HLIDKS LID WODIG COMPUTATIONAL OF COOPERATIVE ASPECTS GAME THEORY CHALKIADAKIS • ELKIND • WOOLDRIDGE SeriesSeries ISSN: ISSN: 1939-4608 1939-4608 COMPUTATIONAL COMPUTATIONAL OF COOPERATIVE ASPECTS GAME OF COOPERATIVE ASPECTS THEORY GAME THEORY CHALKIADAKIS • ELKINDCHALKIADAKIS • WOOLDRIDGE • ELKIND • WOOLDRIDGE
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SeriesSeries Editors: Editors: RonaldRonald J. J. Brachman, Brachman, Yahoo! Yahoo!Yahoo! Research ResearchResearch and and Thomas Thomas G. G. Dietterich, Dietterich, Oregon Oregon State State University University ComputationalComputational AspectsAspects ofof ComputationalComputational Aspects Aspects of of CooperativeCooperative Game Game Theory Theory CooperativeCooperative GameGame Theory Theory GeorgiosGeorgios Chalkiadakis, Chalkiadakis, Technical Technical University University of of Crete, Crete, Greece Greece EdithEdith Elkind, Elkind, Nanyang Nanyang Technological Technological University, University, Singapore Singapore MichaelMichael Wooldridge, Wooldridge, University University of of Liverpool, Liverpool, United United Kingdom Kingdom
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Computational Aspects of Cooperative Game Theory Synthesis Lectures on Artificial Intelligence and Machine Learning
Editors Ronald J. Brachman, Yahoo Research William W. Cohen, Carnegie Mellon University Thomas Dietterich, Oregon State University Computational Aspects of Cooperative Game Theory Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge 2012 Representations and Techniques for 3D Object Recognition and Scene Interpretation Derek Hoiem and Silvio Savarese 2011 A Short Introduction to Preferences: Between Artificial Intelligence and Social Choice Francesca Rossi, Kristen Brent Venable, and Toby Walsh 2011 Human Computation Edith Law and Luis von Ahn 2011 Trading Agents Michael P.Wellman 2011 Visual Object Recognition Kristen Grauman and Bastian Leibe 2011 Learning with Support Vector Machines Colin Campbell and Yiming Ying 2011 iii Algorithms for Reinforcement Learning Csaba Szepesvári 2010 Data Integration: The Relational Logic Approach Michael Genesereth 2010 Markov Logic: An Interface Layer for Artificial Intelligence Pedro Domingos and Daniel Lowd 2009 Introduction to Semi-Supervised Learning Xiaojin Zhu and Andrew B.Goldberg 2009 Action Programming Languages Michael Thielscher 2008 Representation Discovery using Harmonic Analysis Sridhar Mahadevan 2008 Essentials of Game Theory: A Concise Multidisciplinary Introduction Kevin Leyton-Brown and Yoav Shoham 2008 A Concise Introduction to Multiagent Systems and Distributed Artificial Intelligence Nikos Vlassis 2007 Intelligent Autonomous Robotics: A Robot Soccer Case Study Peter Stone 2007 Copyright © 2012 by Morgan & Claypool
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Computational Aspects of Cooperative Game Theory Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge www.morganclaypool.com
ISBN: 9781608456529 paperback ISBN: 9781608456536 ebook
DOI 10.2200/S00355ED1V01Y201107AIM016
A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING
Lecture #16 Series Editors: Ronald J. Brachman, Yahoo! Research William W. Cohen, Carnegie Mellon University Thomas Dietterich, Oregon State University Series ISSN Synthesis Lectures on Artificial Intelligence and Machine Learning Print 1939-4608 Electronic 1939-4616 Computational Aspects of Cooperative Game Theory
Georgios Chalkiadakis Technical University of Crete, Greece
Edith Elkind Nanyang Technological University, Singapore
Michael Wooldridge University of Liverpool, United Kingdom
SYNTHESIS LECTURES ON ARTIFICIAL INTELLIGENCE AND MACHINE LEARNING #16
M &C Morgan& cLaypool publishers ABSTRACT Cooperative game theory is a branch of (micro-)economics that studies the behavior of self-interested agents in strategic settings where binding agreements among agents are possible. Our aim in this book is to present a survey of work on the computational aspects of cooperative game theory. We begin by formally defining transferable utility games in characteristic function form, and introducing key solution concepts such as the core and the Shapley value. We then discuss two major issues that arise when considering such games from a computational perspective: identifying compact representa- tions for games, and the closely related problem of efficiently computing solution concepts for games. We survey several formalisms for cooperative games that have been proposed in the literature, includ- ing, for example, cooperative games defined on networks, as well as general compact representation schemes such as MC-nets and skill games. As a detailed case study, we consider weighted voting games: a widely-used and practically important class of cooperative games that inherently have a natural compact representation. We investigate the complexity of solution concepts for such games, and generalizations of them. We briefly discuss games with non-transferable utility and partition function games. We then overview algorithms for identifying welfare-maximizing coalition struc- tures and methods used by rational agents to form coalitions (even under uncertainty), including bargaining algorithms. We conclude by considering some developing topics, applications, and future research directions.
KEYWORDS game theory, cooperative games, coalitional games, representations, computational complexity, solution concepts, core, Shapley value, coalition formation, coalition struc- ture generation This manuscript was a pleasure to discover, and a pleasure to read—a broad, but succinct, overview of work in computational cooperative game theory. I will certainly use this text with my own students, both within courses and to provide comprehensive background for students in my research group. The authors have made a substantial contribution to the multiagent systems and algorithmic game theory communities. – Professor Jeffrey S. Rosenschein The Hebrew University of Jerusalem, Israel
With the advent of the internet, the computational aspects of cooperative game theory are ever more relevant. This unique and timely book by Chalkiadakis, Elkind, and Wooldridge gives a concise and comprehensive survey of the subject, and serves at the same time as a one-stop introduction to cooperative game theory. – Professor Bernhard von Stengel London School of Economics, UK
In recent years, research on the computational aspects of cooperative game theory has made tremen- dous progress, but previous textbooks have not included more than a short introduction to this important topic. I am excited by the thorough treatment in this new book, whose authors have been and continue to be at the very forefront of this research. Newcomers to the area are well advised to read this book carefully and cover to cover. – Professor Vincent Conitzer Duke University, USA
Cooperative game theory has proved to be a fertile source of challenges and inspiration for computer scientists. This book will be an essential companion for everyone wanting to explore the computa- tional aspects of cooperative game theory. – Prof Makoto Yokoo Kyushu University, Japan An excellent treatise on algorithms and complexity for cooperative games. It navigates through the maze of cooperative solution concepts to the very frontiers of algorithmic game theory research.The last chapter in particular will be enormously valuable for graduate students and young researchers looking for research topics. – Professor Xiaotie Deng University of Liverpool, UK Georgios: for my godmother, Katerina; and my godchildren, Filippos, Antonios, and Maria-Sofia.
Edith: to the memory of my grandfather Gersh.
Michael: for Lily May and Thomas Llewelyn.
xi Contents
Preface ...... xv
Acknowledgments ...... xvii
Summary of Key Notation ...... 1
1 Introduction ...... 5 1.1 Why are Non-Cooperative Games Non-Cooperative? ...... 6 1.2 Computational Problems in Game Theory ...... 8 1.3 The Remainder of This Book ...... 9 1.4 Further Reading ...... 10 2 Basic Concepts ...... 11 2.1 Characteristic Function Games ...... 11 2.1.1 Outcomes ...... 13 2.1.2 Subclasses of Characteristic Function Games ...... 14 2.2 Solution Concepts ...... 17 2.2.1 Shapley Value ...... 17 2.2.2 Banzhaf Index ...... 22 2.2.3 Core and Core-Related Concepts ...... 23 2.2.4 Nucleolus ...... 32 2.2.5 Kernel ...... 33 2.2.6 Bargaining Set ...... 33 2.2.7 Stable Set ...... 34 3 Representations and Algorithms ...... 37 3.1 Combinatorial Optimization Games ...... 38 3.1.1 Induced Subgraph Games ...... 38 3.1.2 Network Flow Games ...... 40 3.1.3 Assignment and Matching Games ...... 42 3.1.4 Minimum Cost Spanning Tree Games ...... 42 3.1.5 Facility Location Games ...... 42 xii 3.2 Complete Representations ...... 43 3.2.1 Marginal Contribution Nets ...... 43 3.2.2 Synergy Coalition Groups ...... 45 3.2.3 Skill-Based Representations ...... 45 3.2.4 Algebraic Decision Diagrams ...... 46 3.3 Oracle Representation ...... 47
4 Weighted Voting Games...... 49 4.1 Definition and Examples ...... 49 4.2 Dummies and Veto Players ...... 51 4.2.1 Power and Weight ...... 52 4.2.2 Computing the Power Indices ...... 55 4.2.3 Paradoxes of Power ...... 57 4.3 Stability in Weighted Voting Games ...... 58 4.3.1 The Least Core, the Cost of Stability, and the Nucleolus ...... 60 4.4 Vector Weighted Voting Games ...... 65 4.4.1 Computing the Dimension of a Simple Game ...... 67
5 Beyond Characteristic Function Games ...... 71 5.1 Non-transferable Utility Games ...... 71 5.1.1 Formal Model...... 72 5.1.2 Hedonic Games ...... 75 5.1.3 Qualitative Games ...... 79 5.2 Partition Function Games ...... 84
6 Coalition Structure Formation ...... 87 6.1 Coalition Structure Generation ...... 87 6.1.1 Dynamic Programming ...... 88 6.1.2 Anytime Algorithms ...... 89 6.2 Coalition Formation by Selfish Rational Agents ...... 92 6.2.1 Coalition Formation Via Bargaining ...... 93 6.2.2 Dynamic Coalition Formation ...... 94 6.2.3 Coalition Formation Under Uncertainty ...... 96 6.3 Coalition Formation and Learning ...... 102 7 Advanced Topics ...... 107 7.1 Links between Cooperative and Non-cooperative Games ...... 107 7.1.1 Cooperation in Normal-Form Games ...... 107 7.1.2 Non-Cooperative Justifications of Cooperative Solution Concepts ..... 109 7.1.3 Program Equilibrium ...... 109 7.2 Using Mechanism Design for Coalition Formation ...... 111 7.2.1 Anonymity-Proof Solution Concepts ...... 112 7.3 Overlapping and Fuzzy Coalition Formation ...... 112 7.4 Logical Approaches to Cooperative Game Theory ...... 114 7.5 Applications ...... 115 7.5.1 Coalitions in Communication Networks ...... 115 7.5.2 Coalitions in the Electricity Grid ...... 116 7.5.3 Core-Selecting Auctions ...... 118 7.6 Research Directions ...... 118
Bibliography ...... 121
Authors’ Biographies ...... 145
Index ...... 147
Preface
Over the past decade, there has been an enormous growth of interest in issues at the intersection of game theory and computer science. This phenomenon has been driven in part by the Internet juggernaut, which so spectacularly arrived in the public consciousness in the late 1990s. On the one hand, the Internet can be understood in purely computational terms: it is a massive, highly dynamic network of computational nodes, exchanging data across physical and wireless connections using carefully defined protocols and procedures. However, a purely computational view of the Internet surely misses aspects that are crucial to understanding it. The Internet is not populated by altruistic or even indifferent entities. The entities that make decisions and act on the Internet are self-interested. They have preferences, goals, and desires, and broadly speaking, they will act to bring about their preferences, goals, and desires as best they can. A traditional computer science analysis has nothing whatsoever to say about these economic aspects of the Internet.This observation motivated researchers to apply techniques from economics, and in particular, techniques from the branch of micro-economics called game theory, to the analysis of networked computer systems like the Internet. Research in this domain has come primarily from two communities: the multiagent systems community (which originated from research in artificial intelligence) [240, 261], and the algorithmic game theory community [195] (which originated from research in theoretical computer science). These two computer science communities have embraced techniques from game theory, and have made considerable progress in understanding both how game-theoretic techniques can be usefully applied in computational settings, and how computational techniques can be brought to bear on game-theoretic problems. Our aim in the present book is to introduce the reader to research in the computational aspects of cooperative game theory.This is an important and extremely active area of contemporary research, of interest to both the multiagent systems and algorithmic game theory communities. Crudely, cooperative game theory (as opposed to non-cooperative game theory) is concerned with settings in which groups of agents can agree to cooperate with each other, for example, by signing a mutually binding contract. Binding agreements may enable the agents to implement solutions that are not considered possible in non-cooperative game theory. Research in the computational aspects of cooperative game theory has two complementary aspects. On the one hand, we study the application of computational techniques to problems in cooperative game theory: how to compute outcomes of a game, for example. On the other hand, we might consider how concepts and techniques from cooperative game theory can be applied to computational problems. For example, as we will discuss, techniques from cooperative game theory can be usefully applied to the analysis of computer networks. READERSHIP Our book is a graduate-level text, and we expect the readership to consist primarily of graduate students and researchers who want to gain an understanding of the main issues and techniques in the field. We expect most readers to have a computer science background, but we hope the text is of interest and value to researchers in economics and game theory who want to gain an understanding of what computer scientists are doing with their ideas. If we have done our job well, then after reading this book, you should have a fighting chance of understanding most of the research papers that we cite in the book.
PRE-REQUISITES We expect the readers of our book to be predominantly computer scientists and artificial intelligence researchers, with little or no background in game theory. For these readers, we provide in chapter 2 a complete (if somewhat terse) summary of the necessary definitions and concepts from cooperative game theory that are used throughout the remainder of the book. Readers from the game theory community will find no surprises in our treatment of their subject, except that in our presentation, coalition structures play a more prominent role than is usually the case in the game theory literature. Specifically, we find it natural to introduce coalition structures from the start, defining the outcome of a cooperative game to be a coalition structure together with a payoff vector. Our reasoning here is that this helps in particular to clarify the story around superadditivity,how superadditivity implies the formation of the grand coalition,and solution concepts that are conventionally formulated with respect to the grand coalition (e.g., the Shapley value). A key concern of our book is, of course, computation, and we make extensive use of techniques from the field of computational complexity in the analysis of cooperative games. A survey of the relevant techniques and concepts from this area is beyond the scope of the present book, and for this we refer the reader to [122, 203]. Throughout the book, we assume a modest level of mathematical ability: the math content is primarily discrete mathematics of the level and type that is usually taught in undergraduate computer science degrees.
WEB SITE The following web site provides some links to relevant web resources, downloadable video lectures to accompany the book, and a complete set of lecture slides: http://web.spms.ntu.edu.sg/˜eelkind/coopbook/
Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge October 2011 Acknowledgments
We are grateful to Mike Morgan for suggesting this volume, and for charming us into agreeing to write it. We thank Haris Aziz, Yoram Bachrach, Craig Boutilier, Paul E. Dunne, Leslie Ann Gold- berg, Paul W. Goldberg, Nick Jennings, Ramachandra Kota, Vangelis Markakis, Dima Pasechnik, Mike Paterson, Maria Polukarov, Valentin Robu, Alex Rogers, Alex Skopalik, and Yair Zick for per- mitting us to adapt material from papers we have co-authored with them. The bulk of the material in this book is based on tutorials that were presented at ECAI-2008 (Patras, Greece), AAMAS- 2009 (Budapest, Hungary), AAMAS-2010 (Toronto, Canada), AAAI-2010 (Atlanta, Georgia), AAMAS-2011 (Taipei, Taiwan), and IJCAI-2011 (Barcelona, Catalonia, Spain): we are grateful to the students who attended these tutorials for their questions, corrections, suggestions, and enthu- siastic feedback. We would also like to thank Stéphane Airiau, Haris Aziz, Piotr Faliszewski, Felix Fischer, Martin Gairing, Reshef Meir, Dima Pasechnik, Tuomas Sandholm, Bernhard von Stengel, Yair Zick, and the anonymous reviewers arranged by the publisher, who carefully read a preliminary draft of the book and gave us detailed and enormously helpful feedback. It goes without saying that any remaining errors and omissions are entirely the responsibility of the authors. Edith Elkind gratefully acknowledges research fellowship RF2009-08 “Algorithmic aspects of coalitional games” from the National Research Foundation (Singapore).
PERSONAL PRONOUNS We find it inappropriate to refer to agents solely as “he” or solely as “she”, and calling agents “it” seems just plain ugly. So, we interleave usages of “he” and “she” (semi-)randomly.
Georgios Chalkiadakis, Edith Elkind, and Michael Wooldridge October 2011
1 Summary of Key Notation
G A game. (Chapter 1)