 6 0.73 0.64 0.60 0.577 0.56 0.55 0.54 7 0.74 0.65 0.60 0.578 0.56 0.55 0.54 8 0.75 0.65 0.60 0.578 0.56 0.55 0.54 9 0.76 0.65 0.60 0.578 0.56 0.55 0.54 10 0.76 0.65 0.60 0.578 0.56 0.55 0.54 .r; n/ p

pn r D Table 16.6 The value r;A givenby(16.13) for the series of local group sizes of 4; 6; 8; 10; 12; 14; 16 as a function of the number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 1 0.91 0.82 0.77 0.74 0.72 0.70 0.68 2 0.85 0.74 0.68 0.64 0.62 0.60 0.59 3 0.82 0.70 0.64 0.60 0.58 0.57 0.56 4 0.80 0.67 0.62 0.59 0.57 0.56 0.55 5 0.79 0.66 0.61 0.58 0.565 0.557 0.545 6 0.78 0.66 0.607 0.58 0.5635 0.5536 0.545 7 0.77 0.65 0.606 0.58 0.5629 0.5525 0.5443 .r; n/ p >pn For a given couple of values any 0 r;A ends up with a victory for A at the top presidential level. Three and four digits are used when required so as to preserve the dynamics from the effects of rounding up

At the other extreme, we build Table 16.6 for the values of the higher magic pn 100 threshold r;A givenby(16.13)fora % A victory. It is worth noting that contrary pn pn 2p n.1 2p / 1 p 1 to the odd case r;A C r;B D c;r C r c;r ¤ when c;r ¤ 2 . Comparing both Tables 16.5 and 16.6, the shrinking of the probabilistic area as a function of increasing the number of levels n is clearly seen, with both magic thresholds getting closer and closer to the separator, from respectively the lower and higher values. When the rounding off to two digits is not enough, we go to three or four digits as explained below in the “Physicist’s corner.” To check the accuracy of the data given in Table 16.5, we insert each one of P .pn / the lower magic thresholds into the associated voting function r r;B ,whichwe iterate until reaching at some j level a value pj 0:004. Such a limit is used since we are considering a two-digit precision. We then compare j to n. For instance, Table 16.7 gives the various corresponding series obtained for the case r D 4 from which the associated values of j are extracted. 320 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

p D pn n D 1; 2; :::; 10 Table 16.7 The variation of 0 r;B as a function of given by (16.12) while climbing up a hierarchy using a local voting group size of r D 4 n p D pn p p p p p p p p p p j 0 4;B 1 2 3 4 5 6 7 8 9 10 1 2 0.30 0.08 0.00 2 3 0.48 0.28 0.07 0.00 3 4 0.59 0.46 0.25 0.05 0.00 4 5 0.66 0.58 0.44 0.23 0.04 0.00 5 6 0.70 0.65 0.57 0.42 0.20 0.03 0.00 6 7 0.73 0.70 0.66 0.58 0.44 0.23 0.04 0.00 7 8 0.74 0.72 0.69 0.63 0.53 0.36 0.14 0.01 0.00 8 9 0.75 0.74 0.72 0.68 0.62 0.52 0.34 0.11 0.01 0.00 9 10 0.76 0.75 0.75 0.73 0.71 0.68 0.59 0.46 0.26 0.06 0.00 10 10 0.76 0.75 0.75 0.73 0.71 0.68 0.59 0.46 0.26 0.06 0.00 P .pn / The successive values are obtained using the voting function r r;B , which is iterated until reaching at some j level a value of pj 0:004, since we are considering a two-digit precision. The number j is compared to the corresponding number n, which has been used to evaluate the pn p lower magic threshold r;B taken as 0

A systematic discrepancy of C1 in the number of levels is observed between n and j ,aswellasatn D 10 for which the result is exact. This means that the value pn n 1 r;B from Table 16.5 becomes accurate if it is associated to a hierarchy with C levels instead of n. In the odd case, we found a systematic deviation of C2. The peculiarity at n D 10 comes from the fact that we are very close to the threshold value pc;rD4 D 0:7676 and a two-digit precision is not enough.

16.6.1 The Physicist’s Corner

• Some note of caution is necessary with respect to Tables 16.5 and 16.6 where not all of the data are given with the same two digit precision. This prompts a comment on the rounding procedure. In general, rounding off results only in losing precision in the result, but without any consequence to the qualitative properties. However, in the case of the magic thresholds, we are dealing with thresholds, which are separators, i.e., points which divide the associated space into two different areas with different qualitative properties. When the corresponding value is a real number as it is, the normal rounding procedure may turn out to be misleading. r 10 n 6 p6 0:57661 For instance, for D at D , 10;B D rounds off normally p6 0:58 0:58 > p6 0:579 0:58 at two digits to 10;B .But, c;10 ,whichin turn puts 0:58, either at the ﬁxed point if the threshold is also rounded off at two

digits, or in the upper area leading to pnc;10 D 1. Both cases are wrong since

indeed the value 0:57661 belongs to the lower area leading to pnc;10 D 0 due to 6 the exact inequality 0:57661 < pc;10. The rounding off limited to two digits has thus inverted the dynamics. 16.6 From the Magic to the Machiavelli Formula 321

p D pn n D 10; 11; 12 Table 16.8 The variation of 0 r;B given by (16.12) for respectively while climbing up a hierarchy with a voting local group size of r D 4 where p0 equals respectively p10 D 0:7622 p11 D 0:7643 p12 D 0:7656 4;B , 4;B , 4;B p0 p1 p2 p3 p4 p5 nj p6 p7 p8 p9 p10 p11 p12 p13 10 11 0.7622 0.7587 0.7529 0.7431 0.7266 0.6982 0.6485 0.5604 0.4081 0.1887 0.0231 0.0000 11 12 0.7643 0.7622 0.7586 0.7528 0.7429 0.7263 0.6978 0.6478 0.5590 0.4058 0.1860 0.0221 0.0000 12 13 0.7656 0.7643 0.7622 0.7587 0.7528 0.7430 0.7265 0.6981 0.6483 0.5600 0.4074 0.1878 0.0228 0.0000

The successive values are then obtained by application of the voting function Pr .p0/,whichis iterated until reaching at some j level a value pj 0:00004, since we are considering a four- digit precision. The number j is compared to the corresponding number n, which has been used to pn p evaluate the lower magic threshold r;B taken as 0

This problem would not occur for an attractor since from both sides, the dynamics leads towards it. Accordingly, the rounding off in the close vicinity of unstable thresholds must be performed carefully in order to preserve the associated dynamics and in particular, its direction. In the present situation, the bad cases are cured by just adding a third digit for most of them. However, for r D 12; 14; 16 at n D 6; 7, a fourth digit must be added, as seen in Table 16.6. • Having made, when necessary, the required adjustment in the rounding off procedure, we come back to the single discrepancy found in Table 16.7 at n D 10. This may be a sign of some serious problem in the formula giving the magic thresholds. It could also be a mismatch of no important signiﬁcance. Or it could be the result of some approximation. Being in the vicinity of the separator, where the dynamics is very sensitive to the initial conditions, we just saw in the previous comment that rounding off could be misleading. It thus favors ﬁrst checking the third possibility, which suggests a mismatch driven by the rounding off approximation. The mismatch occurs in Table 16.7 where a two-digit precision is used throughout. From the results obtained, it seems to be a reasonable choice for reproducing the exact results. Nevertheless, let us now look at the bad case of (r D 4;n D 10) with a higher precision of four digits. To make the investigation more robust, we also examine the additional cases of n D 11; 12 for which a two-digit precision yields a constant j D 10 making the discrepancy increase by one at each higher level. r 4 n 10; 11; 12 p10 0:7622 p11 For D and D we get respectively 4;B D , 4;B D 0:7643 p12 0:7656 , 4;B D .Table16.8 gives the associated successive values p1;p2;p3;p4;p5;p6;p7;p8;p9;p10;p11;p12;p13 to reach 0:0000. Comparing the respective pairs of values .n; j /, the systematic C1 discrepancy found from n D 1 to n D 9 using a two-digit precision is recovered with the four-digit precision. Therefore, what could have been the sign of a problem 322 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

p D pn n D 1; 2; :::; 10 Table 16.9 The variation of 0 r;A as a function of givenby(16.12) while climbing up a hierarchy using a local voting group size of r D 8 n j p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 1 3 0.91 0.96 0.99 1 2 4 0.85 0.89 0.94 0.98 1 3 5 0.82 0.85 0.89 0.94 0.98 0.1 4 6 0.80 0.82 0.85 0.89 0.93 0.98 1 5 7 0.79 0.80 0.82 0.86 0.90 0.95 0.98 1 6 8 0.78 0.79 0.80 0.82 0.85 0.89 0.94 0.98 1 7 10 0.775 0.780 0.787 0.799 0.818 0.846 0.886 0.933 0.975 0.996 1 P .pn / The successive values are obtained using the voting function r r;A , which is iterated until reaching at some j level a value pj 0:996, since we are considering a two-digit precision. The number j is compared to the corresponding number n, which has been used to evaluate the pn p higher magic threshold r;A taken as 0

has made it possible to demonstrate the robustness and the accuracy of the approximate formula for the lower magic threshold. • Since for even-sized groups there exists no symmetry between the two magic thresholds with respect to pc;r, the sum is different from one and so we should also check out the results for the higher magic threshold. Table 16.9 contains the corresponding data for the same group size r D 4. Two remarks can be made about the given data. First, the systematic discrepancy is again of a value of C2 as for the odd case. Second, a mismatch appears nD7 for n D 7, which corresponds to a higher magic threshold prD4;A D 0:775 which is rather close to the separator pc;rD4 D 0:767, while still being above it. Indeed, because it is very close to the separator, the dynamics are very slow, as indicated by the singularities found in the various critical numbers of levels and it is of no interest to try to have valid approximate formulae there. • Before asserting with conﬁdence that we do have powerful formulae for determining the outcome results of a bottom-up vote, we should not forget that all of the above detailed checks were performed for the single group size of r D 4. We should also check the formula’s validity for larger sized groups. Table 16.10 lists the corresponding results for r D 16. Now the discrepancy is of C1 for both magic thresholds. This is due to the fact that for large sizes, the probabilistic area shrinks as well as the asymmetry between A and B, as seen in Fig. 16.15.Butit is always better to have one additional level to be sure of the result. Following the analysis of our “physicist’s corner,” we redeﬁne (16.12)and (16.13)as, pn p nC1p Nr;B D c;r r c;r (16.17) and pn p nC1.1 p /; Nr;A D c;r C r c;r (16.18) pn pn n where Nr;B and Nr;A are the lower and higher magic thresholds associated to an level bottom-up democratic voting hierarchy. These thresholds provide the correct 16.7 Global Size, the Practical Scheme, the Magic Formula 323

n Table 16.10 pc;rD16 0:544 the variation of p0 D prD16;B and p0 D n prD16;A as a function of n D 1; 2; 3; 4 while climbing up a hierarchy using a local voting group size of r D 16 n p D pn p p p p p j 0 16;B 1 2 3 4 5 1 2 0.38 0.11 0.00 2 3 0.49 0.37 0.09 0.00 3 4 0.53 0.50 0.40 0.13 0.00 4 5 0.54 0.53 0.50 0.40 0.15 0.00 p D pn p p p p p nj 0 16;A 1 2 3 4 5 1 2 0.68 0.90 1 2 3 0.59 0.69 0.91 1 3 4 0.56 0.59 0.70 0.93 1 4 5 0.55 0.56 0.60 0.73 0.96 1 P .pn / The successive values are obtained using the voting functions r r;B and P .pn / j r r;A , which are iterated until reaching at some level a value of respectively pj 0:004 and pj 0:996, since we are considering a two-digit precision. The number j is compared to the corresponding number n, which has been used to evaluate the lower and higher magic thresholds. The fact that from 0:40 we get respectively 0.13 (n D 3) and 0.15 (n D 4) using the same formula, the increase is due to the rounding off of the two different values 0.396 and 0.401 qualitative behavior and rather precise quantitative estimates. Their evolution as a function of the number of levels n forthetwoﬁxedsizer D 4 and r D 16 are shown in Fig. 16.16. As for the odd case, the ad hoc change enlarges the probabilistic area when compared to the rough formulae in (16.12)and(16.13) in agreement with the exact results. The various phases of the dynamics to get to power by democratic bottom- up voting via local even groups of size r, within a n level pyramidal hierarchy, are determined by the three regions p pn B • 0 Nr;B ! p pn A • 0 Nr;A ! pn

16.7 Global Size, the Practical Scheme, the Magic Formula

Formulae obtained for the global sizes of the hierarchies in the case of odd-sized voting groups apply without change to the even case. Their dependence on the size r is independent of the corresponding parity. It applies equally to both the revisited practical scheme and the magic formula. 324 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.16 The same phase n n p r,A ,p r,B diagram as in Fig. 16.15 but 1 now using improved (16.17) and (16.18) instead of (16.12) Deterministic A victory and (16.13). The case r D 4 is shown in the upper ﬁgure 0.8 while the lower one shows the case for r D 16.For Probabilistic A Bvictory n D 1 and n D 2, the result is always probabilistic 0.6

Deterministic B victory

0.4

0.2

n 2 4 6 8 10 12

n n p r,A ,p r,B 1

0.8

Deterministic A victory

0.6 Probabilistic A B victory

0.4

Deterministic B victory

0.2

n 2 4 6 8 10 12

16.7.1 Global Size

N n n 1 Accordingly, we use (14.30) where NG;r r .1 C r /, which was found to be n quite an approximation. Table 16.11 gives a series of values for NNG;r for the series of sizes of r D 4; 6; 8; 10; 12; 14; 16, each one being a function of the number of 16.7 Global Size, the Practical Scheme, the Magic Formula 325

N n Table 16.11 Values of NG;r for the series of local group sizes of r D 4; 6; 8; 10; 12; 14; 16 as a function of the number of bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 nrD 4rD 6rD 8rD 10 r D 12 r D 14 r D 16 1 5 7 9 11 13 15 17 2 21 43 73 111 157 211 273 3 85 259 585 1,111 1,885 2,955 4,369 4 341 1,555 4,681 11,111 22,621 41,371 69,905 5 1,365 9,331 37,449 111,111 271,453 579,195 106 6 5,461 55,987 299,593 106 3 106 8 106 7 21,845 335,923 2 106 8 87,381 2 106 9 349,525 10 106 For a given couple of values .r; n/, the total number of agents involved is evaluated. Numbers larger than 106 are rounded off bottom-up hierarchical levels n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10. The divergence occurs more rapidly than for odd sizes, as expected, since using a power of r we are systematically shifting the values from an odd value of r to an even value of r C 1.

16.7.2 The Practical Scheme

Given a support p0 pc;rD4 the results are not very good, as seen from the table. Nevertheless, yielding overestimates guarantees the expected result from the bottom-up voting. In addition, we already saw that increasing the number of digits for the precision of the numerical values increases the number of required levels. Indeed, taking three digits instead of two for p0 D 0:80 yields respectively e e nc;r D 7, then ﬁve digits are required to get nc;r D 5 at p0 D 0:90. But whatever is e done, at p0 D 1 the result is exactly nc;r D 1 making the corresponding nNd D 3 too large. Figure 16.17 shows the variation of both nd and nNd. N 0 p The above formulae were derived from the power law assumption G;r Dj c;r vr p0 j (15.1), which is symmetrical around pc;r, but now the problem is no longer 1 symmetrical around pc;r over 0 p0 1 since pc;r ¤ 2 . Figure 16.17 clearly 326 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

e Table 16.12 Values of the exact numerical estimates nc;r and the corresponding one given by nNd using a group size r D 4 for the series of bottom support p0 D 0:00; 0:10; 0:20; 0:30; 0:40; 0:50; 0:60; 0:70; 0:80; 0:90; 1 p0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0:70pc;r 0.80 0.90 1 e nc;r 11223346 631 nd 0.53 0.81 1.14 1.53 2.01 2.65 3.60 5.42 6.90 4.07 2.94 nNd 11223346 753

Fig. 16.17 The variation of nd,nd both nd and nNd as a function of p0 for r D 4 15

12.5

7.5

2.5

p 0.2 0.4 0.6 0.8 1 0

e Table 16.13 Values of the exact numerical estimates nc;r and the corresponding one given by nNd using a group size r D 10 for the series of bottom support p0 D 0:00; 0:10; 0:20; 0:30; 0:40; 0:50; 0:60; 0:70; 0:80; 0:90; 1 p0 0.00 0.10 0.20 0.30 0.40 0.50pc;r 0.60 0.70 0.80 0.90 1 e nc;r 112223 53211 nd 0.58 0.78 1.03 1.35 1.82 2.69 4.06 2.23 1.59 1.20 0.91 nNd 112223 53221

shows the existence of a problem above pc;r in the vicinity of p0 D 1 whereweget nNd D 3 instead of the exact value of 1. But before going into a deeper analysis of the reason for the discrepancy, it is wise to check the situation for a larger group size. Table 16.13 reproduces Table 16.12 for voting groups of a size of r D 10. The agreement is almost total, with the exception of p0 D 0:80 where nNd D 2 instead of 1. It shows that the approximate formula nNd is rather good despite its weakness for small sizes near p0 D 1, which is anyhow a very improbable situation. 16.7 Global Size, the Practical Scheme, the Magic Formula 327

Fig. 16.18 Same as nd ,nd Fig. 16.17 but extended to the 8 mathematical range 0 p0 2pc;r 7

p 0.2 0.4 0.6 0.8 1 0

16.7.3 The Physicist’s Corner

Looking at Fig. 16.17 prompts two remarks, both connected to symmetry. The ﬁrst remark concerns the symmetry of the upper part of the plot along the vertical line passing at pc;r 0:77. It appears that while the lower left part goes down to p0 D 0, the lower right part p0 >pc;r is just truncated at p0 D 1. Figure 16.18 shows the plot for the unrealistic range 0 p0 2pc;r, which is now perfectly symmetrical 1 around pc;r, the same way in which it occurs for the odd case around pc;r D 2 . In parallel, we need to satisfy a basic symmetry of our problem, which is known in physics as the invariance by time reversal, i.e., the problem is unchanged if A is called B and vice versa. In other words, having, as expected, one level at p0 D 0,we must have the same result at p0 D 1, as against three. But to satisfy this symmetry at the extreme values of p0, required to be consistent, means destroying the symmetry of the formula around pc;r. Of course, all this analysis is of no practical importance since most real situations have a p0 ¤ 0 and p0 ¤ 1. However, satisfying the requirements of symmetry may sometimes improve some results. It is always of a central importance to preserve the symmetry of a problem. Accordingly, the question is how to modify the formula of nd to make nNd yield the value 1 at p0 D 1, i.e., nd.p0 D 1/ < 1? We ﬁrst notice that in the case for r D 10, we do have the right result as shown in Table 16.13. From the expression of nNd at lnjpc;r j lnjpc;r 1j r 4 n .p0 0/ 0:53 n .p0 1/ 2:94 D ,weget d D D ln r and d D D ln r while it is respectively 0.58 and 0.91 at r D 10. 328 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

The above two cases of r D 4 and r D 10 provide the explanation of why the formula works well below pc;r and not quite as well above this value. Indeed since nd.p0 D 0/ < 1 always, it yields automatically the right limit nNd.p0 D 0/ D 1.But in contrast, nd.p0 D 1/ starts at 2.94 for r D 4 and then decreases to 1.51 and 1.11 for respectively r D 6; 8, thus creating the associated overestimates. It becomes lower than 1 only from r D 10 with 0.91. On this basis, we need to renormalize nd to decrease its value, but only above pc;r. The ﬁrst straightforward way to proceed is to withdraw the constant term nd.p0 D 1/ from nd, which by deﬁnition provides the right result at p0 D 1.We nC thus consider d as p 1 1 1 p nC n ln j c;r j c;r ; d d D ln (16.19) ln r ln r p0 pc;r

C which is exactly nc;r from (16.7). It is a nice demonstration of the “magic” of intuitive mathematical manipulation where we succeed in recovering the same result from different approaches. One could ask “what for”? The first answer is simply for the pleasure of tackling a problem using various approximate methods and of finding out the coherence of the approaches. The second answer is more fruitful and arises from the following analysis. nC When we derived the expression for c;r and checked out its validity, we found C; C; that in order to get good results we should use (16.11) with nNc;r N nc;r C 2. But incidentally, we did not focus on the minor fact of obtaining nNc;r.p0 D 0/ D C nNc;r.p0 D 1/ D 2 instead of the exact value of 1. C; At this stage we obtained a formula nNd in a simpler way than nNc;r ,whichis better for p0 pc;r.Soletus keep on with our process. nC N nC 1 Checking the associated results as shown in Table 16.14 with Nd d C , they turned out to be poorer and even worse than the ones given by nNd since they are lower than the exact ones. As mentioned earlier, an overestimate of the number of levels has only a cost in the size of the hierarchy but does not modify the expected result at the top leadership. An underestimate may create unwanted surprises. Therefore, one step further is to weaken the subtracted term, having in mind that we both want to keep it unchanged at p0 D 1, and that around pc;r the behavior is pc;r p0 n .p0 1/ symmetrical. Multiplying d D by pc;r 1 satisfies both requirements since the factor equals 0 at p0 D pc;r and1atp0 D 1. It leads to p p p 1 nCC n c;r 0 ln j c;r j ; d d (16.20) pc;r 1 ln r nCC N nCC 1 p 0:80; 0:90; 1 whose values of Nd d C at 0 D are given in Table 16.14. The associated data are much improved with respectively 7, 3, 1 as compared with the exact estimates of 6, 3, 1. 16.7 Global Size, the Practical Scheme, the Magic Formula 329

e Table 16.14 Values of the exact numerical estimates nc;r and the corresponding one given by nNd using a group size r D 4 for the series of bottom support p0 D 0:00; 0:10; 0:20; 0:30; 0:40, 0:50; 0:60; 0:70; 0:80; 0:90; 1 p0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70pc;r 0.80 0.90 1 e nc;r 11223346 631 nd 0.53 0.81 1.14 1.53 2.01 2.65 3.60 5.42 6.90 4.07 2.94 nNd 11223346 753 nC d 3.97 1.13 0.00 nNC d 421 nCC d 6.49 2.40 0.00 nNCC d 731

Fig. 16.19 The variation of nd,nd,nd ,nd both nd and nNd as a function of 0 p0 pc;r, and of 35 nCC nNCC d and d as a function of pc;r p0 1 for r D 4. 30 To be compared with Fig. 16.17 25

p 0.2 0.4 0.6 0.8 1 0

Figure 16.19 shows the variation of both nd and nNd as a function of 0 p0 p nCC nCC p p 1 r 4 c;r, and of d and Nd as a function of c;r 0 for D . Comparing it with Fig. 16.17 shows that the rescaled behavior obeys the limit constraints. nCC However, the associated formula d is much more involved with its correction n lnjpc;r p0j term than the initial expression d D ln r . As physicists would say, “it is not sexy.” With the above statement, we conclude this parenthesis of initiation to the physicist’s way of “cooking” the formulae by stressing a basic criterion in the guide for good “cooking”. It is always of importance to weigh on the one hand, the exactness of the formula, and on the other, its simplicity. Much more insight can often be extracted from a simple formula, which yields reasonable results, although it is not exact, than from a more precise formula, which is more difﬁcult to manipulate and thus does not make it easy to grasp the meaning of the underlying mechanisms. 330 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

The central point to remember is that the aim is to understand the mechanism, rather than to obtain an exact derivation of the theorems. Once again, this philosophy is certainly the major key that physics can contribute to tackling social and political phenomena within the innovative framework of sociophysics.

16.7.4 The Magic Formula

We can compare the three different expressions obtained for the two magic thresholds. For the lower magic threshold, we have respectively

pn p np r;B D c;r r c;r

(16.14), which was improved by hand to

pn p nC1p Nr;B D c;r r c;r

N 0 .p p /vr (16.17), and the one derived from the power law hypothesis G;r D c;r 0 (15.1), which writes as given by (15.12)

pn p n: r;B D c;r r

Similarly for the higher magic threshold, we have

pn p n.1 p / r;A D c;r C r c;r

(16.15), improved by hand to

pn p nC1.1 p / Nr;A D c;r C r c;r

N 0 .p (16.18), and again the one derived from the power law hypothesis G;r D 0 vr pc;r/ (15.1), to be as given by (15.13)

pnC p n: r;A D c;r C r

In order to check and compare the validity of the above three classes of magic thresholds, we evaluate the corresponding exact estimates using directly the voting function Pr .p0/. Given a value of levels of n D 1; 2; 3; 4; 5; 6; 7; 8; 9; 10, we iterate n Pr .p0/ntimes to reach pn D Pr .p0/. Then we determine the values p0 which yields respectively pn D and pn D 1 . The choice of , which is very small, sets the required accuracy and the associated risk taken. As discussed earlier to reduce the risk taken requires increasing the hierarchy and the subsequent number of involved agents. So the choice is to be made from a balanced consideration between implementing a larger hierarchy and allowing some small risk in the outcome of a vote, with respect to the expected result. 16.7 Global Size, the Practical Scheme, the Magic Formula 331

Table 16.15 Exact estimates of the magic thresholds for both the sizes of r D 4 and r D 16 Magic n D 1nD 2nD 3nD 4nD 5nD 6nD 7nD 8nD 9nD 10 e prD4;A 0.959 0.912 0.867 0.833 0.809 0.794 0.784 0.778 0.774 0.772 e prD4;B 0.110 0.337 0.516 0.623 0.684 0.718 0.738 0.751 0.757 0.758 e prD16;A 0.866 0.663 0.582 0.556 0.548 0.546 0.545 e prD16;B 0.238 0.443 0.513 0.535 0.541 0.543 0.544 0.544 Data are given to within a precision of 103 precision. The probabilistic area shrinks to a width of 14 103 within ten levels for a size of r D 4 against 2 103 within seven levels for groups of a size of r D 16. Due to the lack of symmetry with respect to pc;rD4 D 0:76759 and p D 0:54422 pe C pe ¤ 1 p c;rD16 ,wehave r;A r;B . Data are not shown as soon as the distance to c;r is smaller than the number of digits used

Table 16.16 Similarly to Table 16.15, exact estimates of the magic thresholds for both the sizes of r D 4 and r D 16 but now using the greater precision of 106 Magic n D 1nD 2nD 3nD 4nD 5nD 6nD 7nD 8nD 9nD 10 e prD4;A 0.999 0.987 0.952 0.904 0.861 0.828 0.806 0.791 0.782 0.777 e prD4;B 0.013 0.155 0.379 0.542 0.638 0.692 0.723 0.741 0.752 0.758 e prD16;A 0.927 0.701 0.595 0.560 0.550 0.546 0.545 e prD16;B 0.105 0.378 0.492 0.528 0.539 0.543 0.544 0.544 Nevertheless, data are rounded off to three digits. Increasing the precision makes the probabilistic area larger

Table 16.15 gives exact estimates of the magic thresholds for both the sizes of r D 4 and r D 16 using the accuracy of 103 in the numerical iteration process with D 0:005. To appreciate the sensitivity of the precision effect, Table 16.16 gives the same exact estimates of the magic thresholds for both sizes of r D 4 and r D 16 but using the accuracy of 106 in the calculation with D 0:000005.Data are given to a precision of three digits. The probabilistic area shrinks to a width of 14 103 within ten levels for a size of r D 4 against 2 103 within seven levels for groups of a size of r D 16.Dueto the lack of symmetry with respect to pc;rD4 D 0:76759 and pc;rD16 D 0:54422 we pe pe 1 p have r;A C r;B ¤ . Data are not shown as soon as the distance to c;r is smaller than the number of digits used. Comparing the values from both tables indicates that increasing the accuracy yields smaller lower magic thresholds and larger upper magic thresholds as expected intuitively, thus enlarging the probabilistic area. Figure 16.20 incorporates the magic threshold data for both precisions. The inside dots correspond to Table 16.15 while the outside curves represent the data with higher precision from Table 16.16.The discrepancy shrinks by increasing the number of levels n. The probabilistic area becomes smaller for larger sized voting groups. We can now compare our three classes of magic thresholds to the associated exact estimates as collected from Fig. 16.20 in both of the cases of r D 4 and r D 16. pn The inside curves, previously shown in Fig. 16.15 are given by r;B (16.14)and pn r;A (16.15). The magic thresholds (15.12) and (15.13) extracted from the bottom power law assumption (15.1) are denoted “Magic”. The curves associated to the 332 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

n n n n p r,A ,p r,B p r,A ,p r,B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

n n 0 2 4 6 8 10 0 2 4 6 8 10

Fig. 16.20 Comparison of the higher and lower magic thresholds given respectively by Tables 16.15 (dots)and16.16 (continuous lines). The left part corresponds to r D 4 and the right part to r D 16. Increasing the accuracy enlarges the probabilistic area slightly ad hoc formulae (16.17)and(16.18) are underlined as “1 correction.” The dots represent the exact data from Table 16.15 while the continuous outside lines are from Table 16.16. Several comments arise from the observation of Fig. 16.21. The ﬁrst one relates the accuracy used to evaluate the various approximate magic expressions. Depending on the accuracy, the conclusion on which is the best one changes. In addition, while the “1 correction” curve is better for the lower magic threshold for both r D 4 and r D 16, this is not the case for the upper magic threshold, where the “magic” formula better ﬁts the “1 correction,” especially at r D 4. But indeed, the different deviations occur at n<5for r D 4 and n<3for r D 16. Moreover, the differences are always rather small, indicating our analytical approximations do in fact perform well.

16.7.5 The Physicist’s Corner

Contrary to what was a priori expected, the construction of Fig. 16.21 has turned out to be a laborious and difficult task. I mean, for me. We now review the major obstacles we have been fighting with, since it shows up once more the way physicists practice research, emphasizing both their use of intuitive perception and their obstinacy. • The first disturbing detail: When we built Fig. 15.4, which is the equivalent of Fig. 16.21 for the odd case, the various curves shown associated to the magic thresholds have led to the proud conclusion that (15.12) and (15.13), derived from the power law hypothesis (15.1), were by far the best. This demonstrates the powerful genius of physics in stating the power law hypothesis. 16.7 Global Size, the Practical Scheme, the Magic Formula 333

Fig. 16.21 Comparison of n n p r,A ,p r,B the three classes of magic thresholds to the associated 1 exact estimates in both of the 1 correction cases of r D 4 (upper part) Magic and r D 16 (lower part). The 0.8 inside curves, previously shown in Fig. 16.15,aregiven pn pn by r;B (16.14)and r;A (16.15). The magic thresholds 0.6 (15.12) and (15.12) extracted from the bottom power law assumption (15.1) are denoted “Magic.” The curves 0.4 associated to the ad hoc formulae (16.17)and(16.18) Magic are underlined as “1 correction.” The dots 0.2 represent the exact data from Table 16.15 while the 1 correction continuous partly outside n lines are from Table 16.16 2 4 6 8 10 12

n n p r,A ,p r,B 1

Magic 0.8

1 correction

0.6

0.4

1 correction

0.2

Magic

n 2 4 6 8 10 12

Unfortunately, in the present even case, the similar ﬁgure (only the dots) reveals a distortion to the former conclusion obtained from the symmetrical odd case. Figure 16.21 shows that, while the “magic” expression works quite well at r D 16 for both the lower and upper magic parts, and at r D 4 for the lower threshold, it displays a noticeable discrepancy with the exact estimates 334 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

represented by the dots for the upper term. Instead, the “1 correction” performs better just there. Of course, only purists will be sensitive to such a minor contrariety. • The first search for an explanation: nD3C Examining the figure, the problem arises from the fact that prD4;A D 1 while the corresponding exact estimate is only 0.867. The curvature of the “magic” pnC p n curve is too steep. From the expression r;A D c;r C r , we get respectively 3 1:376 2 0:60 1 0:370 pnC r D , r D and r D , which all make the associated r;A p 0:768 pnC larger than one since c;rD4 D . That is not “physical” since r;A being a 0 pnC 1 probability must satisfy r;A . This is why on the figure all these values are taken to be equal to one. To improve the curve, it would be sufficient schematically to just pick up the n D 3 “magic” point and to pull it down towards the left to the n D 1 dot. This is the situation of all the others three branches. The question is then how to analytically implement this virtual shift based on solid grounding. • The first hint at an improvement: pnC We should remember how the expression r;A (15.13) was derived. It comes N 0 p p vr from the power law hypothesis G;r Dj c;r 0 j (15.1) knowing that by 0 n n N r p p r vr construction G;r D . Equating the two expressions gives j c;r 0 jD , n˙ n ln r p D pc;r ˙ vr D from which r;A;B r is obtained using ln r (15.5). It yields p1C p 1 r;A D c;r C r , which is wrong, as noticed above. p1C 1 Indeed the exact value is r;A D within an infinite precision, but a numerical estimate accounting for a very small risk in the prediction sets the value to p1C 1 0:041 r;A D . For instance Table 16.15 has D . Looking backward to the previous derivation it appears that indeed the power law hypothesis does not include an amplitude as it could do. Accordingly, we write N 0 a.p p /vr ; G;r D 0 c;r (16.21) a N 0 .p and determine the value of the constant from the constraint G;r 0 D vr 1 // D r. It yields a D r.1 pc;r/ , which in turn gives p p vr N 0 r 0 c;r : G;r D (16.22) 1 pc;r

This new expression modiﬁes (15.13) to

p1CC p .1 p /nC1: r;A D c;r C c;r r (16.23)

Unfortunately, using (16.23) instead of (15.13) with D 0:041 does not produce a noticeable change in the results, as can be seen from Figs. 16.22 and 16.23. • An interesting surprise: Nevertheless, looking at (16.23) we note that putting D 0 reproduces exactly pn (16.17), which was obtained by an ad hoc change of the formula r;A (16.15). 16.7 Global Size, the Practical Scheme, the Magic Formula 335

It is a nice surprise to discover that our ad hoc improvement by hand of the formula had indeed a rational grounding. The three expressions for the magic thresholds are now bounded within one single frame, which demonstrates the validity of our mixed approach of combining mathematics, data, intuition, and some esthetics. • Keeping obstinate: Although we were able to unify our formulae, we still did not solve our problem of improving their accuracy. Examining Fig. 16.21, it appears that the exact estimate curves show a light inﬂection point in their curvature, in particular for the lower branch of the case r D 4 at n D 3. In parallel, the discrepancy between the more accurate exact estimates and the approximate magic thresholds seems to be driven by the lower n values. Adding these two remarks to the very fact that the power law hypothesis is by nature essentially accurate near pc;r, which corresponds to larger values of n, we conclude that our analytical derivation must be weak at low values of n.This prompts us to reconsider the amplitude correction of the previous item. But now instead of determining the value of the constant a from the value at n D 1,we should check what happens for using any value of n D k. On this basis, we revisit the former introduction of an amplitude in the power law hypothesis with the aim of determining the value of the constant, both directly from the magic threshold expressions and using an exact estimate associated to any value of levels n D k. • Playing the constant otherwise: From the former derivation, it is more convenient to introduce the constant inside the power law expression as p p vr N 0˙ j 0 c;r j ; G;r D (16.24) a˙

where the index C corresponds to p0 >pc;r and the index corresponds to p

n j p0 pc;r jD a˙r vr ; (16.25)

which becomes ˙ n lnr p0 D pc;r ˙ a˙r vr : (16.26)

ln r 1 r r r Using v D ln r and the property D exp, we get

˙ n p0 D pc;r ˙ a˙r : (16.27) 336 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

pe;k Now choosing one exact estimate r;A;B to determine the constant, we write

pe;k p ak k; r;A;B D c;r ˙ ˙ r (16.28)

which gives ak .pe;k p /k ˙ D˙ r;A;B c;r r (16.29) leading to the expression

pn;k p .pe;k p /nCk: r;A;B D c;r C r;A;B c;r r (16.30)

16.7.6 The Super Magic Formula

Equation (16.30) shows quite naturally that we do have two branches for the magic pe;k p pn;k >p .pe;k p />0 threshold. For a value r;A larger than c;r,weget r;A c;r since r;A c;r pe;k p pn;k

n n n n ppr,A ,p r,B r,A ,p r,B 1 1 1 correction 1 correction Magic Magic 0.8 0.8

0.6 0.6

0.4 0.4

Magic Magic

0.2 0.2

1 correction 1 correction

n n 2 4 6 8 10 12 2 4 6 8 10 12

n n n n p r,A ,p r,B p r,A ,p r,B 1 1 2 correction 3 correction Magic Magic 0.8 0.8

0.6 0.6

0.4 0.4

Magic Magic

0.2 0.2

2 correction 3 correction

n n 2 4 6 8 10 12 2 4 6 8 10 12

Fig. 16.22 Comparison of the upper and lower magic thresholds to exact estimates from Tables 16.15 (dots)and16.16 (continuous lines). The same as in the upper part of Fig. 16.21 but with different expressions for the modiﬁed magic expression. The curves associated to (16.14)and (16.15) have been discarded. The upper left shows the Fig. 16.21 for r D 4 to serve as a test. The upper right incorporates (16.30) with k D 1.Thelower right is for k D 2 and the lower left k D 3

16.7.7 The Physicist’s Corner

Before moving to the next section, I cannot resist coming back to the previous tedious search, which although very enlightening, shows how learning from it can make us more efficient in another problem. A few remarks need to be made. (a) It is worth noting that indeed we manage to analytically implement the intuitive incitement posed in the “first search for an explanation” where we stated that it would be sufficient schematically to just pick up the n D 3 “magic” point and 338 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

n n n n p r,A ,p r,B p r,A ,p r,B 1 1

Magic Magic 0.8 0.8

1 correction 1 correction

0.6 0.6

0.4 0.4

1 correction 1 correction

0.2 0.2

Magic Magic

n n 2 4 6 8 10 12 2 4 6 8 10 12 n n n n p r,A ,p r,B p r,A \ ,p r,B \ 1 1

Magic Magic 0.8 0.8

2 correction 3 correction

0.6 0.6

0.4 0.4

2 correction 3 correction

0.2 0.2

Magic Magic

n n 2 4 6 8 10 12 2 4 6 8 10 12

Fig. 16.23 Same as in Fig. 16.22 for r D 16

to pull it left down to the n D 1 dot. The difference is that we moved the n D 3 “magic” point to match the n D 3 dot instead of the n D 1 dot. (b) We wrongly concentrated on the n D 1 point due to the fact that it is the unique point for which the result can be mathematically exact since for p0 D 0; 1, one level is sufﬁcient to get a deterministic outcome from one voting group. By so doing, we were misled because the whole hierarchical approach is based on introducing some controlled risk in the voting outcome in order to create both a social structure which includes representatives from both opinions and yet ensures that the presidency goes to the global leading opinion. Accordingly, we must use real numbers everywhere and discard the exact integer results of 0 and 1. In other words, if the bottom-up democratic voting process converges quite rapidly towards one of the two attractors, strictly speaking (in the exact mathematical sense), they are never reached. The error associated to the social risk taken is given by the value of the integer ˛ at which we assimilate 10˛ to zero. 16.8 What Happens to the Rare Antidemocratic Events? 339

(c) On the above basis, we get the limit of validity in the value of n of our power pnC p n <1 law hypothesis from the constraint r;A D c;r C r which yields

ln.1 pc;r/ nr > ; (16.33) ln r

whose second term equals 2.94 at r D 4,givingn n4 D 3, which is exactly the value obtained from the graphics analysis. For r D 16, we have 0.67 with n n16 D 1. This explains why we would have been happy with the magic formula if it was only to be applied to large-sized voting groups. But we want one single formula to apply to all values of r. (d) We could have circumvented the problem the physicist’s way at once as follows. Starting from the above natural limit to the magic formula obtained using the n power law hypothesis, we can delimit the problem to the r contribution since we add or subtract it from pc;r, which is the frontier for a deterministic outcome for a number of levels n D 10 at r D 4 and n D 8 at r D 16. Not having used any adjustable parameter, we could have thus written

pn p a n Nr;A;B D c;r ˙ ˙ r (16.34)

a pnD3 pe;3 and then determine ˙ from Nr;A;B D r;A;B.Itgives

a .pe;k p /3; ˙ D˙ r;A;B c;r r (16.35)

kD3 which is exactly a˙ from (16.29). Plugging it back in (16.34) restitutes pn;k k 3 (16.30) the super magic formula r;A;B with D as

pn p .pe;3 p /nC3: Nr;A;B D c;r C r;A;B c;r r (16.36)

We have thus been able to rederive (16.30) in three easy steps.

16.8 What Happens to the Rare Antidemocratic Events?

In the odd-sized voting case, the general frame was democratic and we discovered both the existence of rare “dictatorial” events and the associated nesting strategy to produce “antidemocratic” events. In the even case, the frame is basically antidemocratic from the beginning since the threshold to power is asymmetrically in favor of the opinion already in power. We can thus examine the current status of these rare “antidemocratic” events, emphasizing the minimum number of agents required to reach the presidency instead of “dictatorial” bottom conﬁgurations which are now numerous. We focus on the conﬁgurations which accentuate the dictatorial effect in favor of the opinion B already in power. 340 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.24 A winning bottom distribution of a minority of 4 B-agents against 12 A-agents for a r D 4; n D 2 hierarchy. It is not an “antidemocratic” bottom conﬁguration

Following the odd case, one simple illustration of a rare “antidemocratic” bottom configuration is shown in the hierarchy shown in Fig. 16.24 for a .r D 4;n D 2/. However, with 4 B-agents against 12 A-agents it corresponds to a density of 4 q0 D 16 D 0:25 > qc;4 D 1 pc;4 0:23, which means it is not a rare configuration within the tie driven symmetry breaking, in contrast to the odd case shown in Fig. 15.5. We are thus studying the ultra rare configurations which guarantee a B presidency with the minimum number of B agents at the bottom. To avoid confusion while comparing it to the similar odd case, we stress that there, we looked at the A point of view, i.e., the rare configurations which guarantee an A presidency with the minimum number of A agents at the bottom. However, in the odd case, the situation is totally symmetrical between A and B. Here we focus on the B point of view to keep the presidency. Indeed for the simple one group voting (n D 1) of size 3 the winning 2 configuration of 2A against 1B corresponds to a density p0 D 3 0:67 > p 1 n 2 c;3 D 2 . It is only from D that the rare ultra minority configurations appear 4 1 with the density of p0 D 9 0:44 < pc;3 D 2 . For the even case, the single group configuration .2A; 2B/, which yields a B election, has also a density larger than the 2 required threshold with q0 D 4 D 0:50 > qc;4 0:23. We thus consider the next size hierarchy and check the situation for .r D 4;n D 3/. Looking for a winning bottom configuration, which involves the smaller number of B agents, we single out the one exhibited in Fig. 16.25.Wehave8 8 1 B-agents against 56 A-agents with a density q0 D 64 D 8 D 0:125 < qc;4 0:23. Therefore it is a rare antidemocratic event configuration. Normally in this range of densities the A with 86% should win the presidency even within the handicap of a threshold of 77%. This shows that an ultra minority of 8 B-agents, who are strategically nested, is sufficient to hold up the presidency against the huge majority of 56 A-agents. It is worth stressing that this “antidemocratic” bottom configuration can in principle always occur by chance under a random selection of the voting agents. However, the associated probability is very low, as appears from calculating it. The first 16.8 What Happens to the Rare Antidemocratic Events? 341

Fig. 16.25 The optimized distribution of 8 B-agents against 56 A-agents within the voting bottom groups of a .r D 4; n D 3/ hierarchy, which in turn ensures B with the certainty of winning the presidency by democratic means

56 8 contribution comes from the probability p0 q0 of having 8 B-agents and 56 A-agents where q0 D 1 p0. The second contribution arises from the number of distinct rearrangements of this minority winning conﬁguration within the 16 voting groups of a size of 4. It scores to .62 6/2 6 D 279936. The ﬁnal result is

56=8 8 56 Q4;3 D 279936q0.1 q0/ ; (16.37) whose variation as a function of p0 D 1q0 is shown in Fig. 16.26. The first term in the higher index gives the number of A agents and the second one the number of B agents, the sum being equal to rn D 43 D 64. Its peak reaches the maximum value 9:44 106 0 p 7 0:875 at 0 D 8 D . This proves how infinitesimal is the chance of one of this family of “antidemocratic” bottom configurations occurring.

16.8.1 The Minimum Number of Bottom Agents to Win the Presidency

Knowing there exists a minimum number of bottom well-located agents to win the presidency with certainty makes the determination of this number of both a strategic importance and a fundamental ingredient to ensure the democratic functioning of the bottom-up voting hierarchy. 342 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

56 8 q 4,3

8·10-6

6·10-6

4·10-6

2·10-6

p0 0.2 0.4 0.6 0.8 1

56=8 Fig. 16.26 The probability Q4;3 .p0/ of bottom “antidemocratic” rearrangements for a .r D 4; n D 3/ hierarchy with the 56-A agents against 8 B-agents (16.37). The maximum value is 9:44 106 0 p D 7 D 0:875 at 0 8

In the odd case, we found that this critical number is given by (15.19) with lc;r;n D rC1 n m 1 1 1 n 2 and the associated density is r;n D 2n C r from (15.20). To generalize ma rC1 to the even case, it is sufﬁcient to substitute to the local majority 2 with 8 ˆ rC1 odd sizes A opinion <ˆ 2 ma r C 1 even sizes A opinion (16.38) ˆ 2 : r 2 even sizes B opinion; since the opinion B beneﬁts from ties and not A for even sizes. We thus have

n lc;r;n D ma (16.39) and ma n ; c;r;n D r (16.40) 16.8 What Happens to the Rare Antidemocratic Events? 343

rrc,r,n c,r,n 1 1

0.8 0.8

0.6 0.6

r 4

0.4 0.4

r 8

0.2 0.2

r 36 r n n 0 2 4 6 8 10 0 2 4 6 8 10

Fig. 16.27 The evolution of c;r;n for ﬁxed even values of the voting group size as a function of the number of levels n.Theleft part is associated to opinion A with r D 4; 8; 36 while the right part represents opinion B, which is independent of r as seen in (16.41)

N 0 rn since G;r D is the total number of bottom agents. In turn, we get 8 1 1 1 n 1 1 n A ˆ 2n C r 2n C r odd sizes opinion < 1 1 2 n 1 1 2n A c;r;n D ˆ 2n C r 2n C r even sizes opinion (16.41) :ˆ 1 n B : 2 even sizes opinion Equation (16.41) shows that the nasty density for opinion B, which benefits from tie configurations, is independent of the size of r, making any increase in n sufficient to quickly reduce it to zero. It thus allows very easy B nesting with negligible manpower. On the other hand, for the A opinion, the correction with respect to B 2n is r , which maybe significant for small hierarchies. It is twice the odd-sized value, whichisthensymmetricalforAandB. The series of Figs. 16.27 and 16.28 exhibit the variation of the nasty densities for both a fixed size and number of levels, as a function of respectively the number of levels and the size. It is worth emphasizing that increasing the size r waves the tie effect as seen from Figs. 16.29 and 16.30. This is expected intuitively since tie configurations weigh less and less for large-sized voting groups. Accordingly, the difference between opinions A and B smoothes off with increasing r. Densities provide an important measure of the nesting task for each opinion but could, however, become misleading while dealing with real manpower. Given a .r D 4;n D 5/ hierarchy, to say opinion B needs only a density c;4;5 0:03 does not show up what it means in terms of the actual required number of agents, which is lc;4;5 D 32. For opinion A, we have respectively approximately 0:24 and 243 for a total number of 1; 024 bottom agents. This data becomes .0:01; 128/ and .0:13; 2187/ for a .r D 4;n D 7/ hierarchy. 344 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.28 The evolution of rc,r,n c;r;n forﬁxedvaluesofthe 1 number of levels with n D 1; 3; 7 as a function of the even voting group sizes r. The left part is associated to 0.8 opinion A while the right part r 36 represents opinion B, which is a constant that is independent of r as seen in 0.6 (16.41) r 4

0.4

r 8

0.2

r n 0 2 4 6 8 10

rc,r,n rc,r,n 1 1

n 1 n 1

0.8 0.8

0.6 0.6

n 3 n 7

0.4 0.4 n 3

n 7

0.2 0.2

r r 0 5 10 15 20 0 5 10 15 20

Fig. 16.29 Both parts from Fig. 16.27 in the same frame. It appears clearly that increasing the size smoothes off the advantage of opinion B over opinion A

Along the same lines, it is worth stressing that contrary to the possible feeling from (16.38), the difference of only one single agent in the determination of ma may yield huge differences in the current associated numbers of agents since it is taken to the power n. One illustration is given in Fig. 16.31 for a voting group size r D 4 as a function of n. 16.8 What Happens to the Rare Antidemocratic Events? 345

Fig. 16.30 Both parts from rc,r,n Fig. 16.28 in the same frame. 1 Increasing the number of levels does not smooth off the n 1 advantage of opinion B over opinion A as quickly as for an 0.8 increasing size

0.6

0.4 n 3

n 7

0.2

r 0 5 10 15 20

Fig. 16.31 The variation of lc,4,n lc;4;n for respectively opinion A and B. While ma is respectively 3 and 2, the 6000 associated minimum numbers of bottom agents diverges quite rapidly as a function of 5000 Opinion A the number of levels n 4000

3000 Opinion B

2000

1000

n 2 3 4 5 6 7 8 346 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

16.8.2 The Associated Number of Different Bottom Nasty Conﬁgurations

n We have calculated the minimum number of agents lc;r;n D ma (16.39) with ma given by (16.38), required at the bottom to win the presidency once they are strategically located. However, to reveal the immense possibilities of geometric nesting, we need to evaluate all possible winning configurations. Not every bottom permutation will conserve the nasty character. The purpose is not to describe a precise real situation, but to show up all the possibilities which exist in real organizations of completely diverting the dynamics of a democratic system at the expense of the majority of the people. An infinitesimally small group that knows how to “play” the structure against the democratic system itself can actually seize the organization. To discover these possibilities provides hints on how to thwart these “geometric coups.” In particular, using random redistributions of people from their different committees may be a very efficient and simple trick to neutralize any nasty attempt. To make the calculation, we notice that the combinatorial dynamics is similar to the so-called Russian dolls, which are enclosed one inside another, the first smaller “doll” being the group of size r. The associated number of different configurations is then all the permutations inside the group with ! r rŠ al ;r;1 D : (16.42) c ma maŠ.r ma/Š

Adding one hierarchical level creates r bottom different groups of size r each. To win the presidency at .n D 2/, the opinion A needs to get the majority at a level .n D 1/, i.e., to have ma elected representatives. This implies having ma bottom groups in which it holds ma agents in each. The associated total number of conﬁgurations accounting for all independent permutations inside each group r ma r is therefore ma . It must then be multiplied by ma to account for the various distributions of these ma groups themselves among the bottom r groups. The total scores to !1Cma r a : lc ;r;2 D ma (16.43)

Adding one more hierarchical level generates r similar small hierarchies, from which opinion A must win the number ma. It thus gives 0 1 !1Cma ma !maCma2 r r @ A ; ma D ma (16.44) 16.8 What Happens to the Rare Antidemocratic Events? 347

Fig. 16.32 The divergence of alc,3,n al ;3;n from (16.46)asa c 5·1012 function of n.Itdiverges quite rapidly 4·1012

3·1012

2·1012

1·1012

n 2 3 4 5

r which must be multiplied by ma ending with

!1CmaCma2 r a : lc ;r;3 D ma (16.45)

Extending successively the procedure to n levels results in

! 1man r 1ma a ; lc ;r;n D ma (16.46)

1man 2 n where the exponent 1ma D 1 C ma C ma C ::: C ma , the sum of the geometric n1 series un D ma . Equation (16.46) diverges as a function of n since it is a power of a power as seen from Fig. 16.32.

16.8.3 The Actual Probability of a Nasty Bottom Conﬁguration

Such a divergence in the number of possible nasty bottom configurations sounds counterintuitive since we do not expect to have so many of these events which from the beginning were defined as rare events. But what matters is not only the number n lc r lc of nasty bottom configurations but also the associated probability p0 .1p0/ to n n have lc;r;n A agents and .r lc;r;n/ B agents at the bottom with a total of r agents. And this probability falls to zero very quickly with n. To evaluate how fast this happens, we calculate its maximum value by taking first p lc its derivative with respect to 0. Equating the derivative to zero yields the value rn at which the maximum occurs. Inserting this value back into the probability gives its maximum llc .rn l / c c ; rnrn (16.47) 348 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.33 The variation of maxi proba the maximum of the -14 n lc r lc 6·10 probability p0 .1 p0/ r D 3 -14 given by (16.47)for as 5·10 a function of n. It goes to zero quite rapidly 4·10-14

3·10-14

2·10-14

1·10-14 n 2 3 4 5 whose variation for r D 3 as a function of n is shown in Fig. 16.33. For instance it is 6:12 10530 at n D 5 against 6:18 1014 for the corresponding number of nasty configurations, proving that the product goes to zero. On this basis, the mathematics has restored the expected intuition and we obtain for the probability of the very rare victory of the smallest bottom number of A agents the formula ! 1man 1ma n r n P lc =.r lc / plc .1 p /r lc ; r;n D ma 0 0 (16.48) which is by nature of finite probability, positive, and smaller or equal to one. The n lc =.r lc / series of Figs. 16.34–16.36 show the variations of Pr;n for voting group sizes of r D 3 and r D 4. The A opinion view point is shown in Fig. 16.34 for r D 3 and in Fig. 16.35 for r D 4. The B opinion view point, which benefits from local tie configurations, is given in Fig. 16.36. For each figure, the series of cases n D 2;3; 4;5 are presented. In each series, the width is seen to shrink with increasing n and simultaneously the height is drastically reduced, indicating that a nasty bottom configuration is an extremely rare event indeed. Besides, for the B opinion with n D 2, the maximum is shifted toward a lower and lower p0 for the A opinion and towards a larger and larger p0 for the B opinion.

16.9 All Bottom Minorities and Majorities Winning the Presidency

In the previous section, we singled out the nasty bottom configurations which correspond to the minimum number of agents required to win the presidency in a .r; n/ hierarchy. However, there exist many more nasty configurations. They include all the winning bottom configurations which have a minority of agents and this number must not be the minimum one we have calculated. It runs from lc;r;n 16.9 All Bottom Minorities and Majorities Winning the Presidency 349

n lc r lc n P lc r lc 3,2 P 3,3

0.05 0.00015

0.04 0.000125 0.0001 0.03 0.000075 0.02 0.00005 0.01 0.000025 p p0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

n n lc r lc lc r lc P 3,4 P 3,5

4·10-11 4·10-27

3·10-11 3·10-27

2·10-11 2·10-27

1·10-11 1·10-27

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

n lc =.3 lc / Fig. 16.34 The variations of P3;n for the A opinion for n D 2; 3; 4; 5

n n lc r lc lc r lc P P 4,2 4,3

0.00025 8·10-16 0.0002 6·10-16 0.00015 -16 0.0001 4·10

0.00005 2·10-16

p0 p 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

n n lc r lc lc r lc P 4,4 P 4,5

-208 4·10-58 5·10

4·10-208 3·10-58 3·10-208 2·10-58 2·10-208 -58 1·10 1·10-208

p p0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

n lc =.4 lc / Fig. 16.35 The variations of P4;n for the A opinion for n D 2; 3; 4; 5

rnC1 rn2 up to 2 for odd sizes and 2 for even sizes. Beyond these upper limits the winning conﬁgurations are democratically balanced since they eventually attribute the president to the actual bottom majority. 350 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

n n lc r lc lc r lc Q 4,2 Q 4,3 0.0012 0.2 0.001

0.15 0.0008

0.1 0.0006 0.0004 0.05 0.0002

p p 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0

l rn l n c c lc r lc Q 4,5 Q 4,4 4·10-29 1.5·10-10 -10 1.25·10 3·10-29 1·10-10 -29 7.5·10-11 2·10 5·10-11 1·10-29 2.5·10-11 p 0.2 0.4 0.6 0.8 1 0 p 0.2 0.4 0.6 0.8 1 0

n lc =.4 lc / Fig. 16.36 The variations of P4;n for the B (with tie) opinion for n D 2; 3; 4; 5

To enumerate all these cases by hand, and in particular the associated numbers of conﬁgurations, is tedious and laborious. Nevertheless, we are fortunate since we realized in the study of the odd case that the task may be completed in a simple manner due to the insight of our scheme. Given a .r; n/ hierarchy all opinion A winning bottom conﬁgurations are given at once by n iterations

Pr .Pr .Pr :::Pr .p0/// D Pr;n.p0/ (16.49) of Pr , which ends up as a polynomial of the form

Xrn l rnl Pr;n.p0/ D al;r;np0.1 p0/ ; (16.50) lD0

n since they are N0;r D r agents at the bottom. Notice that the coefficients al;r;n are not binomial coefficients. Moreover, all of them with an index l lower than lc;r;n givenby(16.39) are zero. The first nonzero coefficient is alc ;r;n given by (16.46). We can then extract all the terms associated to an A minority, i.e., with l D lc;r;n;.lc;r;nC mi ma 1/; :::; .ma 1/. Denoting them as Pr;n .p0/, the rest of the terms give Pr;n .p0/ D mi Pr;n.p0/ Pr;n .p0/, which accounts for all possible winning bottom A majorities with l D ma; .ma C 1/; :::; r n. To grasp the meaning and the implications of these abstract formulae, we now complete a detailed investigation by revisiting the odd-sized case r D 3 more thoroughly and compare it to the even case r D 4 with respectively n D 1; 2; 3; 4; 5 16.9 All Bottom Minorities and Majorities Winning the Presidency 351 levels. To keep the comparison consistent with the previous study of r D 3 we present the opinion A viewpoint for the odd case and both points of view for the even case.

16.9.1 The Odd Case r D 3

3 2 We ﬁrst note that for a .r D 3; n D 1/ hierarchy P3;1.p0/ D p0 C 3p0.1 p0/, which contains no nasty conﬁguration. We thus have

mi P3;1 .p0/ D 0 (16.51) and ma 3 2 P3;1 .p0/ D p0 C 3p0.1 p0/: (16.52) Going to the next hierarchy .r D 3; n D 2/,weget

mi 4 5 P3;2 .p0/ D 27p0.1 p0/ (16.53) and

ma 9 8 7 2 6 3 5 P3;2 .p0/ D p0 C 9p0.1 p0/ C 36p0.1 p0/ C 84p0.1 p0/ C 99p0 (16.54)

mi ma with P3;2.p0/ D P3;2 .p0/ C P3;2 .p0/ obtained previously in (15.16). Going one more step to a .r D 3; n D 3/ hierarchy things get a little bit more involved with

mi 13 14 12 15 P3;3 .p0/ D 7902198p0 .1 p0/ C 3677346p0 .1 p0/ 11 16 10 17 C1203903p0 .1 p0/ C 266085p0 .1 p0/ 9 18 8 19 C35721p0.1 p0/ C 2187p0.1 p0/ (16.55) and

ma 27 26 P3;3 .p0/ D p0 C 27p0 .1 p0/ 25 2 24 3 C351p0 .1 p0/ C 2925p0 .1 p0/ 23 4 22 5 C17550p0 .1 p0/ C 80730p0 .1 p0/ 21 6 20 7 C296010p0 .1 p0/ C 888030p0 .1 p0/ 19 8 18 9 C2217888p0 .1 p0/ C 4651104p0 .1 p0/ 17 10 16 11 C8170200p0 .1 p0/ C 11833992p0 .1 p0/ 15 12 14 13 C13706514p0 .1 p0/ C 12156102p0 .1 p0/ (16.56) 352 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

mi ma with P3;3.p0/ D P3;3 .p0/ C P3;3 .p0/. To be brave, we go one step further to .r D 3; n D 4/ to ﬁnd

mi 40 41 P3;4 .p0/ D 96816061789880676474474p0 .1 p0/ 39 42 C74786898680035624413000p0 .1 p0/ 38 43 C53186116783067545636584p0 .1 p0/ 37 44 C34779320820830833177716p0 .1 p0/ 36 45 C20890925482203350804052p0 .1 p0/ 35 46 C11516318680717783266840p0 .1 p0/ 34 47 C5820567495608318152248p0 .1 p0/ 33 48 C2693960516309155121214p0 .1 p0/ 32 49 C1140018377621141274750p0 .1 p0/ 31 50 C440188071179034237336p0 .1 p0/ 30 51 C154677716202730221432p0 .1 p0/ 29 52 C49299395699483046756p0 .1 p0/ 28 53 C14193786978375451524p0 .1 p0/ 27 54 C3672937443088525896p0 .1 p0/ 26 55 C849018001892607144p0 .1 p0/ 25 56 C173992810974047730p0 .1 p0/ 24 57 C31318108552450674p0 .1 p0/ 23 58 C4893247126810728p0 .1 p0/ 22 59 C653669385748488p0 .1 p0/ 21 60 C73174775709348p0 .1 p0/ 20 61 C6677021181060p0 .1 p0/ 19 62 C476949165624p0 .1 p0/ 18 63 C25011739224p0 .1 p0/ 17 64 C856151451p0 .1 p0/ 16 65 C14348907p0 .1 p0/ (16.57) and

ma 81 80 P3;4 .p0/ D p0 C 81p0 .1 p0/ 16.9 All Bottom Minorities and Majorities Winning the Presidency 353

79 2 C3240p0 .1 p0/ 78 3 C85320p0 .1 p0/ 77 4 C1663740p0 .1 p0/ 76 5 C25621596p0 .1 p0/ 75 6 C324540216p0 .1 p0/ 74 7 C3477216600p0 .1 p0/ 73 8 C32164253550p0 .1 p0/ 72 9 C260887834350p0 .1 p0/ 71 10 C1878392407320p0 .1 p0/ 70 11 C12124169174520p0 .1 p0/ 69 12 C70724320184700p0 .1 p0/ 68 13 C375382930211100p0 .1 p0/ 67 14 C1823288518168200p0 .1 p0/ 66 15 C8144022047817960p0 .1 p0/ 65 16 C33594090932900178p0 .1 p0/ 64 17 C128447993942153874p0 .1 p0/ 63 18 C456703956493346376p0 .1 p0/ 62 19 C1514333777515065576p0 .1 p0/ 61 20 C4694429511817935660p0 .1 p0/ 60 21 C13636146230899820172p0 .1 p0/ 59 22 C37189035618820241112p0 .1 p0/ 58 23 C95394744492184206072p0 .1 p0/ 57 24 C230517806428115839926p0 .1 p0/ 56 25 C525478011132629654838p0 .1 p0/ 55 26 C1131324528953561521464p0 .1 p0/ 54 27 C2302606510058762476824p0 .1 p0/ 53 28 C4433630861775194339436p0 .1 p0/ 52 29 C8079483583057041053964p0 .1 p0/ 51 30 C13935212780308578219816p0 .1 p0/ 50 31 C22739954358565376424072p0 .1 p0/ 354 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

49 32 C35078954168854500383700p0 .1 p0/ 48 33 C51085725992094070371636p0 .1 p0/ 47 34 C70103695810372706072952p0 .1 p0/ 46 35 C90439120615885306407000p0 .1 p0/ 45 36 C109385469174567263779188p0 .1 p0/ 44 37 C123664942950917211585684p0 .1 p0/ 43 38 C130275662321061769352616p0 .1 p0/ 42 39 C127491473152722338267400p0 .1 p0/ 41 40 C115576228634515184339946p0 .1 p0/ (16.58)

mi ma with P3;4.p0/ D P3;4 .p0/ C P3;4 .p0/.Togoto.r D 3; n D 5/ is straightforward using formal calculus but out of proportion to print the results as can be easily figured out from (16.58). 3 To sum up, we found that lc;3;2 D 4 with a4;3;2 D 27 D 3 , lc;3;3 D 8 with 7 15 a8;3;3 D 2; 187 D 3 , lc;3;4 D 16 with a16;3;4 D 14; 348; 907 D 3 and lc;3;5 D 32 31 with a32;3;5 D 617; 673; 396; 283; 947 D 3 . These results are independent of the formulae derived explicitly for lc;3;n with alc ;3;n in (16.39)and(16.46) and thus allows the checking of their validity. In particular the data given in terms of power of 3 would have hinted at the existence of an analytical expression. Indeed we have n 2n1 lc;3;n D 2 and alc ;3;n D 3 , which reproduces the results obtained from P3;n.p0/. To get a more precise feeling of the respective amplitudes of the various contributions given above, we plot them as a function of p0. Figure 16.37 shows mi ma the variation of both PrD3;n (left part) and PrD3;n (right part) for the four values mi n D 2;3;4;5 of levels. Note the slight asymmetry of PrD3;n towards the left ma (p0 <0:5)andPrD3;n towards the right (p0 >0:5). The net bottom minority mi contributions are small with a maximum at PrD3;5 D 0:175. Increasing the number of levels n shrinks the width of the distribution but simultaneously increases its height. It is different from the nasty bottom configuration with the smaller number of agents where the maximum decreases and is shifted towards the left. The bottom majority contributions are thus the leading ones and are rather close to the usual function Pr .p0/ besides the notable slight asymmetry towards p0 >0:5. The left part of Fig. 16.38 compares the contributions from the minority versus the majority bottom distributions. It enhances the minor weight of the minority configurations. Both are added and shown on the right part of the figure. Increasing the number of levels sharpens the S shape of the voting functions. Finally, Fig. 16.39 compares the various curves PrD3;n with n D 2;2;4;5 to the 5 function PrD243 which corresponds to a single voting group of size 3 D 243.The latest is the steepest and shown as a dashed line. They are perfectly symmetrical 1 around pc;r D 2 . It enhances our on going proof that with increasing r and n the “exact” Pall .p0/ is recovered. 16.9 All Bottom Minorities and Majorities Winning the Presidency 355

ma mi P 3,n P 3,n 0.175 1 r 3,n 5 0.15 0.8 0.125 r 3,n 2

0.6 0.1 r 3,n 2 0.075 0.4 r 3,n 5 0.05 0.2 0.025

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

mi ma Fig. 16.37 PrD3;n (left part)andPrD3;n (right part) as a function of p0 for four values of levels mi ma n D 2; 3; 4; 5. Note the slight asymmetry of PrD3;n towards the left (p0 <0:5)andPrD3;n towards mi mi the right (p0 >0:5). The maximum of PrD3;n is at PrD3;5 D 0:175. The dashed lines correspond to .r D 3; n D 4/

mi ma P3,n P 3,n,P 3,n 1 1

0.8 0.8

r 3,n 2

0.6 0.6

0.4 0.4

r 3,n 5

0.2 0.2

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

min maj Fig. 16.38 Comparison of PrD3;n and PrD3;n as a function of p0 for four values of levels n D min maj 2; 3; 4; 5 (left part). Variation of PrD3;n D PrD3;n C PrD3;n as a function of p0 for ﬁve values of levels n D 1; 2; 3; 4; 5 (left part). Note that the symmetry around (pc;3 D 0:5) has been restored by adding majority and minority contributions 356 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.39 Comparison of P3,n ,P243 PrD3;n as a function of p0 for 1 four values of levels n D 2; 3; 4; 5 with PrD35D243

0.8

r 3,n 2

0.6

0.4

r 3,n 5

0.2 r 243

p0 0.2 0.4 0.6 0.8 1

16.9.2 The Even Case r D 4: The Challenging View Point

We now carry out the same analysis as above with the even case r D 4. In this case, we must discriminate between the two asymmetrical view points of the challenging opinion A and the running power B opinion. We start with the A opinion for which 4 3 P4;1.p0/ D p0 C 4p0.1 p0/.Asfor.r D 3; n D 1/,the.r D 4;n D 1/ has no A nasty conﬁguration with mi P4;1 .p0/ D 0 (16.59) and ma 4 3 P4;1 .p0/ D p0 C 4p0.1 p0/: (16.60) Going to the next hierarchy .r D 4;n D 2/, we again have an absence of a nasty conﬁguration for A. We have mi P4;2 .p0/ D 0 (16.61) and

ma 16 15 14 2 P4;2 .p0/ D p0 C 16p0 .1 p0/ C 120p0 .1 p0/ 13 3 12 4 C560p0 .1 p0/ C 1604p0 .1 p0/ 11 5 10 6 C2352p0 .1 p0/ C 1216p0 .1 p0/ 9 7 C256p0.1 p0/ (16.62) 16.9 All Bottom Minorities and Majorities Winning the Presidency 357

ma with P4;2.p0/ D P4;2 .p0/. One additional step at .r D 4;n D 3/ creates the ﬁrst possibility of the occurrence of A nasty conﬁgurations, though things get a little bit more involved with

37 27 36 28 P4;3.p0/ D 67108864.1 p0/ p0 C 2030043136.1 p0/ p0 35 29 34 30 C29746003968.1 p0/ p0 C 280636686336.1 p0/ p0 33 31 32 32 C1912283332608.1 p0/ p0 C 10011331657728.1 p0/ p0 31 33 30 34 C41835977048064.1 p0/ p0 C 143170691284992.1 p0/ p0 29 35 28 36 C408582813459456.1p0/ p0 C 985381632995584.1p0/ p0 27 37 26 38 C2028373704426496.1p0/ p0 C3590244700558848.1p0/ p0 25 39 24 40 C5492698566778368.1p0/ p0 C7285071592716480.1p0/ p0 23 41 22 42 C8382603575932416.1p0/ p0 C8356157963246592.1p0/ p0 21 43 20 44 C7196216134132800.1p0/ p0 C5339381373830448.1p0/ p0 19 45 18 46 C3409957620624192.1p0/ p0 C1877799597216288.1p0/ p0 17 47 16 48 C895609342390464.1 p0/ p0 C 372302448529860.1 p0/ p0 15 49 14 50 C135857845615680.1 p0/ p0 C 43825248783072.1 p0/ p0 13 51 12 52 C12572140857408.1 p0/ p0 C 3220716904848.1 p0/ p0 11 53 10 54 C738078471360.1 p0/ p0 C 151125161056.1 p0/ p0 9 55 8 56 C27526401088.1 p0/ p0 C 4425885432.1 p0/ p0 7 57 6 58 C621216192.1 p0/ p0 C 74974368.1 p0/ p0 5 59 4 60 C7624512.1 p0/ p0 C 635376.1 p0/ p0 3 61 2 62 C41664.1 p0/ p0 C 2016.1 p0/ p0 1 63 64 C64.1 p0/ p0 C p0 : (16.63)

From which we extract all nasty conﬁgurations with

mi 37 27 36 28 P4;3 .p0/ D 67108864.1 p0/ p0 C 2030043136.1 p0/ p0 35 29 34 30 C29746003968.1 p0/ p0 C 280636686336.1 p0/ p0 33 31 32 32 C1912283332608.1 p0/ p0 C 10011331657728.1 p0/ p0 : (16.64)

This expression includes a contribution with a global tie .32=32/, normally attributed only to B but locally. To win, A requires a minimum of 27 A agents over 64, well located at the bottom. Although the B opinion has the advantage of 358 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures being in power and as such is favored by the local tie cases, the A opinion can also win with a minority at the bottom. The winning bottom A minorities cover the range 27 l 31. As noted, A can win with a global tie of l D 32 and from l D 33 upto l D 64 it wins with a bottom majority. The counterintuitive cases correspond to a winning A bottom minority with an eventual A majority in the surrounding population, i.e., with p0 >0:50.It is nevertheless very unlikely to occur. In the light of the above case .r D 4;n D 3/, it is not hard to imagine how mi ma messy is the explicit enumeration of PrD4;n and PrD4;n for the follow-up increase in the number of levels of the hierarchy with respectively .r D 4;n D 4/ and .r D 4;n D 5/. It goes rapidly out of hand, although it is a manageable task. We will thus skip the writing of the formulae and only present the associated graphs. mi The series of Fig. 16.40 show the evolution of PrD4;n as a function of increasing n with n D 2;3;4;5. It mainly shows that the various contributions are extremely small and very narrow. In solid terms, it implies an almost zero probability to occur from a random selection of bottom agents. mi The above ﬁnding of the negligible contribution from PrD4;n leads to the ma conclusion that PrD4;n contributes the quasi total part of PrD4;n. Figure 16.41 compares the various curves PrD4;n with n D 2;2;4;5 to the function PrD1024, which corresponds to a single voting group of size 45 D 1; 024. The last one is the steepest and is shown as a dashed line. It shows a large value of r, waves almost entirely the tie effect but yet not totally and Pall .p0/ is not recovered exactly. It is worth noting that the series of curves PrD4;n are very far from PrD1024 indicating that the tie effect at the expense of opinion A is still very strong even with ﬁve hierarchical levels.

16.9.3 The Even Case r D 4: The Running Power View Point

We have seen that the tie effect is very efficient for deriving the challenging A opinion from a democratic balance to reach the presidency by making any nasty bottom configuration a much rarer event than for the odd case. It is thus of special importance to examine the view point of the running B opinion, which benefits from 4 3 2 2 all local ties. We start with Q4;1.p0/ D .1 p0/ C 4.1 p0/ p0 C 6.1 p0/ p0 for a .r D 4;n D 1/ hierarchy. As for the challenging A opinion, there exists no nasty configuration, only a tie configuration giving

mi Q4;1.p0/ D 0 (16.65) and

ma 4 3 2 2 Q4;1.p0/ D Q4;1.p0/ D .1 p0/ C 4.1 p0/ p0 C 6.1 p0/ p0 : (16.66) 16.9 All Bottom Minorities and Majorities Winning the Presidency 359

mi P 4,n mi 1 P 4,n

6·10-7 0.8

5·10-7

0.6 4·10-7

3·10-7 0.4

2·10-7

0.2 1·10-7

p0 p 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 0

mi P 4,n mi P 4,n

-13 6·10-7 1.75·10

1.5·10-13 5·10-7 1.25·10-13 4·10-7 1·10-13 3·10-7 7.5·10-14

-7 2·10 5·10-14

-14 1·10-7 2.5·10

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

mi mi Fig. 16.40 PrD4;nD2 (upper left part)andPrD4;nD3 (upper right part) as a function of p0. While mi it is exactly zero for the left part at n D 2, it is very small at n D 3 for the right part. PrD4;nD4 mi (lower left part)andPrD4;nD5 (lower right part) as a function of p0. The width and the peak are drastically reduced by increasing the number of levels n

Going to the next hierarchy .r D 4;n D 2/, contrary to the A opinion, some nasty conﬁgurations do occur. We have

mi 12 4 11 5 Q4;2.p0/ D 216p0 .1 p0/ C 2016p0 .1 p0/ 10 6 9 7 C6792p0 .1 p0/ C 11184p0.1 p0/ (16.67) and

ma 8 8 7 9 6 10 Q4;2.p0/ D 12870p0.1 p0/ C 11440p0.1 p0/ C 8008p0.1 p0/ 5 11 4 12 3 13 C4368p0.1 p0/ C 1820p0.1 p0/ C 560p0.1 p0/ 2 14 15 C120p0.1 p0/ C 16a.1 p0/ C .1 p0/ (16.68) 360 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.41 Comparison of P4,n,P1024 PrD4;n as a function of p0 for 1 four values of levels n D 2; 3; 4; 5 with PrD45D1024 shown as a dashed line. PrD1024 is very 0.8 close to the vertical line of PrDall but yet with a r 1024 noticeable difference. The 0.6 series of curves PrD4;n are very far from PrD1024

0.4

r 4,n 5

0.2 r 4,n 2

p0 0.2 0.4 0.6 0.8 1

mi ma with Q4;2.p0/ D Q4;2.p0/ C Q4;2.p0/. The B opinion can thus conserve the presidency with only four bottom agents, out of 16, but strategically located. Altogether there exist four nasty conﬁgurations with many associated different distributions as seen from (16.67). We keep on going one step further with the evaluation of Q4;3.p0/ for (r D 4;n D 3/ from which we obtain

mi 56 8 55 9 Q4;3.p0/ D 279936p0 .1 p0/ C 14183424p0 .1 p0/ 54 10 53 11 C348053760p0 .1 p0/ C 5517310464p0 .1 p0/ 52 12 51 13 C63497798208p0 .1 p0/ C 564717954816p0 .1 p0/ 50 14 49 15 C4030451175744p0 .1 p0/ C 23661154247040p0 .1 p0/ 48 16 47 17 C116224488549720p0 .1 p0/ C 483760832893056p0 .1 p0/ 46 18 45 19 C1723889193801792p0 .1 p0/ C 5309920504998528p0 .1 p0/ 44 20 43 21 C14280344408820672p0 .1 p0/ C 33911780743802880p0 .1 p0/ 42 22 41 23 C71991290479991328p0 .1 p0/ C 138338824016067264p0 .1 p0/ 40 24 39 25 C243364033876949640p0 .1 p0/ C 395545870184687424p0 .1 p0/ 38 26 37 27 C597967608426639840p0 .1 p0/ C 844608604770890176p0 .1 p0/ 36 28 35 29 C1117784911352244304p0 .1p0/ C1388409711926838336p0 .1p0/ 34 30 33 31 C1620144839839062432p0 .1p0/ C 1777048240088494272p0 .1p0/ (16.69) 16.9 All Bottom Minorities and Majorities Winning the Presidency 361 and

ma 32 32 31 33 Q4;3.p0/ D 1832614129610932806p0 .1 p0/ C 1777088163782209728p0 .1 p0/

30 34 29 35 C1620287729893661088p0 .1p0/ C1388818264994293824p0 .1p0/

28 36 27 37 C1118770290955196752p0 .1p0/ C 846636978408207808p0 .1p0/

26 38 25 39 C601557853127198688p0 .1 p0/ C 401038568751465792p0 .1 p0/

24 40 23 41 C250649105469666120p0 .1 p0/ C 146721427591999680p0 .1 p0/

22 42 21 43 C80347448443237920p0 .1 p0/ C 41107996877935680p0 .1 p0/

20 44 19 45 C19619725782651120p0 .1 p0/ C 8719878125622720p0 .1 p0/

18 46 17 47 C3601688791018080p0 .1 p0/ C 1379370175283520p0 .1 p0/

16 48 15 49 C488526937079580p0 .1 p0/ C 159518999862720p0 .1 p0/

14 50 13 51 C47855699958816p0 .1 p0/ C 13136858812224p0 .1 p0/

12 52 11 53 C3284214703056p0 .1 p0/ C 743595781824p0 .1 p0/

10 54 9 55 C151473214816p0 .1 p0/ C 27540584512p0 .1 p0/

8 56 7 57 C4426165368p0 .1 p0/ C 621216192p0 .1 p0/

6 58 5 59 C74974368p0 .1 p0/ C 7624512p0 .1 p0/

4 60 3 61 C635376p0 .1 p0/ C 41664p0 .1 p0/

2 62 1 63 C2016p0 .1 p0/ C 64p0.1 p0/

64 C.1 p0/ : (16.70)

Already this simple .r D 4;n D 3/ hierarchy generates many B opinion nasty bottom conﬁgurations, which result from the natural local inertia of power related to the tie breaking at the beneﬁt of the current ruler to favor its staying in power. From (16.69), the minimum number of B agents required at the bottom to win the presidency with certainty is only 8 as against 56 A agents. Again we should not forget that these agents must be strategically located. Overall, the B nasty minority can vary from 8 up to 31 agents. Afterwards, it wins with a global tie and then with bottom majorities. Equations (16.69)and(16.70) are both impressive and beautiful looking, rather like a sailing ship with all its sails deployed in the wind. Esthetics is always a criterion in theoretical research, although only implicit and silent. Indeed, an ugly formula shows that something is missing, if not in the validity at least in the formulation. The notion of esthetics is rather subjective, but I am pretty sure that 362 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.42 Both 64 Qb 4,3 ,Q4 probabilities Q4.p0/ and Q4;3.p0/ as a function of p0. 1 The ﬁrst one is contained within the second one. The arrows show that three 0.8 iterations on the inside curve are identical to one iteration on the outside curve 0.6

0.4

0.2

p0 0.2 0.4 0.6 0.8 1 most theoreticians are guided by it at some level in their work. It certainly plays a role in the magic process of getting to the formal solution of a problem. At least for me, it is an essential ingredient. In addition, (16.69)and(16.70) also show how a problem can be very involved and sometimes unsolvable if not tackled from the right point of view. The direct detailed calculation of all the winning bottom configurations from the bottom geometry would be hopeless, against the simplicity of the iterative process. An illustration is given in Fig. 16.42 for .r D 4;n D 3/ with a comparison of Q4.p0/ and Q4;3.p0/ as a function of p0. As seen from the figure Q4.p0/ is contained within tQ4;3.p0/ as expected. The arrows show that three iterations on the inside curve are identical to one iteration on the outside one. As before for the A opinion, we will not write explicitly the formulae for mi ma mi ma Q4;4.p0/, Q4;4.p0/, Q4;5.p0/ and Q4;5.p0/ since this would need a whole fleet of sailing ships. Nevertheless, we present the associated plots in Figs. 16.43 and 16.44 as a function of p0 for the series of levels n D 2;3;4;5. The left part of Fig. 16.43 shows the variation of the probability of a B opinion nasty minority bottom configuration with increasing n. The situation is totally different from that of the A opinion. It is never negligible. On the contrary, it may even become certain. For n D 4;5, the width is slightly reduced but is more or less comprised between 0.40 and 0.80 and the associated peaks reach 1.00 around p0 D 0:62 or so as seen from the figure. The right part is more balanced and similar to the A situation opinion. To win from a bottom majority is indeed no different for the two competing opinions. 16.9 All Bottom Minorities and Majorities Winning the Presidency 363

mi ma Q 4,n Q 4,n 1 1

0.8 0.8

0.6 0.6

r 4,n 5 r 4,n 2

0.4 0.4

r 4,n 2 r 4,n 5

0.2 0.2

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

mi ma Fig. 16.43 QrD4;n (left part)andQrD4;n (right part) as a function of p0 for four values of levels mi ma n D 2; 3; 4; 5. Note the slight asymmetry of QrD4;n towards the right (p0 >0:5)andQrD3;n mi toward the left (p0 <0:5). The maximum of QrD4;n is at 1 and for both reaches n D 4; 5.The dashed lines correspond to .r D 3; n D 4/

Q4,n mi ma Q 4,n ,Q 4,n 1 1

0.8 0.8

0.6 0.6 r 4,n 2

0.4 0.4 r 4,n 5

0.2 0.2

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

min maj Fig. 16.44 Comparison of QrD4;n and QrD4;n as a function of p0 for four values of levels min maj n D 2; 3; 4; 5 (left part). Variation of QrD4;n D QrD4;n C QrD4;n as a function of p0 for four values of levels n D 2; 3; 4; 5 (left part). The nasty contributions are similar in amplitude to those of the democratic ones

The contributions from nasty minority versus majority bottom distributions are compared on the left part of Fig. 16.44. While for the A opinion the minority configuration’s weight is almost zero, here the figure enhances the major weight of the minority configurations. Indeed, while these nasty contributions are secondary when p0 <0:50, i.e., B has the majority, they become very important when B is in 364 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

Fig. 16.45 Comparison of Q4,n ,Q1024 QrD4;n as a function of p0 1 for four values of levels n D 2; 3; 4; 5 with QrD45D1024 0.8

r 1024

0.6

r 4,n 2

0.4

r 4,n 5

0.2

p0 0.2 0.4 0.6 0.8 1

the minority p0 >0:50. They are quite massive in the range of 0:50 < p0 <0:77 or so, which is consistent with the value of the critical threshold to power pc;rD4 0:77. The right part of Fig. 16.44 shows the addition of both majority and minority contributions. Increasing n sharpens the jump between a B or an A victory. It is worth emphasizing that the value of the critical threshold pc;rD4 is independent of n as seen in the ﬁgure. Finally the series of functions QrD4;n with n D 2;3;4;5 are compared to QrD45D1024 in Fig. 16.45. While increasing n makes the n D 5 associated curve shape more similar to that of QrD45D1024, there exists an incompressible shift due to the tie breaking which is by nature independent of n. This distance is a function of the voting group size r. Increasing further from the current r D 4 will reduce the distance, due to the reduction in tie possibilities. In order to see how huge the tie breaking effect is, we show both series of functions QrD4;n and PrD4;n with n D 2;3;4;5 together with QrD45D1024 and PrD45D1024 in Fig. 16.46. The tremendous breaking in democratic balance produced by an a priori very reasonable mechanism of power inertia is striking. To end this section, we check again the validity of our formulae (16.38), (16.39) and (16.46) now for the even case. We have for the minimum numbers of bottom n agents to win the presidency lc;r;n D ma with ma givenby(16.38) and for the 1man a r 1ma associated numbers of different bottom winning conﬁgurations lc ;r;n D ma . 16.9 All Bottom Minorities and Majorities Winning the Presidency 365

Q4,n ,Q1024 P4,n ,P1024 1 1

0.8 0.8

r 1024 r 1024

0.6 0.6

r 4,n 2

0.4 0.4

r 4,n 5 r 4,n 5

0.2 0.2 r 4,n 2

p0 p0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 16.46 Comparison of QrD4;n and QrD45D1024 (left part) with PrD4;n and PrD45D1024 as a function of p0 for n D 2; 3; 4; 5

We compare their respective values at r D 4 to the corresponding results extracted directly from PrD4;n and QrD4;n for the series n D 1; 2; 3.Itisworth stressing that lc;r;n is the minimum number of agents to win without any assumption about being a minority or majority. The inequality rn l man < c;r;n D 2 (16.71)

l > rn implies a minority, which was deﬁned as being nasty. In contrast, when c;r;n 2 , this means a majority, which thus satisﬁes the democratic balance at least at the hierarchical bottom. For a voting group size of r D 4 we have ma D 3 and ma D 2 for opinions respectively A and B. We obtain for opinion A lc;rD4;nD1 D 3 with alc ;rD4;nD1 D

4, lc;rD4;nD2 D 9 with alc ;rD4;nD2 D 256, lc;rD4;nD3 D 27 with alc ;rD4;nD3 D 67; 108; 864, which are identical to the corresponding values from (16.59), (16.61) 41 and (16.64). The inequalities lc;rD4;nD1 D 3> 2 D 2 and lc;rD4;nD2 D 9> 42 8 n 1; 2 2 D conﬁrm the absence of the A nasty minority for D . On the contrary, l 27 < 43 32 c;rD4;nD3 D 2 D signals the occurrence of nasty conﬁgurations from n D 3 and above.

The equivalent results for opinion B are lc;rD4;nD1 D 2 with alc ;rD4;nD1 D 6, lc;rD4;nD2 D 4 with alc ;rD4;nD2 D 216, lc;rD4;nD3 D 8 with alc ;rD4;nD3 D 279;936, which are identical to the corresponding values extracted from (16.65), (16.67)and 41 (16.69). The equality lc;rD4;nD1 D 2 D 2 D 2 conﬁrms the absence of the B nasty n 1 l 4< 42 8 l 8< 43 minority for D .However, c;rD4;nD2 D 2 D and c;rD4;nD3 D 2 D 32 prove the occurrence of nasty conﬁgurations already from n D 2 and above. 366 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

16.9.4 The Physicist’s Corner

The above remark about the nasty character of a winning bottom configuration using the inequality (16.71) could lead to the idea of avoiding them playing with both r and n since we found that they did not exist for opinion A with .r D 4;n D 1; 2/. The intuition could be that given a number of levels, increasing or reducing the voting size may suppress the nasty configurations. In order to check, this we have to solve (16.71). ma . r 1/n For the A opinion we plug in D 2 C to reach the condition 2 r> ; 1 1 (16.72) 2 n 1 which is impossible for n D 1 whatever the size of r. It yields r>4:83at n D 2 and r > 3:40 at n D 3, which means that no nasty configuration is possible with .r D 4;n D 2/ but occurs from r D 6 and above. At n D 3,wehaver > 3:40 showing that nasty configurations do occur with a hierarchy of .r D 4;n D 3/. These findings confirm our former results, but demonstrate that our intuition was wrong. There is no way of avoiding nasty configurations by increasing either r or n besides the unique limiting cases of n D 1 and .r D 4;n D 2/. r n Looking at opinion B, we plug ma D . 2 / into (16.71) to reach

2n1 >1; (16.73) which is independent of r and always satisfied for n D 1. Nasty configurations are unavoidable for B as soon as n D 2 above. To complete the analysis, we look at what happens for odd sizes for which ma D . rC1 /n rnC1 rn 2 and 2 should be substituted to 2 in the right part of (16.71). However, rC1 n rnC1 the new equation . 2 / < 2 is not solvable analytically in an easy formulation due to the C1 correction on the right side. In formal mathematical language it is said that the equation appears to involve variables that need to be solved in an essentially nonalgebraic way. Of course it is easily solved graphically or numerically. And here comes the physicist’s know how. The C1 correction arises from the fact that for odd cases, by definition, r is an odd integer and is thus not divisible by 2. To also determine the majority inside one local voting group within the global bottom configuration, we need to add 1 and then divide by 2 to get the corresponding integer, which is the minimum majority. But keeping in mind that only integer values are meaningful, we can extend the n rnC1 calculations to real numbers. Accordingly, to request having am lower than 2 rn 1 is the same as being lower than 2 since there is only a 2 difference. In other words, to request the minimum number of agents for a .r D 3; n D 2/ hierarchy to be lower than 5 is the same as requesting it to be lower than 4.5 since this number, being an 16.10 The Worrying Power of Geometric Nesting or How to Make... 367 integer, satisfies automatically the first constraint if the second one is valid. Once this change is done, that is without consequence, we get the condition 1 r> ; 1 1 (16.74) 2 n 1 which is always satisfied except at n D 1. It is in contrast to (16.74), which has one additional case at .r D 4;n D 2/.

16.10 The Worrying Power of Geometric Nesting or How to Make Certain a Very Rare Event

The investigation in the previous chapter led to the discovery of the existence of winning bottom minorities, or the nasty configurations. In particular they were found to be numerous and unavoidable. However, their respective probabilities of occurring are very very small. With the noticeable exception of even-sized voting groups where the tie breaking symmetry benefits the ruling opinion, a tremendous advantage is opened to stick to power very easily and by chance. It is essential to emphasize one more time that all the calculations for the associated probabilities are performed under the assumption of a random selection of agents in the various local voting groups. Such a setback of any hierarchical organization, even the most democratic ones, can be tolerated in comparison to all the benefits of using hierarchical organizations. However, a severe problem arises from the possibilities of the antidemocratic practice of geometrical nesting as discovered in the odd case. While the associated results apply to the challenging A opinion since winning a local election A requires a local majority, they are expected to become much more efficient in the even case for the ruling opinion. It is therefore the collective global monitoring of a given ultra small group which, by adequate geometric nesting, may transform an extremely rare event into a certain one. The above plan corresponds, using the minimum bottom minority size to reach the presidency to achieving a “democratic” holdup of an institution. It implies that the other opinion is not aware of the ongoing nesting attitude. The existence of nasty configurations opens a tremendous framework for democratic manipulation. Being aware of them produces a scheme for making certain a very rare improbable event. For instance, we saw that only 8 B agents strategically located in a .r D 4; n D 3/ hierarchy are sufficient to ensure against 56 A agents the preservation of the B presidency. In such a case with the 8 B agents selecting their respective locations, we are no longer dealing with randomness and probability but with the subtle manipulation of a democratic system so as to twist the democratic bottom-up voting dynamics to the advantage of the minority against that of a huge majority. 368 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

16.10.1 The Sudden and Unexpected Taking Over of Large Institutions

Making a very rare event into a certain one using strategic nesting illustrates the mechanisms by which small activist groups take power in various organizations including local unions, to the great surprise of everyone. When they do occur, these sudden take-overs of sometimes very old democratic institutions by ultra minority groups always come as a big unexpected shock. No one either understands or foresees such a drastic turning upside down of an otherwise stable leadership. All kinds of explanations are given afterwards, but they usually miss the central point since by nature it is very difﬁcult to prove it is not the result of a rare, still democratic event. In other words, once such a nesting minority holds the right positions, it is almost impossible to move its members away. The whole hierarchical scheme is based on anonymous locations of agents in the various voting groups. Anonymous means that a given agent is not asked its opinion before it becomes part of a group. It eventually reveals its opinion once it votes, but then its choice does not modify its belonging to the group. It is the paradox of the situation, that if a few agents can coordinate their location to take advantage of the global voting structure, then the structure itself cannot be aware of it. And once it is aware of it, it cannot do anything. To identify the manipulation and to neutralize it would oblige the structure to perform antidemocratic procedures, which in turn, would destroy its own democratic construction.

16.10.2 The Scary Lobbying

In addition to strategic geometric nesting, our results cast new light upon the well- known practice of lobbying. It has proven itself to be a very important and efficient tool in influencing decision making of large bodies. Many corporations allocate substantial budgets to their lobbying policy. Our study shows up in solid terms the dramatic power of well-targeted lobbying in institutional political and social decision making bodies. Indeed, lobbying is much easier than trying to place its own agents in the strategic locations. A given group does not need to focus on placing its own agents, but it may well instead identify these positions and then try to convert the few agents well positioned to support its opinion. The convincing process can be either by using arguments or other kinds of bribes. This converting procedure may turn out to be much more efficient than direct nesting since there exist many possible equivalent positions as seen from the above analysis with the numerous different nasty configurations associated to a nasty minority. However, our analysis indicates that blind or random lobbying is usually not very 16.10 The Worrying Power of Geometric Nesting or How to Make... 369 effective. Even a massive lobbying will not produce the expected goal with certainty. Only an understanding of the underlying geometry of the organization can allow efficient lobbying.

16.10.3 A Striking Idealized Illustration

To illustrate this discussion, consider the following idealized example of the presidential election in the United States. To make it clear without losing the essence of the purpose, we assume that there exists 101 states, with each one having exactly two millions voters. Each State has then one delegate to vote for the president. Accordingly, the B president is elected from 51 votes in its favor. This scheme can be achieved if B holds in each one of the 51 States only one million and one votes, and zero votes elsewhere. This adds up to ﬁfty-one million and ﬁfty-one votes, which scores a percentage of approximately 25%. The actual distribution of States and delegates is more complicated but the same evaluation can be performed and leads to the victory of a small minority. Of course our example is totally unrealistic since Democrats or Republicans or others are not going to move to concentrate living in the selected states following the optimization of the votes. However, our example is perfectly reachable for organizations where far fewer people are involved. It is given for the purpose of grasping the main features of geometric nesting. And in real systems, with smaller groups and more levels, we saw that densities of nasty agents is often negligible.

16.10.4 Hint to Restore the Democratic Functioning

The previous comments could create a very negative feeling about the whole hierarchical construction. It appears to be too fragile for easy manipulation by active minorities. It would thus lead to the conclusion that the one vote system with the whole population voting at once is the best and unique democratic model to designate the President of an organization. But that would be a wrong conclusion. First, we should keep in mind all the advantages of bottom-up hierarchies. Amongst them there are: much less people are involved, the inclusion of representatives from the minority at almost all levels of the hierarchy, some stability of the leadership for conducting necessary but unpopular policies, possibilities of more frequent consultations and the guarantee to satisfy the democratic balance at the Presidential level. All these are so, provided that our results are taken into account to build the hierarchy. Second, there exists a very easy and simple procedure to neutralize all possibilities of either nasty geometric nesting or nasty lobbying. Indeed, while the discovery and the determination of all bottom distributions leading to the presidency has 370 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures provided a clear data base for manipulation, it also indicates the remedy for solving this location paradox and for suppressing the associated democratic totalitarian option. A simple and efficient tip in order to avoid these nesting strategies would be a random redistribution of agents among the various bottom groups from time to time, and without notice. Such a reshuffling guarantees restoring the democratic character of the hierarchy, provided that it is done randomly. Any elaborated bias in the location of agents would thus be avoided. The setting of the reshuffling procedure is essential. The task of implementing such a procedure should not be given to the current leadership since this would provide it with the legal means of using the strategic nesting described above to its own advantage. An independent body involving the various opinions should be in charge of the reshuffling. Maybe it should even be carried out using some mathematical procedure to ensure the independence of the process. Randomness appears to be a key feature of the democratic functioning of bottom- up democratic voting. To enforce randomness with certainty may not be an easy task, but to reach a certain level of randomness is more realistic, and will still be sufficient for doing the job of breaking up any strategic correlations set up by an operational active ultra minority. If the organization is big, reshuffling bottom agents may be impossible, for instance if they are located in voting groups that are spread throughout a country. Nevertheless, reshuffling only at the higher hierarchical levels will also achieve the goal and is a much more manageable task to achieve.

16.11 Softening the Inertia Principle

While dealing with even-sized voting groups, we made the assumption that at a tie the elected representative belongs automatically to the current ruling opinion. This is an easy and powerful way to account for the well-known inertia of power and the series of unwritten advantages of being in power. We investigated in detail the effect of the apparently wise saying that “to make a change of leadership you need a majority,” and therefore at a tie we keep to what is already there. The consequences on the democratic balance were found to be disastrous. Accordingly, such a tie-breaking could be softened to temper the net effect on the hierarchical leadership. We have used so far (17.3) ! Xr r m rm p1 Pr .p0/ D p0 .1 p0/ ; r m mD 2 C1 16.11 Softening the Inertia Principle 371 which is identical to the earlier (17.3) with ! ! r X r r m rm r r P .p / p .1 p / k p 2 .1 p / 2 r 0 m 0 0 C r 0 0 r 2 mD 2 C1 with k D 0 since p0 and Pr .p0/ are the probabilities for A. It was then shown that k 1 P .p / P .p / k D 2 makes rD2w 0 D rD2w1 0 .However, can cover the whole range 0 k 1.Atk D 1, it is the A opinion, which beneﬁts from ties. Thus, the 0

4 3 2 2 p1 P4.p0/ D p0 C 4p0.1 p0/ C 6kp0.1 p0/ ; (16.76) which has the two attractors pB D 0 and pA D 1 unchanged but its separator pc;r now becomes a function of k in addition to being a function of r, p .6k 1/ 13 36k C 36k2 pc;4;k D (16.77) 6.2k 1/

p 1 k 1 with c;4 D 2 at D 2 obtained directly from (16.76). Figure 16.47 shows the variation of pc;4;k as a function of k. Given a value p0 there exists a range 0 kpc;4;k.Atk D kA, p0 D pc;4;k.ThevaluekA is a function of p0.For p0 0:77,wehavekA D 0 and kA D 1 when p0 0:23. It thus appears that given a hierarchy .r; n/ with some p0, varying only the value of k may shift the current president. It is worth examining the effect of the value of k on the number of levels required to ensure the election of a president whose support is above its critical threshold to 372 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

p Fig. 16.47 Variation of c;4 Pc,4,k as a function of k.Fora 1 collective risk aversion .k D 1/, pc;4 0:77 while for a collective novelty attraction .k D 0/, 0.8 pc;4 0:23. In the case where there is no collective A President .k D 1 / p D 1 bias 2 , c;4 2 .An 0.6 initial support of ps D 0:40 is shown to lead to an extremism in favor of (pS D 1) for the range of 0.4 bias 0 k 0:36.In contrast, the extremism is B President against (pS D 0)forthe 0:36 k 1 whole range 0.2

k 0.2 0.4 0.6 0.8 1

jp p j nN NŒn C 1 n D ln c;r 0 power. Using the formula d d with d ln r (15.8), which was found to be also rather good for even sizes, we need to substitute pc;r;k to pc;r to get

ln j pc;r;k p0 j nd;k D ; (16.78) ln r which applies for the whole range 0 p0 1. Figure 16.48 shows the variations of the number of levels nd;k for three values k D 0:10; 0:50; 0:80. The same initial support is shown to lead to different outcomes as a function of k as given in Table 16.17.Avaluep0 D 0:35 leads to respectively a B victory, a B victory, an A victory with nd;k D 2:01; 4:68; 6:99 and nNd D 3; 5; 7. Avaluep0 D 0:78 leads to three A victories with nd;k D 6:48; 3:14; 1:66 and nNd D 7; 4; 2. The practical question is to justify a value of inertia k that is different from zero when B is the ruling opinion. In some innovative high tech enterprises and ﬁnancial institutions, the priority is given to novelty or to youth to the detriment of old practices and people, making k D 1 instead of k D 0. Moreover, in addition to the inertia of the ruling opinion, there often exists a local inertia of power. This means that a local group voting at a tie will not elect a running opinion candidate but instead a candidate having an opinion that was held in this precise position beforehand, even if it belonged to the overall challenging opinion. Taking into account such competition between the inertia of local power and of global presidential power yields an overall average bias in favor of both opinions with a probability of k in favorofAand.1 k/ in favor of B. 16.11 Softening the Inertia Principle 373

Fig. 16.48 Number of nd required nd;k hierarchical 25 levels for three values k D 0:10; 0:50; 0:80. Associated values are shown for k D 0:10; 0:50; 0:80. 20 Speciﬁc points are shown for k 0.50 k 0.10 p0 D 0:35 and p0 D 0:78. They correspond respectively to a B victory, a B victory, an 15 k 0.80 A victory with nd;k D 2:01; 4:68; 6:99,and to three A victories with 10 nd;k D 6:48; 3:14; 1:66

p 0.2 0.4 0.6 0.8 1 0

Table 16.17 Values of n ;k d p0 k D 0:10 k D 0:50 k D 0:80 and nNd for three values k D 0:10; 0:50; 0:80 at 0.35 2.01,3 4.68,5 6.99,7 p0 D 0:35 and p0 D 0:78 0.78 6.48,7 3.14,4 1.66,2

16.11.1 The Physicist’s Corner

While renormalization group techniques are a mathematical trick to evaluate long wavelength fluctuations in collective phenomena, here we give a real meaning to each renormalization step. Instead of integrating out short range fluctuations, we build a voting procedure which associates an elected person to each local cell. Moreover, in contrast to critical phenomena where the focus is on the unstable fixed point, here we are studying the dynamics for reaching the stable fixed points. We are thus applying some physical tools to build a new quantitative model to describe the dynamics of political voting. Emphasis is on other aspects of the renormalization group transformation rather than on physics. We are not just making a metaphor between physics and politics. We are building a new and innovative tool for studying social and political hierarchies. As in the renormalization group scheme, we start from small-sized local cells. Here degrees of freedom are individual opinions. Each cell is constituted randomly from the population. Once formed, it elects a representative, either an A or a B using a local majority rule within the cell. These elected people (the equivalent of the superspin rescaled to an Ising one in real space renormalization) constitute the first hierarchical level of the hierarchy, 374 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures known as level-1. Here this level is real and not fictitious as in the renormalization group scheme. The process is then repeated again and again, each time starting from the people elected at one level to form new cells which in turn elect new representatives to build up a higher level. At the top level is the president.

16.11.2 Three Competing Opinions: It Becomes Even More Counterintuitive

Up to now, we have treated very simple cases to single out main trends produced by democratic voting aggregating over several levels. In particular we have shown how these thresholds become nonsymmetrical. Such asymmetries are indeed always present in most realistic situations, in particular when more than two groups are competing. Let us consider for instance the case of three competing groups A, B, and C [3]. Assuming a three-cell case, now the (A B C) configuration is unsolved using majority rule as was the case for the previous (A A B B) configuration. For the A B case we made the bias in favor of the group already in power, such as giving an additional vote to the committee president. For multigroup competitions, the situation is different. Typically the bias results from agreement between the parties. For instance, in most cases the two largest parties, say A and B are hostile towards each other while the smallest one C could compromise with either one of them. Then the (A B C) configuration gives an election of C. In such a case, we need 2A or 2B to elect respectively an A or a B. Otherwise a C is elected. Therefore, the elective function for A and B is the same as for the AB r D 3 model. This means that the critical threshold to full power for AandBis50%. Accordingly, for initial A and B supports, which are lower than 50%, the C gets full power provided that the number of levels is larger than some minimum limit [3]. Generalization is possible to as many groups as required. However, the analysis becomes much more involved very quickly and must be solved numerically. But the mean features of voting flows towards a fixed point are preserved. A thorough analysis is presented in Chap. 17.

16.12 Communist Collapse and French FN Victory

Our model sheds new light on the sudden and quick auto-collapse of Eastern European communist parties. Such an astonishing and crucial historical event marked the end of the last century, Communist parties have always seemed to be eternal. Then they collapsed all at once, taking everyone by surprise. Most explanations given were based on a 16.12 Communist Collapse and French FN Victory 375 hierarchical opportunistic change from the communist leaderships. The end of the Soviet army threat was an additional reason for the Eastern European countries. Our hierarchical model may provide a different new insight into such a unique event since communist organizations are based, at least in principle, on the concept of democratic centralism which is a tree-like hierarchy similar to our bottom- up model. Suppose for instance that the critical threshold to power within these organizations was of the order of 77% like in our size 4 case. We could then consider that the internal opposition to the orthodox leadership grew continuously over several decades and eventually became massive, yet without any visible change within the top strata of the organizations. At some point of this long and invisible internal growing opposition, the critical threshold was reached. At this point, a little further increase produced at once a surprising shift in the top leadership as exhibited in our series of Figs. 7.4–7.7. From the outside, the decades long increase of opposition was unnoticeable. Indeed it looked like nothing was changing. Once the threshold was passed, the brutal shift appeared to be instantaneous [16]. Therefore, what looked like a sudden and punctual decision of the top leadership could indeed be the result of a very long and solid phenomenon within the communist parties. Such an explanation is not opposed to the very many additional features which were instrumental in these collapses. It only singles out some trend within the internal mechanism of these organizations, which in turn made them extremely stable. Using our model, we predicted a political scenario which could happen in France, and which eventually did occur with respect to the extreme right party of the National Front. We highlighted the conditions for its success back in 1997 [14, 15] and it happened along these line in 2000 with its leader winning the presidential ﬁrst round. He eventually lost in the second ﬁnal run [17–19].

16.12.1 Hierarchies Are Everywhere

Although the model is only an extreme simpliﬁcation of real hierarchies, it does succeed in allowing us to grasp some of the essential and surprising mechanisms of majority rule voting. Being very generic, it allows applications to a large spectrum of different social and political organizations. Several counterintuitive results are obtained, which in turn provide paradoxical and unexpected explanations to a series of social features and historical events. In particular, we could quote the empirical difﬁculty in changing leaderships in well-established institutions. The hierarchical model we have presented, although being rather formal and ideal, may make it possible to shed new light on a large variety of human organizations, which are based totally or in part on hierarchical structures. This general scheme could be relevant for state institutions, political parties, social movements, and industrial and company structures. Hierarchies are everywhere, in whatever kind of human society, making our model a useful tool for revisiting various “democratic” procedures whose architectures are very rarely disputed. 376 16 Dictatorship Paradoxes of Democratic Voting in Hierarchical Structures

In today’s global world, many economic and financial institutions are spread worldwide and it is often unpractical to gather all the executives together for one vote all at once, making a hierarchical structure a functioning requirement. Using the proposed model, I was able to warn in the late 1990s against the dangers of the extreme right party in France [14]. I succeeded in predicting the scenario of the 2000 presidential election in which the extreme right leader, against all expectations, reached second place in the first stage of the national vote [15,16]. He eventually ran and lost in the second final two candidate race.

References

1. S. Galam, “Majority rule, hierarchical structures and democratic totalitarism: a statistical approach”, J. of Math. Psychology 30, 426-434 (1986) 2. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943-951 (1990) 3. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191-200 (1991) 4. S. Galam, “Real space renormalization group and social paradoxes in hierarchical organi- sations”, Models of self-organization in complex systems (Moses) Akademie-Verlag, Berlin V.64, 53-59 (1991) 5. S. Galam, “Paradoxes de la rgle majoritaire dans les systmes hierarchiques”,´ Revue de Bibliologie, 38, 62-68 (1993) 6. S. Galam, “Application of Statistical Physics to Politics”, Physica A 274, 132-139 (1999) 7. S. Galam, “Real space renormalization group and totalitarian paradox of majority rule voting”, Physica A 285, 66-76 (2000) 8. S. Galam and S. Wonczak, “Dictatorship from Majority Rule Voting”, Eur. Phys. J. B 18, 183-186 (2000) 9. S. Galam, “Democratic Voting in Hierarchical Structures”, Application of Simulation to Social Sciences, G. Ballot and G. Weisbush, Eds. Hermes, Paris, 171-180 (2000) 10. S. Galam, “Building a Dictatorship from Majority Rule Voting”, ECAI 2000 Modelling Artiﬁcial Societies, C. Jonker et al, Eds., Humboldt U. Press (ISSN: 0863-0957), 23-26 (2001) 11. S. Galam,“How to Become a Dictator”, Scaling and disordered systems. International Work- shop and Collection of Articles Honoring Professor Antonio Coniglio on the Occasion of his 60th Birthday. F. Family. M. Daoud. H.J. Herrmann and H.E. Stanley, Eds., World Scientiﬁc, 243-249 (2002) 12. S. Galam, “Dictatorship effect of the majority rule voting in hierarchical systems”, Self- Organisation and Evolution of Social Systems, Chap. 8, Cambridge University Press, C. Hemelrijk (Ed.) (2005) 13. S. Galam, “Stability of leadership in bottom-up hierarchical organizations”, Journal of Social Complexity 2 62-75 (2006) 14. S. Galam, “Le dangereux seuil critique du FN”, Le Monde, Vendredi 30 Mai, 17 (1997) 15. S. Galam, “Crier, mais pourquoi”, Liberation,´ Vendredi 17 Avril, 6 (1998) 16. S. Galam, “Le vote majoritaire est-il totalitaire ?”, Pour La Science, Hors serie,´ Les Mathematiques´ Sociales, 90-94 July (1999) 17. S. Galam, “Citation in front page of the Figaro in an editorial from Jean dOrmesson”, Le Figaro, Mardi 4 Juin, 1 (2002) 18. S. Galam, “Risque de raz-de-maree´ FN, Entretien, France Soir, La Une et 3, Mercredi 5 Juin (2002) 19. S. Galam, “Le FN au microscope”, Le Minotaure 6, 88-91, Avril (2004) Chapter 17 Bottom-Up Democratic Voting in a Three-Choice Competition

The use of bottom-up democratic voting in hierarchical structures was shown to produce some major and unexpected biases which in turn might drive toward the stable establishment of democratic dictatorships. However, this demonstration was performed in the case of two competing parties A and B. If, as is often the case, political competitions end up as a binary race via the building up of ad hoc coalitions, it is of interest to investigate the dynamics of these “internal” coalitions. To achieve such a search, we now consider the case of three competing parties A, B, and C, still using bottom-up democratic voting hierarchical organizations. The instrumental mechanism in the production of these dictatorship trends was shown to be embodied in the use of even local voting groups. While this restriction could be viewed as a limit to the robustness of our ﬁndings, we show that they resist an extension of the model to more realistic situations. Indeed, it is found that extending the competition to three different parties yields even more unbalanced situations with strong asymmetries for winning the top leadership for the respective parties A, B, and C. The results highlight the overrepresentation of small in-between parties in many hierarchical structures. Shifting alliances with either of two opposite strong parties gives them a weight of elected representatives at top levels that is much higher than their support at the bottom of the hierarchy. Earlier studies considered the extreme case of two big parties and a tiny one where an alliance is set between one of the big parties and the tiny one [1, 2]. An extensive study has been carried out in the frame of opinion dynamics using three competing opinions. The equations are identical but the social meaning is totally different [3]. In this case, it is neither a hierarchy that is built nor agents that are elected. It is the same people involved who eventually change their opinion.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 379 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 17, © Springer Science+Business Media, LLC 2012 380 17 Bottom-Up Democratic Voting in a Three-Choice Competition

17.1 Two Competing Parties in Short

17.1.1 The General Frame

Two parties A and B are competing to run a bottom-up democratic hierarchical structure. It may be a political group, a firm, a society, or any human organization. Each member of the corresponding frame has an opinion. At the moment of the vote, the respective proportions in favor of A and B are p0 and .1 p0/. At the bottom of the hierarchy are positioned people who have been selected at random from the associated population. Each person supports either party A or party B. They are randomly organized in small groups of size r. Then, in a second step, each one of these r-sized groups elects a representative according to a local majority rule. Following the local majority, the associated representative is a supporter of either the A or B parties. For a two party competition, a local majority always exists for odd-sized groups. However, for even-sized groups a tie may occur, resulting in a no majority situation. In these cases, mechanisms of different kinds can be evoked to justify that an A supporter is elected with a probability of k and a B supporter with a probability .1 k/.Thevalueofk is a direct function of both the psychosociological nature and the history of the population that is voting. The above elected agents constitute the first hierarchical level of the hierarchy denoted by level 1 with respect to the bottom level denoted 0. The local group voting process used at the bottom level is then repeated again identically to build a second level where representatives are democratically elected from the democratically elected representatives of the first level of the bottom-up hierarchy. The process can be repeated a certain number of times n using the same voting scheme to move from a level l to the above level .l C 1/. Each time, starting from the elected people at a given level, new groups are formed, which in turn elect new representatives to build up a higher level. At the top level is the president. However, since each one of these votes reduces the number of associated representatives by a factor r from level l to level .l C 1/, it is necessary to have selected enough local bottom groups in order to ensure an n-level hierarchy. Therefore, rn local groups are required at the bottom level 0 to have a succession of n levels. This yields respectively rn1 elected representatives at level 1, rn2 elected representatives at level 2, rn1 elected representatives at level l, and rnn D 1 elected representative at level n.

17.1.2 Predicting the Results of Democratic Elections

Once the above structure has been set, the question arises, “Is it possible to predict the corresponding voting results?” 17.1 Two Competing Parties in Short 381

More precisely, knowing the overall proportions of support for A and B with the population concerned, “is it possible to predict the outcomes at each one of the hierarchical levels?”. The answer is yes, it is possible to make solid predictions, knowing those bottom supports p0 and 1 p0, the voting group size r and the number of levels n. However, to make a solid prediction implies a deterministic prediction. The respective proportions pl and .1pl / of the respective supporters of A and B can be evaluated at each level l of the n levels as a function of p0, r,andn. Indeed, from the voting majority rule plus the probabilistic result at a tie, the probability plC1 to have an A supporter elected at level .l C 1/ can be calculated explicitly from plC1, the proportions of A supporter representatives at level l. The voting function plC1 D Pr .pl / can be written, ! ! r X n h io r r m r m r r r r P .p / p .1 p / kı N p 2 .1 p / 2; r l m l l C 2 2 r 0 0 (17.1) r 2 mDNŒ2 r rŠ k 0 k 1 NŒx x where m mŠ.rm/Š, is a real with , Integer part of and ıfxg is the Kronecker function, i.e., ıfxgD1 if x D 0 and ıfxgD0 if x ¤ 0. r r 1 For odd-sized r, there exists no tie contribution as seen from f 2 NŒ2 gD 2 , which makes the Kronecker function always equal to zero and in turn cancels the r r last term. On the contrary, an even r size makes f 2 NŒ2 gD0 with the last term r r r 2 contributing as k r p0 .1 p0/ 2 . 2 A more readable manner to write (17.1) is to split it into two different expressions with ! Xr r P .p / pm.1 p /rm; r l D m l l (17.2) rC1 mD 2 for odd sizes r,and ! ! r X r r m r m r r P .p / p .1 p / k p 2 .1 p / 2; r l m l l C r l l (17.3) r 2 mD 2 C1 when r is even. A broken democratic balance is thus produced by the votes at a tie. Monitored by k k>1 the value of , the A party is favored over the B party when 2 and vice versa 1 1 for k<2 . The democratic balance is restored only at k D 2 . Depending on the current values of p0, r and n, pn is either within the range 0

1.0

0.8

0.6

0.4

0.2

5 10 15 20 25 30 35

Fig. 17.1 Any small proportion p0 grows while climbing up the hierarchy to eventually reach 1 provided the number of levels n is sufﬁciently large. Four values are shown p0 D 103;102;101;3 101 with k D 0:7

17.1.3 The Bottom-Up Voting Dynamics

To determine the exact voting dynamics driven by (17.1) bottom-up democratic voting, we need to solve the associated fixed point equation to find out the attractors and separators, which monitor it. Solving Pr .pl / D pl , two fixed points pB D 0 and pA D 1 are found. They are invariant with size change.

17.1.3.1 The Very Special Pair Case

The case r D 2 is of a special interest since it corresponds to a second degree polynomial making the existence of an extra fixed point impossible. Equation (17.1) is written 2 2 p1 P2.p0/ D p0 C 2kp0.1 p0/; (17.4) from which we conclude that out of the two fixed points pB D 0 and pA D 1, one must be an attractor of the dynamics while the other one is unstable. Indeed, when 1 k>2 , any small proportion p0 would grow while climbing up the hierarchy to eventually reach the value of 1 provided the number of levels n is sufficiently large, as seen in Fig. 17.1. 17.1 Two Competing Parties in Short 383

1.0

0.8

0.6

0.4

0.2

5 10 15 20

Fig. 17.2 Same initial four values of p0 as in Fig. 17.1 but with a stronger bias k D 0:9

3 2 1 1 Four different values p0 D 10 ;10 ;10 ;3 10 are used with k D 0:7. After 30 voting levels, every case ends up with an elected president who is a member 3 of the A party despite the frame support being totally negligible, with p0 D 10 . However, no hierarchy will have 30 levels. Most have ten levels or so. For n D 10, 1 1 both small supports p0 D 10 ;3 10 have a signiﬁcant chance to win the presidency with a probability of p10 equal to 0.85 and 0.97, respectively. Considering a more extreme bias in favor of party A with a value k D 0:9 leads to more dramatic dictatorship effects as shown in Fig. 17.2. The same four 3 2 1 1 different values p0 D 10 ;10 ;10 ;3 10 are used. Now, only 15 voting levels are enough to guarantee an A member elected president for all cases. Within eight levels, the chance to win the presidency is respectively p0 D 0:10; 0:65; 1; 1 3 2 1 1 for p0 D 10 ;10 ;10 ;310 . The same results are shown in Fig. 17.3,which is another way of representing the dynamics of voting while climbing the hierarchy democratically. These are very worrying results! Such a radical effect in favor of the status quo of pair interactions has been substantiated in some nice work by Kulakowski and Nawojczyk [4]onthevotesof married couples in the United States. Figure 17.3 shows another extreme situation where the bias at a tie is in favor of party B with a probability of .1 k/ D 0:70, which is much stronger than in favor of party A with a probability of only k D 0:30. A bottom support for A at the bottom of p0 D 0:85 shrinks to p8 D 0:20 after climbing up to the eighth level (Fig. 17.4). 384 17 Bottom-Up Democratic Voting in a Three-Choice Competition

pl 1 1.0

0.8

0.6

Ais0 elected A is 100 elected

0.4

0.2

pl 0.2 0.4 0.6 0.8 1.0

Fig. 17.3 The variation of plC1 as a function of plC1 for k D 0:9. The evolution from p0 D 0:10 is shown. Eight levels are required to reach 1

pl 1 1.0

0.8

0.6

Ais0 elected Ais100 elected

0.4

0.2

pl 0.2 0.4 0.6 0.8 1.0

Fig. 17.4 The variation of plC1 as a function of plC1 for k D 0:3. The evolution from p0 D 0:85 is shown. Eight levels shrink the bottom support of p0 down to p8 D 0:20 17.1 Two Competing Parties in Short 385

17.1.3.2 The Odd Cases

While we saw that using the smallest size of r D 2 for group voting is very unbalanced with respect to both parties, the next smallest size of r D 3 ensures a democratically balanced situation between A and B. In this case, we have

3 2 plC1 P3.pl / D pl C 3pl .1 pl /; (17.5) where the voting function P3.pn/ denotes a simple majority rule and pn is the proportion of A elected representatives at level n. p 0 p 1 p 1 Besides the two stable ﬁxed points B D and A D , a third one c;3 D 2 appears. Being unstable, it determines the critical threshold to have repeating votes p p > 1 p to drive toward either full power at A for 0 2 or total disappearance at B when 1 p0 < 2 . It is democratic since it is the bottom majority which is secure at the top provided there are enough hierarchical levels. For instance, a bottom support of p0 D 0:45 yields successively p1 D 0:42, p2 D 0:39, p3 D 0:34, p4 D 0:26, p5 D 0:17, p6 D 0:08 down to p7 D 0:02 before p8 D 0:00. A strong minority of 45% is self-eliminated within eight levels. Fewer levels give a chance to the minority of running the organization. The situation is perfectly symmetrical with respect to A and B. Both share the same threshold to full power. The bottom-up voting structure is wonderful since it simultaneously allocates local power to the minority party and ensures the organization presidency to the majority opinion given enough numbers of levels. It costs very little in terms of manpower since only a small part of the associated population is needed to implement and activate the hierarchy. The above dynamics is invariant for an increase in the size r as long as odd values are used. Only the number of levels and people involved change. The ﬁrst one decreases while the second one increases.

17.1.3.3 The Regular Even Cases

The pair size case stands apart from any other even size since it has only two fixed points. Already, at r D 4, the next up even size, three fixed points are obtained. The odd case two attractors pB D 0 and pA D 1 are recovered but now the unstable p 1 fixed point c;r no longer sits at 2 . The dynamics turns nondemocratic with an asymmetry in the threshold values for respectively rulers and nonrulers. For r D 4, the probability to have a representative A elected given by (17.1) is written,

4 3 2 2 plC1 P4.pl / D pl C 4pl .1 pl / C 6kpl .1 pl / ; (17.6) 386 17 Bottom-Up Democratic Voting in a Three-Choice Competition while the probability to have a B elected is

4 3 2 2 1 P4.pl / D .1 pl / C 4.1 pl / pl / C 6.1 k/pl .1 pl / ; (17.7) where the last term embodies the bias in favor of B. The threshold for A is thus, p 1 13 p C 0:77; c;4 D 6 (17.8)

1 which is drastically shifted from 2 . Simultaneously, the B threshold to stay in power is down to about 23%. To take over power, the A needs to go over 77% of the bottom support while to stick to power, the B only needs to keep their support above 23%. The bottom-up voting is no longer democratic. In addition, the asymmetry of the tie makes the number of levels to democratic self-elimination even smaller than in the democratic r D 3 case. For r D 4,the above r D 3 series starting from p0 D 0:45 becomes p1 D 0:24, p2 D 0:05 and p3 D 0:00. Instead of eight levels, three are enough to make the A disappear from the elected representatives. But now, even with a bottom majority the challenging party A can get self-eliminated, as seen from the case p0 D 0:70 which yields p1 D 0:66, p2 D 0:57, p3 D 0:42, p4 D 0:20, p5 D 0:03,andp6 D 0:00.A population of 70% is thus self-eliminated democratically after six bottom-up levels. An a priori reasonable bias in favor of the ruling party has turned a majority rule democratic voting system into a totalitarian outcome. To get to power, the challenging party must pass over 77% of the bottom support to reach the presidency. Such ﬁgures are virtually out of reach in any normal democracies with two competing opinions. Such an overwhelming support would be often the signature of a dictatorship or a totalitarian regime. Increasing the size of even groups weakens the breaking of the democratic symmetry between the two parties as expected sincep increasing the size makes the p 1C 13 p 1 chance of getting a tie less probable. From c;4 D 6 for size 4 we have c;r D 2 for r!1. Larger voting group sizes also reduce the number of levels necessary to get to the stable ﬁxed points, which in turn guarantee a deterministic result.

17.1.4 From Theoretical Principles to Reality

We saw that given a bottom-up voting hierarchy using groups of size r for each one of two competing parties, a threshold pc;r exists to decide on its required initial weight to ensure a successful outcome while climbing up the hierarchy. Figure 17.5 sums up the results for a bottom support of p0 D 0:65 using voting groups of respective sizes of r D 3; 4; 5; 6. The bottom-up voting outcomes are shown as a function of the hierarchical level. Both cases for r D 3; 5 lead to an A ruling from the ﬁfth level for r D 3 and the fourth level for r D 5. On the contrary, r D 4;6 17.1 Two Competing Parties in Short 387

Fig. 17.5 The evolution of a pl bottom p0 D 0:65 climbing 1 up the hierarchy is shown as a r 5 function of the hierarchical level for voting group sizes of 0.8 r 3 r D 3; 4; 5; 6. The cases r D 3; 5 leads to A ruling from the fifth level for r D 3 0.6 and the fourth level for r 6 r D 5. The cases r D 4; 6 leads to B ruling although the A party has a stronger 0.4 majority at the bottom. However, the B party rules with certainty from the fifth 0.2 level using r D 4 and only r 4 from the ninth one for r D 6 Level l 2 4 6 8 10 produces a B ruling although the A party has a strong majority at the bottom with 65%. However, the B party is ruling with certainty from the fifth level using r D 4 and only from the ninth one for r D 6. However, if to be above the threshold guarantees an increasing weight in terms of the elected representatives, to be certain to win at the presidency level, a minimum number of levels is required and that number is a function of the bottom support p0. The good news is that in addition to knowing the value of pc;r it is possible to calculate the critical number of levels which ensures running the organization, provided that p0 >pc;r. To achieve this goal, we expand the voting function pl D Pr .pl1/ around the unstable fixed point pc;r to get

pl pc;r C .pl1 pc;r/r ; (17.9)

dPr .pn/ r jp Pr .pc;r/ D pc;r where dpn c;r with . Rewriting (17.9)as

pl pc;r .pl1 pc;r/r ; (17.10) it is possible to iterate the process from level l down to level 0 to reach,

l pl pc;r .p0 pc;r/r ; (17.11) from which we get the equation

l pl pc;r C .p0 pc;r/r ; (17.12) 388 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.6 The variation of nc the critical number of levels 20 nc;r as a function of p0 for nc 34 r 4 three group sizes 17.5 r D 4; 10; 30 r 10

15 r 30 12.5

7.5

2.5

p 0.2 0.4 0.6 0.8 1 0 which in turn yields, 1 pl pc;r l ln : (17.13) ln r p0 pc;r

Equation (17.13) links pl the proportion of A elected representatives at level l given an A bottom support of p0 with respect to the practical question of determining the critical number of levels to ensure a deterministic outcome, i.e., to be certain of which party will rule the organization. Certainty means that it is located at one of the two attractors pA and pB. Accordingly, for p0 >pc;r we want to determine the value of l to reach pl D pA while for p0

C Two different critical numbers of levels nc;r and nc;r can thus be obtained from (17.14). The ﬁrst one corresponds to p0 >pc;r with att D 1 pc;r and the second one has att Dpc;r for p0

17.1.5 From Reality to Implementation

The above results are of interest but not very useful in practice since most bottom- up hierarchies already exist and are not modiﬁed before an election. Therefore the number of levels is given and not open to changing. Therefore, to take advantage of (17.14) we have to reverse its variables and parameters. Thus, the question “How many levels are needed to self-eliminate an opinion?” must be set to, “Given n levels, what is the minimum bottom support to take over the presidency with certainty?” In other words for the ruling party, the strategic question is “Given n levels, what is the critical overall support above which the ruling opinion is automatically reelected?” which means not worrying about the current ruling policy even if it is unpopular as long as the bottom support is above that value. For the challenging party it is “Given n levels what is the critical overall support required to ensure a power shift?” To implement this operative question, we rewrite (17.12) to extract the value of p0 with l p0 D pc;r C .pl pc;r/r : (17.15)

Plugging pl D 0 means that at level l the A party is certain to get no elected representative. Therefore, given an n-level hierarchy using voting groups of size r pr;n the critical value of bottom support 0;B at and below which the B party wins the presidency is given by pr;n p .1 n/: 0;B D c;r r (17.16) p 1 pr;n At the other extreme, putting n D into (17.15) gives the bottom threshold 0;A from which it is the challenging A party which wins the presidency. We get pr;n pr;n n ; 0;A D 0;B C r (17.17) pr;n where 0;B is given by (17.16). From both (17.16)and(17.17), three regimes are obtained, of which two are p pr;n pr;n

17.2 Three Competing Parties

From the above study of two competing parties, we could be inclined to temper the astonishing associated results by stressing that they directly relied on the broken tie hypothesis, which indeed required the existence of even-sized voting groups. However, tie breaking is a much more general feature of voting, which is not restricted to even sized groups. Indeed, tie situations may occur for odd sizes when three-parties A, B, and C are competing. For instance, three voting group sizes yield tie situations for A B C configurations. In such cases, the tie cannot be resolved by giving an advantage to the ruling party. For two competing parties, it makes sense that the challenging party must have at least half plus one votes to win the election. But for three-parties at a tie, each party has one third of the votes, which is less than 50 votes, so that an agreement among two parties is necessary to win the election. This corresponds to the alliance set up among the various parties during an election. As soon as two parties make an alliance to join their votes, the tie is resolved in favor of one of their respective representatives. The voting rules are shown in Fig. 17.7 for the most general case. For each configuration of three agents with a local majority of 2 or 3, the elected representative belongs to this very majority. Accordingly, 2 A, 2 B, 2 C elects respectively an A, B, or C, whatever the third agent. On the contrary, all six configurations with 1 A, 1B, 1C elect a representative according to the current voting agreement running between two parties, independently of the third one. To account for all possible cases of alliances, which might be global or local, we introduce a probability ˛ to have an A elected, a probability

Fig. 17.7 The voting rules for groups of size 3 17.2 Three Competing Parties 391

ˇ to have a B elected and a probability to have a C elected under the constraint ˛ C ˇ C D 1. Therefore, only two out of these three parameters are independent. Without any loss of generality, we select them as ˛ and ˇ with D 1 ˛ ˇ. Denoting pa;lC1;pb;lC1;pc;lC1 the respective proportions of elected representatives of party A, B, and C at level .l C 1/, we have at the lower level l just below, the proportions pa;l ;pb;l ;pc;l of A, B, and C representatives. These proportions obey the constraint pa;l C pb;l C pc;l D 1 since we assume that every agent supports one party. The problem has therefore only two independent variables. We choose them as pa;l and pb;l and thus pc;l D 1 pa;l pb;l . The associated voting functions for respectively A, B, and C are thus,

3 2 pa;lC1 Pa;3.pa;l ;pb;l / D pa;l C 3pa;l .1 pa;l / C 6˛pa;l pb;l .1 pa;l pb;l /; (17.18) 3 2 pb;lC1 Pb;3.pa;l ;pb;l / D pb;l C 3pb;l .1 pb;l / C 6ˇpa;l pb;l .1 pa;l pb;l /; (17.19)

pc;lC1 D .1 pa;lC1 pb;lC1/: (17.20)

17.2.1 Two Competing Parties, a One-Dimensional Problem

When having a situation with two competing parties A and B with respective support pa;0 and pb;0 the problem is a two-dimensional problem with the associated proportions pa;l and pb;l of the elected representatives at each hierarchical level l of the hierarchy. A point showing the respective ratios of A and B can be put in a two-dimensional ﬂow diagram. However, the constraint pa;l C pb;l D 1 restricts the area available to these representative points to the line pb;l D 1 pa;l in the plane .pa;l ;pb;l / as seen in Fig. 17.8. To implement this property, we perform a change of coordinate system. The source coordinate system is deﬁned by the origin O and the unit vectors .i ; j /. The new coordinate system is obtained by a translation of the origin from point O to point A whose coordinates are .1; 0/ and a =2 counterclockwise rotation of the unit vectors to get .u; v/ asshowninFig.17.8. Accordingly, a point Pl associated to a ratio pa;l and pb;l is represented simultaneously by

OP l pa;l i C pb;l j D OA C qa;l u C qb;l v; (17.21) where its coordinates are respectively .pa;l ;pb;l / in the coordinate system .O; i ; j / and .qa;l ;qb;l / in the coordinate system .A; u; v/. The coordinate transformations 392 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.8 The two party problem reduces to a one-dimensional problem

are readily obtained by expressing .i ; j / in the coordinate system .vecu; v/.We thus obtain, 8 p p < i 2 2 ; D 2 u C 2 v p p (17.22) : 2 2 i D 2 u C 2 v; which after substitution into (17.21) yields the proportion transformations ( p qa;l D 2.1 pa;l /; p (17.23) qb;l D 2.1 pa;l pb;l / D 0; p which reduces the problem to the range 0 qa;l 2 where qa;l is no longer the proportion of A elected at level l but the coordinate of the point whose projections .i ; j / on the axes represents the A and B proportions of thep elected representatives. Indeed, as seen from Fig. 17.8 the largest value of qa;l is 2 which corresponds to both the length of .AB/ and the point B whose coordinates are .1; 0/. 17.2 Three Competing Parties 393

17.2.1.1 The Physicist’s Corner

The above transformation to change the coordinate system while dealing with a one-dimensional problem could indeed look like a rather tedious and useless manipulation. And from a pragmatic point of view it is. Since we have pa;l Cpb;l D 1 for any value of l, we can at once realize that the problem is governed by a single free parameter either pa;l or pb;l , which in turn makes the problem right away appear as being one-dimensional. So why did I do this mathematical digression? The reason is to illustrate how a geometrical feature of a given problem may be disclosed by ﬁnding the right coordinate system. Once the feature has been identiﬁed, the whole mathematical treatment becomes much simpler. Accordingly, the adequate coordinate system is not automatically the one which is associated with the immediate parameters of the problem. In addition, going through the transformation for a two-parameter one- dimensional system paves the way to implement the similar transformation for the more complicated system of having a three-parameter two-dimensional system as is shown in the following paragraph.

17.2.2 Three Competing Parties, a Two-Dimensional Problem

While for two competing parties the constraint pa;l C pb;l D 1 reduces the problem to one dimension, for three competing parties the constraint becomes pa;l C pb;l C pc;l D 1, which in turn reduces the problem to two dimensions. Accordingly, although we are in three-dimensional space .pa;l ;pb;l ;pc;l/ all points describing the various possible distributions of ratios among elected representatives from the three-parties, are conﬁned onto the triangle pa;l C pb;l C pc;l D 1 as seen in Fig. 17.9. Following the above procedure, we need to make a change of coordinate system from the source .O; i ; j ; k/. The origin O is translated to C with coordinates .0;0;1/ and two successive rotations of the unit vectors. The ﬁrst one is =4 clockwise along the k axis and the second one is of an angle along the line AB as shown in Fig. 17.9. The new unit vectors are .u; v; w/. Accordingly, a point Pl associated to a ratio pa;l ;pb;l ;pc;l is represented simultaneously by

OP l pa;l i C pb;l j C pc;lk D OC C qa;l u C qb;l v C qc;lw; (17.24) where its coordinates are respectively .pa;l ;pb;l ;pc;l/ in the coordinate system .O; i ; j ; k/ and .qa;l ;qb;l ;qc;l/ in the coordinate system .C; u; v; w/. Expressing .u; v; w/ in the coordinate system .i ; j ; k/,weget 8 p p ˆ u D 2 cos./i C 2 cos./j sin./k; <ˆ 2 2 p p v D 2 i C 2 j ; (17.25) ˆ 2 2 :ˆ p p p 3 i 3 j 3 k: w D 3 C 3 C 3 394 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.9 The three-party problem reduces to a two-dimensional problem

p p p After substitution into (17.24) using sinŒ D 2= 3 and cosŒ D 1= 3,the proportion transformations are obtained with 8 p 1 2 1 ˆ pa;l p qa;l qb;l p qc;l; ˆ D 6 2 C 3 < p 1 2 1 pb;l D p qa;l C qb;l C p qc;l; (17.26) ˆ 6 2 3 :ˆ 2 1 p 1 p q p q : c;l D 6 a;l C 3 c;l Inverting the associated matrix yields 8 p ˆ q 6 .p p /; <ˆ a;l D 2 a;l C b;l 1 qb;l p . pa;l pb;l /; (17.27) ˆ D 2 C :ˆ qc;l D1 C pa;l C pb;l C pc;l D 0; which are the respective proportions of the elected representatives of each party. Equation (17.27) demonstrates the reduction of the three dimensional space onto ABC .C; ; / the two-dimensional triangle of Fig.p 17.9p spannedp byp pu v p. Here, the A; B; C . 3= 2; 2=2/ . 3= 2; 2=2/ coordinates of pare respectively p , p p and .0; 0/. We thus have 2=2 qa;l 2=2 and 0 qb;l 3= 2 with the constraint 0 qa;l C qb;l 1. 17.2 Three Competing Parties 395

17.2.2.1 The Physicist’s Corner

It is worth emphasizing that the point Pl belongs to the two-dimensional equilateral triangle ABC shown in Fig. 17.9. Its coordinates .qa;l ;qb;l ;qc;l/ correspond to the proportions of the elected representatives of each party at the hierarchical level l. However, its coordinates .qa;l ;qb;l / in the coordinate system .C; u; v/ are not the associated proportions .pa;l ;pb;l / but indeed combines the two of them as seen from (17.27). Projecting the triangle ABC onto the plane spanned by .i ; j / yields the isosceles 0 triangle AOB where Pl , the projection of Pl , has the coordinates .pa;l ;pb;l /,which are the proportions of the elected representatives of parties A and B as shown in Fig. 17.24.

17.2.3 The Three-Party Bottom-Up Voting Flow

To build the voting flow diagram associated to the general voting function given by (17.18)–(17.20) we must first determine all the associated fixed points, i.e., solving the set of equations 8 <ˆ pa;lC1 D pa;l ; pb;l 1 D pb;l ; (17.28) :ˆ C pc;lC1 D pc;l: It is worth noting that putting one proportion equal to zero makes the corresponding two other fixed point equations identical to the two party case fixed point equation. For instance, pc D 0 satisfies pc;lC1 D pc;l, which in turn reduces 3 2 pa;lC1 D pa;l and pb;lC1 D pb;l to the form of (17.5) with plC1 DD pl C 3pl .1 pl /. This makes it possible to recover the corresponding fixed points. We thus have pc D 0 with pa D 0; 1=2; 1 and pb D 0; 1=2; 1. All associated combinations .pa ;pb ;pc / are valid solutions of (17.28) provided their sum is equal to one. This leaves only the two solutions .pa ;pb ;pc / D .0; 1; 0/; .1=2; 1=2; 0/ which, with permutations on a, b, c, yields .0;1;0/;.0;0;1/;.1=2;0;1=2/; .0;1=2;1=2/amounting to a total of six fixed points. We denote them: •Afor.1;0;0/ •Bfor.0;1;0/ •Cfor.0;0;1/ •O0 for .1=2; 1=2; 0/ •Efor.1=2;0;1=2/ •Ffor.0;1=2;1=2/ in the system of coordinates .O; i; j; k/. Their respective projections onto the two- dimensional plane .O; i; j / are: • A with .1;0;0/ • B with .0;1;0/ • O with .0;0;0/ 396 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.10 The six ﬁxed pointsA,B,C,E,F,O0 are shown together with their projectionA,B,O,A0,B0,O0 onto the plane .O; i ; j /

•O0 with .1=2; 1=2; 0/ •A0 with .1=2;0;0/ •B0 with .0;1=2;0/ as shown in Figs. 17.10 and 17.11. We can also include the associated flows to each one of the two party combinations along the three one dimensional lines A-O0-B, A-E-C, B, F, C together with the projections onto the .O; i ; j / as reproduced in Figs. 17.12 and 17.13. Having enumerated six analytical solutions, more solutions have to be found numerically. Since (17.28) combines three equations of the third degree, this yields in principle nine solutions. Three more are thus to be discovered. However, to be “physical”, i.e., relevant to our problem, any additional solution must be located within the triangle ABC of Fig. 17.14 to represent an eventual set of proportions. However, before solving the equations numerically, which could be a tedious task, it is worth first investigating the stability of all six fixed points A, B, C, O0,E,F shown in Fig. 17.11. This will allow us to reduce the search for numerical solutions by determining the .˛; ˇ/ range where extra fixed points are expected inside the triangle ABC. To achieve this task, we perform a Taylor expansion of (17.18)and(17.19) around a fixed point .pa ;pb /. It is not necessary to consider (17.20) since the third dimension pc;lC1 is determined from the two others. Deriving both .pa;lC1;pb;lC1/ with respect to pa;l and pb;l we get 8 @Pa;3.pa;l ;pb;l / 2 < da;a 6p 6pa;l 6˛pb;l .1 2pa;l pb;l /; @pa;l D a;l C C (17.29) : @Pa;3.pa;l ;pb;l / da;b 6˛pa;l .1 pa;l 2pb;l /; @pb;l D and 8 @Pb;3.pa;l ;pb;l / < db;a 6ˇpb;l .1 2pa;l pb;l /; @pa;l D (17.30) : @Pb;3.pa;l ;pb;l / 2 db;b 6p 6pb;l 6ˇpa;l .1 pa;l 2pb;l /: @pb;l D b;l C C 17.2 Three Competing Parties 397

Fig. 17.11 The six ﬁxed pointsA,B,C,E,F,O0 are shown on the plane ABC together with their projection A, B, O, A0,B0,O0 onto the plane ABO

Fig. 17.12 The six ﬁxed pointsA,B,C,E,F,O0 are shown together with their projections A, B, O, A0,B0, O0 onto the plane .O; i ; j /. The respective stability of each point is indicated by arrows 398 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.13 The same as in Fig. 17.12 but respectively in the ABC and ABO planes

Then, plugging in (17.29)and(17.30) the coordinates .pa ;pb / we get 8 < pa;lC1 D Pa;3.pa;l ;pb;l / Pa;3.pa ;pb / C da;a.pa;l pa / C da;b.pb;l pb /; : pb;lC1 D Pb;3.pa;l ;pb;l / Pb;3.pa ;pb / C db;a.pa;l pa / C db;b.pb;l pb /; (17.31) which yields ( pa;lC1 pa da;a.pa;l pa / C da;b.pb;l pb /; (17.32) pb;lC1 pb db;a.pa;l pa / C db;b.pb;l pb /; which can be cast in the matrix form ! ! pa;lC1 pa pa;l pa D M ; (17.33) pb;lC1 pb pb;l pb where ! da;a da;b M D : (17.34) db;a db;b 17.2 Three Competing Parties 399

Fig. 17.14 ThetriangleABC is projected onto the .i ; j / plane yielding the isosceles 0 triangle AOB. The point Pl is the projection of Pl whose coordinates .pa;l ;pb;l / are the proportions of the elected representatives of parties A and B

In a second step, we have to diagonalize the matrix M to ﬁnd out its associated eigenvalues and eigenvectors with ! a 0 M D (17.35) 0b and the associated two eigenvectors ! ! Va;a V b;a Va D ;Vb D ; (17.36) Va;b Vb;b

such that M Va D a Va and M Vb D b Vb . Plugging in (17.34) the A, B, C, O0, E, F six ﬁxed-point coordinates we get respectively ! 00 MA D MB D MC D MO D ; (17.37) 00 400 17 Bottom-Up Democratic Voting in a Three-Choice Competition ! 3 .1 ˛/ 3 ˛ 2 2 MO0 D ; (17.38) 3 ˇ 3 .1 ˇ/ 2 2 ! 3 3 2 2 ˛ ME D MA0 D ; (17.39) 3 0 2 ˇ ! 3 2 ˛0 MF D MB0 D ; (17.40) 3 3 2 ˇ 2 which in turn yield the respective eigenvalues ( A B C O a D a D a D a D 0 ; (17.41) A B C O b D b D b D b D 0 ( O0 3 a D 2 0 ; (17.42) O 3 .1 ˛ ˇ/ b D 2 ( E A0 3 a D a D 2 ; (17.43) E A0 3 b D b D 2 ˇ ( F B0 3 a D a D 2 : (17.44) F B0 3 b D b D 2 ˛

For the ﬁrst series of the zero matrix (17.37), any vector is an eigenvector and in particular along i and j, which gives, ! 1 Va;A D Va;B D Va;C D Va;O D ; (17.45) 0 ! 0 Vb;A D Vb;B D Vb;C D Vb;O D : (17.46) 1 For the other series, the associated eigenvectors are, ! ! 1 ˛ ˇ Va;O0 D ;Vb;O0 D ; (17.47) 1 1 17.2 Three Competing Parties 401 ! ! 1 ˛ 1ˇ Va;E D Va;A0 D ;Vb;E D Vb;A0 D ; (17.48) 0 1 ! ! 1˛ 0 ˇ Va;F D Va;B0 D ;Vb;F D Vb;B0 D : (17.49) 1 1

We are now in a position to keep on constructing the representative flow driven A B C O A B by bottom-up democratic elections. Having a D a D a D a D b D b D C O b D b D 0 indicates that the fixed points A, B, C are stable in all directions as well as O within the projection onto the .O; i; j / plane. On the contrary, the result O0 E A0 F B0 3 0 0 0 a D a D a D a D a D 2 makes the O ,A,B fixed points unstable along the i direction and O0 E, F unstable along the transformed direction in the ABC plane obtained using the transformation (17.27). Indeed the corresponding eigenvectors in the plane .C; u; v/ are, 0 q 1 3 @ 2 A Wa; D Wa; D Wa; D Wa; D ; (17.50) A B C O 1 p 2 0 q 1 3 @ 2 A Wb; D Vb; D Wb; D Wb; D : (17.51) A B C O 1 p 2 For the other series, the associated eigenvectors are less “sexy” in their expression with

! 0 q 1 0 3 . ˛Cˇ / @ 2 ˇ A Wa;O0 D ;Wb;O0 D ; (17.52) p ˛Cˇ 2 . p / 2ˇ

0 q 1 0 q 1 3 3 1C˛Cˇ @ 2 A @ 2 1Cˇ A Wa;E D V1;A0 D ;Wb;E D V2;A0 D ; (17.53) 1 1˛Cˇ p p 2 2.1Cˇ/

0 q 1 0 q 1 3 1 3 . 1 ˛ ˇ/ @ 2 A @ ˇ 2 C C A Wa;F D W1;B0 D ;Wb;F D W2;B0 D : (17.54) 1 1˛Cˇ p p 2 2ˇ

All the first eigenvectors, which correspond to the 3/2 eigenvalue, are along the sides of the triangle ABC or its projection OAB as seen in Fig. 17.13. In addition, 402 17 Bottom-Up Democratic Voting in a Three-Choice Competition for each one of them, the stability along the other eigenvector direction, i.e., the direction toward the interior of the triangle ABC, depends on ˛ and ˇ as seen from O0 E A0 F B0 their respective second eigenvalue b ;b D b ;b D b (17.42)–(17.44). We thus have: • Stability for O0, i.e., the flow goes toward the side of the triangle, when D 2 .1 ˛ ˇ/ < 3 . The direction is unstable otherwise and the flow goes toward .˛ ˇ/ > 1 the interior of the triangle when C 3 . • Stability for (E, A0), i.e., the flow goes toward the side of the triangle, when ˇ<2 3 . The direction is unstable otherwise and the flow goes toward the interior 2 of the triangle when ˇ>3 . 0 ˛<2 • Stability for (F, B ), i.e., the flow goes toward the side of the triangle, when 3 . The direction is unstable otherwise and the flow goes toward the interior of the ˛>2 triangle when 3 . As can be seen from Fig. 17.13, the basin of attraction of each stable fixed points C, A, B and O, A, B include respectively the triangle CEF, AEO0,BFO0 and OA0B0, AA0O0,BB0O0. Depending on the value of the biases ˛ and ˇ, the strategic question is to determine which party benefits from the inside triangles A0B0O0 and EFO0. In other words, which part of that triangle coalesces with each one of the three incompressible basins of attraction? From the above findings, it is seen that if one of the flows, which goes toward the interior of the triangle, i.e., either ˛, ˇ,or is larger than two thirds, then the two other values must be simultaneously lower than two thirds since ˛ C ˇ C D 1. This means that if one direction toward the inside of the triangle is unstable, both others are necessarily stable. However, while only one direction can be unstable at a time, the three directions can be simultaneously stable, i.e., ˛<2=3, ˇ<2=3,and <2=3. One straightforward way to resolve the question directly is to extract all possible nine solutions from (17.28). However, the difficulty is that it is not possible to find all the solutions analytically. To be more precise, I was not able to do it, at least with compact and tractable expressions. Some intuitive and hand waving arguments are thus welcome to simplify the search for the eventual relevant solutions.

17.2.4 The Physicist’s Corner

Always trying to avoid a huge amount of hard and fastidious work is a cornerstone of the physicist’s philosophy in tackling any given problem. By doing so, a better understanding of the problem is required, so that a more profound output can be expected. Solving (17.28) by investigating the nature and properties of each solution for all sets of values of (˛, ˇ) is typical of such a “trap” to avoid. 17.2 Three Competing Parties 403

Fig. 17.15 The triangles CEF, AEO0,BFO0 present a steady slope towards their respective attractors C, A, B. From any point of these triangles the “water” pours down toward the associated attractor. The question is to determine the landscape inside the triangle EFO0

However, to be successful in being lazy and efficient at the same time needs some intuitive hunches in grasping qualitatively how the flow should work. One way of achieving this goal is to look at the problem within the analogy of water pouring on a geographical topology of which we want to determine the shape. We need to know the ground landscape so as to be able to determine the various paths that the water follows in order to ultimately reach one of the three sinks, i.e., the attractors. Let us focus on the CAB triangle, the actual one in which the dynamics is contained. We know already that each one of the triangles CEF, AEO0,BFO0 presents a steady slope toward its attractor, respectively C, A, B as drawn schematically in Fig. 17.15. From any point of these triangle, the “water” pours down towards the associated attractor. Knowing what is happening beyond the lines EF, EO0,FO0, the next step is to figure out the topology inside the EFO0 triangle. If the three fixed points E, F, O0 are sinks, i.e., they are all stable (see Fig. 17.15), we clearly see that in order to have them supplied with pouring water from inside the triangle EFO0, a source of water is necessary there. This implies the existence of an additional fixed point (see Fig. 17.16), located within EFO0 at a peak, the highest part of the whole landscape. Such a fixed point, must then be unstable in all directions so as to accomplish its “job”. And so far, this has not been extracted from (17.28). On this basis, we can infer that, when simultaneously ˛<2=3, ˇ<2=3,and <2=3, or for the last constraint ˛ C ˇ > 1=3, the fixed point equation (17.28) must have a seventh solution G with three real coordinates within the range 0 and 404 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.16 In the case where the three ﬁxed points E, F, O0 are sinks, a source of water is necessary inside triangle EFO0. This implies the existence of an additional ﬁxed point located at a crest, the highest point for the whole landscape

1, and with a sum smaller than one. Solving the equation within these restrictions dramatically reduces the domain of search. Nevertheless, it still has to be done numerically for given specific values of ˛ and ˇ. On the contrary, if one of the fixed points E, F, O0 is a source, i.e., the associated bias respectively ˇ, ˛, , is larger than two thirds, the two others are sinks, i.e., with biases smaller than two thirds, as shown in Fig. 17.17. From topology continuity no additional fixed point inside EFO0, neither a sink nor a crest, is necessary. Moreover, it would be incompatible with the topology. It is not feasible, being forbidden by the ground landscape produced by the E, F, O0 fixed point properties (see Fig. 17.18). Therefore, when ˛,orˇ or is larger than two thirds, we know that a priori, there exists no seventh fixed point from (17.28).

17.3 The Bottom-Up Voting Flow Diagram

Given the previous calculations, we are in a position to solve numerically the problem of three competing parties with any set of local alliances to obtain the full ﬂow diagram of the voting dynamics driven by bottom-up democratic elections. Before embracing the general problem set by (17.18)–(17.20), we focus on a rather frequent case of two large opposing parties, each having less than 50% support and a small one in between whose alliance is required to reach the hierarchy presidency. 17.3 The Bottom-Up Voting Flow Diagram 405

Fig. 17.17 When one of the ﬁxed points E, F, O0 is a source, i.e., the associated bias is larger than two thirds and then the two others are sinks with biases smaller than two thirds. From topology continuity, no additional ﬁxed point inside EFO0, neither a sink nor a crest, is necessary. Moreover, it would be incompatible with the topology

Fig. 17.18 When ˛,orˇ or is larger than two thirds, a priori there exists no seventh ﬁxed point inside EFO0 406 17 Bottom-Up Democratic Voting in a Three-Choice Competition

17.3.1 The Frequent Case of Two Large Opposing Parties with a Small One in Between

Many situations are characterized by the competition between two big parties A and B, of which none has the absolute majority. While they usually defend very different policies on key issues, there often exists a third party C, small in terms of support, whose emphasis is to ensure some speciﬁc interest of a given minority. Each one of the large parties will seek to get the support of C in order to ensure the hierarchy presidency. To encompass the implications of such local alliances, we consider the case of a voting group size of 3 making the assumption that at any local tie, i.e., an a (A B C) conﬁguration, it is a C agent who is elected whether a coalition is set up with A or B. In addition, 2 A or 2 B are still necessary to elect respectively an A or a B. Also, 2 C elect a C. The voting equations are written

3 2 pa;lC1 Pa;3.pa;l / D pa;l C 3pa;l .1 pa;l /; (17.55) 3 2 pb;lC1 Pb;3.pb;l / D pb;l C 3pb;l .1 pb;l /; (17.56) 3 2 pc;lC1 Pc;3.pc;l/ D pc;l C 3pc;l.1 pc;l/ C 6pa;l pb;l pc;l ; (17.57) where (17.18)and(17.19) are identical to the voting function for two competing parties. Therefore, for both parties A and B the critical threshold to win the hierarchy presidency requires a bottom support of above 50%. The associated necessary number of levels to reach the presidency with certainty is also unchanged with respect to the bottom-up voting dynamics. This is why local alliances with C are made. Accordingly, the bottom-up voting increases the proportion of C representatives while decreasing those of A and B. To illustrate this feature, let us build the ﬂow diagram associated to (17.55)–(17.57). In the case of two competing parties, we already saw that to start with a bottom support of even 45% leads to a quick decrease in the elected representatives while climbing up the hierarchy. This means that simultaneously C will get stronger and stronger percentages and eventually the presidency. Then it is enough to have both pa;0 <1=2 and pb;0 <1=2 to boost C even with pc;0 of the order of a few percent. Figure 17.19 shows the domain of attraction for the bottom small party C. Whatever the bottom support, it gains bottom up support thanks to the agreement with either one of the two large parties. Four different bottom supports for A and B are shown with respectively pa;0 and pb;0 equal to .0:40; 0:52/; .0:40; 0:4/; .0:15; 0:32/; .0:55; 0:32/. Only the last one with pa;0 D 0:55 > 1=2 drives the A party toward the presidency as the ﬁrst one with pb;0 D 0:52 > 1=2 does so for B. Any bottom support within the rectangle leads toward increasing C power. Another illustration is shown in Fig. 17.20 pa;0 D 0:48; pb;0 D 0:46; pc;0 D 0:06. Although the minority party C has only 6% of support at the bottom, its representative power will grow dramatically while climbing democratically the 17.3 The Bottom-Up Voting Flow Diagram 407

Fig. 17.19 The alliance set B follows .˛ D 0; ˇ D 0; 1 D 1/. As soon as pa;0 <1=2and pb;0 <1=2,C will grow in power representation, climbing up 0.8 the hierarchy. Four bottom supports are considered fpa;0 D 0:40; pb;0 D 0.6 0:52; pc;0 D 0:18g, fpa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20g, fpa;0 D 0:20; pb;0 D 0.4 0:32; pc;0 D 0:48g, fpa;0 D 0:55; pb;0 D 0:32; pc;0 D 0:13g.TheC domain of power increase is 0.2 double that for A and B

A 0.2 0.4 0.6 0.8 1

Fig. 17.20 The respective 1 variation of power sharing for A, B, C from a bottom p 0:48; p support a;0 D b;0 D pc,0 0.06 0:46; pc;0 D 0:06. The tiny 0.8 minority party C starts from 6% of support and yet it gets more and more power while climbing democratically the 0.6 hierarchy levels. At the seventh level, it eventually reaches the majority. This is 0.4 what is known in French as pa,0 0.48 “Un jeu de dupes.”

0.2

pb,0 0.46 0 0 2 4 6 8 10 12

hierarchy levels. At the seventh level, it reaches the majority. This is what is known in French as, “Un jeu de dupes,” or in English, “pulling the wool over your eyes.” From the above results, it appears that it is not very efﬁcient for either one of the large parties to make a systematic alliance with the minority party C since indeed it is equivalent to putting it in power while the aim of the agreement was to allow the corresponding large parties to get the presidency. We thus need to revisit the 408 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.21 The same as in 1 Fig. 17.20 but here the set of alliances between A and C is different. From the bottom up pa,0 0.48 to level 5, A votes for C in the 0.8 case of a tie, but from level 6 and up it is party C, that votes for A in the case of a tie. Now A eventually reaches the 0.6 presidency although C gained a substantial increase in power 0.4 pc,0 0.06

0.2

pb,0 0.46 0 0 2 4 6 8 10 12 alliance scheme. Consider now that parties A and C are still making an alliance, but in two steps. At the first hierarchy levels, A will vote for C in case of a (A B C) tie, thus boosting the C representation. But then, after for instance the completion of five levels, starting at level 6, it will be C that will vote for A in the case of a tie. Starting with the same above bottom supports pa;0 D 0:48; pb;0 D 0:46; pc;0 D 0:06, the new alliance scheme yields the series pa;1 D 0:47; pb;1 D 0:44; pc;1 D 0:09, pa;2 D 0:45; pb;2 D 0:41; pc;2 D 0:14, pa;3 D 0:43; pb;3 D 0:37; pc;3 D 0:20, pa;4 D 0:40; pb;4 D 0:31; pc;4 D 0:29,andpa;5 D 0:35; pb;5 D 0:22; pc;5 D 0:43. This means an increase of 37% for C and a 13% loss for A. But then from level 6 and above, A grows steadily toward the presidency as shown in Fig. 17.21. pa;6 D 0:48; pb; D 0:13; pc;6 D 0:39, pa;7 D 0:62; pb;7 D 0:04; pc;7 D 0:34, pa;8 D 0:73; pb;8 D 0:01; pc;8 D 0:26, pa;9 D 0:83; pb;9 D 0:00; pc;9 D 0:13, pa;10 D 0:92; pb;10 D 0:00; pc;10 D 0:08, pa;11 D 0:98; pb;11 D 0:00; pc;11 D 0:02, pa;12 D 1; pb;12 D 0:00; pc;12 D 0:00. The strategy reveals itself to be rather efficient for both A and C. It is worth stressing the fact that pa;6 D 0:48 D pa;0 is a pure coincidence resulting from the rounding up. Indeed, pa;6 D 0:484 and could be quite different from pa;0 depending on both the pa;0 values. The above two schemes of alliances demonstrates that the alliance must be dealt with great care. Figure 17.22 shows the case of an error from A with a reverse in the votes only from level 8. It will be too late for A and C will eventually reach the presidency. A mistake at the level at which the alliance direction has to be switched is crucial. For instance, within the above figures, selecting level 7 instead of six turns out to be a disaster for A as seen from the following series with pa;6 D 0:28; pb; D 0:13; pc;6 D 0:59, pa;7 D 0:32; pb;7 D 0:04; pc;7 D 0:64, pa;8 D 0:30; pb;8 D 0:01; pc;8 D 17.3 The Bottom-Up Voting Flow Diagram 409

Fig. 17.22 The same as in 1 Fig. 17.21 but with apparently a slight difference. C votes for A at a tie only from level pc,0 0.06 7. The result is disastrous for 0.8 A and great for C

0.6

0.4 pa,0 0.48

0.2

pb,0 0.46 0 0 2 4 6 8 10 12 14

0:69, pa;9 D 0:22; pb;9 D 0:00; pc;9 D 0:78, pa;10 D 0:13; pb;10 D 0:00; pc;10 D 0:87, pa;11 D 0:05; pb;11 D 0:00; pc;11 D 0:95, pa;12 D 0:00; pb;12 D 0:00; pc;12 D 1, which are plotted in Fig. 17.22. The reversing voting bias came in too late to make A win. Although pa;7 D 0:32 > pa;6 D 0:28, pa;8

17.3.2 Some General .˛;ˇ;/Case Snapshots

The above simple case of three competing parties highlighted the complexity and the subtleties of the possible local alliances. We now show a series of snapshots of them to enhance various voting dynamics. To emphasize the effect of the alliances, we consider four different bottom situations with respectively pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53.Werepresentthe associated bottom-up voting process for six hierarchical levels given a set of alliances characterized by speciﬁc values for .˛;ˇ;/. We stress that taking noninteger values for .˛;ˇ;/ means that we consider averages over different voting groups at all levels. In particular it makes it possible to take into account local defections with respect to global party agreements. The series of ﬁgures represent the two dimensional projection OAB of the plane ABC as shown in Fig. 17.10. In each case, the two frontiers which determine the respective basins of attraction of each pure attractor are calculated numerically and 410 17 Bottom-Up Democratic Voting in a Three-Choice Competition

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 17.23 The set of local and global alliances ends up with the effective ˛ D 0; ˇ D 0; D 1 (left)and˛ D 0; ˇ D 0:20; D 0:80 (right) with four respective bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown. Note that for the left part the last two set of bottom values are different from those in Fig. 17.19 drawn on the ﬁgures. The origin O is the projection of the C attractor. All ﬁxed points are also included. We start with .˛ D 0; ˇ D 0; D 1/. Then keeping ˛ D 0, we take successively .ˇ D 0:20; D 0:80/, .ˇ D 0:40; D 0:60/, .ˇ D 0:60; D 0:40/, .ˇ D 0:80; D 0:20/, .ˇ D 1; D 0/. All cases are shown in Figs. 17.23 and 17.24. When an attractor is not reached within six levels, it means that the presidency winner is probabilistic, like for .˛ D 0; ˇ D 1; D 0/ with pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. More levels would be necessary to make it deterministic. Moving to ˛ D 0:20, we consider successively .ˇ D 0; D 0:80/, .ˇ D 0:20; D 0:60/,.ˇ D 0:40; D 0:40/, .ˇ D 0:60; D 0:20/, .ˇ D 0:80; D 0/ reproduced in Figs. 17.25 and 17.26. We then put ˛ D 0:40 with successively .ˇ D 0; D 0:60/, .ˇ D 0:20; D 0:40/, .ˇ D 0:40; D 0:20/, .ˇ D 0:60; D 0/, shown in Fig. 17.27. With ˛ D 0:60, we show successively .ˇ D 0; D 0:40/, .ˇ D 0:20; D 0:20/, .ˇ D 0:40; D 0/ in Fig. 17.28. At ˛ D 0:80, we have only .ˇ D 0; D 0:20/ and .ˇ D 0:20; D 0/ in Fig. 17.29. The last case of this survey is thus .˛ D 1; ˇ D 0; D 0/ shown in Fig. 17.30, which is the symmetrical case of .˛ D 0; ˇ D 0; D 1/ from Fig. 17.23 with a permutation between A and C. We also add .˛ D 1; ˇ D 0; D 0/,whichis the third extreme alliance case to illustrate the drastic differences obtained from the same bottom respective supports. 17.4 The “Golden Triangle” to Win the Presidency 411

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 1 1 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 17.24 The set of local and global alliances ends up with the effective ˛ D 0; ˇ D 0:40; D 0:60 (upper left), ˛ D 0; ˇ D 0:60; D 0:40 (upper right), ˛ D 0; ˇ D 0:80; D 0:20 (lower left), ˛ D 0; ˇ D 1; D 0 (lower right) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown

17.4 The “Golden Triangle” to Win the Presidency

In the above Section, we saw how slight differences in local alliances may induce drastic changes. It is therefore of central interest to have the map of the frontiers which are determined by an effective average value of .˛;ˇ;/. Indeed, what matters is the positioning of the frontiers within what we could call the “golden triangle” to win the presidency. It includes all the central parts of the triangle ABC where none of .pa;0;pb;0;pc;0/ are larger than 50%, which is most likely the case in any democratic organization. In the following lines, we draw all the frontiers 412 17 Bottom-Up Democratic Voting in a Three-Choice Competition

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 17.25 The set of local and global alliances ends up with the effective ˛ D 0:20; ˇ D 0; D 0:80 (left), ˛ D 0:20; ˇ D 0:20; D 0:60 (right) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown associated with the series of the above cases. It is worth stressing that these frontiers are independent of pa;0;pb;0;pc;0. They depend only on the values of .˛;ˇ;/, which in turn result from both the alliances set up by the parties as well as the discipline to apply these party agreements by local representatives. In addition, we also draw the corresponding global voting flows. The voting flow shown in the series of Figs. 17.31–17.38 is self-explicative and does not require additional comments. It puts a new quantitative light on the dramatic effects of setting up alliances together with their local reliability that needs to be implemented. We have only considered the same set of values for .˛;ˇ;/ throughout the hierarchy but as seen in the previous section, different settings can be finalized by the parties involved in an alliance as a function of the hierarchical level. References 413

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

B 1

0.8

0.6

0.4

0.2

A 0.2 0.4 0.6 0.8 1

Fig. 17.26 The set of local and global alliances ends up with the effective ˛ D 0:20; ˇ D 0:40; D 0:40 (upper left), ˛ D 0:20; ˇ D 0:60; D 0:20 (upper right), ˛ D 0:20; ˇ D 0:80; D 0 (lower) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown

References

1. S. Galam, “Social paradoxes of majority rule voting and renormalization group”, J. of Stat. Phys. 61, 943-951 (1990) 2. S. Galam, “Political paradoxes of majority rule voting and hierarchical systems”, Int. J. General Systems 18, 191-200 (1991) 3. S. Gekle, L. Peliti, and S. Galam, “Opinion dynamics in a three-choice system”, Eur. Phys. J. B45, 569-575 (2005) 4. K. Kulakowski and M. Nawojczyk, “The Galam Model of Minority Opinion Spreading and the Marriage Gap”, Int. J. Mod. Phys. C 19, 611-615 (2008) 414 17 Bottom-Up Democratic Voting in a Three-Choice Competition

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 17.27 The set of local and global alliances ends up with the effective ˛ D 0:40; ˇ D 0; D 0:60 (upper left), ˛ D 0:40; ˇ D 0:20; D 0:40 (upper right), ˛ D 0:40; ˇ D 0:40; D 0:20 (lower left), ˛ D 0:40; ˇ D 0:60; D 0 (lower right) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown References 415

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 B 1

0.8

0.6

0.4

0.2

A 0.2 0.4 0.6 0.8 1

Fig. 17.28 The set of local and global alliances ends up with the effective ˛ D 0:60; ˇ D 0; D 0:40 (upper left), ˛ D 0:60; ˇ D 0:20; D 0:20 (upper right), ˛ D 0:60; ˇ D 0:40; D 0 (lower) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown 416 17 Bottom-Up Democratic Voting in a Three-Choice Competition

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

Fig. 17.29 The set of local and global alliances ends up with the effective ˛ D 0:80; ˇ D 0; D 0:20 (left), ˛ D 0:80; ˇ D 0:20; D 0 (right) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown References 417

B B 1 1

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

A A 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

B 1

0.8

0.6

0.4

0.2

A 0.2 0.4 0.6 0.8 1

Fig. 17.30 The set of local and global alliances ends up with the effective ˛ D 1; ˇ D 0; D 0 (upper left), ˛ D;ˇ D 1; D 0 (upper right), ˛ D 0; ˇ D 0; D 1 (lower) with the four bottom values pa;0 D 0:40; pb;0 D 0:52; pc;0 D 0:18, pa;0 D 0:40; pb;0 D 0:40; pc;0 D 0:20, pa;0 D 0:25; pb;0 D 0:32; pc;0 D 0:43, pa;0 D 0:15; pb;0 D 0:32; pc;0 D 0:53. Six hierarchical levels are shown 418 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.31 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ 0; ˇ 0:56; 0:44/ D D D E F

AB O'

Fig. 17.32 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0:33; ˇ 0:56; 0:11/ D D E F G

AB O' References 419

Fig. 17.33 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0:42; ˇ 0:52; 0:06/ D D F E G

AB O'

Fig. 17.34 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0:52; ˇ D 0:42; D 0:06/ F E G

AB O' 420 17 Bottom-Up Democratic Voting in a Three-Choice Competition

Fig. 17.35 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0:18; ˇ D 0:42; D 0:40/ E F

AB O'

Fig. 17.36 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0:25; ˇ 0:68; 0:07/ D D E F

AB O' References 421

Fig. 17.37 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ D 0; ˇ D 0; D 1/ E F

AB O'

Fig. 17.38 Two-dimensional C voting ﬂow diagram for a three-party competition. The arrows indicate the direction of a voting to a higher hierarchical level with the proportions given by the arrows’ tails. Here, the set of voting alliances is .˛ 0:68; ˇ 0; 0:32/ D D D E F

AB O' Chapter 18 So Sorry, That’s the End of the Tour!

That’s it; we have reached the end point of our journey through the wonderful world of sociophysics, which I hope, you have found beautiful and enlightening. It is time to say goodbye (see Fig. 18.1) but before parting, I would like to reassess the particularities of this book. The book constitutes a real challenge both in its form and its content. It is a unique book of its kind, having mixed humor, personal stories, philosophy, ethics, politics, an initiation in doing research the way physicists do, an introduction of the first rather incredible tools of sociophysics, and finally a lot of heavy and serious stuff. Then there is its unusual and hopefully pleasant combination of cartoons, graphs, and equations, which yields a sensation of lightness embedded in a solid grounding of seriousness. At least, that was my purpose, as seen from moving from Fig. 18.2 to Fig. 18.3. In building a methodology for sociophysics, we have discovered that it could be well possible that most of our paradigms of freedom and democracy have to be revisited in order to reach their explicit goals of effective individual freedom and real collective democracy. It appears that our knowledge of the specific laws which govern our own behavior as human beings is much more underdeveloped than our understanding of both inert matter and biological entities. The theories I have presented here are the first tunes of a symphony yet to come, and not the first bricks of a new imposing building. It is aimed at making us more educated about ourselves and the social world in which we live. In particular we have singled out our internal biases, which manipulate our rationality, so making us puppets in the hands of our irrationality. However, we must also keep in mind that all the results are to be taken as trends for tackling the reality from a different viewpoint, but not as a kind of absolute truth to be implemented by force. A personal remark: I did not cite my numerous colleagues and friends, who are contributing a lot of great work in the field of sociophysics, not because of an arrogant posture or by ignorance or even by a sentiment of contempt, but because I have presented my own construction of the field, which indeed has been done

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 423 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 18, © Springer Science+Business Media, LLC 2012 424 18 So Sorry, That’s the End of the Tour!

Fig. 18.1 I say bye-bye...

Fig. 18.2 Human behavior perceived with today’s tools 18 So Sorry, That’s the End of the Tour! 425

Fig. 18.3 Human behavior with the tools of sociophysics

Fig. 18.4 Searching for the mother of sociophysics mostly alone. It brings me to envision my accomplished work as one from a solitary father. I hope I will soon encounter a mother, as in Fig. 18.4,soastoatlasthave some fun; it is high time! Chapter 19 I Thank You

Before going to the acknowledgments per se, I would like to pay tribute to the physicists who have been instrumental in my education as a physicist, a very few being adamantly opposed to the very idea of sociophysics and the others being more neutral. I will not disclose their respective names! However, I must confess to their credit that the ones who adamantly opposed the idea of sociophysics never allowed this opposition to interfere with our exchanges regarding more standard physics. In chronological order, I cite Yaacov Bar Hen, Michel Cassien, the late Pierre- Giles De-Gennes, Claude Cohen-Tannudji, Jean-Pierre Hansen, Amnon Aharony, Michael Fisher, and Mark Azbel. I have not been in touch with most of them for many years. At the same time, I am happy to acknowledge the physicists who, in one way or another, directly or indirectly, have been helpful to me in developing the field of sociophysics. It includes in alphabetical order, Alain Mauger, Luca Peliti, Pierre Pfeuty, Silvio R. Salinas, Eugene H. Stanley, Dietrich Stauffer, and Alexander Voronel. I would also like to thank my current younger and ambitious competitors Dirk Helbing, Frank Schweitzer, Johannes Schneider, and Kasia Sznajd-Weron, who are very instrumental in developing the field of sociophysics, although it is sometimes at my expense for the first two. They are all keeping me young! I must also thank Hans Koelsch who a few years ago, the number of which I am ashamed to reveal, suggested to me to write a book on sociophysics. He was obstinate and enthusiastic in pushing me along with this book project but then, when I eventually started to write that book, his career development took him away into other positions. Without him, I think this book would still be on the bookshelf of my “in another life” projects. I would also like to thank my copyeditor Paul Montgomery, an English experimental physicist based in Strasbourg, who has been my first dedicated reader and commentator. Initially written in English, my manuscript had a strong French accent. Paul has been quite successful in softening it a bit, adding a nice topping of an English flavor and yet leaving in my own particular literary style.

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 427 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3 19, © Springer Science+Business Media, LLC 2012 428 19 I Thank You

Last but not least I want to thank Aude, my sister Sylvie, my brother Eric, as well as my Mom Giuliana, to whom this book is dedicated, for putting up with me throughout these many years hoping they will continue (see Fig. 18.3). I end with a nostalgic thought for my late father, Sion, without whom, I would be presumably in jail for the selling of oversized clothing to thin men! Index

A Axelrod, R., 139 Active open space (AOS), 159–160 Azbel, M., 427 Afanasyev, V., 58 Aharony, A., 152, 157, 427 Alain Mauger, universal formula, 164 Albert, R., 70 B Amaral, L.A.N., 58 Ball, P., 61, 76 Amblard, F., 169 Barabasi,´ A.L., 70 Amengual, P., 169 Behera, L., 169 Asimov, I., 4, 27 Bennett, D.S., 139 Asimov’s foundation syndrome Bernardes, A.T., 82 fall of empires parenthesis, 10–11 Bilateral exchanges, 142 future prediction, 5–7 Blondel, V.D., 169 Hari Seldon and psychohistory, 3–5 Borghesi, C., 82, 97, 170, 191 historical vs. ahistorical sciences, 7 Bottom minorities and majorities mathematician and physicist, 9–10 democratic balance, 349 small and large groups, 8–9 even case r D 4 Atoms and humans challenging view point, 356–360 atom-individual connection running power view point, 358–365 naive question, 31–32 nasty configurations, 348 series of questions, 32–36 nonzero coefficient, 350 bare methodology, 34–36 odd case r D 3, 351–356 basic philosophy, 21–23 physicist corner, 366–367 deterministic behavior, 29, 30 Bottom-up democratic voting, 96 general procedure and provocative flow diagram hypothesis, 38–39 ˛; ˇ; case, 409–417 mathematical rigor, 27–28 democratic elections, 404 microscopic and macroscopic levels local alliances, 404 interplay, 30–31 two large opposing parties, 406–409 model study, 37–38 golden triangle, presidency winning, physics-like approach implementation, 411–412, 418–421 28–29 three competing parties toy model, 24 associated voting functions, 391 XY3Wz agents, 24–26 even-sized voting groups, 390 Ausloos, M., 83, 169 physicist’s corner, 402–405

S. Galam, Sociophysics: A Physicist’s Modeling of Psycho-political Phenomena, 429 Understanding Complex Systems, DOI 10.1007/978-1-4614-2032-3, © Springer Science+Business Media, LLC 2012 430 Index

Bottom-up democratic voting (cont.) endless dynamics and complexity, 143, three competing parties, two- 144 dimensional problem, 393–395 five countries dynamics, 145–149 three-party bottom-up voting flow (see infernal dynamics, 142 Three-party bottom-up voting flow) negative bond addition, 149 tie situations, 390 real life sets, 143, 145 two competing parties, one-dimensional situation, 142–143 problem, 391–393 two-country problem voting rules, 390 cost evaluation, 140 two competing parties definition, 139 bottom evolution, 386, 387 Ising variables and, 140–141 bottom-up voting outcomes, 386 Cohen-Tannudji, C., 427 critical number of levels, 388 Collective choice (C), 110, 112 democracy, 389 Contrarian model, 82 democratic elections result prediction, Contrarians, 170. See also Opinion dynamics 380–382 modeling dynamics, 382–386 Contucci, P., 169 elected representatives, 380, 387, 388 Critical behavior, 31 leadership, 389 Critical phenomena, 42 local majority rule, 380 Critical thresholds, 209 odd- and even-sized groups, 380 self-elimination, 389 unstable fixed point, 387 D variables and parameters, 389 Davis, J., 106 Bricmont, J., 61 de Arcangelis, L., 169 Buldyrev, S.V., 58 Decision making theory, 56–57 Burnstein, E., 109 Deffuant, G., 169 De-Gennes, P.-G., 427 de la Lama, M.S., 170 C Democratic balance, 247 Callen, E., 46, 61 applications, practical scheme, 247 Cassien, M., 427 fluctuations and group sizes Castellano, C., 70, 71, 82, 94, 169 deterministic outcome, 229, 231 Caticha, N., 169 overamplification, 229 Caution, sociophysics models probabilistic outcome, 229, 230 error making, 203, 205 probability, 229 reality aspects, 203–204 single, random group voting scheme Cellular automata simulation, 196 result, 230, 231 Centre de Recherche en Epist´ emologie´ truncated binomial expansion, 229 Appliquee´ (CREA), 64 three random selected agents, 238–239 Chakrabarti, B.K., 70 Democratic bottom-up voting Chakraborti, A., 70 achievements, 135–137 Chaotic behavior, 30 critical thresholds and, 209 Chatterjee, A., 70 democratic balance breaking, 211–212 Chirac, J., 198 democratic-driven dictatorship twist, Chopard, B., 8, 70, 97, 169, 196 215–219 Classical terrorism, 160–162 eastern European communist collapse, Coalition-fragmentation vs. global coalitions 220–221 achievements, 153 elaborate hierarchy, 122, 123 binary to a multiple coalitions, 151–152 inertia effect incorporation, 123–125 pair bonds superposing, 149–151 key configurations, 215, 220 physical model overlap, 152 local majority rules, 122–124, 210 three-country problem overall process, 210–211 Index 431

physical model overlap, 132–133 competing opinions, 374 presidency formula, 126–127 even-sized voting groups, 370 probabilistic to deterministic, 125–126 hierarchical leadership, 370–371 simplest hierarchy, 121–122 inertia of power, 372 simulations, 127–131 nd;k hierarchical levels, 372, 373 strategic key, 214–215 pc;4 variation, 371, 372 support value calculation, 213–214 physicist corner, 373–374 three competing parties, 220 magic formula two to three competing parties exact estimates, voting function, 330, competition and tie, 129–131 331 two-dimensional voting flow diagrams, lower and highet magic threshold, 134–135 330–332 voting dynamics, 132 numerical iteration process, 331 voting functions, 132 physicist’s corner, 332–336 voting group sizes, 212–213 probabilistic area shrink, 331 Democratic-driven dictatorship twist, 215–219 three classes, 332, 333 Democratic threshold, 247 magic-Machiavelli formula de Oliveira, P.M.C., 70, 75, 93, 94 associated voting function, 319 de Oliveira, S.M., 70, 75, 93, 94 lower and upper magic thresholds, Deterministic voting, 125–126 317 Dictatorship paradoxes. See also Democratic r (pc /variation, 317, 318 bottom-up voting phase diagram, 317, 318 bottom minorities and majorities (see physicist’s corner, 320–324 pn Bottom minorities and majorities) r;A value, 319 pn communist collapse and French FN victory r;B value, 317, 319 p pn critical threshold, 375 0 D r;B variation, 319, 320 Eastern European communist parties, probabilistic area shrinking, 319 374 probabilistic outcome, 317 economic and financial institutions, 376 self-elimination, 316 majority rule voting, 375 systematic discrepancy, 320 tree-like hierarchy, 375 two-digit precision, 319, 320 dynamics, repeated democratic voting upper probabilistic region, 317 attraction range, 312, 313 perfect democratic structure-dictatorship elected representatives density flow amplitude difference, 306, 307 diagram, 312, 313 associated series, 311 logarithmic singularity, 313 democratic outcome, 305 physicist’s corner, 313–316 dynamics, 310 unstable fixed-point separator, 313 elected representatives, 312 upper hierarchical levels, 312 five level hierarchy, 310 global size, 324–325 leadership position, 312 inertial effect majority-minority status, 306–307 even-sized groups, 299 majority rule voting, 312 formidable machinery, 300 nonmonotonic double valued behavior, inertia of power, 299 308 probabilistic outcome, 299 odd-case treatment, 306 social, economic, and political precedent democratic bottom-up organizations, 298 hierarchy, 308, 310 structural manipulations, 298 repeated bottom-up voting, 305 substantial dissatisfaction, 297 self-elimination, 307, 312 tie breaking vote, 300 single level hierarchies, 308 tie probability, 299, 300 single voting groups, 306 top leaderships, 298 two level hierarchy, 308, 309 inertia principle softening practical scheme associated voting dynamics, 371 discrepancy, 326 432 Index

Dictatorship paradoxes. See also Democratic nasty configurations, 367 bottom-up voting (cont.) scary lobbying, 368–369 exact numerical estimates, 325, 326 striking idealized illustration, 369 physicist’s corner, 327–330 sudden and unexpected taking over, power law assumption, 325 large institutions, 368 rare antidemocratic events winning bottom minorities, 367 actual probability of nasty bottom Dissatisfaction function (F), 102–104 configuration, 347–350 Droz, M., 8, 70, 97, 169, 196 associated number of bottom nasty configurations, 346–347 4 B-agents against 12 A-agents, 340 E 8 B-agents against 56 A-agents Eastern European communist collapse, optimized distribution, 340–341 220–221 bottom antidemocratic rearrangements Econophysics, 29, 58 probability, 341, 342 Emblematic founding models dictatorial bottom configurations, 339 bottom-up democratic voting and terrorism, minimum number of bottom agents (see 96 Minimum bottom agents) coalitions vs. fragmentation, 96–97 nesting strategy, 339 development and decision making, 95 ultra rare configurations, 340 major real political events, 94 16 voting groups, 341 public opinion, 97 winning bottom configuration, 340 sociopolitical phenomena, 93–94 super magic formula Epsilon expansion, 43 physicist’s corner, 337–339 upper vs. lower magic thresholds, 336–338 F tie break voting, single random group Father of sociophysics, 62, 77 voting scheme Fisher, M.E., 43, 74, 427 democratic balance, 302 Flags, 165–167 inertia of power, 300 Floaters, 169, 170 opinion voting dynamics, 302 Florian, R., 82, 139, 151–153 randomly selected four agents, 300, Fortunato, S., 70, 71, 82, 94, 169 301 Fractal dimensions, 44 voting function, 301 Frustration effects, 152 winning election probability, 301–302 voting group size variation asymmetric truncated binomial G expansion, 302 Galam–Mauger formula, 164–165 binomial coefficient, 303 Galam, S., 7, 8, 46–49, 51, 53, 54, 56, 57, democratic balance, 303 61, 62, 70, 74, 75, 78, 81–84, 86, error bar, 305 87, 93–96, 101, 102, 105–107, fixed points values, 303 121–123, 125, 136, 139, 151–153, initial probabilistic value, 304 157, 164–167, 169, 170, 175, 188, odd and even size discrepancy, 191, 192, 196, 198, 199, 374–376 305 Gefen, Y., 47, 48, 53, 61, 78, 95, 101, 102, 105 odd-sized voting group, 303 Gekle, S., 97, 196 series of values of p1, 303, 304 Geometry, terrorism series of values of r; 303, 304 open spaces, 158–159 worrying power, geometric nesting reverse situation, 160, 161 democratic bottom-up voting dynamics, and terrorist movement, 156–158 367 VOP and AOS, 159–160 democratic functioning restore, Ghirlanda, S., 169 369–370 Global size hierarchical organizations, 367 bottom-up democratic hierarchy, 263, 268 Index 433

fixed voting group, 264 Havlin, S., 58 linear curves, 268 Hegselmann, R., 169 Logarithm, 268 Helbing, D., 427 nM evolution, 270, 271 Hen, Y.B., 427 N n G;r evolution, 264, 265 Herrmann, H.J., 82, 169 N n G;r values, 264 Heterogeneous agents, 188–189 region of divergence, 271 Hirtreiter, C., 82, 169 series of curves ar;n, 268, 269 Human behavior, tools, 423–425 three-dimensional behavior, 270, 271 Hypothetical agents, 24–26 Global warming, 7, 14, 191–193 Goldberger, A.L., 58 Gonzalez, M.C., 82, 169 I Grand unification, 24 Individual freedom, 423 Group decision making Inertia effect, 123–125 achievements, 118–119 Inertia principle, 174 elements, 113–114 Inflexible effect, 191–193 emblematic illustrations Intergovernmental panel on climate change minority effect, 116–117 (IPCC), 51 two half/half balanced opposite social Irrationality, 423 representations, 114–115 Ising ferromagnetic model, 101 See also Group two half/half unbalanced opposite social decision making representations, 115–116 Italian communist party, 223 group choice anticipation, 109–111 leader effect, 117–118 J physical model overlap, 118, 119 Jacob, B., 78 random symmetry breaking choice, Jacobs, F., 97, 170, 191 108–109 Jan, N., 169 Serge Moscovici, 106–107 Journal of mathematical psychology, 47, 53 social pressure, 111–112 social representational state, 112–113 strike phenomenon K achievements, 106 Kertsz, J., 82 F vs. M, 103–104 Killer geometries, 8, 9 metastability limits, 104 King Solomon, 77 minimum dissatisfaction principle, Koelsch, H., 427 102–103 Kogut, J., 42, 74 model, 102 Krausse, U., 169 M vs. H, 105 Kronecker function, 107, 233, 381 novel counterintuitive social highlights, Kulakowski, K., 169, 185, 383 105–106 physical model overlap, 104–105 states of, 101 L symmetrical individual vs.the symmetrical Lambiotte, R., 169 group, 107–108 Lavicka, H., 169 Group internal conflict function (G), 108, 113 Leader effect, 117–118 Group size mixing Lebowitz, J., 54 general update equation, 185–186 Lehir, P., 84, 85 variations, 186–188 Leschhorn, H., 58 Li, H., 70 Local fluctuations, 276 H Local majority model and biases existence Haken, H., 78 local updates, 175 Hansen, J.-P., 427 mimicking cognitive processes, 173–174 Hari Seldon, 3, 4 opinion flow diagrams, 175, 181 434 Index

Local majority model and biases existence Masselot, A., 8, 70, 97, 169, 196 (cont.) Master equation approach, 78 polarization process, 174 Math, J., 61 schematic illustration, 175–180 Mattis, D.C., 152 thresholds and extreme cases, 182–184 Mattis model, 152 Local majority rules, 122–123, 210 Mauger, A., 70, 82, 97, 156, 157, 164–167, 427 Lopez,´ J.M., 170 Maurus, V., 51, 87 Loreto, V., 70, 71, 94, 169 Metastability phenomenon, 103, 104 Miguel, M.S., 169 Minimum bottom agents M bottom-up voting hierarchy, 341 Maass, P., 58 large-sized voting groups, 343 Magic formula lc;4;n variation, 344, 345 bottom-up hierarchical levels, 256 nasty density, 343 coexistence region, 258 odd and even case, 342 democratic balance, 261 c;r;n evolution, 343, 344 deterministic outcome, 257 Minimum dissatisfaction principle, 102–103 fixed and constant hierarchical levels, 256 Minority effect, 116–117 magic threshold, 256–257, 259–260 Minority spreading model, 83, 84 minority tendency, 256 Mobilia, M., 169, 198 phase diagram, 257, 258, 262 Modern theory of phase transitions, 44, 46 physicist like arrangement, 260 Monte Carlo scheme, 232 n p0 D pr;B variation, 260, 261 Monte-Carlo simulations, 164 probabilistic area, 262 Moscovici, S., 56, 95, 106, 107, 109, Mother, probabilistic outcome, 257 sociophysics, 425 probability, 256 pyramidal hierarchy, 263 qualitative behavior and quantitative N estimates, 262 National Center for Scientific Research systematic discrepancy, 260, 261 (CNRS), 51 Taylor expansion, 259, 263 National Front, 375 unstable fixed point separator, 260 Nawojczyk, M., 169, 185, 383 Magic-Machiavelli formula Neau, D., 169 associated voting function, 319 Nonthreshold dynamics, 184–185 lower and upper magic thresholds, 317 Nucleation phenomenon, 105 r (pc /variation, 317, 318 phase diagram, 317, 318 physicist’s corner, 320–324 O n pr;A value, 319 One-person-one-argument principle, 169 n pr;B value, 317, 319 Opinion dynamics modeling n p0 D pr;B variation, 319, 320 achievements, 198–199 probabilistic area shrinking, 319 applications, 199–200 probabilistic outcome, 317 conservative public opinion self-elimination, 316 collective choice, 172 systematic discrepancy, 320 external and internal mechanisms, two-digit precision, 319, 320 172–173 upper probabilistic region, 317 refusal, 171 Majewski, J., 70 group size mixing Malarz, K., 166, 196 general update equation, 185–186 Mandelbrot, B.B., 44 variations, 186–188 Mantegna, R.N., 58, 70, 71 heterogeneous agents and the contrarian Mao Zedong statement, 156 effect, 188–190 Martins, A.C.R., 169 inflexible effect and the global warming Ma, S.-k., 101, 109 issue, 191–193 Index 435

local majority model and biases existence probabilistic outcome, 229, 230 local updates, 175 probability, 229 mimicking cognitive processes, single, random group voting scheme 173–174 result, 230, 231 opinion flow diagrams, 175, 181 truncated binomial expansion, 229 polarization process, 174 former communist organizations schematic illustration, 175–180 anti-communists, 224 thresholds and extreme cases, cold war, 223, 224 182–184 dictatorships, 223, 224 nonthreshold dynamics, 184–186 Italian communist party, 223 overview, 169–170 lack of democracy, 223 physics systems overlap, 197 one group voting scheme limitations reshuffling effect and rare event nucleation, agents voting scheme, 231 196 democratic majority rule principle, 232 three opinions, 195–196 error bars, 231 thresholdless case, 193–194 Monte Carlo scheme, 232 thresholdless driven coexistence, 189–192 probability, 232 Order parameter (M), 103 random character, agents selection, Orthodox physics, 53 231–232 Ott, J., 70 random number generator, 232 simple voting process, 225–226 single random small group voting scheme, P 227–228 Pajot, S., 169 three random selected agents Paternity claiming, sociophysics, 61–63 bottom-up evolution, 239, 243 Pathria, R.K., 101 democratic balance, 238–239 Pekalski, A., 70 democratic bottom up hierarchy, Peliti, L., 97, 196, 427 236–238 Peng, C.-K., 58 density variation, 238, 239 Percolation theory. See Terrorism and difference amplitude, 234, 235 percolation, passive supporters elected local representatives, 234, 235 Pereira, C.B., 169 higher ranking representative, 235 Perfect democratic structure probabilistic outcome, 239 dynamics, repeated democratic voting probability, 234, 236 attractor, 241 voting machine, 238 deterministic event, 240 zero, comments, 243–245 elected representatives flow diagram, Petroni, F., 169 241, 242 Pfeuty, P., 46, 75, 427 r (pc /variation, 241, 242 Phase transitions, 8, 42–43 properties, 240 Physical and human fundamental aspect separator, 241 digression, 294–295 Taylor expansion validity limit, 242, magic formula, 277–280 243 phase transitions and critical phenomena even-sized voting groups correlation length, 273, 274 identity, 234 global organization, 273 Kronecker function, 233 intrinsic equality, 274 odd values voting groups, 232 minimum global size, 275–276 symmetry, 233, 234 power law behavior, 274 tie configurations, 232–233 universal metamorphosis, 273 voting function, 233 vr exponent variation, 274, 275 fluctuations, group sizes, and democratic practical scheme, 276–277 balance radical efficiency turning nasty deterministic outcome, 230, 231 antidemocratic bottom configuration, overamplification, 229 288 436 Index

Physical and human fundamental aspect (cont.) Physicist’s corner associated number of A-agents, 288 ad hoc arrangement, 250 democratic dynamics, 289 approximate analytical formula, 255 geometric coups, 289 approximation level, 267 nesting strategy, 287 bottom minorities and majorities, 366–367 phase diagram, winning presidential compact exact formula, 265 election, 286, 287 corrections values, 267, 268 probabilistic phase, 286 degree of polynomial, 251, 252 c;r;n evolution, 289, 290 deterministic outcome, 252 uncertain phase, 286 dynamics, repeated democratic voting zero density proportion, 287 degree of polynomial, 316 randomness, 290–291 deterministic outcome, 316 rare antidemocratic bottom configurations exact numerical estimates, 315, 316 8 A-agents optimized distribution, 285 fitting function, 314 19 B-agents, 285 pc;r behavior, 314 controlled deterministic event, 284 pc;r variation, 313, 314 nesting distribution, 285 tie effect, 315 probability, 285, 286 unstable fixed-point separator, 315 pyramidal organization, 284 exact numerical estimates, 253 rare dictatorial events vs. antidemocratic higher order derivatives, 251 agent selection fluctuation, 281 hypothesis, 256 agents random distribution, 280 inertia principle softening, 373–374 associated democratic voting phase iterations, 252, 253 diagram, 280 linear dependence vs. logarithmic, 266 associated total probability, 281 magic formula bottom antidemocratic rearrangements ad hoc change, 334 probability, 281, 282 constant, 335–336 bottom configurations, 281 –1 correction, 334 democratic configurations, 282 light inflection point, 335 global majority, 279 magic expression, 333 iteration process, 282 mathematics, data, intuition, and level-by-level repeated voting scheme, esthetics, 335 283 numerical estimation, 334 minority strength, 280 power law hypothesis, 332, 335 P3.p0/ and P3;2.p0/ probabilities, 283 three classes, 332, 333 physicist corner, 283–284 virtual shift, solid grounding, 334 repeated majority rule voting, 279 magic-Machiavelli formula strategic nesting, 283 even-sized groups, 322 winning bottom distribution, 281 four-digit precision, 321 zero probability, 280 n level pyramidal hierarchy, 323 voting groups geography and multisize pc;r = 16 results, 322, 323 combination phase diagram, 323, 324 n associated phase diagram, 294 p0 D pr;A variation, 322 n committees, 291 p0 D pr;B variation, 321 districts, 292 qualitative properties, 320 elected A representative, 293 rounding off procedure, 320, 321 heterogeneity, 291 systematic discrepancy, 322 hierarchical bottom-up structure, 293 two-digit precision, 321 localities, 292 unstable thresholds, 321 national representation, 292 mathematical tribulations, 250 n presidency, 292 NG;r evolution, 267 pyramidal structuring, 291 nonzero derivative, 251 states, 292 odd-sized voting groups, 256 Index 437

practical scheme Redner, S., 169, 198 exact numerical estimates, 328, 329 Reif, F., 101 intuitive mathematical Renormalization group techniques, 133, 273, manipulationwhere, 328 373 invariance, 327 Reshuffling effect, 196 limit constraints, 329 social and political phenomena, 330 unrealistic range plot, 327 S smoother sensitivity, 266 Salinas, S.R., 427 super magic formula Salinger, M.A., 58 bottom-up democratic voting process, Sa Martins, J.S., 70, 75, 93, 94 338 Sa Martins, S.A., 170 intuitive incitement implementation, Saramaki, J., 169 337–338 Schelling, T.C., 61, 76 large-sized voting groups, 339 Schneider, J.J., 77, 82, 169, 427 one voting group, 338 Schweitzer, F., 70, 169, 427 power law hypothesis, 339 Science fiction, 3–6 Taylor expansion validity limit, 252–255 Shapiro, D., 46, 61 three competing parties, 395, 402–405 Shapir, Y., 46–48, 53, 61, 78, 95, 101, 102, two competing parties, one-dimensional 105 problem, 393 Slanina, F., 169 Pierre and Marie Curie University (UPMC), 44 Social pressure, 111–112 Plasmasociology, 78 Social representational state, 112–113 Polarization, 109, 174 Sociophysics Political window, 157 emblematic founding models (see Potts variables, 151 Emblematic founding models) Presidency, formula, 126–127 emergence, 70–71 Prince, P.A., 58 epistemological foundations Psychohistory. See Sociophysics emerging reasons, 72 Public opinion, 97 negative statements, 73–74 collective choice, 172 positive statements, 72–73 external and internal mechanisms, 172–173 essential challenges, 69–70 refusal, 171 fatherhood papers on, 77 seminar series, 78–79 R future, 71 Radical efficiency human and social freedom discovery, antidemocratic bottom configuration, 288 16–18 associated number of A-agents, 288 IPCC, 87–88 democratic dynamics, 289 new paradigms, 11–13 geometric coups, 289 origin, 74–75 nesting strategy, 287 academic freedom, 51–52 phase diagram, winning presidential creative history, 60–61 election, 286, 287 development in 1990s, 41 probabilistic phase, 286 future frustration, 45 c;r;n evolution, 289, 290 literature papers, 46–47 uncertain phase, 286 manifesto of, 47–48 zero density proportion, 287 orthodox physics research, 53 Radomski, J.P., 8, 196 paternity claiming, 61–63 Random field Ising ferromagnetic model, 107 phase transitions, 42–43 Random symmetry breaking choice, 108–109 research papers and acknowledgement, Rare event nucleation, 196 54–55 Real-space renormalization group techniques, rising sun, 58–60 132, 133 shadow of loneliness, 57–58 438 Index

Sociophysics (cont.) physical model overlap, 104–105 social scientists and, 55–57 states of, 101 story illustration, 48–51 Super bonds, 150 TAU program, 46 Superspin, 132, 373 twenty-first century beginning, 64–65 Sznajd, J., 82, 169 well established field, 63–64 Sznajd-Weron, K., 82, 169, 427 Wilson and Fisher implementations, 43–44 positive achievements T event prediction, 82–83 Taylor expansions list, 81–82 dynamics, repeated democratic voting, precise event prediction 240 fail prediction, 86 magic formula, 259, 263 minority spreading model, 83 physicist’s corner, 250–256 risk taking, 84–85 three-party bottom-up voting flow, 396 true prediction, 85–86 Tel-Aviv University (TAU), 44, 46 road map establishment, 86–87 Terrorism and percolation, passive supporters science fiction and Asimov’s foundation achievements, 168 syndrome Galam–Mauger universal formula, 164–166 fall of empires parenthesis, 10–11 geometry future prediction, 5–7 open spaces, 158–159 Hari Seldon and psychohistory, 3–5 reverse situation, 160, 161 historical vs. ahistorical sciences, 7 and terrorist movement, 156–158 mathematician and physicist, 9–10 VOP and AOS, 159–160 small and large groups, 8–9 local vs. global soviet-like rewriting history AOS, 163 preconceptions, 75 classical terrorism, 160–162 publications and Schelling model, 76 Mao Zedong statement, 156 reasons, 77 modeling and reducing, 155 weaknesses, 79–81 neutralization, 163 zero risk path, 13–16 percolation thresholds, 164 Sociopolitical phenomena, 93–94 physics model overlap, 167 Sokal, A., 61 reasons, 156, 157 Solomon, S., 76, 77, 169 various flags, 165–167 Sousa, A.O., 82, 169, 196 Tessone, C.J., 169 Spin glasses, 152 Three competing parties Stair functions, 249 associated voting functions, 391 Standoff, R.-S., 86 even-sized voting groups, 390 Stanley, H.E., 58, 62, 70, 71, 427 physicist’s corner, 402–405 Stanley, M.H.R., 58 three-party bottom-up voting flow Statistical physics, 8, 28 bottom-up democratic elections, 401 Stauffer, D., 54, 61, 70, 75–77, 82, 94, 109, coordinates, 398, 399 157, 169, 170, 427 eigenvalues and eigenvectors, 399–401 Stenflo, L., 78 fixed point equations, 395 Strike phenomenon isosceles triangle, 399 achievements, 106 matrix form, 398 F vs. M, 103–104 six fixed points, 396–398 metastability limits, 104 stability, 402 minimum dissatisfaction principle, stable fixed points, 402 102–103 Taylor expansion, 396 model, 102 voting function, 395 M vs. H, 105 tie situations, 390 novel counterintuitive social highlights, two competing parties, one-dimensional 105–106 problem, 391–393 Index 439

two-dimensional problem, 393–395 leadership, 389 voting rules, 390 local majority rule., 380 Three-country problem odd- and even-sized groups, 380 endless dynamics and complexity, 143, 144 self-elimination, 389 five countries dynamics, 145–149 unstable fixed point, 387 infernal dynamics, 142 variables and parameters, 389 negative bond addition, 149 Two-country problem real life sets, 143, 145 cost evaluation, 140 situation, 142–143 definition, 139 Three-party bottom-up voting flow Ising variables and, 140–141 bottom-up democratic elections, 401 coordinates, 398, 399 eigenvalues and eigenvectors, 399–401 V fixed point equations, 395 Vicente, R., 169 isosceles triangle, 399 Vignes, A., 97 matrix form, 398 Vinokur, A., 109 six fixed points, 396–398 Virtual open spaces (VOSs), 159–160 stability, 402 Viswanathan, G.M., 58 stable fixed points, 402 Voronel, A., 49, 427 Taylor expansion, 396 Voting model, 82 voting function, 395 Thresholdless driven coexistence, 189–192 W Toral, R., 169 Weidlich, W., 46, 61, 70, 78 Toy model, 24 Weisbuch, G., 169 Truncated binomial expansion, 229 Wilhelmsson, H., 78 Turner, J.C., 109 Wilson, K.G., 42, 43, 74 Two competing parties Wio, H.S., 169, 170 bottom evolution, 386, 387 Wonczak, S., 96, 127 bottom-up voting outcomes, 386 critical number of levels, 388 democracy, 389 Y democratic elections result prediction, Young, A.E., 152 380–382 dynamics odd cases, 385 Z regular even cases, 385–386 Zavalloni, M., 107, 109 very special pair case, 382–384 Zedong, M., 156, 164 elected representatives, 380, 387, 388 Zucker, J.D., 95, 107