Friedrich Pukelsheim Proportional Representation Apportionment Methods and Their Applications With a Foreword by Andrew Du MEP Second Edition Proportional Representation Friedrich Pukelsheim

Proportional Representation Apportionment Methods and Their Applications

Second Edition

With a Foreword by Andrew Duff MEP

123 Friedrich Pukelsheim Institut furR Mathematik Universität Augsburg Augsburg, Germany

ISBN 978-3-319-64706-7 ISBN 978-3-319-64707-4 (eBook) DOI 10.1007/978-3-319-64707-4

Library of Congress Control Number: 2017954314

Mathematics Subject Classification (2010): 91B12

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This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland The just, then, is a species of the proportionate :::. For proportion is equality of ratios, and involves four terms at least :::; and the just, too, involves at least four terms, and the ratio between one pair is the same as that between the other pair; for there is a similar distinction between the persons and between the things. As the term A, then, is to B, so will C be to D, and therefore, alternando,asA is to C, B will be to D :::. This, then, is what the just is—the proportional; the unjust is what violates the proportion. Hence one term becomes too great, the other too small, as indeed happens in practice; for the man who acts unjustly has too much, and the man who is unjustly treated too little, of what is good :::. This, then, is one species of the just. Aristotle, Nichomachean Ethics, Book V, Chapter 3. Translated and Introduced by Sir David Ross, 1953. The World’s Classics 546, Oxford University Press. Foreword to the First Edition

The virtue of parliamentary democracy rests on the representative capability of its institutions. Even mature democratic states cannot take the strength of its representative institutions for granted. Newer democracies seek practicable ways and means on which to build lasting structures of governance which will command the affinity of the people they are set up to serve. The debate about the structural reform of parliamentary democracies is never far away. Nor should it be. The powers and composition of parliamentary chambers, their rules and working methods, the organization and direction of the political parties which compete for votes and seats, the electoral systems (who to register, how to vote, how to count), and the size and shape of constituencies—all these and more are rightly subject to continual appraisal and are liable to be reformed. Electoral reform is a delicate business: handled well, it can be the basis on which new liberal democracies spread their wings; it can refresh the old, tired democracies. Handled badly, electoral reform can distort the people’s will, entrench the abuse of power, and sow the seeds of destruction of liberty. Electoral systems are central to the debate in emerging democracies, and the relatively new practice of election observation by third parties highlights the need for elections to be run not only fairly but also transparently. Voting and counting should be simple, comprehensible, and open to scrutiny—qualities which are too often lacking even in old established democracies. Electoral reform is also very difficult to achieve. Those who must legislate for it are those very same people who have a vested interest in the status quo. That turkeys don’t vote for Christmas is amply demonstrated in the UK, where reform of the House of Lords has been a lost cause for over a century. Advocates of reform need to stack up their arguments well, be persistent, and enjoy long lives. Friedrich Pukelsheim has written a definitive work on electoral reform. He takes as his starting point the simple premise that seats won in a parliamentary chamber must represent as closely as possible the balance of the votes cast in the ballot box. Rigorous in his methodology, the author knows that there is no single perfect electoral system: indeed, in their quirky details, every system affects the exact outcome of an election. We are fortunate indeed that this professor of mathematics

vii viii Foreword to the First Edition is a profound democrat. He ably brings to the service of politicians the science of the mathematician. Dr Pukelsheim was an indispensable participant at the meeting in Cambridge in 2011, chaired by Geoffrey Grimmett, which devised “CamCom”—the best consensual solution to the problem of how to apportion seats in the European Parliament. As the Parliament’s rapporteur for electoral procedure, I am happy that our ideas are now taken forward in this publication.

The European Parliament

The European Parliament presents unusual challenges both to the scientist and practitioner. It is one chamber of the legislature of the European Union with a lot of power but little recognition. It reflects a giant historical compromise between the international law principle of the equality of states and the democratic motto of “One person, one vote.” Proportional representation at the EU level needs to bear in mind not only party but also nationality. The European Parliament is the forum of the political single market where the different political cultures and constitutional practices of the 28 member states meet up. MEPs are constitutionally representatives of the Union’s citizens, but they are elected not by a uniform electoral procedure but by different procedures under which separate national political parties and candidates fight it out, largely untroubled by their formal affiliation to European political parties.1 Efforts to make more uniform the election of the world’s first multinational parliament to be directly elected by universal suffrage have been frustrated. Voter turnout, as we know, has declined at each election to the European Parliament from 62% in 1979 to 43% in 2009, although these overall figures disguise sharp contrasts among the states and between elections. The long financial and economic crisis since 2008 has brought to a head a crisis of legitimacy for the European Parliament. If the euro is to be salvaged, and the EU as a whole is to emerge strengthened from its time of trial, transnational democracy needs to work better. Banking union and fiscal union need the installation of federal government. That federal government must be fully accountable to a parliament which connects directly to the citizen and with which the citizen identifies. The parliament must be composed in a fair and logical way best achieved in accordance with a settled arithmetical formula and not as a result of unseemly political bartering which borders on gerrymandering and sparks controversy. It is probable that in spring 2015 there will be a new round of EU constitutional change. This will take the form of a Convention in which heads of government and the European Commission will talk things through with members of the European and national parliaments. Part of the complex negotiations must include the electoral

1Article 14(2), Treaty on European Union. Foreword to the First Edition ix reform of the European Parliament. This will be the chance to progress CamCom for the apportionment of state seats alongside an ambitious proposal for the creation of a pan-European constituency for which a certain number of MEPs would be elected from transnational party lists.2 There is no reason to doubt that the notion of degressive proportionality,which strikes mathematicians as odd, will survive these negotiations because it expresses quite well the broadly understood belief that in a federal polity, the smaller need to be protected from subordination to the larger. CamCom copes logically with degressive proportionality in a way which should satisfy even the austere requirements of the Bundesverfassungsgericht at Karlsruhe. Nevertheless, as Friedrich Pukelsheim recognizes, fully fledged CamCom means radical adjustments to the number of MEPs elected in several states. It is important, therefore, that changes to the electoral system for one chamber of the legislature are balanced by changes to the electoral system in the other. Here, the Jagiellonian Compromise, which uses the square root as the basis for weighing the votes of the member states in the Council, deserves a good hearing. In June 2013, the Council and European Parliament eventually agreed that the new member state of Croatia should have 11 MEPs in the Parliament which were elected in May 2014. We worked hard to ensure that the reapportionment of seats would not contradict the logic of CamCom. There is a first, albeit clumsy, legal definition of degressive proportionality. More importantly, the European Union has now formally decided to pursue the objective of a formulaic approach to the future distribution of seats in the Parliament, coupled with a commitment to revisit the matter of qualified majority voting (QMV) in the Council. The decision of the European Council, now agreed by the European Parliament, lays down that a new system will be agreed in good time before the 2019 elections which in the future will make it possible, before each fresh election to the European Parliament, to allocate the seats between Member States in an objective, fair, durable, and transparent way, translating the principle of degressive proportionality as laid down in Article 1, taking account of any change in their number and demographic trends in their population, as duly ascertained thus respecting the overall balance of the institutional system as laid down in the Treaties. So perhaps CamCom and JagCom are destined to surface together in the next EU treaty. Legislators who care to understand the maths should start with this book.

Cambridge, UK Andrew Duff MEP September 2013

2For a full exposition of this proposal, see Spinelli Group, A Fundamental Law of the European Union, Bertelsmann Stiftung 2013. Preface to the Second Edition

In this second edition, the text of the first edition has been completely revised and expanded. Many empirical election results that serve as examples have been updated. Four changes deserve particular mention. Chapter 1 contains a review of the European Parliament elections of 2014. Chapter 9 has been enlarged by a novel and stringent proof of the Coherence Theorem of Balinski and Young. The presentation of double-proportional divisor methods, formerly condensed into a single chapter, now extends over two chapters. Chapter 14 illustrates the practice of double proportionality, while Chap. 15 explains the required theory. Chapter 16 is entirely new. It assembles biographical sketches and authoritative quotes from individuals who coined the development of apportionment methodol- ogy. The author is grateful to the attentive readers of the first edition who provided feedback and valuable criticisms. Particular thanks are due to a team of expert scholars and devoted friends who worked through most or all of the draft for this second edition. Their critical advice and numerous suggestions were vital to improve and clarify the exposition. I would like to thank Wolfgang Bischof, Hochschule Rosenheim (DE); Xavier Mora, Universitat Autònoma de Barcelona (ES); Grégoire Nicollier,HauteEcole d’Ingénierie, Sion (CH); Antonio Palomares, Universidad de Granada (ES); and Ben Torsney, University of Glasgow (UK).

Augsburg, Germany Friedrich Pukelsheim June 2017

xi Preface to the First Edition

Proportional representation systems determine how the political views of individual citizens, who are many, mandate the Members of Parliament, who are but a few. The same techniques apply when in the Parliament, the political groups are to be represented in a committee of a size much smaller than the Parliament itself. There are many similar examples all showing that proportional representation inevitably culminates in the task of translating numbers into numbers—large numbers of those to be represented into small numbers of those serving as representatives. The task is solved by procedures called apportionment methods. Apportionment methods and their applications are the theme of this work. A more detailed Outline of the Book follows the Table of Contents. By profession a mathematician rather than a politician, I have had the privilege of getting involved in several proportional representation reform projects in recent years. These include the introduction of a double-proportional electoral system in several Swiss cantons since 2006, the amendment of the German Federal Election Law during 2008–2013, and the discussion of the future composition of the European Parliament. The practical challenges and the teaching experience of many lectures and seminars on the subject of proportional representation and apportionment methods have shaped my view and provided the basis for this book. Apportionment methods may become quite complex. However, these complex- ities are no ends in themselves. They are reflections of the historical past of a society, its constitutional framework, its political culture, its identity. On occasion, the complexities are due to partisan interests of the legislators responsible. This mélange turns the topic into a truly interdisciplinary project. It draws on such fields as constitutional law, European law, political sciences, medieval history, modern history, discrete mathematics, stochastics, and computational algorithms, to name but a few. I became increasingly fascinated by the interaction of so many disciplines. My fascination grew when I had the pleasure of conducting student seminars jointly with colleagues from the humanities on topics of common interest. These experiences made me realize that proportional representation and apportionment methods are a wonderful example to illustrate the universitas litterarum, the unity of arts and sciences.

xiii xiv Preface to the First Edition

In retrospect, I find it much easier to conduct an interdisciplinary seminar than to author an interdisciplinary textbook. Nevertheless, I hope that this book may prove a useful reference work for apportionment methods, for scholars of constitutional law and political sciences as well as for other electoral system designers. The many apportionment methods studied span a wide range of alternatives in Germany, the European Union, and elsewhere. The book not only describes the mechanics of each method but also lists the method’s properties: biasedness in favor of stronger parties at the expense of weaker parties, preferential treatments of groups of stronger parties at the expense of groups of weaker parties, optimality with respect to goodness-of- fit or stability criteria, reasonable dependence on such variables as house size, vote ratios, size of the party system, and so on. These properties are rigorously proved and, whenever possible, substantiated by appropriate formulae. Since the text developed from notes that I compiled for lectures and seminars, I am rather confident that it can be utilized for these purposes. The material certainly suffices for a lecture course or a student seminar in a curriculum of mathematics, quantitative economics, computational social choice, or electoral system design in the political sciences. I have used parts of the text with particular success in classes for students who are going to be high-school teachers. The chapters presuppose readers with an appreciation for rigorous derivations and with a readiness to accept arguments from scientific fields other than their own. Most chapters can then be mastered with a minimum knowledge of basic arithmetic. Three chapters involve more technically advanced approaches. Chapters 6 and 7 use some stochastic reasoning and Chap. 14 discrete optimization and computer algorithms. The subject of the book is restricted to the quantitative and procedural rules that must be employed when a proportional representation system is implemented; as a consequence, the book does not explicate the qualitative and normative foundations that would be called for when developing a comprehensive theory of proportional representation. As in all sciences, the classification of quantitative procedures starts with the basic methods that later get modified to allow for more ambitious settings. The basic issue is to calculate seat numbers proportionally to vote counts. This task is resolved by divisor methods or by quota methods. Later, geographical subdivisions of the electoral region come into play, as do guarantees for small units to obtain representation no matter how small they are, as do restrictions for stronger groups to limit their representation lest they unduly dominate their weaker partners. In order to respond to these requirements, the basic methods are modified into variants that may achieve an impressive degree of complexity. When teaching the topic, I soon became convinced that its intricacies can be appreciated only by contemplating real data, that is, data from actual elections in the real world, rather than imaginary data from contrived elections in the academic ivory tower. My Augsburg students responded enthusiastically and set out to devise an appropriate piece of software, BAZI. BAZI has grown considerably since 2000 and has proved an indispensable tool for carrying out practical calculations and theoretical investigations. I would like to encourage readers of this book to use the program to retrace the examples and to form their own judgment. BAZI is freely available from the website www.uni-augsburg.de/bazi. Preface to the First Edition xv

Acknowledgments

My introduction to the proportional representation problem was the monograph of Michel Balinski/Peyton Young (1982). In their book, the authors recount the appor- tionment history in the House of Representatives of the USA, and then proceed to establish a Theory of Apportionment. This seminal source was soon complemented by the treatise of Klaus Kopfermann (1991) who adds the European dimension to the proportional representation heritage. Svante Janson’s (2012) typescript proved invaluable for specific mathematical questions. These books provide the foundations on which the results of the present work are based. Several colleagues and friends read parts or all of the initial drafts of this book and proposed improvements. I have benefited tremendously from the critical comments and helpful suggestions of Paul Campbell, Rudy Fara, Martin Fehndrich, Dan Felsenthal, Svante Janson, Jan Lanke,andDaniel Lübbert. Throughout the project, I had the privilege to rely on the advice and inspiration of my colleagues Karl Heinz Borgwardt, Lothar Heinrich,andAntony Unwin in Augsburg University Institute for Mathematics. A special word of thanks is due to my nonmathematical Augsburg colleagues who helped me in mastering the interdisciplinary aspects of the topic. I wish to thank Günter Hägele (Medieval History, University Library), Thomas Krüger (Medieval History), Johannes Masing (Constitutional Law, now with the University of Freiburg im Breisgau), Matthias Rossi (Constitutional Law), and Rainer-Olaf Schultze (Political Sciences). The largest debt of gratitude is due to the current and former members of my workgroup at Augsburg University. Many of them contributed substantially to this work through their research work and PhD theses. Moreover, they helped in orga- nizing lectures and seminars, in sorting the material, in optimizing the terminology, and in polishing the presentation. For their cooperation, I am extremely grateful to Olga Birkmeier, Johanna Fleckenstein, Christoph Gietl, Max Happacher, Thomas Klein, Sebastian Maier, Kai-Friederike Oelbermann, Fabian Reffel,andGerlinde Wolsleben. I would like to thank Andrew Duff, MEP, for graciously consenting to contribute the foreword to this book. Finally, I wish to acknowledge the support of the Deutsche Forschungsgemeinschaft.

Augsburg, Germany Friedrich Pukelsheim October 2013 Notations

N Set of natural numbers f0; 1; 2; : : :g (Sect. 3.1) btc; bbtcc Floor function (Sect. 3.2); rule of downward rounding (Sect. 3.4) dte; t Ceiling function, rule of upward rounding (Sect. 3.5) hti; hh tii Commercial rounding (Sect. 3.6); rule of standard rounding (Sect. 3.7) s.n/; n 2 N Signpost sequence .always s.0/ D 0/ (Sect. 3.8) ŒŒ t ; Œ t General rounding rule, general rounding function (Sect. 3.9) sr.n/ D n  1 C r Stationary signposts (n  1) with split parameter r 2 Œ0I 1 (Sect. 3.10) esp.n/ Power-mean signposts with power parameter p 2 Œ1I 1 (Sect. 3.11) h 2 N House size (Sect. 4.1) ` 2f2;3;:::g Number of parties entering the apportionment calculations (Sect. 4.1) k 2f2;3;:::g Number of districts (Sect. 15.1) v D .v1;:::;v`/ Vector of vote weights vj 2 Œ0I1/ for parties j Ä ` (Sect. 4.1) vC D v1 CCv` Component sum of the vector v D .v1;:::;v`/ (Sect. 4.1) wj D vj=vC Vote share of party j (Sect. 4.1) ` ` N .h/ Set of seat vectors x 2 N with component sum xC D h (Sect. 4.1) A Apportionment rule (Sect. 4.2); apportionment method (Sect. 4.4) A.hI v/ Set of seat vectors for house size h and vote vector v (Sect. 4.2) nCDfn; n C 1g Upward tie, increment option (Sect. 4.8) nDfn  1; ng Downward tie, decrement option (Sect. 4.8) vC=h Votes-to-seats ratio, also known as Hare-quota (Sect. 5.2)

xvii xviii Notations

wj h Ideal share of seats for party j Ä ` (Sect. 7.1) x  y Majorization of vectors (Sect. 8.2) A.hI v/  B.hI v/ Majorization of sets of vectors (Sect. 8.3) A  B Majorization of apportionment methods (Sect. 8.4) a D .a1;:::;a`/ Minimum requirements (Sect. 12.3) b D .b1;:::;b`/ Maximum cappings (Sect. 12.3) WD Definition  End-of-proof -ward Suffix of adjectives: the downward rounding, etc. -wards Suffix of adverbs: to round downwards, etc.  Multipurpose eye-catcher in tables Outline of the Book

Chapters 1 and 2: Apportionment Methods in Practice

The two initial chapters present an abundance of apportionment methods used in practice. Chapter 1 reviews the European Parliament elections of 2014, providing a rich source of examples. Chapter 2 deals with the German Bundestag election of 2009; emphasis is on the interplay between procedural steps and constitutional requirements. The chapters introduce concepts of proportional representation sys- tems that prove crucial beyond their specific use in European or German elections. Concepts and terminology in the initial chapters set the scene for the methodology that is developed in the sequels.

Chapters 3–5: Divisor Methods and Quota Methods

A rigorous approach to apportionment needs to appeal to rounding functions and rounding rules. They are introduced in Chap. 3. Chapters 4 and 5 discuss the two dominant classes of apportionment methods: divisor methods and quota methods. Usually, the input vote counts are much larger than the desired output seat numbers. Therefore, vote counts are scaled down to interim quotients of a fitting magnitude. Then, the interim quotients are rounded to integers. Divisor methods use a flexible divisor for the first step and a specific rounding rule for the second. Quota methods employ a formulaic divisor—called quota—for the first step and a flexible rounding rule for the second.

xix xx Outline of the Book

Chapters 6–8: Deviations from Proportionality

Many apportionment methods deviate from perfect proportionality in a systematic fashion. Chapters 6 and 7 investigate seat biases. A seat bias is the average of the deviations between actual seat numbers and the ideal share of seats, assuming that all vote shares are equally likely or that they follow an absolutely continuous distribution. Chapter 8 compares two apportionment methods by means of the majorization ordering. The majorization relation allows comparison of two appor- tionment methods. The relation indicates whether one method is more beneficial to groups of stronger parties—and hence more disadvantageous to the complementary group of weaker parties—than the other method.

Chapters 9–11: Coherence, Optimality, and Vote Ranges

Chapter 9 explores the idea that a fair division should be such that every part of it is fair, too. This requirement is captured by the notion of coherence. Divisor methods are coherent; quota methods are not. Chapter 10 evaluates goodness-of- fit criteria to assess the deviations of actual seat numbers from ideal shares of seats. Particular criteria lead to particular apportionment methods. Chapter 11 reverses the role of input and output. Given a seat number, the range of vote shares leading to the given seat number is determined. The results elucidate situations when a straight majority of votes fails to lead to a straight majority of seats. As a corrective, many electoral laws include an extra majority preservation clause. Three majority clauses are discussed, and their practical usage is illustrated by example.

Chapters 12 and 13: Practical Implementations

Many electoral systems impose restrictions on seat numbers. Chapter 12 shows how to handle minimum requirements as well as maximum cappings. The practical relevance of restrictions is shown by examples, such as the allocation of the seats of the European Parliament between the Member States of the Union. Chapter 13 discusses the 2013 amendment of the German Federal Election Law. The system realizes practical equality of the success values of all voters in the whole country. Mild deviations from proportionality, due to direct-seat restrictions, are incurred when assigning the seats of a party to its lists of nominees. Outline of the Book xxi

Chapters 14 and 15: Double Proportionality

Double proportionality aims at a fair representation of the geographical division of the electorate as well as of the political division of the voters. The methods achieve this two-way fairness by apportioning seats to districts proportionally to population figures and to parties proportionally to vote counts. The core is the sub-apportionment of seats by district and party in such a way that, for every district, the seats sum to the given district magnitude and, for every party, the seats sum to their overall proportionate due. To this end, two sets of electoral keys are required: district divisors and party divisors. While it is laborious to determine the electoral keys, their publication makes it rather easy to verify the double- proportional seat apportionment. Chapter 14 explains double-proportional divisor methods by example; Chap. 15 adjoins the necessary theory.

Chapter 16: Biographical Digest

Homage is paid to selected individuals who contributed to the genesis of apportion- ment methods for use in proportional representation systems. Contents

1 Exposing Methods: The 2014 European Parliament Elections ...... 1 1.1 The 28 Member States of the 2014 Union ...... 1 1.2 Austria, Belgium, Bulgaria; Electoral Keys ...... 3 1.3 Cyprus, Czech Republic, Germany; Table Design ...... 7 1.4 Denmark, Estonia, Greece; Alliances and Indeps ...... 11 1.5 Spain, Finland, France, Croatia; Vote Patterns and Vote Categories ...... 13 1.6 Hungary, Ireland, Italy; Quotas ...... 19 1.7 Lithuania, Luxembourg, Latvia; Residual Fits ...... 24 1.8 Malta, the Netherlands, Poland; Nested Stages...... 27 1.9 Portugal, Romania, Sweden; Method Overview ...... 28 1.10 Slovenia, Slovakia, United Kingdom; Local Representation ..... 33 1.11 Diversity Versus Uniformity...... 37 2 Imposing Constitutionality: The 2009 Bundestag Election ...... 41 2.1 The German Federal Election Law...... 41 2.2 Countrywide Super-Apportionment 2009 ...... 44 2.3 Per-Party Sub-Apportionments 2009 ...... 46 2.4 Negative Voting Weights ...... 48 2.5 Direct and Universal Suffrage ...... 49 2.6 Free, Equal, and Secret Ballots...... 51 2.7 Equality of the Voters’ Success Values ...... 52 2.8 Equality of Representative Weights ...... 54 2.9 Satisfaction of the Parties’ Ideal Shares of Seats ...... 55 2.10 Continuous Fits Versus Discrete Apportionments...... 56 3 From Reals to Integers: Rounding Functions and Rounding Rules ...... 59 3.1 Rounding Functions ...... 59 3.2 Floor Function ...... 60 3.3 Ties and the Need for Rules of Rounding ...... 60 3.4 Rule of Downward Rounding ...... 61

xxiii xxiv Contents

3.5 Ceiling Function and Rule of Upward Rounding ...... 62 3.6 Commercial Rounding Function ...... 63 3.7 Rule of Standard Rounding...... 63 3.8 Signpost Sequences ...... 65 3.9 Rounding Rules ...... 66 3.10 Stationary Signposts ...... 68 3.11 Power-Mean Signposts ...... 68 3.12 Simple Rounding Does Not Suffice! ...... 69 4 Divisor Methods of Apportionment: Divide and Round ...... 71 4.1 House Size, Vote Weights, and Seat Numbers...... 71 4.2 Apportionment Rules ...... 73 4.3 The Five Organizing Principles ...... 75 4.4 Apportionment Methods...... 77 4.5 Divisor Methods ...... 77 4.6 Max-Min Inequality ...... 80 4.7 Jump-and-Step Procedure and the Select Divisor ...... 82 4.8 Uniqueness, Multiplicities, and Ties ...... 83 4.9 Tie Resolution Provisions ...... 85 4.10 Primal Algorithms and Dual Algorithms ...... 86 4.11 Adjusted Initialization for Stationary Divisor Methods ...... 87 4.12 Universal Initialization ...... 88 4.13 Bad Initialization...... 89 4.14 Highest Comparative Scores ...... 91 4.15 Authorities ...... 92 5 Quota Methods of Apportionment: Divide and Rank ...... 95 5.1 Quota Methods ...... 95 5.2 Hare-Quota Method with Residual Fit by Greatest Remainders ...... 96 5.3 Greatest Remainders and the Select Split...... 97 5.4 Shift-Quota Methods...... 97 5.5 Max-Min Inequality ...... 98 5.6 Shift-Quota Methods and Stationary Divisor Methods ...... 100 5.7 Authorities ...... 100 5.8 Quota Variants ...... 101 5.9 Residual Fit Variants ...... 102 5.10 Quota Method Variants ...... 103 6 Targeting the House Size: Discrepancy Distribution ...... 107 6.1 Seat Total and Discrepancy ...... 107 6.2 Universal Divisor Initialization...... 108 6.3 Recommended Divisor Initialization...... 109 6.4 Distributional Assumptions ...... 111 6.5 Seat-Total Distributions...... 113 6.6 Hagenbach-Bischoff Initialization ...... 114 Contents xxv

6.7 Discrepancy Probabilities: Formulas...... 116 6.8 Discrepancy Probabilities: Practice ...... 118 6.9 Discrepancy Representation by Means of Rounding Residuals...... 121 6.10 Invariance Principle for Rounding Residuals ...... 122 6.11 Discrepancy Distribution ...... 124 7 FavoringSomeattheExpenseofOthers:SeatBiases...... 127 7.1 Seat Excess of a Party ...... 127 7.2 Rank-Order of Parties by Vote Shares ...... 128 7.3 Vote Share Thresholds ...... 128 7.4 Seat Bias Formula...... 129 7.5 Biasedness Versus Unbiasedness...... 130 7.6 House Size Recommendation ...... 132 7.7 Cumulative Seat Biases by Subdivisions into Districts ...... 133 7.8 Total Positive Bias: The Stronger Third, the Weaker Two-Thirds ...... 134 7.9 Alliances of Lists ...... 135 7.10 Proof of the Seat Bias Formula...... 139 7.11 Proof of the Seat Bias Formula for List Alliances...... 144 7.12 Seat Biases of Shift-Quota Methods ...... 145 8 Preferring Stronger Parties to Weaker Parties: Majorization ...... 149 8.1 Bipartitions by Vote Strengths...... 149 8.2 Majorization of Two Seat Vectors...... 150 8.3 A Sufficient Condition via Pairwise Comparisons ...... 151 8.4 Majorization of Two Apportionment Methods ...... 151 8.5 Majorization of Divisor Methods ...... 152 8.6 Majorization-Increasing Parameterizations...... 153 8.7 Majorization Paths ...... 154 8.8 Majorization of Shift-Quota Methods...... 156 9 Securing System Consistency: Coherence and Paradoxes ...... 159 9.1 The Whole and Its Parts ...... 159 9.2 The Sixth Organizing Principle: Coherence ...... 160 9.3 Coherence and Completeness of Divisor Methods ...... 161 9.4 Coherence Theorem ...... 162 9.5 House Size Monotonicity...... 163 9.6 Vote Ratio Monotonicity ...... 165 9.7 Retrieval of an Underlying Signpost Sequence...... 167 9.8 Proof of the Coherence Theorem in Sect. 9.4...... 168 9.9 Coherence and Stationary Divisor Methods ...... 172 9.10 Coherence and the Divisor Method with Standard Rounding .... 175 9.11 Violation of Coherence: New States Paradox ...... 177 9.12 Violation of House Size Monotonicity: Alabama Paradox ...... 179 xxvi Contents

9.13 Violation of Vote Ratio Monotonicity: Population Paradox ...... 181 9.14 Violation of Voter Monotonicity: No-Show Paradox...... 183 10 Appraising Electoral Equality: Goodness-of-Fit Criteria ...... 185 10.1 Optimization of Goodness-of-Fit Criteria ...... 185 10.2 Voter Orientation: DivStd ...... 186 10.3 Curtailment of Overrepresentation: DivDwn ...... 189 10.4 Alleviation of Underrepresentation: DivUpw ...... 191 10.5 Parliamentary Orientation: DivGeo ...... 193 10.6 Party Orientation: HaQgrR ...... 195 10.7 Stabilization of Disparity Functions ...... 198 10.8 Success-Value Stability: DivStd...... 199 10.9 Representative-Weight Stability: DivHar ...... 200 10.10 Unworkable Disparity Functions...... 201 10.11 Ideal-Share Stability: DivStd ...... 202 10.12 Ideal Share of Seats Versus Exact Quota of Seats ...... 203 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses...... 207 11.1 Vote Share Variation for a Given Seat Number...... 207 11.2 Vote Share Bounds: General Divisor Methods ...... 208 11.3 Vote Share Bounds: Stationary Divisor Methods...... 209 11.4 Vote Share Bounds: Modified Divisor Methods ...... 211 11.5 Vote Share Bounds: Shift-Quota Methods ...... 212 11.6 Overview of Vote Thresholds ...... 213 11.7 Preservation of a Straight Majority and Majority Clauses ...... 215 11.8 House-Size Augmentation Clause ...... 217 11.9 Majority-Minority Partition Clause ...... 217 11.10 The 2002 German Conference Committee Dilemma ...... 219 11.11 Residual-Seat Redirection Clause ...... 220 11.12 Divisor Methods and Ideal Regions of Seats ...... 221 12 Truncating Seat Ranges: Minimum-Maximum Restrictions ...... 225 12.1 Minimum Representation for Electoral Districts ...... 225 12.2 Quota Method Ambiguities ...... 226 12.3 Minimum-Maximum Restricted Variants of Divisor Methods ... 227 12.4 Direct-Seat Restricted Variant of DivDwn...... 228 12.5 Proportionality Loss ...... 232 12.6 Direct-Seat Restricted Variant of DivStd ...... 232 12.7 Composition of the EP: Legal Requirements ...... 234 12.8 Cambridge Compromise...... 236 12.9 Power Compromise ...... 238 12.10 Jagiellonian Compromise...... 242 13 Proportionality and Personalization: BWG 2013 ...... 247 13.1 The 2013 Amendment of the Federal Election Law ...... 247 13.2 Apportionment of Seats Among Parties ...... 248 13.3 Assignment of Candidates to Seats ...... 249 Contents xxvii

13.4 Initial Adjustment of the Bundestag Size ...... 251 13.5 Alternative House Size Adjustment Strategies ...... 252 14 Representing Districts and Parties: Double Proportionality ...... 259 14.1 Double Proportionality: Practice ...... 259 14.2 The 2016 Parliament Election in the Canton of Schaffhausen.... 260 14.3 Discordant Seat Assignments ...... 264 14.4 Winner-Take-One Modification ...... 265 14.5 Double Proportionality in Swiss Cantons: The Reality ...... 265 14.6 Double Proportionality in the European Union: A Vision ...... 266 14.7 Degressive Compositions and Separate District Evaluations ..... 267 14.8 Compositional Proportionality for Unionwide Lists ...... 268 15 Double-Proportional Divisor Methods: Technicalities ...... 275 15.1 Row and Column Marginals, Weight and Seat Matrices ...... 275 15.2 Double-Proportional Divisor Methods ...... 276 15.3 Cycles of Seat Transfers ...... 278 15.4 Uniqueness of Double-Proportional Seat Matrices...... 279 15.5 Characterization of Double-Proportional Seat Matrices ...... 281 15.6 Existence of Double-Proportional Seat Matrices ...... 285 15.7 A Dual View ...... 287 15.8 Alternating Scaling Algorithm ...... 288 15.9 Tie-and-Transfer Algorithm ...... 292 16 Biographical Digest...... 297 16.1 Thomas Jefferson 1743–1826 ...... 297 16.2 Alexander Hamilton 1755–1804 ...... 298 16.3 Daniel Webster 1782–1852 ...... 299 16.4 Thomas Hare 1806–1891 ...... 300 16.5 Henry Richmond Droop 1832–1884 ...... 301 16.6 Eduard Hagenbach-Bischoff 1833–1910 ...... 302 16.7 Victor D’Hondt 1841–1901 ...... 305 16.8 Joseph Adna Hill 1860–1938...... 306 16.9 Walter Francis Willcox 1861–1964 ...... 308 16.10 Ladislaus von Bortkiewicz 1868–1931 ...... 309 16.11 Edward Vermilye Huntington 1874–1952 ...... 310 16.12 Siegfried Geyerhahn 1879–1960 ...... 312 16.13 Jean-André Sainte-Laguë 1882–1950...... 313 16.14 George Pólya 1887–1985 ...... 315 16.15 Horst Friedrich Niemeyer 1931–2007 ...... 317

Notes and Comments ...... 319

References...... 331

Index ...... 339 Chapter 1 Exposing Methods: The 2014 European Parliament Elections

Abstract The multitude of apportionment methods that is available for the transla- tion of vote counts into seat numbers is exemplified. The examples are taken from the 2014 European Parliament elections. For each of the 28 Member States the computational steps to convert the parties’ vote counts into the parties’ seat numbers are explained. The vote patterns on the ballot sheets are described, and the list positions of the successful candidates among the parties’ nominees are identified. The exposition is interspersed with conceptual remarks concerning proportional representation systems at large.

1.1 The 28 Member States of the 2014 Union

The 2014–2019 European Parliament (EP) was elected in 2014. In this chapter we review the apportionment methods that are applied by the 28 Member States of the European Union. Their 28 electoral laws are so different that we refer to the event in the plural: “elections”. The differences between Member States illustrate the many apportionment methods that are being used to translate vote counts into seat numbers. We choose the present tense for our review, since the properties of the apportionment methods do not depend on the particular instance of their application. The Union’s Interinstitutional Style Guide identifies the Member States in three ways, by a two-letter code, by a short name, and by an official name. Table 1.1 lists the Member States in the sequence of the two-letter codes; they are language independent, while short names and official names are not. Table 1.1 also includes the seat allocations assigned to the Member States for the 2014 elections. Sections 1.2–1.10 discuss the Member States in batches of three or four. Our presentation follows the sequence of the two-letter codes in Table 1.1,yetwe prefer to quote short names in the section headings. Every section is supplemented with conceptual remarks. The remarks pertain to the Member States at large. They comment on the procedures used in the Union, the presentation chosen in this book, and the problems addressed in the sequel. At the 2014 EP elections political parties campaigned under their domestic names. The party tabs shown in the tables are taken from the web page www.europarl.europa.eu/elections2014-results/en/seats-member-state-absolut.html where all tabs are expanded to the parties’ full names in their Member States.

© Springer International Publishing AG 2017 1 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_1 2 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.1 The 2014 European Union Two-letter code Short name Official name 2014 seats AT Austria Republic of Austria 18 BE Belgium Kingdom of Belgium 21 BG Bulgaria Republic of Bulgaria 17 CY Cyprus Republic of Cyprus 6 CZ Czech Republic Czech Republic 21 DE Germany Federal Republic of Germany 96 DK Denmark Kingdom of Denmark 13 EE Estonia Republic of Estonia 6 EL Greece Hellenic Republic 21 ES Spain Kingdom of Spain 54 FI Finland Republic of Finland 13 FR France French Republic 74 HR Croatia Republic of Croatia 11 HU Hungary Hungary 21 IE Ireland Ireland 11 IT Italy Italian Republic 73 LT Lithuania Republic of Lithuania 11 LU Luxembourg Grand Duchy of Luxembourg 6 LV Latvia Republic of Latvia 8 MT Malta Republic of Malta 6 NL The Netherlands Kingdom of the Netherlands 26 PL Poland Republic of Poland 51 PT Portugal Portuguese Republic 21 RO Romania Romania 32 SE Sweden Kingdom of Sweden 20 SI Slovenia Republic of Slovenia 8 SK Slovakia Slovak Republic 13 UK United Kingdom United Kingdom of Great Britain and 73 Northern Ireland Sum 751 The 28 Member States are listed alphabetically by their two-letter codes. The seat allocations are those pertaining to the 2014 elections

In order to integrate domestic parties from a European viewpoint we subsume them under the Political Groups they joined immediately after the election. This is illegal, strictly speaking. Political Groups implement the parliamentary business, and are barred from campaigning in elections. Although invisible at the 2014 EP elections, European Parties do exist. They were invented for funneling money from the Union’s budget into political channels. This book is about apportionment methodology, however, not about money. Neglecting mundane subtleties, we auda- ciously replace the invisible European Parties by the visible Political Groups. 1.2 Austria, Belgium, Bulgaria; Electoral Keys 3

Table 1.2 Political groups in the 2014 EP Acronym Political group in the 2014 EP EPP Group of the European People’s Party (Christian Democrats) S&D Group of the Progressive Alliance of Socialists and Democrats in the EP ECR European Conservatives and Reformists ALDE Alliance of Liberals and Democrats for Europa GUE/NGL Gauche unitaire européenne/Nordic Green Left GREENS/EFA The Greens / European Free Alliance EFDD Europe of Freedom and Direct Democracy Group NI Members of the EP not belonging to any political group In the 2014 EP seven Political Groups were formed. We treat the non-attached Members of the EP as an eighth group, NI

The seven Political Groups that formed immediately after the 2014 EP elections are listed in Table 1.2, together with their acronyms. The eighth and last line “NI” refers to those Members of the EP not joining a group. The tag “NI” originates from the French term “Non-inscrits”. The previous Parliament used the label “NA”, short for the English term “Non-attached”. The tag “NA” nicely conforms to standard statistical jargon for items that are “Not Available”, “Not Applicable”, “Not Active” or, for the present purpose, “Not Attached” to a Political Group.

1.2 Austria, Belgium, Bulgaria; Electoral Keys

Austria has an allocation of 18 seats to fill, of the overall 751 EP seats. The Austrian voters cast 2,823,561 valid votes. However, not all of these become effective to participate in the apportionment calculations. The Austrian law sets an electoral threshold of 4% of the valid votes. This means that a valid vote becomes effective to enter into the apportionment calculations only when cast for a party drawing at least 4% of all valid votes. Four percent of the Austrian valid votes is 112,942.4 vote shares (that is, a fractional vote index rounded to one decimal place). Hence a valid vote becomes effective provided it is cast for a party that draws at least 112,943 votes. Of the nine parties campaigning, four fail the threshold, and the 184,780 votes for these parties turn ineffective. The apportionment of the 18 seats among the remaining five parties is shown in Table 1.3. The penultimate column in the table contains the final seat numbers and could be labeled “Seats”. Rather, we label the column with the acronym of the apportionment method used, since our focus is on the procedure to calculate the seat numbers. Austria employs the “divisor method with downward rounding” whence the column is labeled “DivDwn”. The apportionment methods form the main theme of this book and are treated in greater detail later. The present chapter provides just introductory descriptions, to set the scene. 4 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.3 EP election 2014, Austria EP2014AT-LV1 Political group Vot es Quotient DivDwn MEPs’ list positions ÖVP EPP 761;896 5:9 5 1, 2, 3, 4, 5 SPÖ S&D 680;180 5:2 5 1, 2, 3, 4, 5 FPÖ NI 556;835 4:3 4 1, 2, 3, 4 GRÜNE GREENS/EFA 410;089 3:2 3 1, 2, 3 NEOS ALDE 229;781 1:8 1 1 Sum (Divisor) 2;638;781 (130,000) 18 Five parties with 2;638;781 votes participate in the apportionment calculations. With the divisor method with downward rounding, DivDwn, every 130,000 votes justify roughly one seat. As for downward rounding, “roughly” means that interim quotients are rounded downwards. Preference votes, though allowed, do not affect the assignment of seats

Belgium subdivides the country into three electoral areas which we call districts. Of its allocation of 21 seats, the Belgian electoral provisions assign 12 seats to District 1: Nederlands kiescollege, eight to District 2: Collège électoral français, and one to District 3: Deutschsprachiges Wahlkollegium. Since Belgium does not use an electoral threshold, all valid votes become effective. Seats are apportioned using the divisor method with downward rounding except that now the method is applied three times, separately in each of the three districts. The apportionment is exhibited in Table 1.4. The establishment of separate districts caters to the three distinct language groups in Belgium, and secures minority representation. Note that the three divisors vary significantly. To justify roughly one Belgian seat, 280,500 votes are needed in District 1 and 200,000 in District 2, while 10,000 suffice in District 3. Bulgaria tells a story of its own. The threshold demands that a party secures at least as many votes as are stipulated by the quotient of valid votes per seats. With 2,239,430 valid votes and an allocation of 17 seats, the quotient is found to be 131,731.2 vote shares. Twenty-five parties miss the threshold, and their 364,669 votes are discarded as ineffective. The threshold of 131,732 votes amounts to 5.8% of votes cast. This percentage level violates the norm set by the European Union. The norm stipulates that the threshold must not exceed 5% of votes cast. Relative to the 2,361,966 votes cast in Bulgaria, 5% are reached with 118,099 votes or more. Yet the Bulgarian provisions demand a higher threshold, 131,732 votes. Remarkably, nobody complained. No complaint, no redress. The apportionment method used is the “Hare-quota method with residual fit by greatest remainders”, abbreviated as “HaQgrR”. The method relies on a quantity which for historical reasons is called “Hare-quota”. It simply is the votes-to-seats ratio, or more precisely, the ratio of effective votes and seat total. In the present instance it turns out to be 1;874;761=17 D 110;280:1. Division of the Hare-quota into the parties’ vote counts produces the interim quotients shown in the “Quotient” column of Table 1.5. 1.2 Austria, Belgium, Bulgaria; Electoral Keys 5

Table 1.4 EP election 2014, Belgium EP2014BE-LVm Political group Vot es Quotient DivDwn MEPs’ list positions District 1: Dutch electoral college N-VA ECR 1;123;355 4:005 4 1*, 2*, 3*, 4 OPEN VLD ALDE 859;099 3:1 3 1*, 2, 12 CD&V EPP 840;783 2:997 2 1*, 2* SP.A S&D 555;348 1:98 1 1* GROEN GREENS/EFA 447;391 1:6 1 1* VLAAMS BELANG NI 284;856 1:02 1 1* 1Other – 101;237 – 0 Sum (Divisor) 4;212;069 (280,500) 12 District 2: French electoral college PS S&D 714;645 3:6 3 1*, 2*, 3 MR ALDE 661;332 3:3 3 1*, 2*, 3* ECOLO GREENS/EFA 285;196 1:4 1 1* CDH EPP 277;246 1:4 1 1* 8Others – 501;627 – 0 Sum (Divisor) 2;440;046 (200,000) 8 District 3: German language community CSP EPP 11;710 1:2 1 1* ECOLO GREENS/EFA 6429 0:6 0 4Other – 20;457 – 0 Sum (Divisor) 38;596 (10,000) 1 Belgium establishes three electoral districts, each evaluated by the divisor method with downward rounding. No threshold is imposed; all valid votes become effective. Parties with too few votes fail to obtain representation (labeled “Others”). Seat assignments that are solely determined by preference votes (see Sect. 1.5 for details) are marked with an asterisk (*)

Table 1.5 EP election 2014, Bulgaria EP2014BG-LV1 Political group Vot es Quotient HaQgrR MEPs’ list positions GERB EEP 680;838 6:174 6 1*, 2, 3, 4, 5, 6 BSP S&D 424;037 3:845 4 15*, 1*, 2, 3 DPS ALDE 386;725 3:507 4 1*, 3, 4, 5 Coal. BWC et al. ECR 238;629 2:164 2 1*, 2* RB EEP 144;532 1:311 1 2* Sum (Split) 1;874;761 (0.5) 17 With the Hare-quota method with residual fit by greatest remainders, HaQgrR, every 1;874;761=17 D 110;280:1 votes justify one seat, thus allotting 15 seats. The two residual seats are given to the parties whose quotients have the greatest remainders, that is, with remainders above the split point 0:5. Seats based on preference votes are marked with an asterisk (*)

Each quotient is decomposed into its integral part and its fractional part. The integral parts are instrumental for the first stage of the apportionment, called “main apportionment”. Every full satisfaction of the quota goes along with the 6 1 Exposing Methods: The 2014 European Parliament Elections apportionment of a seat. The strongest party is apportioned 6 seats, while the weaker parties gain 3, 3, 2 and 1 seats. Thus the main apportionment allocates 15 seats and accounts for 15 110;280:1 D 1;654;200:9 vote shares. The second stage of the apportionment, the “residual apportionment”, takes care of the 1;874;761  1;654;200:9 D 220;560:1 remaining vote shares and of the two remaining seats. For the strongest party there remain 680;838  6 110;280:1 D 19;157:6 vote shares, for the Others, 93,196.8, 55,884.8, 18,068.9, and 34,251.9. In terms of the Hare-quota the remaining vote shares correspond to the interim quotients’ fractional parts 0.174, 0.845, 0.507, 0.164, and 0.311. The two greatest claims are 93,196.8 and 55,884.8 vote shares, or equivalently in terms of fractional parts, 0.845 and 0.507 quota fractions. These two parties are awarded one residual seat each. Parties with remainders below the split of 0.5 gain no residual seat. Electoral Keys: The Select Divisor An electoral key is a number enabling an easy check of a published apportionment. A quick confirmation of a seat apportionment enhances its acceptance by the electorate. Our tables always quote an electoral key. The electoral key for a divisor method is a “divisor”. Austria provides an instructive example. A divisor of 130,000 means that every 130,000 votes justify roughly one seat. The qualification “roughly” is needed because, literally, the measure of 130,000 votes would justify 761;896=130;000 D 5:9 seat shares for the strongest party, 5.2 seat shares for the second-strongest party, and so on. These seat shares are listed in the “Quotient” column in Table 1.3. They are interim quantities deserving only passing attention. Members of Parliament are human beings, whence interim quotients must be rounded to whole numbers. Different rounding procedures induce different divisor methods. The method used in Austria is the divisor method with downward rounding, DivDwn. Downward rounding means that interim quotients are rounded downwards to the integer below. A practitioner might simply say that fractional parts are neglected. The “Quotient” column in Table 1.3 shows the interim quantities which, when rounded downwards, produce the seat numbers that are displayed in the “DivDwn” column. There is some leeway on which divisor to select. Sometimes this is expressed by speaking of flexible divisors, or sliding divisors. For the Austrian example, Table 1.3 quotes the select divisor 130,000. However, any value between 126,982.7 and 136,036 would do as well. If the value is less than 761;896=6 D 126;982:7, the strongest party’s quotient rises above six, such as 761;896=126;982 D 6:00003. The party would be awarded a sixth seat whence the seat total would exceed the allocation of 17 seats. If the value is greater than 680;180=5 D 136;036, the second-strongest party loses a seat. The seat total would stay below 17 seats. The allocation of exactly 17 seats is ensured if and only if the divisor belongs to the “divisor interval” Œ126;982:7I 136;036. We exploit the flexible nature of divisors by selecting from the divisor interval a value that eases communication. The “select divisor” is determined from the divisor interval by calculating its midpoint and reducing the midpoint to as few digits as permitted by the divisor interval. In this way the select divisor 1.3 Cyprus, Czech Republic, Germany; Table Design 7 exhibits a maximum number of trailing zeros. For example, in Austria the divisor interval Œ126;983I 136;036 has midpoint 131,509.5 leading to the select divisor 130,000. Electoral Keys: The Select Split The electoral key for a quota method with residual fit by greatest remainders is called a “split”. The split is a remainder value separating the parties that are awarded one of the residual seats from those that are not. The term “quota” is indicative of a fixed divisor. Hare-quota methods employ the Hare-quota, the votes-to-seats ratio. We use the term “divisor” when the divisor is flexible, as opposed to “quota” when it is fixed. The distinction is technical jargon, but useful. Divisor methods employ a flexible divisor and a fixed rounding rule. Quota methods combine a fixed divisor with a flexible rounding rule. The prime example of how quota methods may round interim quotients to whole numbers is the residual fit by greatest remainders. The residual fit step means that the rounding rule varies from case to case. For instance, in Bulgaria the second- and third-greatest remainders of the quotients in Table 1.5 are 0:507 and 0:311. Hence every splitting point—“split”, for short—in the “split interval” Œ0:311I 0:507 separate the parties that receive one of the residual seats from those that do not. Publication of a split facilitates the check on whether a particular quotient is rounded downwards or upwards. A party receives one of the residual seats if and only if its remainder exceeds the split. Quoting a split disposes of the labor of again ranking all parties by the remainders of their interim quotients when verifying the seat numbers. Our strategy on which “select split” to quote is normally the same as with divisors. In the Bulgarian example the midpoint of the split interval, 0:409,is reduced to 0.4, in order to minimize the number of its digits. Normally the select split is obtained in this way. The only exceptions are instances when the split interval includes the value one half. In these instances we quote the select split 0.5. The exception applies to Bulgaria.

1.3 Cyprus, Czech Republic, Germany; Table Design

Cyprus stipulates an electoral threshold at the peculiar level of 1.8% of valid votes. Thus the 258,914 valid votes entail a threshold of 4661 votes. Eleven parties fail the threshold, and 8979 votes become ineffective. The 249,935 effective votes are evaluated by means of the “Hare-quota variant-3 method with residual fit by greatest remainders”, abbreviated as “HQ3grR”. The variant uses a quota based on valid votes rather than effective votes. The ratio of valid votes and seats is rounded downwards to obtain the quota. We refer to this quota as the Hare-quota variant- 3, HQ3. In Cyprus it amounts to b258;914=6cDb43;152:3cD43;152.See Table 1.6. 8 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.6 EP election 2014, Cyprus EP2014CY-2CV Political group Vot es Quotient HQ3grR MEPs’ list positions DISY EPP 97;732 2:265 2 2, 3 AKEL GUE/NGL 69;852 1:619 2 6, 5 DIKO S&D 28;044 0:650 1 4 KS EDEK S&D 19;894 0:461 1 2 1Other – 17;549 0:407 0 2Others – 16;864 – 0 Ineffective votes 8979 Sum (Split) 258;914 (0.43) 6 The Hare-quota variant-3 method with residual fit by greatest remainders, HQ3grR, relies on variant-3 of the Hare-quota, HQ3 = b258;914=6cDb43;152:3cD43;152.Themain apportionment allocates three seats. The three residual seats go to the parties whose quotients have a remainder greater than the split quoted, 0.43

Table 1.7 EP election 2014, Czech Republic EP2014CZ-LV2 Political group Vot es Quotient DivDwn MEPs’ list positions ANO 2011 ALDE 244;501 4:9 4 1*, 2, 3, 4 Coal. TOP 09+STAN EPP 241;747 4:8 4 2*, 1*, 5*, 3 CSSDˇ S&D 214;800 4:3 4 1*, 2*, 3, 4 KSCMˇ GUE/NGL 166;478 3:3 3 1*, 4*, 2* KDU-CSLˇ EPP 150;792 3:02 3 2*, 1*, 3 ODS ECR 116;389 2:3 2 1*, 2* SVOBODNI EFDD 79;540 1:6 1 1* Sum (Divisor) 1;214;247 (50,000) 21 The electoral threshold of 5% of the valid votes amounts to 75,775 votes. Thirty-one parties fail the threshold, and their 301,245 votes turn ineffective. The remaining four parties share the 21 Czech seats using the divisor method with downward rounding. Seats based on preference votes are marked with an asterisk (*)

The Czech Republic has an electoral threshold of 5% of valid votes. With 1,515,492 valid votes the threshold amounts to 75,775 votes. Thirty-one parties miss it, and the 301,245 votes cast for them turn ineffective. The 21 Czech seats are apportioned according to the remaining 1,214,247 effective votes. The apportionment is performed using the divisor method with downward rounding. See Table 1.7. Germany has no electoral threshold. In previous EP elections there used to be a 5% threshold. Remarkably, somebody complained. In November 2011 the German Federal Constitutional Court ruled that the 5% threshold is unconstitutional. Of course, the Court does not bind Other constitutional courts in the Union nor the Court of Justice of the European Union. The ruling does not mean that the Union’s permission for domestic provisions to include a 5% threshold violates the Union’s 1.4 Denmark, Estonia, Greece; Alliances and Indeps 9 primary law. It only says that it is unconstitutional to apply the Union’s permission at the German EP election. Germany has 96 seats to apportion. Parties present their candidates on a federal list, except for CDU. CDU submits fifteen state-lists, one for each German state where the party campaigns. As a consequence the seat apportionment proceeds in two stages. The first stage, the “super-apportionment”, evaluates the valid votes in the whole country. The second stage is the CDU “sub-apportionment”. It apportions the 29 CDU seats to the 15 CDU state-lists. Both stages use the “divisor method with standard rounding”, abbreviated as “DivStd”. Interim quotients are rounded upwards when their fractional parts are greater than one half, and downwards when less. If fractional parts are exactly equal to one half, lots must be drawn. See Table 1.8. Table Design In designing the tables we follow some general rules. For divisor methods, the “Votes” column contains counts of effective votes. For quota methods, the electoral key requires the total of valid votes whence a line with the ineffective votes is added. The tables rank parties in the order of decreasing vote counts. Electoral bureaus sometimes rely on the party ranking that pertained to the previous legislative period. Formerly this helped to prepare the record sheets for the upcoming election. Nowadays computers make it easy to sort parties by their current vote counts. All tables conclude with a bottom line showing column sums. A column sum provides a primitive check on whether column entries got corrupted by copying, pasting, or Other editorial operations. The bottom line also records in parentheses the electoral key used, either the select divisor or the select split. In the “Quotient” column, the number of decimal digits shown depends on the apportionment method used. For divisor methods, a single digit mostly suffices to decide whether a quotient is rounded upwards or downwards. Otherwise we exhibit as many digits as are needed for a clear decision. For quota methods with a residual fit by greatest remainders, the task is to rank the fractional parts of the quotients. In practice three digits suffice. Multiple subtables in a table correspond to multiple calculations. For example, the two subtables in Table 1.8 show the super-apportionment in the whole of Germany, and the districtwise sub-apportionment within CDU. The three subta- bles in Table 1.4 exhibit the apportionments in the three Belgian districts. The nature of a subtable is context-dependent. In the German example, the super- apportionment needs to be completed before its results can be handed down to the sub-apportionment. In the Belgian example, the calculations in the three districts are independent of one another. 10 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.8 EP election 2014, Germany EP2014DE-LV0 Political group Vot es Quotient DivStd CDU EPP 8;812;653 29:48 29 SPD S&D 8;003;628 26:8 27 GRÜNE GREENS/EFA 3;139;274 10:503 11 DIE LINKE GUE/NGL 2;168;455 7:3 7 AFD ECR 2;070;014 6:9 7 CSU EPP 1;567;448 5:2 5 FDP ALDE 986;841 3:3 3 FREIE WÄHLER ALDE 428;800 1:4 1 PIRATEN GREENS/EFA 425;044 1:4 1 TIERSCHUTZPARTEI GUE/NGL 366;598 1:2 1 NPD NI 301;139 1:0 1 FAMILIE ECR 202;803 0:7 1 ÖDP GREENS/EFA 185;244 0:6 1 DIE PARTEI NI 184;709 0:6 1 11 Others – 512;442 – 0 Sum (Divisor) 29;355;092 (298,900) 96 Sub-apportionment within CDU Schleswig-Holstein 334;121 1:1 1 Mecklenburg-Vorpommern 210;268 0:7 1 Hamburg 135;780 0:45 0 Lower Saxony 1;174;739 3:9 4 Bremen 43;353 0:1 0 Brandenburg 233;468 0:8 1 Saxony-Anhalt 245;010 0:8 1 Berlin 232;274 0:8 1 North Rhine-Westphalia 2;439;979 8:1 8 Saxony 559;899 1:9 2 Hesse 564;294 1:9 2 Thuringia 290;703 1:0 1 Rhineland-Palatinate 661;339 2:2 2 Baden-Württemberg 1;542;244 5:1 5 Saarland 145;182 0:48 0 Sum (Divisor) 8;812;653 (300,000) 29 The divisor method with standard rounding is used, DivStd. In the super-apportionment, every 298,000 votes justify roughly one seat. As for standard rounding, “roughly” means that interim quotients are rounded downwards or upwards according as their fractional part is less than one half or greater. The CDU with its fifteen district lists calls for a sub-apportionment, with divisor 300,000 1.4 Denmark, Estonia, Greece; Alliances and Indeps 11

1.4 Denmark, Estonia, Greece; Alliances and Indeps

Denmark introduces a new twist into the exposition: “alliances”. Alliances are also referred to as list apparentements, party coalitions, or electoral cartels. They are treated as entities that participate in the seat apportionment with the aggregate vote counts of their partners. This initial stage is called super-apportionment. Thereafter every alliance requires an additional step to apportion its seats among its partners. These sub-apportionments, one for each alliance, constitute the final stage. In the Danish election two sub-apportionments are called for, one for Alliance I with three partners, and one for Alliance II with two partners. See Table 1.9. Estonia apportions its six seats on the basis of 328,493 valid votes. There is no electoral threshold. The apportionment uses the divisor method with downward rounding. It turns out that 38,000 votes justify roughly one seat. The Estonian election features an “indep”, an independent candidate who stands at the election with no party affiliation. See Table 1.10. Greece employs a unique quota method, the “Hare-quota variant-3 method with Greek residual fit”, abbreviated as “HQ3-EL”. An electoral threshold of 3% of valid votes applies. With 5,715,985 valid votes the threshold amounts to 171,480 votes. Thirty-six parties fail the threshold, and their 978,433 votes are discarded as

Table 1.9 EP election 2014, Denmark EP2014DK-1CV Political group Vot es Quotient DivDwn MEPs’ list positions Alliance I 833;499 5:6 5 O.(DF) ECR 605;889 4:04 4 1, 3, 2, 6 Alliance II 588;102 3:9 3 N. GUE/NGL 183;724 1:2 1 1 1Other – 65;480 – 0 Sum (Divisor) 2;276;694 (150,000) 13 Sub-apportionment within Alliance I = A.(S)+F.(SF)+B.(RV) A.(S) S&D 435;245 3:3 3 1, 3, 2 F.(SF) GREENS/EFA 249;305 1:9 1 1 B.(RV) ALDE 148;949 1:1 1 1 Sum (Divisor) 833;499 (130,000) 5 Sub-apportionment within Alliance II = V.(V)+C.(KF) V. (V) ALDE 379;840 2:4 2 1, 13 C.(KF) EPP 208;262 1:3 1 1 Sum (Divisor) 588;102 (160,000) 3 Two alliances are registered. The statewide super-apportionment treats each of them as a transient entity. Thereafter each alliance requires a sub-apportionment in order to allot the seats of the alliance among its partners. All three apportionment calculations use the divisor method with downward rounding 12 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.10 EP election 2014, Estonia EP2014EE-1CV Political group Vot es Quotient DivDwn MEPs’ list positions ER ALDE 79;849 2:1 2 1, 2 KE ALDE 73;419 1:9 1 12 IRL EPP 45;765 1:2 1 1 SDE S&D 44;550 1:2 1 1 Indep I. Tarand GREENS/EFA 43;369 1:1 1 19 Others – 41;541 – 0 Sum (Divisor) 328;493 (38,000) 6 There is no electoral threshold. An independent candidate (flagged “Indep” in our tables) is restricted to win one seat at the most. Except for this restriction an indep enters the seat apportionment calculations as do political parties. The Estonian indep Indrek Tarand, with quotient 1.1, is apportioned a seat ineffective. Seven parties pass the threshold and take part in the allocation of 21 seats. The apportionment calculations consist of three stages: a phase-1 apportion- ment, a phase-2 apportionment, and a residual fit by greatest phase-2 remain- ders. The phase-1 apportionment is based on the Hare-quota variant-3, HQ3 = b5;715;985=21cD272;189. It assigns 13 seats. The remaining eight seats are handled by the phase-2 apportionment. Phase-2 introduces the “unused voting power”, UVP, as follows. In phase-1 the strongest party is allocated five seats and uses up 5 272;189 D 1;360;945 votes of its total of 1,518,376 votes, leaving an UVP of 1;518;376  1;360;945 D 157;431 votes. The unused voting powers of the Other parties are found similarly. Now another quota is introduced with reference to the aggregate UVP. It is DQ5 Db1;199;095=.8 C 1/cD133;232,whichis categorized in Sect. 1.6 as variant-5 of the Droop-quota. The resulting quotients of UVP per DQ5 are shown in the penultimate column of Table 1.11. They entail the apportionment of another five seats. The last three seats are allotted by greatest phase-2 remainders among the parties without a phase-2 seat (that is, TO POTAMI and KKE), and finally by greatest remainders among the parties that did gain a phase-2 seat (X.A.). See Table 1.11. Alliances and Indeps Alliances are formations responding to a notorious defect of the divisor method with downward rounding. The method is well-known for awarding stronger parties an overproportional share of seats, at the expense of weaker parties who must tolerate an underproportional share of seats. Alliances were introduced with the intention to allow weaker parties to become stronger by joining together. However, once a legal provision allows formation of alliances, stronger parties may also join alliances. Thereby they grow yet stronger. In the light of empirical data it becomes evident that alliances fail to serve their purpose of neutralizing the notorious seat biases of the divisor method with downward rounding. 1.5 Spain, Finland, France, Croatia; Vote Patterns and Vote Categories 13

Table 1.11 EP election 2014, Greece EP2014EL- Political 4CV group Vot es Quotient-1 Phase-1 UVP Quotient-2 HQ3-EL SY.RI.ZA. GUE/NGL 1;518;376 5:578 5 157;431 1:182 6 N.D. EPP 1;298;948 4:772 4 210;192 1:578 5 X.A. NI 536;913 1:973 1 264;724 1:987C1 3 ELIA DA S&D 458;514 1:685 1 186;325 1:399 2 TO S&D 377;622 1:387 1 105;433 0:791C1 2 POTAMI KKE NI 349;342 1:283 1 77;153 0:579C1 2 ANEL ECR 197;837 0:727 0 197;837 1:485 1 Ineffective 978;433 votes Sum (quotas: HQ3, DQ5) 5;715;985 (272,189) 13 1;199;095 (133,232) 21 The Greek method HQ3-EL is inimitable. Based on the Hare-quota variant-3, HQ3 = b5;715;985=21cD272;189, phase-1 hands out 13 seats. Phase-2 evaluates the unused voting power “UVP”. Using the Droop-quota variant-5, DQ5 = b1;199;095=.8C1/cD133;232, phase-2 adjoins five seats. The last three seats are allotted by greatest phase-2 remainders among the parties without a phase-2 seat, and finally by greatest remainders among the Other parties

The term “indep” designates an independent candidate who stands at the election without being affiliated with a political party. It is the sole neologism in this book. An indep is an antipode to an alliance. An alliance gathers a large ensemble of nominees, an indep boasts a maximum of individuality. Since an individual cannot fill more than one seat, the calculations carry along a maximum restriction of one seat per indep. In case the restriction is active we earmark the relegated interim quotient by a dot (). The Romanian indep Mircea Diaconu in Table 1.26 provides an example.

1.5 Spain, Finland, France, Croatia; Vote Patterns and Vote Categories

Spain has no electoral threshold. Every voter casts a vote for a party list of candidates (list vote, LV). The 54 Spanish seats are apportioned among parties using the divisor method with downward rounding. Within a party its seats are assigned to the top nominees on the party’s candidate list. Note that the delegates of two parties (IP, CEU) join distinct Political Groups. See Table 1.12. Finland has no electoral threshold either. Every voter casts a vote for the candidate of his or her choice (candidate vote, CV). These votes are evaluated twice, for the candidate’s party and for the candidate proper. The 13 Finish seats are apportioned among the parties based on the parties’ vote counts. The divisor method with downward rounding is used. Within a party its seats are assigned to the candidates with the most candidate votes. See Table 1.13. 14 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.12 EP election 2014, Spain EP2014ES-LV0 Political group Vot es Quotient DivDwn PP EPP 4;098;339 16:5 16 PSOE/PSC S&D 3;614;232 14:6 14 IP GUE/NGL:5, GREENS/EFA:1 1;575;308 6:4 6 PODEMOS GUE/NGL 1;253;837 5:1 5 UPYD ALDE 1;022;232 4:1 4 CEU ALDE:2, EPP:1 851;971 3:4 3 EPDD GREENS/EFA 630;072 2:5 2 C’S ALDE 497;146 2:005 2 LPD GUE/NGL 326;464 1:3 1 PRIMAVERA EUROPEA GREENS/EFA 302;266 1:2 1 29 Others – 1;176;782 – 0 Sum (Divisor) 15;348;649 (248,000) 54 The last three characters “LV0” of the table’s label indicate the vote pattern. Each voter casts a vote for a party’s list of candidates (list vote, LV), plus zero (0) preference votes. The seats of a party are assigned to the top nominees in the party’s candidate list. Note that the representatives of IP and CEU join distinct Political Groups

Table 1.13 EP election 2014, Finland EP2014FI-1CV Political group Vot es Quotient DivDwn MEPs’ list positions KOK EPP 390;376 3:9 3 17, 9, 19 KESK ALDE 339;895 3:4 3 16, 19, 4 PS ECR 222;457 2:2 2 3, 17 SDP S&D 212;781 2:1 2 2, 8 VIHR GREENS/EFA 161;263 1:6 1 8 VA S GUE/NGL 161;074 1:6 1 6 SFP (RKP) ALDE 116;747 1:2 1 19 8Others – 123;701 – 0 Sum (Divisor) 1;728;294 (100,000) 13 The vote pattern code “1CV” means that each voter casts one vote for a candidate (candidate vote, CV). The vote is evaluated twice, once for the candidate’s party and once for the candidate proper. The seats of a party are assigned to its candidates with the most votes. In Finland candidates are listed alphabetically whence list positions are uninformative

France subdivides the country into eight districts. The 74 French seats are allocated to the districts well ahead of the election, with district magnitudes 10, 9, 9, 10, 13, 5, 15, and 3. The electoral threshold is set at 5% of valid votes, voix exprimées, and pertains to each district separately. In all districts the seat apportionment is carried out using the divisor method with downward rounding. The results are displayed in Table 1.14. In District 8: Outre-Mer the apportionment of its three seats follows particular rules to secure a fair geographical representation. The district is further subdivided into three sections: Atlantique, Océan Indien,andPacifique. Every section is 1.5 Spain, Finland, France, Croatia; Vote Patterns and Vote Categories 15

Table 1.14 EP election 2014, France EP2014FR-LV0 Political group Vot es Quotient DivDwn District 1: Nord-Ouest FN NI 914;222 5.1 5 UMP EPP 509;939 2.8 2 PS+PRG S&D 320;250 1.8 1 ALTERNATIVE ALDE 255;108 1.4 1 EUROPE ÉCOLOGIE GREENS/EFA 194;595 1.1 1 FG GUE/NGL 173;531 0.96 0 Sum (Divisor) 2;367;645 (180,000) 10 District 2: Ouest UMP EPP 535;059 3.02 3 FN NI:1, EFDD:1 526;019 2.97 2 PS+PRG S&D 425;722 2.4 2 ALTERNATIVE ALDE 334;963 1.9 1 EUROPE ÉCOLOGIE GREENS/EFA 282;167 1.6 1 FG GUE/NGL 141;341 0.8 0 Sum (Divisor) 2;245;271 (177,000) 9 District 3: Est FN NI 703;809 4.1 4 UMP EPP 551;809 3.2 3 PS+PRG S&D 321;563 1.9 1 ALTERNATIVE ALDE 223;280 1.3 1 EUROPE ÉCOLOGIE GREENS/EFA 155;694 0.9 0 FG GUE/NGL 127;269 0.7 0 Sum (Divisor) 2;083;424 (170,000) 9 District 4: Sud-Ouest FN NI 726;797 3.6 3 UMP EPP 544;551 2.7 2 PS+PRG S&D 462;737 2.3 2 EUROPE ÉCOLOGIE GREENS/EFA 337;554 1.7 1 ALTERNATIVE ALDE 253;069 1.3 1 FG GUE/NGL 252;197 1.3 1 Sum (Divisor) 2;576;905 (200,000) 10 District 5: Sud-Est FN NI 935;182 5.03 5 UMP EPP 743;343 3.996 3 PS+PRG S&D 394;114 2.1 2 EUROPE ÉCOLOGIE GREENS/EFA 309;168 1.7 1 ALTERNATIVE ALDE 280;091 1.5 1 FG GUE/NGL 197;754 1.1 1 Sum (Divisor) 2;859;652 (186,000) 13 (continued) 16 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.14 (continued) EP2014FR-LV0 Political group Vot es Quotient DivDwn District 6: Massif-Central/Centre FN NI 356;098 2.4 2 UMP EPP 314;959 2.1 2 PS+PRG S&D 233;079 1.6 1 ALTERNATIVE ALDE 146;482 0.98 0 FG GUE/NGL 110;087 0.7 0 EUROPE ÉCOLOGIE GREENS/EFA 101;331 0.7 0 Sum (Divisor) 1;262;036 (150,000) 5 District 7:Ile-de-France UMP EPP 667;991 4.8 4 FN NI 521;093 3.7 3 PS+PRG S&D 437;678 3.1 3 ALTERNATIVE ALDE 367;513 2.6 2 EUROPE ÉCOLOGIE GREENS/EFA 296;766 2.1 2 FG GUE/NGL 198;534 1.4 1 Sum (Divisor) 2;489;575 (140,000) 15 District 8: Outre-Mer UMP EPP 76;168 1.5 1 PS+PRG S&D 55;214 1.1 1 UOM GUE/NGL 52;017 1.04 1 FN NI 29;241 0.6 0 ALTERNATIVE ALDE 24;059 0.5 0 EUROPE ÉCOLOGIE GREENS/EFA 19;167 0.4 0 Sum (Divisor) 255;866 (50,000) 3 France establishes eight districts. Each district is evaluated separately, based on list votes with zero (that is, no) preference votes cast in that district. The divisor method with downward rounding is used throughout. Special restrictions pertain to the three geographical sections of District 8: Outre-Mer guaranteed representation in the EP; that is, one seat at the least. To this end the candidate lists of the parties must include at least one nominee from every section. The seats of the strongest party are filled with the nominees from the sections where the strongest party performs best. The seat of the second-strongest party goes to that section among the remaining sections where the second-strongest party scores best. In the event that the third-strongest party gains a seat it is allocated to the remaining section. Croatia has 11 seats to fill. On the ballot sheet every voter casts a vote for a party list of candidates plus, optionally, one vote for one of the candidates on this list (preference vote). A total of 950,980 ballots are cast, of which 921,904 are valid. There is an electoral threshold of 5% of ballots cast, 47,549. Twenty-one parties fail the threshold, and their 113,913 votes are discarded. This leaves 807,991 effective votes for four candidate lists to enter into the apportionment calculations. The divisor method with downward rounding is used. 1.5 Spain, Finland, France, Croatia; Vote Patterns and Vote Categories 17

Table 1.15 EP election 2014, Croatia EP2014HR-LV1 Political group Vot es Quotient DivDwn MEPs’ list positions HDZ+HSP AS EPP:5, ECR:1 381;844 6:01 6 6*, 1*, 5*, 2, 3, 4 Coal. SDP et al. S&D :2, ALDE:2 275;904 4:3 4 5*, 1, 2, 3 ORAH GREENS/EFA 86;806 1:4 1 1 1Other – 63;437 – 0 Sum (Divisor) 807;991 (63,500) 11 The vote pattern code “LV1” indicates that each voter casts a vote for the list of candidates of a party plus up to one preference vote for a particular candidate. Candidates with sufficiently many preference votes bypass the prespecified list ordering and jump to the top of the list. Seats that are based on preference votes are marked with an asterisk (*)

The assignment of seats to candidates makes allowance for the preference votes via a “ten percent bypass rule”. Candidates whose preference votes meet or exceed 10% of their party’s preference vote total jump to the top of the list. In Table 1.15 they are flagged by an asterisk (*). For example the strongest list, jointly submitted by the parties HDZ, HSS, HSP AS, BUZ, ZDS, and HDS, ranks candidate Ruža Tomašic´ on position six. As she draws the most preference votes, 107,206, she finishes at the top of the list. Since the list gets six seats in total, she would have been assigned a seat anyway. For the second strongest list, jointly submitted by the parties SDP, HNS, IDS, and HSU, the result is more spectacular. The list ranks candidate Tonino Picula fifth, but his preference votes put him first to become a MEP. With only four seats for this list, he would not have made it into the EP without preference votes. Vote Patterns The last three characters of the table labels indicate the pattern of how people express their votes to eventually elect parliamentary representatives. In the 2014 EP elections we meet three groups of vote patterns. A first group is coded LVx: Votes are cast for a candidate list of a party, plus optionally zero up to x preference votes for particular candidates of this party. The next group is abbreviated xCV: One up to x votes are cast for each of one up to x candidates of the same party, or of several parties. Another group, STV, designates a single transferable vote scheme. Vote pattern LV0 says that every voter casts a list vote, and that’s it. The list vote is cast for a prespecified list of candidates of a party. There are no (0) preference votes to back particular candidates. Voters cannot influence the list ranking submitted to the electoral authorities by party headquarters. For this reason these lists are often paraphrased as “closed lists”, or “rigid lists”. When a party is apportioned n seats, they are assigned to the candidates on list positions one through n. Since the assignment of seats to candidates is predetermined, there is no need to include in the tables a “MEPs’ list positions” column. Vote pattern LV0 is employed in seven Member States: Germany, Spain, France, Hungary, Portugal, Romania, and United Kingdom. 18 1 Exposing Methods: The 2014 European Parliament Elections

Vote patterns LV1, LV2, and LVm mean that every voter casts a vote for a candidate list of a party plus, optionally, one (1), two (2), or multiple (m) preference votes for particular candidates on this list. Candidates who draw sufficiently many preference votes—as detailed by the Member State’s provisions—jump to the top of the list. Such lists are also termed “flexible lists”. Candidates who bypass the prespecified list ranking are flagged in the “MEPs’ list positions” column of a table by an asterisk (*). Vote pattern LV1 is utilized in six Member States: Austria, Bulgaria, Croatia, the Netherlands, Sweden, and Slovenia; LV2 in two Member States: Czech Republic and Slovakia; and LVm in one Member State: Belgium. Vote patterns 1CV, 2CV, 3CV, 4CV, 5CV, 6CV, and mCV indicate that every voter casts one vote (or respectively two, three, four, five, six, or an arbitrary number m of votes) for particular candidates. With a single candidate vote (1CV) the vote is counted twice, once for the candidate’s party and once for the candidate proper. With two or more candidate votes, all countries except Luxembourg accept the ballot as valid only if the marked candidates all belong to the same party. The ballot is counted once in favor of this party, and it is counted in favor of each candidate marked. The sequence in which a party lists its candidates has no procedural consequences for the assignment of seats, which is why such lists are also known as “open lists”. Nevertheless the electorate is possibly influenced, in one way or the Other, by the sequence in which candidates are presented. Finnish party lists are alphabetical, Italian party lists place one or several Spitzenkandidaten at the top of the list in the hope of attracting the voters’ attention. Ten Member States make use of candidate votes: Cyprus (2CV), Denmark (1CV), Estonia (1CV), Greece (4CV), Finland (1CV), Italy (3CV), Lithuania (5CV), Luxembourg (6CV), Latvia (mCV), and Poland (1CV). Single transferable vote (STV) schemes are applied in three Member States: Ireland, Malta, and United Kingdom in District 12: Northern Ireland.Avoter ranks all candidates by expressing a first preference, a second preference, a third preference and so on. When a count of first preferences reaches the pertinent quota, the candidate is assigned a seat and the surplus votes beyond the quota are transferred to candidates with lower-order preferences. The precise rules for the transfer of surplus votes are detailed in the Member State’s provisions. From a retrospective viewpoint the resulting seat numbers are very similar to what is obtained by reinterpreting first preferences as a single candidate vote (1CV) and aggregating them by Political Groups. Vote Categories For the design and analysis of proportional representation systems the fundamental ingredients are the votes that count towards the success of political parties. In the sequel we restrict attention to these “party votes”. That a vote has the complexion of a party vote is directly visible when voters cast a simple list vote (LV0). When the electorate submits candidate votes, such as xCV, the party vote is implied indirectly. Passing over these subtleties we assume that the counts of party votes as reported by the domestic bureaus are all of a comparable quality. In the sequel our use of the term “votes” invariably means party votes. 1.6 Hungary, Ireland, Italy; Quotas 19

An electoral system includes various categories of voters. The all-embracing reference set is the entire citizenry. The citizens who have the franchise to vote form the electorate. Those of the electorate who go to the polls are the voters. Just as the term “electorate” is a singular, many experts also refer to a singular, “the voter”, when they really mean many people who vote. We find the typifying singular misleading. If there were only one voter we would not have to deal with numbers. The “votes cast” are categorized into “valid votes” or good votes, versus “invalid votes” or rejected votes. The valid votes subdivide into the “effective votes” that enter into the apportionment calculations, and the “ineffective votes” that, though valid, are nevertheless discarded. For the purpose of analyzing the diversity of apportionment methods we mostly rely on effective votes, and refer to valid votes or votes cast only when needed. Weak parties without a seat nor affiliated with a Political Group are subsumed under the label “Others”.

1.6 Hungary, Ireland, Italy; Quotas

Hungary is allocated 21 seats. There is an electoral threshold of 5% of valid votes, that is, 115,974 votes. Two parties miss the threshold, and 21,398 votes become ineffective. Four lists take part in the seat apportionment. The calculations are conducted using the divisor method with downward rounding. Every 96,000 votes justify roughly one seat. See Table 1.16. Ireland has 11 seats to fill. The country is divided into three districts: Dublin (3 seats), Midlands–North-West (4), and South (4). There is no electoral threshold. In each district the Droop-quota bvC=.h C 1/cC1 is calculated, where vC is the sum of votes and h the number of seats in this district. A single transferable vote (STV) scheme is used. In brief it works as follows. On the ballot sheet voters mark the candidates as first preference, second preference, and so on downwards. When the first-preference tally for a candidate reaches the Droop-quota, the candidate is declared elected. The ballots in excess of the Droop-quota are transferred to the

Table 1.16 EP election 2014, Hungary EP2014HU-LV0 Political group Vot es Quotient DivDwn FIDESZ+KDNP EPP 1;193;991 12.4 12 JOBBIK NI 340;287 3.5 3 MSZP S&D 252;751 2.6 2 DK S&D 226;086 2.4 2 EGYÜTT+PM GREENS/EFA 168;076 1.8 1 LMP GREENS/EFA 116;904 1.2 1 Sum (Divisor) 2;298;095 (96,000) 21 The threshold of 5% of valid votes shuts out two parties and turns 21,398 votes ineffective. For the six remaining lists every 96,000 votes justify roughly one seat, where interim quotients are rounded downwards 20 1 Exposing Methods: The 2014 European Parliament Elections candidate who is next according to the voter’s preferences. The decision which ballots are in excess is random. We label the system with the acronym STVran. See Table 1.17. The first preference counts in Table 1.17 are insufficient to reconstruct subsequent vote transfers, and to double-check the final seat apportionment. The accumulation of lower-order preferences bears on the result in District 1: Dublin. The district magnitude is three seats. Candidate Nessa Childers finishes fifth, but is elected by means of the votes that are transferred to her during the evaluation process. Alternatively, if first preferences are interpreted as candidate votes (vote pattern 1CV) and aggregation is by Political Groups, then the divisor method with standard rounding would have equipped S&D with the seat now occupied by Nessa Childers. Though not identical, the two approaches produce rather similar results. Italy has an electoral threshold of 4% of the valid votes. With 27,448,906 valid votes the threshold requires 1,097,957 votes. Minority parties can evade the threshold by registering an electoral alliance with a party that campaigns in the whole country. In 2014, the Südtiroler Volkspartei (SVP) is allied with the Partito democratico (PD). Five parties fail the threshold, their 1,686,908 votes become ineffective. The remaining 25,761,998 effective votes participate in the countrywide apportionment of seats to parties. The apportionment uses the “Hare-quota variant-1 method with residual fit by greatest remainders”, “HQ1grR”. Variant-1 of the Hare- quota is the integral part of the votes-to-seats ratio, 352,902. For the assignment of seats to candidates the country is subdivided into five districts. The 73 Italian seats are allocated between the districts in proportion to population figures. District 1: Italia nord-occidentale features 20 seats, District 2: Italia nord-orientale 14, District 3: Italia centrale 14, District 4: Italia meridionale 17, and District 5: Italia insulare 8. Within each district provisional apportionments are calculated using the Hare-quota method with residual fit by greatest remainders (HaQgrR). When confronting the five districtwise apportionments with the earlier country- wide apportionment it is seen that for every party the sum of its district seats differs from its countrywide seats, except for FI where the sum fits. The law stipulates that the countrywide seat apportionment is decisive and takes precedence. Therefore the districtwise apportionments are adjusted in order to compensate for the differences. The adjustment appears to be a makeshift way of searching for compensating seat transfers. Luckily the search strategy finds a solution for the 2014 data (Table 1.18). There are three parties for which the districtwise sum exceeds the countrywide apportionment by one seat: PD+SVP, NCD+UDC+PPI, and L’ALTRA EUROPA. For PD+SVP the smallest remainder that is rounded upwards occurs in Italia meridionale: 0.424. Here the compensation procedure retracts the seat; that is, this remainder is rounded downwards. The same happens for NCD+UDC+PPI in Italia centrale (remainder 0.525), and for L’ALTRA EUROPA in Italia nord- orientale (0.544). Now the sums of the districtwise seats of the three parties fit their countrywide seat numbers. For M5S the districtwise sum falls short of the countrywide apportionment by one seat, and for LN by two seats. Among these two parties and the three districts 1.6 Hungary, Ireland, Italy; Quotas 21

Table 1.17 EP election 2014, Ireland EP2014IE-STV Party Political group First pref STVran District 1: Dublin Lynn Boylan SF GUE/NGL 83;264 1 Brian Hayes FG EPP 54;676 1 Mary Fitzpatrick FF ECR 44;283 0 Eamon Ryan GP GREENS/EFA 44;078 0 Nessa Childers Indep S&D 35;939 1 Paul Murphy SP GUE/NGL 29;953 0 Emer Costello LAB S&D 25;961 0 Five further candidates – – 34;421 0 Sum (Droop-quota) (88,144) 352;575 3 District 2: Midlands-North-West Luke ‘Ming’ Flanagan Indep GUE/NGL 124;063 1 Matt Carthy SF GUE/NGL 114;727 1 Mairead Mcguinness FG EPP 92;080 1 Marian Harkin Indep ALDE 68;986 1 Pat Gallagher FF ECR 59;562 0 Thomas Byrne FF ECR 55;384 0 Jim Higgins FG EPP 39;908 0 Ronan Mullen Indep – 36;326 0 Lorraine Higgins LAB S&D 31;951 0 Mark Dearey GP GREENS/EFA 9520 0 Four further candidates – – 13;938 0 Sum (Droop-quota) (129,290) 646;445 4 District 3: South Brian Crowley FF ECR 180;329 1 Liadh Ni Riada SF GUE/NGL 125;309 1 Seán Kelly FG EPP 83;520 1 Simon Harris FG EPP 51;483 0 Deirdre Clune FG EPP 47;453 1 Diarmuid P. O’Flynn Indep – 30;323 0 Phil Prendergast LAB S&D 30;317 0 Kieran Hartley FF ECR 29;987 0 Grace O’Sullivan GP GREENS/EFA 27;860 0 Six further candidates – – 50;917 0 Sum (Droop-quota) (131,500) 657;498 4 Ireland uses the single transferable vote (STV) scheme wherein voters rank candidates from first preference to last preference. Surplus votes are transferred to lower preferences (not shown in the table) by a randomized decision rule (STVran). Party seats are nearly proportional to the party votes that are obtained by aggregating first preferences 22 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.18 EP election 2014, Italy EP2014IT-3CV Political group Vot es Quotient HQ1grR PD+SVP S&D :31, EPP:1 11;341;268 32.137 32 M5S EFDD 5;807;362 16.456 17 FI EPP 4;614;364 13.075 13 LN NI 1;688;197 4.784 5 NCD+UDC+PPI EPP 1;202;350 3.407 3 L’ALTRA EUROPA GUE/NGL 1;108;457 3.141 3 Sum (Split) 25;761;998 (0.43) 73 Vot es Quotient HaQgrR MEPs’ list positions Sub-apportionment within District 1: Italia nord-occidentale PD 3;240;825 8:618 9 1, 3, 2, 4, 5, 11, 15, 10, 19 M5S 1;470;247 3:910 4 3, 17, 7, 20 FI 1;294;490 3:442 3 1, 4, 11 LN 933;644 2:483 2 1, 3 NCD+UDC+PPI 276;748 0:736 1 1 L’ALTRA EUROPA 305;078 0:811 1 3 Sum (Split) 7;521;032 (0.5) 20 Sub-apportionment within District 2: Italia nord-orientale PD+SVP 2;620;651 6:804 7 1, 6, 3, 2, 4, 11; SVP: 1 M5S 1;081;564 2:808 3 2, 1, 7 FI 738;911 1:919 2 1, 14 LN 565;951 1:469 1 " 2 1, 2 NCD+UDC+PPI 175;394 0:455 0 L’ALTRA EUROPA 209;424 0:544 1 # 0 Sum (Split) 5;391;895 (0.5) 14 Sub-apportionment within District 3: Italia centrale PD 2;657;892 6:976 7 1, 2, 9, 6, 4, 5, 3 M5S 1;243;070 3:263 3 1, 4, 12 FI 841;276 2:208 2 1, 13 LN 122;509 0:322 0 " 1 1 NCD+UDC+PPI 200;117 0:525 1 # 0 L’ALTRA EUROPA 269;286 0:707 1 1 Sum (Split) 5;334;150 (0.5) 14 Sub-apportionment within District 4: Italia meridionale PD 2;024;687 6:424 7 # 6 2, 1, 4, 7, 5, 17 M5S 1;388;908 4:407 4 " 5 1, 11, 9, 2, 14 FI 1;281;891 4:067 4 1, 5, 14, 4 LN 43;393 0:138 0 NCD+UDC+PPI 378;867 1:202 1 1 L’ALTRA EUROPA 240;017 0:762 1 1 Sum (Split) 5;357;763 (0.42) 17 (continued) 1.6 Hungary, Ireland, Italy; Quotas 23

Table 1.18 (continued) Sub-apportionment within District 5: Italia insulare PD 797;213 2.957 3 2, 1,4 M5S 623;573 2.313 2 1, 4 FI 457;796 1.698 2 7, 4 LN 22;700 0.084 0 NCD+UDC+PPI 171;224 0.635 1 1 L’ALTRA EUROPA 84;652 0.314 0 Sum (Split) 2;157;158 (0.5) 8 The statewide super-apportionment is followed by sub-apportionments in five districts. The Hare-quota method with residual fit by greatest remainders is used throughout. In order to comply with the super-apportionment, the within-district sub-apportionments are supplemented by compensating seat transfers (marked with a dot, ) where a seat is being retracted, the largest remainder that is rounded downwards occurs for LN in Italia nord-orientale, 0.469. Here the compensation procedure creates a seat; that is, this remainder is rounded upwards. The same occurs to M5S in Italia meridionale (remainder 0.407), and for LN in Italia centrale (0.322). At this stage every party has its district seats summing to its countrywide seat number, and every district verifies its district magnitude. Therefore the compensation procedure terminates. In the table the remainders whose rounding is reversed are earmarked with a dot (). Finally seats are assigned to candidates separately within each district. The seats of a party are filled with those of its candidates who receive the most candidate votes. Quotas Here is an overview over various quota definitions used now and in former time, where h (“house size”) denotes the number of seats to be apportioned:   effective votes effective votes HaQ D ; DrQ D C 1; h h C 1     effective votes effective votes HQ1 D _ 1; DQ1 D _ 1; h h C 1     effective votes effective votes HQ2 D ; DQ2 D ; h h C 1     valid votes effective votes HQ3 D _ 1; DQ3 D _ 1; h h C 1     valid votes unused voting power HQ4 D ; DQ5 D C 1 _ 1: h residual seats 24 1 Exposing Methods: The 2014 European Parliament Elections

The functions bc, de,andhi indicate downward rounding, upward rounding, and commercial rounding, respectively; they are discussed in Chap. 3. Being employed as fixed divisors, quotas must be non-zero. The definitions force the quota to stay positive, due to t _ 1 WD maxft;1g1. Yet another quota, the unrounded Droop- quota DQ4, is introduced in Sect. 5.8. As an example we consider District 3: South in Table 1.17. With 657,498 votes and 4 seats, the Hare-quota amounts to 164,374.5 vote shares. Its variant-1 rounds the number downwards to 164,374. Variant-2 rounds upwards: 164,375. The default Droop-quota divides by 4 C 1 D 5, rounds the quotient 131,499.6 downwards, and finally adds 1, giving 131,500 votes. Variant-1 of the Droop-quota equals 131,499 votes, while variant-3 uses commercial rounding and yields 131,500. “Commercial rounding” hti rounds t downwards when its first digit after the decimal point is 0, 1, 2, 3, or 4; it rounds upwards when the first digit after the decimal point is 5, 6, 7, 8, or 9.

1.7 Lithuania, Luxembourg, Latvia; Residual Fits

Lithuania is allocated 11 seats. Voters mark a party and may cast up to five preference votes. A total of 1,211,279 ballots are cast. There is an electoral threshold of 5% of ballots cast: 60,564. Of the 1,144,131 valid ballots, 1,063,650 become effective. For the apportionment of seats to parties the “Hare-quota variant-2 method with residual fit by greatest remainders” is applied, abbreviated as “HQ2grR”. Variant-2 of the Hare-quota is the votes-to-seats ratio rounded upwards, 96,696. The assignment of seats to candidates is solely based on the candidates’ preference votes. See Table 1.19. Luxembourg has six seats to fill. Voters have six candidate votes which may span across party lines. In all, 1,172,614 candidate votes are marked on 203,772 valid ballot sheets. On average, there are 5.75 marks per ballot. The vote numbers

Table 1.19 EP election 2014, Lithuania EP2014LT-5CV Political group Vot es Quotient HQ2grR MEPs’ list positions TS-LKD EPP 199;393 2:062 2 3, 1 LSDP S&D 197;477 2:042 2 2, 1 LRLS ALDE 189;373 1:958 2 2, 3 TT EFDD 163;049 1:686 2 1, 3 DP ALDE 146;607 1:516 1 1 LLRA (AWPL) ECR 92;108 0:953 1 1 LVZS GREENS/EFA 75;643 0:782 1 1 Sum (Split) 1;063;650 (0.6) 11 Voters choose a party and may vote for up to five of its candidates. The Hare-quota variant-2 method with residual fit by greatest remainders is used, HQ2grR. Variant-2 of the Hare-quota is the votes-to-seats ratio rounded upwards, HQ2 = d1;063;650=11eDd96;695:5eD96;696 1.7 Lithuania, Luxembourg, Latvia; Residual Fits 25 would need to be scaled by the marks-per-ballot average, 5.75, in order to mirror the number of people who back the parties. There is no electoral threshold. The seat apportionment is performed using the divisor method with downward rounding. See Table 1.20. Latvia is allocated eight seats. Voters select a party and may mark any of the party’s candidates whom they support (a “plus”), or cross out the candidates of whom they disapprove (a “crossing-out”). Of the 445,225 ballots cast, 440,288 are valid. There is an electoral threshold of 5% of votes cast: 22,262. It gives rise to 49,277 ineffective votes, cast for nine parties. The remaining 391,011 effective votes arise from five parties. The divisor method with standard rounding is used to apportion the seats among parties. The assignment of a party’s seats to its candidates is based on their individual balance of pluses minus crossing-outs. See Table 1.21. Residual Fits In the 2014 EP elections all quota methods finish with a residual fit by greatest remainders, grR, except for Greece (-EL). Other procedures are available. In the 2009 EP elections Lithuania employed the “full-seat restricted residual fit by greatest remainders”, gR1. The restriction excludes parties with fewer votes than a full quota. Yet another rule was practiced in the Swiss Canton Solothurn 1896–1917: all remaining seats go to the strongest party. We abbreviate

Table 1.20 EP election 2014, Luxembourg EP2014LU-6CV Political group “Votes” Quotient DivDwn MEPs’ list positions CSV/PCS EPP 76;736 3:8 3 1, 2, 3 DÉI GRÉN/LES VERTS GREENS/EFA 30;597 1:5 1 1 DP/PD ALDE 30;108 1:5 1 1 LSAP/POSL S&D 23;895 1:2 1 1 5Others – 42;437 – 0 Sum (Divisor) 203;773 (20,000) 6 Voters vote for up to six candidates who may belong to different parties. The total number of valid ballot sheets is 203,772. On average a ballot sheet contains 1;172;614=203;772 D 5:75 votes. The raw vote counts (not shown) are scaled by 5.75 and rounded to the table’s “Votes” in order to reflect the number of individuals supporting the parties

Table 1.21 EP election 2014, Latvia EP2014LV-mCV Political group Vot es Quotient DivStd MEPs’ list positions V. EPP 204;979 4:1 4 1, 2, 3, 4 Coal. NA ECR 63;229 1:3 1 1 SASKAN¸ASDP S&D 57;863 1:2 1 4 Coal. ZZS EFDD 36;637 0:7 1 3 LKS GREENS/EFA 28;303 0:6 1 1 Sum (Divisor) 391;011 (50,000) 8 Voters choose a party and may vote for any of the party’s candidates whom they support, a “plus”, or cross out the candidates whom they do not support, a “crossing-out”. The assignment of seats to candidates follows the within-party ranking of the candidates by their tallies of pluses minus crossing-outs 26 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.22 EP election 2014, Malta EP2014MT-STV Party Political group First pref STVran Alfred Sant PL/MLP S&D 48;739 1 Roberta Metsola PN/NP EPP 32;360 1 Miriam Dalli PL/MLP S&D 23;479 1 David Casa PN/NP EPP 19;582 1 Marlene Mizzi PL/MLP S&D 14;057 1 Clint Camilleri PL/MLP S&D 13;484 0 Joseph Cuschieri PL/MLP S&D 10;461 0 Francis Zammit Dimech PN/NP EPP 8660 0 Therese Comodini Cachia PN/NP EPP 7859 1 Raymond Bugeja PN/NP EPP 7846 0 Lino Bianco PL/MLP S&D 7268 0 Norman Vella PN/NP EPP 7099 0 Charlon Gouder PL/MLP S&D 6719 0 Deborah Schembri PL/MLP S&D 5983 0 Stefano Mallia PN/NP EPP 5663 0 Kevin Cutajar PN/NP EPP 5415 0 Helga Ellul PN/NP EPP 2976 0 Jonathan Shaw PN/NP EPP 2087 0 Ivan Grixti PL/MLP S&D 1595 0 Mario Farrugia Borg PL/MLP S&D 1297 0 Kevin Plumpton PN/NP EPP 1238 0 Peter Cordina PL/MLP S&D 868 0 Fleur-Anne Vella PL/MLP S&D 512 0 Nine further candidates – – 16;604 0 Sum (Droop-quota) (35,979) 251;851 6 Malta uses the single transferable vote (STV) scheme. The decision of which votes exceed a candidate’s Droop-quota and are transferred to lower preferences is random (STVran). Party seats are by and large proportional to the party votes that are obtained by aggregating first preferences by parties this “winner-take-all” imperative by WTA. In summary, here is an overview over viable residual fits: grR The remaining seats are apportioned, one by one via greatest remainders, among all parties participating in the apportionment process. gR1 The remaining seats are apportioned, one by one via greatest remainders, among those parties drawing at least one full quota of votes. WTA Winner-take-all: All remaining seats go to the strongest party. -EL The remaining seats are apportioned as in Greece, see Sect. 1.4. 1.8 Malta, the Netherlands, Poland; Nested Stages 27

1.8 Malta, the Netherlands, Poland; Nested Stages

Malta applies a single transferable vote scheme, STVran. Candidate Therese Comodini Cachia has fewer first preference votes than her fellow party member Francis Zammit Dimech, but is assigned a seat due to the votes transferred to her during the evaluation process. If first preferences were interpreted as candidate votes (vote pattern 1CV) and aggregated by Political Groups, then the divisor method with standard rounding would have given the seat to candidate Dimech, not Cachia. Other than that the two approaches would give the same result. See Table 1.22. The Netherlands have an electoral threshold given by the quotient of valid votes per seats, 4;753;746=26 D 182;837. The threshold renders 290,332 votes for ten parties ineffective. There are two alliances. Thus the super-apportionment is followed by two sub-apportionments. The super-apportionment uses the divi- sor method with downward rounding, which is notorious for yielding biased seat numbers that favor stronger parties at the expense of weaker parties. The sub-apportionments apply the Hare-quota method with residual fit by greatest remainders, which is known to produce unbiased seat numbers. See Table 1.23.

Table 1.23 EP election 2014, the Netherlands EP2014NL-LV1 Political group Vot es Quotient DivDwn MEPs’ list positions Alliance A 1;086;609 7.1 7 Alliance B 778;357 5.1 5 D66 ALDE 735;825 4.8 4 1*, 3*, 2*, 4 PVV NI 633;114 4.1 4 10*, 1*, 2*, 3 VVD ALDE 571;176 3.7 3 1*, 2*, 6* SP GUE/NGL 458;079 2.97 2 1*, 3* PVDD GUE/NGL 200;254 1.3 1 1* Sum (Divisor) 4;463;414 (154,000) 26 Sub-apportionment within Alliance A = CDA+CU-SGP CDA EPP 721;766 4.650 5 1*, 25*, 3*, 2*, 7* CU-SGP ECR 364;843 2.350 2 1*, 2* Sum (Split) 1;086;609 (0.5) 7 Sub-apportionment within Alliance B = PVDA+GROENLINKS PVDA S&D 446;763 2.870 3 1*, 2*, 3 GROENLINKS GREENS/EFA 331;594 2.130 2 1*, 2* Sum (Split) 778;357 (0.5) 5 The super-apportionment uses the divisor method with downward rounding, which is biased in favor of stronger parties. The sub-apportionment within an alliance applies the Hare-quota method with residual fit by greatest remainders, which is well-known to be unbiased. Seats based on preference votes are marked with an asterisk (*) 28 1 Exposing Methods: The 2014 European Parliament Elections

Poland also conducts a two-stage evaluation, but for a different reason. The electoral threshold amounts to 353,475 votes, 5% of the 7,069,485 valid votes. Seven parties miss it, and 897,649 votes turn ineffective. Five parties pass the threshold and hence participate in the apportionment process. First all 51 Polish seats are apportioned in proportion to the parties’ statewide vote counts. This stage uses the (biased) divisor method with downward rounding. The second stage consists of a sub-apportionment for each party to allocate its seats to the district lists in proportion to the votes in that district. All sub-apportionments use the (unbiased) Hare-quota method with residual fit by greatest remainders. See Table 1.24. Nested Apportionments The Netherlands and Poland implement two-stage appor- tionment calculations. The “super-apportionment” at the top level is followed by several “sub-apportionments” on a lower level. The two states apply two-stage systems for quite different reasons. In the Netherlands, political parties join into an alliance in order to increase their weights in the top-level calculations. In Poland, the super-apportionment secures an evaluation of the parties’ successes with respect to the countrywide electorate. In the sub-apportionments the vote counts of a party are geographically disaggregated in order to secure regional representativeness. The selective use of apportionment procedures, such as DivDwn versus HaQgrR in the Netherlands and Poland, is due to the fact that some methods are notorious for being biased in favor of stronger parties at the expense of weaker parties (DivDwn, DrQgrR). Other apportionment methods are known to be unbiased (DivStd, HaQgrR). Weaker parties might be reluctant to join an alliance that would expose them to a disadvantageous bias. An alliance becomes more attractive when promising its partners to treat them fairly. Nevertheless alliances lose their appeal when seen with the eyes of the voters. Without alliances people know that a vote for party A directly helps party A. In an alliance with partners A and B, a vote for party A may be redirected to support party B. In the presence of alliances the dedication of the votes is determined not directly by the voters, but indirectly by “the system”.

1.9 Portugal, Romania, Sweden; Method Overview

Portugal is allocated 21 seats. There is no electoral threshold. The divisor method with downward rounding is used. Every 116,000 votes justify roughly one seat. Eleven parties are too weak to obtain representation. See Table 1.25. Romania has 33 seats to fill. There are two electoral thresholds, both referring to the number of valid votes, 4,840,033. The first threshold applies to parties and amounts to 5% of the valid votes, 242,002. The second threshold applies to indeps and is the Hare-quota variant-4, d4;840;033=33eD146;668. Four parties and five indeps fail the thresholds. The 139,728 votes for them are discarded. Five parties 1.9 Portugal, Romania, Sweden; Method Overview 29

Table 1.24 EP election 2014, Poland EP2014PL-1CV Political group Vot es Quotient DivDwn PO EPP 2;271;215 19.6 19 PIS ECR 2;246;870 19.4 19 SLD S&D 667;319 5.8 5 KNP NI 505;586 4.4 4 PSL EPP 480;846 4.1 4 Sum (Divisor) 6;171;836 (116,000) 51 Vot es Quotient HaQgrR List pos. Vot es Quotient HaQgrR List pos. Sub-apportionment within PO Sub-apportionment within PIS Gdansk´ 218;962 1:832 2 1, 10 117;620 0.995 1 1 Bydgoszcz 100;430 0:840 1 3 96;663 0.817 1 1 Olsztyn 105;541 0:883 1 1 140;342 1.187 1 1 Warszawa 1 308;468 2:581 2 1, 2 216;773 1.833 2 1, 5 Warszawa 2 75;369 0:631 1 1 163;775 1.385 1 4 Łód´z 149;474 1:250 1 1 177;654 1.502 1 1 Poznan´ 192;801 1:613 2 2, 1 142;675 1.206 1 1 Lublin 64;889 0:543 0 164;578 1.392 1 2 Rzeszów 73;381 0:614 1 1 196;247 1.660 2 1, 3 Kraków 232;330 1:944 2 1, 2 307;624 2.601 3 2, 1, 4 Katowice 337;478 2:823 3 1, 3, 5 234;515 1.983 2 1, 2 Wrocław 252;513 2:112 2 1, 2 179;432 1.517 2 1, 2 Gorzów 159;579 1:335 1 1 108;972 0.921 1 1 Wielkopolski Sum (Split) 2;271;215 (0.6) 19 2;246;870 (0.51) 19 Sub-apportionment within SLD Sub-apportionment within KNP Gdansk´ 35;164 0:263 0 30;324 0.240 0 Bydgoszcz 74;833 0:561 1 1 20;753 0.164 0 Olsztyn 41;422 0:310 0 28;412 0.225 0 Warszawa 1 57;010 0:427 0 49;794 0.394 1 1 Warszawa 2 24;647 0:185 0 27;671 0.219 0 Łód´z 35;344 0:265 0 29;202 0.231 0 Poznan´ 74;695 0:560 1 1 40;540 0.321 0 Lublin 21;248 0:159 0 27;482 0.217 0 Rzeszów 18;761 0:141 0 28;474 0.225 0 Kraków 62;748 0:470 0 72;393 0.573 1 1 Katowice 79;543 0:596 1 1 73;573 0.582 1 1 Wrocław 78;557 0:589 1 1 47;615 0.377 1 1 Gorzów 63;347 0:475 1 1 29;353 0.232 0 Wielkopolski Sum (Split) 667;319 (0.472) 5 505;586 (0.35) 4 (continued) 30 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.24 (continued) EP2014PL-5subs Vot es Quotient HaQgrR List pos. Sub-apportionment within PSL Gdansk´ 14;817 0.123 0 Bydgoszcz 32;507 0.270 0 Olsztyn 36;221 0.301 0 Warszawa 1 19;098 0.159 0 Warszawa 2 61;259 0.510 1 1 Łód´z 29;615 0.246 0 Poznan´ 61;431 0.511 1 1 Lublin 70;055 0.583 1 2 Rzeszów 28;927 0.241 0 Kraków 58;541 0.487 1 1 Katowice 18;480 0.154 0 Wrocław 28;087 0.234 0 Gorzów Wielkopolski 21;808 0.181 0 Sum (Split) 480;846 (0.4) 4 The super-apportionment uses the divisor method with downward rounding, which produces seat numbers that are biased in favor of stronger parties and at the expense of weaker parties. The within-party sub-apportionments by districts applies the Hare-quota method with residual fit by greatest remainders, which yields unbiased seat numbers

Table 1.25 EP election 2014, Portugal EP2014PT-LV0 Political group Vot es Quotient DivDwn PS S&D 1;034;249 8:9 8 Coal. PSD+CDS-PP EPP 910;647 7:9 7 Coal. PCP+PEV GUE/NGL 416;925 3:6 3 MPT ALDE 234;788 2:02 2 B.E. GUE/NGL 149;764 1:3 1 11 Others – 294;399 – 0 Sum (Divisor) 3;040;772 (116,000) 21 There is no electoral threshold. The divisor method with downward rounding is used. Every 116,000 votes justify roughly one seat of the Portuguese allocation of 21 seats and one indep enter into the apportionment calculation. The apportionment method used is the divisor method with downward rounding. See Table 1.26. Sweden uses a threshold of 4% of valid votes. With 3,716,778 valid votes it amounts to 148,672 votes. Eleven parties fail the threshold, and their 109,911 votes become ineffective. The apportionment of the 20 Swedish seats among the nine eligible parties is carried out using the “Swedish 1952 modification of the divisor method with standard rounding”, abbreviated as “Div0.7”. The modification applies to interim quotients below one. In the interval Œ0I 1 the standard rounding point 0.5 is replaced by the rounding point 0.7. That is, if a quotient stays below 0.7 it is rounded downwards to zero, if greater, it is rounded upwards to one. Our 1.9 Portugal, Romania, Sweden; Method Overview 31

Table 1.26 EP election 2014, Romania EP2014RO-LV0 Political group Vot es Quotient DivDwn Coal. PSD+UNPR+PC S&D 2;093;234 16.1 16 PNL EPP 835;531 6.4 6 PDL EPP 680;853 5.2 5 Indep M. Diaconu ALDE 379;582 2.9 1 UDMR EPP 350;689 2.7 2 PMP EPP 345;973 2.7 2 Sum (Divisor) 4;685;862 (130,000) 32 The divisor method with downward rounding is used. The independent candidate Mircea Diaconu attracts so many votes that the quotient 2.9 would have justified two seats. However, an indep can occupy at most one seat ()

Table 1.27 EP election 2014, Sweden EP2014SE-LV1 Political group Vot es Quotient Div0.7 MEPs’ list positions S S&D 899;074 5:499 5 1*, 2, 3, 4, 5 MP GREENS/EFA 572;591 3:502 4 1*, 2*, 3, 4 M EPP 507;488 3:1 3 3*, 1*, 2* FP ALDE 368;514 2:3 2 1*, 2* SD EFDD 359;248 2:2 2 1*, 2* C ALDE 241;101 1:47 1 3* V GUE/NGL 234;272 1:4 1 1* KD EPP 220;574 1:3 1 1* FI S&D 204;005 1:2 1 1* Sum (Divisor) 3;606;867 (163,500) 20 The divisor method with standard rounding is used, except that a quotient below unity is rounded downwards to zero if less than 0.7, and upwards to one if greater. This modification is indicated by the acronym Div0.7. Seats that are based on preference votes are marked with an asterisk (*) acronym for this method is Div0.7. In 2014 all interim quotients are greater than one, whence the special rounding prescription in the interval Œ0I 1 remains dormant. See Table 1.27. Method Overview Before turning to the last batch of states we classify the seat apportionment methods that are used in the 2104 EP elections. A frequent choice are divisor methods. They follow the motto “Divide and round”. The most popular procedure in this group is the divisor method with downward rounding, DivDwn. It is the method of choice in seventeen of the 28 Member States: Austria, Belgium, Czech Republic, Denmark, Estonia, Spain, Finland, France, Croatia, Hungary, Luxembourg, the Netherlands, Poland, Portugal, Romania, Slovenia, and United Kingdom. 32 1 Exposing Methods: The 2014 European Parliament Elections

Why is the divisor method with downward rounding so popular? First, the divisor method with downward rounding originates from the beginning of the proportional representation movement in the second half of the nineteenth century. Endorsed by famous protagonists of the movement, the Belgian Victor D’Hondt and the Swiss Eduard Hagenbach-Bischoff, the method is distinguished by history. Second, its technical instructions are elementary. Whoever calculates the interim quotients of votes and divisor may drop their pen when reaching the decimal point! The omitted fractional parts do not matter since they are rounded downwards to zero anyway. Superficially, downward rounding conveys the impression that no rounding operation is going on at all. Third, the method enjoys features that presumably appeal to parliamentary actors and party officials. The divisor method with downward rounding yields seat numbers that are biased in favor of stronger parties at the expense of weaker parties. Stronger parties are a major player in forging a majority. When they have to bow to a proportional representation system, the divisor method with downward rounding is a favorite choice to secure an advantage. Advantages are especially boosted by subdividing the electoral region into multiple districts since every separate evaluation of a district generates a bias bonus for the big players. The unbiased alternative in the class of divisor methods is the divisor method with standard rounding: DivStd. It relies on the rounding rule that has stood the test of time. Partners are treated fairly when fractional parts are rounded downwards or upwards according as they are smaller or greater than one half. This is true for commerce and business as much as it is true for electoral systems. The divisor method with standard rounding is used in Germany and Latvia and, with a minor modification, in Sweden. Another frequent choice for the seat apportionment are quota methods. They are captured by the motto “Divide and rank”. It is tempting to believe that there is a definitive number of votes to justify a seat, the quota. However, the EP elections demonstrate that there is a bewildering variety of quotas. Bulgaria, the Netherlands and Poland use the Hare-quota, HaQ. Italy employs the variant HQ1, Lithuania the variant HQ2, and Cyprus the variant HQ3. Ireland, Malta and the United Kingdom utilize the Droop-quota, DrQ. The variant DQ3 is put into practice in Slovakia. Quota methods solve the seat apportionment problem in a somewhat eclectic manner. The third category of apportionment methods comprises STV schemes. In Ireland and Malta surplus votes are transferred to lower-order preferences in a random fashion, STVran. In the Northern Ireland district of the United Kingdom, yet to be discussed, surplus votes are redistributed in a deterministic fashion by the fractions of votes in favor of lower preferences: STVfra. The available empirical data demonstrate that STV schemes closely resemble systems which interpret first preferences as single candidate votes (vote pattern 1CV). 1.10 Slovenia, Slovakia, United Kingdom; Local Representation 33

1.10 Slovenia, Slovakia, United Kingdom; Local Representation

Slovenia is allocated 8 seats. There are 419,661 votes cast. The electoral threshold is set at 4% of votes cast, 16,787. The threshold removes eight parties; their 60,601 votes become ineffective. The divisor method with downward rounding is used. Every 30,000 votes justify roughly one seat. See Table 1.28. Slovakia uses an electoral threshold of 5% of valid votes. Since there are 560,603 valid votes the threshold is 28,031. It excludes 21 parties, and deems 121,081 votes as ineffective. The allocation of the 13 Slovakian seats uses the “Droop- quota variant-3 method with residual fit by greatest remainders”, abbreviated as “DQ3grR”. Variant-3 of the Droop-quota equals DQ3 Dh439;522=14iD h31;394:4iD31;394.SeeTable1.29.

Table 1.28 EP election 2014, Slovenia EP2014SI-LV1 Political group Vot es Quotient DivDwn MEPs’ list positions SDS EPP 99;643 3:3 3 1*, 2*, 3 Coal. NSi+SLS EPP 66;760 2:2 2 8*, 1* VERJAMEM GREENS/EFA 41;525 1:4 1 1* DESUS ALDE 32;662 1:1 1 1* SD S&D 32;484 1:1 1 2* 3Others – 68;396 – 0 Sum (Divisor) 341;470 (30,000) 8 There is an electoral threshold of 4% of votes cast discarding 60,601 votes and eight parties. The divisor method with downward rounding is used. Every 30,000 votes justify roughly one seat. Seats based on preference votes are marked with an asterisk (*)

Table 1.29 EP election 2014, Slovakia EP2014SK-LV2 Political group Vot es Quotient DQ3grR MEPs’ list positions SMER-SD S&D 135;089 4:303 4 1*, 2*, 5*, 4* KDH EPP 74;108 2:361 2 1*, 2* SDKÚ-DS EPP 43;467 1:385 2 2*, 1* OL’ANO ECR 41;829 1:332 1 4* NOVA ECR 38;316 1:220 1 2* SAS ALDE 37;376 1:191 1 3* SMK-MPK EPP 36;629 1:167 1 1* MOST-HID EPP 32;708 1:042 1 2* Sum (Split) 439;522 (0.37) 13 The Droop-quota variant-3 with residual fit by greatest remainders is used, DQ3grR. Variant-3 of the Droop-quota equals Dh439;522=.13 C 1/iDh31;394:4iD31;394. Twelve seats are allotted in the main apportionment, one seat in the residual apportionment. All seats are assigned by preference votes as indicated by asterisks (*) 34 1 Exposing Methods: The 2014 European Parliament Elections

The United Kingdom subdivides its area into 12 districts. The 73 seats are allocated to the districts prior to the election. Each district is evaluated separately. There is no electoral threshold. Eleven districts apply the divisor method with downward rounding. See Table 1.30. The twelfth district, Northern Ireland, uses the single transferable vote scheme with fractional vote transfer: STVfra. The pertinent quota is the Droop-quota. See Table 1.31. Local Representation Several Member States divide their country into two or more electoral districts. Electoral districts are meant to strengthen the local ties between the electorate and their representatives. Districts may be handled in various ways. Belgium, France, Ireland, and the United Kingdom allocate their seats to the districts prior to election day. After the election the districts are evaluated separately, one at a time. Germany and Poland integrate districts through nested evaluations, in the form of super- and sub-apportionments. Yet another solution is a double- proportional apportionment method discussed in Chap. 14. Local representativeness is the historical origin of parliamentary elections. In former times the country was divided into single-seat “constituencies”. In each constituency the electorate voted a candidate into Parliament to represent them and their constituency. Personalization was the natural ideal at a time when political parties were non-existent. In modern times political parties have come to play a dominant role in the political structure of a country, and hence also in electoral systems. They are the institutional link mediating between the many voters and the few Members of Parliament. A candidate who is nominated by a party communicates to the electorate through personal standing as well as through party affiliation. Current proportional representation systems shift the focus towards the electorate’s division along party lines. The aim is to map the voter support of a party into this party’s parliamentary seats. Yet electoral systems allow many ways to maintain the original intention of personalization through local representativeness. There is a hierarchy of localization terminology to be found in proportional representation systems. We use the following classification. The “electoral region” is the largest possible territorial extension where the election takes place. The electoral region may be composed of various “electoral areas”. An electoral area may be subdivided into “electoral districts”. An electoral district is split into “electoral sections”. For EP elections the electoral region is the whole territory of the 28 Member States. Each Member State constitutes an electoral area. Some Member States subdivide their area into electoral districts: Belgium, France, Germany, Ireland, Poland, and the United Kingdom. The French Outre-Mer district is split into three electoral sections. 1.10 Slovenia, Slovakia, United Kingdom; Local Representation 35

Table 1.30 EP election 2014, United Kingdom, Districts 1–11 EP2014UK-LV0 Political group Vot es Quotient DivDwn District 1: East Midlands UKIP EFDD 368;734 2.6 2 CONS ECR 291;270 2.1 2 LAB S&D 279;363 1.995 1 GP GREENS/EFA 67;066 0.5 0 LDP ALDE 60;772 0.4 0 4Others – 53;516 – 0 Sum (Divisor) 1;120;721 (140,000) 5 District 2: East of England UKIP EFDD 542;812 3.9 3 CONS ECR 446;569 3.2 3 LAB S&D 271;601 1.9 1 GP GREENS/EFA 133;331 0.95 0 LDP ALDE 108;010 0.8 0 5Others – 72;023 – 0 Sum (Divisor) 1;574;346 (140,000) 7 District 3: London LAB S&D 806;959 4.2 4 CONS ECR 495;639 2.6 2 UKIP EFDD 371;133 1.95 1 GP GREENS/EFA 196;419 1.03 1 LDP ALDE 148;013 0.8 0 12 Other – 182;312 – 0 Sum (Divisor) 2;200;475 (190,000) 8 District 4: North East LAB S&D 221;988 2.02 2 UKIP EFDD 177;660 1.6 1 CONS ECR 107;733 0.98 0 LDP ALDE 36;093 0.3 0 GP GREENS/EFA 31;605 0.3 0 3Others 35;573 – 0 Sum (Divisor) 608;652 (110,000) 3 District 5: North West LAB S&D 594;063 3.96 3 UKIP EFDD 481;932 3.2 3 CONS ECR 351;985 2.3 2 GP GREENS/EFA 123;075 0.8 0 LDP ALDE 105;487 0.7 0 6Others – 97;861 – 0 Sum (Divisor) 1;754;403 (150,000) 8 (continued) 36 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.30 (continued) EP2014UK-LV0 Political group Vot es Quotient DivDwn District 6: South East UKIP EFDD 751;439 4.1 4 CONS ECR 723;571 3.9 3 LAB S&D 342;775 1.9 1 GP GREENS/EFA 211;706 1.2 1 LDP ALDE 187;876 1.02 1 10 Others – 120;683 – 0 Sum (Divisor) 2;338;050 (184,000) 10 District 7: South West and Gibraltar UKIP EFDD 484;184 2.95 2 CONS ECR 433;151 2.6 2 LAB S&D 206;124 1.3 1 GP GREENS/EFA 166;447 1.01 1 LDP ALDE 160;376 0.98 0 3Others – 49;160 – 0 Sum (Divisor) 1;499;442 (164,000) 6 District 8: West Midlands UKIP EFDD 428;010 3.3 3 LAB S&D 363;033 2.8 2 CONS ECR 330;470 2.5 2 LDP ALDE 75;648 0.6 0 GP GREENS/EFA 71;464 0.5 0 6Others – 90;582 – 0 Sum (Divisor) 1;359;207 (130,000) 7 District 9: Yorkshire and the Humber UKIP EFDD 403;630 3.1 3 LAB S&D 380;189 2.9 2 CONS ECR 248;945 1.9 1 GP GREENS/EFA 102;282 0.8 0 LDP ALDE 81;108 0.6 0 5Others – 80;547 – 0 Sum (Divisor) 1;296;701 (130,000) 6 District 10: Wales LAB S&D 206;332 1.9 1 UKIP EFDD 201;983 1.8 1 CONS ECR 127;742 1.2 1 PL-PW GREENS/EFA 111;864 1.02 1 GP GREENS/EFA 33;275 0.3 0 LDP ALDE 28;930 0.3 0 5Others – 22;934 – 0 Sum (Divisor) 733;060 (110,000) 4 (continued) 1.11 Diversity Versus Uniformity 37

Table 1.30 (continued) EP2014UK-LV0 Political group Vot es Quotient DivDwn District 11: Scotland SNP GREENS/EFA 389;503 2.8 2 LAB S&D 348;219 2.5 2 CONS ECR 231;330 1.7 1 UKIP EFDD 140;534 1.004 1 GP GREENS/EFA 108;305 0.8 0 LDP ALDE 95;319 0.7 0 3Others – 30;273 – 0 Sum (Divisor) 1;343;483 (140,000) 6 EP election 2014, United Kingdom, Districts 1–11: The United Kingdom allocates its 73 seats to twelve districts which are evaluated separately. Districts 1–11 use the divisor method with downward rounding. District 12 is shown in Table 1.31

Table 1.31 EP election 2014, United Kingdom, District 12 EP2014UK-STV Party Political group First pref STVfra District 12: Northern Ireland Martina Anderson SF GUE/NGL 159;813 1 Diane Dodds DUP NI 131;163 1 Jim Nicholson UUP ECR 83;438 1 Alex Attwood SDLP S&D 81;594 0 Henry Reilly UKIP EFDD 24;584 0 Ross Brown GP GREENS/EFA 10;598 0 Mark Brotherston CONS ECR 4144 0 Three further candidates – – 130;791 0 Sum (Droop-quota) (156,532) 626;125 3 In the United Kingdom, District 12: Northern Ireland applies a single transferable vote (STV) scheme. Surplus votes are transferred in fractions as given by the distribution of lower preferences (STVfra). See Table 1.30 for Districts 1–11

1.11 Diversity Versus Uniformity

Ever since its inception the EP has announced its intention to unify the procedures that the Member States employ for EP elections. Electoral systems encompass more than just counting votes. They determine who stands at the election, how they register, if they are given access to the media, whether they are reimbursed for their expenses, which ballot design is submitted to the voters and much more. But even when the view is narrowed down to what happens with the resulting vote counts, the multitude of procedures in the 28 Member States is perplexing. Often a country is treated as a single electoral district. Some Member States establish multiple electoral districts which are evaluated separately. Others form districts, too, but handle them through nested calculations of a super-apportionment followed by several sub-apportionments. Some states admit alliances, Others do not. 38 1 Exposing Methods: The 2014 European Parliament Elections

Table 1.32 Electoral indices of the 28 Member States Member Vot e Electoral Apportionment state pattern(s) threshold method(s) AT LV1 4% of valid votes DivDwn BE*3 LVm None DivDwn BG LV1 Quotient of valid votes per seats HaQgrR CY 2CV 1.8% of valid votes HQ3grR CZ LV2 5% of valid votes DivDwn DE/16 LV0 None DivStd DK+2 1CV None DivDwn EE 1CV None DivDwn EL 4CV 3% of valid votes HQ3-EL ES LV0 None DivDwn FI 1CV None DivDwn FR*8 LV0 5% of valid votes, per district DivDwn HR LV1 5% of votes cast DivDwn HU LV0 5% of valid votes DivDwn IE*3 STV None STVran IT/5 3CV 4% of valid votes HQ1grR, HaQgrR LT 5CV 5% of votes cast HQ2grR LU 6CV None DivDwn LV mCV 5% of votes cast DivStd MT STV None STVran NL+2 1CV Quotient of valid votes per seats DivDwn, HaQgrR PL/13 1CV 5% of valid votes DivDwn, HaQgrR PT LV0 None DivDwn RO LV0 5% of valid votes DivDwn SE LV1 4% of valid votes Div0.7 SI LV1 4% of votes cast DivDwn SK LV2 5% of valid votes DQ3grR UK*12 LV0, STV None DivDwn, STVfra Belgium, France, Ireland, and the United Kingdom establish multiple districts (marked *) for separate evaluation. Germany, Italy and Poland subdivide the country into districts (marked /) for nested evaluations via super- and sub-apportionments. Denmark and the Netherlands feature list alliances (marked +); each alliance entails a sub-apportionment of its seats among its partners

Some states forgo an electoral threshold or even declare it unconstitutional. Others install a threshold but differ in its reference base, votes cast or valid votes or some Other key number. The seat apportionment methods vary to such an extent that it is even hard to tell how many of them are in use. The states’ electoral systems excel in diversity, not in uniformity. See Table 1.32. Does it matter? After all, the Union may be viewed as a timely political construction allowing the citizenries of its Member States to preserve their domestic identities and idiosyncrasies in a diverse world. On the Other hand, all parliaments 1.11 Diversity Versus Uniformity 39 in this world derive their political legitimization from the way in which they get elected, and uniformity is always part of the underlying electoral principles. We will have more to say about electoral principles in the next chapter. As far as the EP is concerned, the Union’s electoral principles are enshrined in its primary law and, to some extent, promise that all citizens of the Union are treated equally. Electoral equality can be reliably assessed only when the European Parties start functioning on the Union level and give rise to a political system in which the many domestic parties agree to find their place. Certainly this scenario does not apply to the 2014 elections. In order to be able to present illustrative calculations from a unionwide viewpoint we replace the invisible European Parties in the European Union by the visible Political Groups in the EP. To this end our tables mention for every domestic party the Political Group it joined. In Spain, Italy and Croatia some parties split their seats between two Political Groups; we split their votes accordingly. In Ireland, Malta, and the Northern Ireland district of the United Kingdom, where STV schemes are used, we aggregate only the first preferences that are shown in Tables 1.17, 1.22, and 1.31. Domestic parties not affiliated to a Political Group nor obtaining a seat, labeled “Others”, are omitted. The vote counts thus aggregated induce the unionwide seat apportionment in Table 1.33. If, in the future, votes are cast for properly campaigning European Parties, a single unionwide apportionment would faithfully reflect the political division of the Union’s electorate. Yet it would miss out on the geographical dimension that the Union is composed of 28 Member States. Divisor methods can be adapted to honor both dimensions: the geographical distribution of the Union’s citizens across Member States, and the political division of the electorate that is expressed by their party votes. These double-proportional methods are the topic of Chap. 14. All seat apportionment methods must obey the electoral principles decreed by a country’s constitution or, in the case of the EP, by the Union’s primary law. The Court of Justice of the European Union has yet to specify to what extent the

Table 1.33 Actual seats by Political Groups versus hypothetical seat apportionment, 2014 EP election Political group Actual seats Vot es Quotient DivStd Difference S&D 191 40;189;841 200:9 201 10 EPP 221 40;170;109 200:9 201 20 ALDE 67 13;169;384 65:8 66 1 GUE/NGL 52 11;985;225 59:9 60 8 ECR 70 11;933;741 59:7 60 10 GREENS/EFA 50 11;534;372 57:7 58 8 EFDD 48 10;822;471 54:1 54 6 NI 52 10;224;602 51:1 51 1 Sum (Divisor) 751 150;029;745 (200,000) 751 32–32 The hypothetical apportionment is based on the groups’ aggregate vote counts. The two seat vectors differ by a transfer of thirty-two seats 40 1 Exposing Methods: The 2014 European Parliament Elections

Union’s primary law binds the electoral provisions of the Member States. As for the European Union, the legal ramifications of the European Electoral Act are still in flux. Therefore Chap. 2 turns to a particular Member State, Germany, in order to study the electoral principles in its constitution, the German Basic Law. The exposition is exemplified with the data from the 2009 Bundestag election. However, the German Federal Election Law was amended for the 2013 election. The discussion of the current law is postponed until Chap. 13. Chapter 2 serves a more general function: to elucidate the interplay between constitutional principles and electoral systems. Chapter 2 Imposing Constitutionality: The 2009 Bundestag Election

Abstract Electoral systems cater to constitutional demands and political goals, and to procedural rules and practical manageability. The diverse requirements are exemplified by the 2009 election of the German Bundestag. The Federal Election Law provides citizens with two votes: a first vote to elect a constituency representative by plurality, and a second vote to mirror the electorate’s division along party lines by proportionality. As Germany is a federation of sixteen states, the federal subdivision is also incorporated. The Bundestag electoral system illustrates the five electoral principles that underlie Europe’s electoral heritage: to elect the members of parliament by direct and universal suffrage in a free, equal, and secret ballot.

2.1 The German Federal Election Law

The Bundestag (Federal Diet), Germany’s parliament, is one of the five constitu- tional organs of the country. The other four are the Federal President (Bundespräsi- dent), the Federal Government (Bundesregierung), the Federal Council (Bundesrat), and the Federal Constitutional Court (Bundesverfassungsgericht). The electoral system for the Bundestag is governed by the Federal Election Law (Bundeswahlgesetz). The law’s aim is to implement “a proportional representation system that is combined with the election of persons” (“eine mit der Personenwahl verbundene Verhältniswahl”). The law’s intentions and instructions grow out of the country’s history. Parliamentary representation in Germany began with the North-German Con- federation 1867–1871. It was established through the impetus and under the leader- ship of Prussian Prime Minister Otto von Bismarck. Impressed by the effectiveness of the franchise in Napoléon III’s Second Republic in France, Bismarck had the members of the Confederate Reichstag (Diet) elected in single-seat constituencies by straight majority, with a second-round runoff if the straight majority was missed in the first round. Like other designers of electoral systems, Bismarck wanted to secure a safe majority for the ruling government. Trusting that the masses would support monarchic needs more enthusiastically than bourgeoisie and nobility, Bismarck extended the franchise to universal manhood suffrage, practically a novelty in Europe.

© Springer International Publishing AG 2017 41 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_2 42 2 Imposing Constitutionality: The 2009 Bundestag Election

With the formation of the Imperial Reich in 1871 in the aftermath of waging a war against France, the confederate electoral system was copied into the new Constitution. There was a noticeable exception, however. In order to obligate the members of the Imperial Reichstag to serve the Reich’s interests and not just the needs of their local constituencies, and also to shield them from the evolving influence of political parties that Bismarck distrusted wholeheartedly, the Constitution’s Article 29 obliged the Members of the Reichstag to represent “the whole people” (“das ganze Volk”). The constitutional obligation that every Member of Parliament represents the whole people has since been upheld, in the Weimar Republic 1919–1933 as well as in the present Federal Republic. In the Confederate Reichstag 1867–1871 and in the Imperial Reichstag 1871– 1919, candidates stood at the election, and eventually were elected, on the basis of their personal standing. As parliamentary politics grew in its importance, the members of the Reichstag informally assembled in groups, and later formally aligned in political parties. In addition, the sheer number of voters under a universal manhood franchise necessitated a formal coordination of party politics. Eventually candidates were perceived as party deputies more than as individuals. They were nominated as one among many, on lists of nominees submitted by a party, and identified with this party. The developing party system met with the disapproval by those for whom the purpose of an election was to elect representatives of the people rather than deputies of political parties. The issue was strongly felt and widely discussed during the Weimar Republic. By then a proportional representation system had been adopted with every 60,000 votes justifying one seat, called the “automatic system”. With a large electoral turnout the Reichstag had many seats. When people stayed away from the polls the number of seats decreased. The house size of the Weimar Reichstag varied between a low of 459 seats in the elections 1920–1922, and a high of 647 seats in 1933. Variability of the house size was deemed inefficient, in particular when the Reichstag of one legislative period handed business over to its successor. Yet the bigger issue was how parties filled the seats they won. The law decreed strict adherence to the lists of nominees which the parties had submitted prior to the election. Voters could not express their preferences for particular candidates, nor delete or add names. A party’s list of nominees was definitive for the sequence by which seats were filled. Critics maintained that the rigid lists put voters at the mercy of party bosses rather than encouraging the election of committed individuals. Strong candidates with a leadership personality, unwilling to submit to the intervention of a party’s nomination assembly, would turn away from the political life of the Republic. Such problems had been foreseen much earlier. Siegfried Geyerhahn (Sect. 16.12 [1]) had proposed a system to combine proportional representation with the election of persons. His system rests on two pillars. First, he recommends a subdivision of the electoral region into equal-sized constituencies, half as many as there were seats to fill. Second, he proposes a vote pattern that might be called a “double-evaluated single vote”. On their ballot sheets voters mark a candidate of a party. These marks 2.1 The German Federal Election Law 43 are counted twice. The first count is countrywide and by parties, to determine the number of seats a party deserves proportionally. The second count is per constituency and by candidates. In each constituency, the candidate with the most votes is declared elected. Across the whole electoral region, the non-elected candidates of a party are ranked by the number of votes they gain in their constituencies. This generates a retrospective ranking of candidates based on the voters’ ballots, as an alternative to rigid lists of nominees which are submitted by the parties in advance. The number of constituency seats that are won by the candidates of a party is subtracted from the party’s overall seat number. The remaining seats are filled in the sequence given by the retrospective ranking of the party’s candidates. Geyerhahn also addressed the possibility that a party would win more con- stituency seats than entitled to by proportionality. Nowadays a surplus seat is called an “overhang seat” (Überhangmandat). Dismissed by Geyerhahn as a rare eventuality, the occurrence of overhang seats has become a common event. They have emerged in all Bundestag elections over the last 30 years, and in steadily increasing numbers. Overhang seats have been an inherent difficulty of the Federal Election Law up to the 2009 election. Geyerhahn’s pamphlet was published in 1902 in a prestigious series edited by prominent law scholars. The author did not enter into an academic career, though, and his name is absent from the debate during the Weimar Republic. A system similar to his was proposed in a 1925 newspaper article by Richard Thoma,a renowned law professor. Thoma opted for a ballot with two votes, a first vote for the election of a constituency representative, and a second vote for the election of a party list. A year later Thoma’s contribution triggered a response article by Wilhelm Heile. After the Second World War the Allied Powers installed a Parliamentary Council with the remit to draft a constitution for a new, democratic Germany. The Council decided that the constitution was to include just those electoral principles that were deemed fundamental, but no procedural particulars. Indeed Article 38 of the 1949 Basic Law stipulates that the Members of the Bundestag are elected by direct and universal suffrage in a free, equal, and secret ballot. The Members of the Bundestag are representatives of the whole people. They are not bound by orders or directives, and shall submit solely to their conscience. The task of designing an electoral system for the Bundestag was relegated to a subcommittee of the Parliamentary Council, the Committee on Electoral Procedure. The committee invited the expert witness Richard Thoma to review the electoral systems past and present. Wilhelm Heile, who had commented on Thoma’s 1925 newspaper article, was among the deputy committee members. After extensive deliberations the committee proposed an electoral system with a double-evaluated single vote, with rigid party lists to be registered before the election, and with a separate apportionment in every state of the Federal Republic. The common ground with Geyerhahn’s pamphlet is striking, but regrettably we lack evidence on how the proposal came about. It eventually found its way into the Federal Election Law, and was used in the first two Bundestag elections in 1949 and 1953. 44 2 Imposing Constitutionality: The 2009 Bundestag Election

In 1956 the law was amended substantially by introducing the two-votes ballot design. Furthermore, the seat apportionment calculations were arranged in two stages. The first stage is the super-apportionment: the apportionment of all Bundestag seats among the eligible parties proportionally to their countrywide vote counts. The second stage consists of the per-party sub-apportionments: the apportionment of the countrywide seats of a party among its state-lists of nominees proportionally to the parties’ statewide vote counts. Current Bundestag ballot sheets, printed and issued by the election authorities, consist of a single piece of paper displaying two columns. The left column, in black print, is the voter’s “first vote” (Erststimme). This is the vote to elect a constituency representative. The right column, in blue print, is the voter’s “second vote” (Zweitstimme). The second vote serves to elect one of the candidate lists that parties register in each of the sixteen states. The header of the right column includes a small-print hint, awkwardly worded, telling voters that the second vote is the “decisive vote for the distribution of the seats altogether among the distinct parties” (“maßgebende Stimme für die Verteilung der Sitze insgesamt auf die einzelnen Parteien”). The two-votes electoral system has grown into an export hit of democratic Germany. For instance New Zealand converted to it from first-past-the-post plurality in 1993, and the Scottish Parliament has been using it since its establishment in 1998. When adapted, the name may change, of course. New Zealand’s acronym MMP is indicative of a “mixed member proportional” parliament. The Scottish acronym AMP is short for an “additional member proportional” parliament. In Germany, the name “two-votes system” (Zweistimmensystem) is input-oriented. The name emphasizes that the vote pattern offers voters a dual choice, in fulfillment of the law’s aim to implement a proportional representation system that is combined with the election of persons.

2.2 Countrywide Super-Apportionment 2009

The Federal Election Law introduces an electoral threshold consisting of three components. A valid second vote becomes “effective” (that is, enters into the apportionment calculations) provided it is cast for a party (1) gaining at least 5% of the countrywide valid second votes, or (2) winning at least three constituency seats by first votes, or (3) representing a national minority. The three components honor second votes (1), first votes (2), and minority representation (3). The historical record of Bundestag elections provides some examples. (1) In 2009 twenty-seven parties participated at the election. Twenty-one parties gained less than 5% of the total of second votes and dropped out from the seat apportionment. (2) In 1994 the PDS, the successor of the communist party in the former German Democratic Republic, won four direct seats by first votes. They participated in the overall seat apportionment, although they gained only 4.4% of the second votes. (3) 2.2 Countrywide Super-Apportionment 2009 45

Table 2.1 Super-apportion- 17BT2009 Second votes Quotient DivStd ment of 598 seats by second votes: election to the 17th Super-apportionment Bundestag 2009 CDU 11;828;277 173.4 173 SPD 9;990;488 146.497 146 FDP 6;316;080 92.6 93 LINKE 5;155;933 75.6 76 GRÜNE 4;643;272 68.1 68 CSU 2;830;238 41.502 42 Sum (Divisor) 40;764;288 (68,196) 598 Every 68,196 votes justify roughly one seat. The seat apportionment is performed using the divisor method with standard rounding, DivStd. The notional house size of 598 seats is subsequently modified

In 1949 the SSW, a party recognized to represent the Danish minority, was entitled to a seat. It received just 0.3% of the second votes. The functioning of the Federal Election Law is illustrated with the election of the 17th Bundestag 2009. We begin with the proportionality part of the system, the decisive second votes. The 40,764,288 effective second votes are cast for six parties. SPD, FDP, LINKE, and GRÜNE present candidate lists in all sixteen states. CDU stands in fifteen states, but not in Bavaria. CSU campaigns in Bavaria, only. For every party, the second votes for their state-lists are aggregated into a countrywide count of second votes. The law decrees a notional Bundestag size of 598 seats. Therefore the apportionment calculations begin with the allocation of 598 seats proportional to second votes. This stage is called the “super-apportionment” (Oberzuteilung). The divisor method with standard rounding, DivStd, is used to translate second votes into seats. See Table 2.1. The 2009 numbers easily reveal the pertinent divisor interval. A divisor much smaller than 68,196 leads to more seats than 598. SPD, closest to acquiring the next seat, determines the lower limit: 9;990;488=146:5 D 68;194:46. A divisor much larger than 68,196 yields fewer seats. The party securing the last seat is CSU. Hence the upper limit is 2;830;238=41:5 D 68;198:50. Any number in the interval Œ68;194:46I 68;198:50 is a viable divisor. The midpoint 68,196.48 is reduced to as few digits as the interval permits to obtain the select divisor: 68,196. The Bavarian CSU results are readily finalized. The super-apportionment awards 42 “proportionality seats” to the CSU. However, the party wins 45 “direct seats”, since in all 45 Bavarian constituencies the CSU candidates get the most first votes. Beyond the 42 proportionality seats, three extra seats are needed; they are referred to as “overhang seats”. The law stipulates that the notional house size of 598 seats is enlarged by the three overhang seats. The other five parties call for a sub- apportionment each, to assign the party’s overall seats to that party’s state-lists of nominees. 46 2 Imposing Constitutionality: The 2009 Bundestag Election

2.3 Per-Party Sub-Apportionments 2009

Parties nominate their candidates on “state-lists” (Landeslisten), that is, a separate list of candidates for each state. Thus a state functions as a lower-level electoral district. The names for such districts vary greatly. It is Land in Germany, kiescollege and circonscription électorale in Belgium, circonscription in France, constituency in Ireland, circoscrizione in Italy, okre¸gach in Poland, and electoral region in the United Kingdom. To evade language barriers we refer to these units uniformly as “districts”. Districts are usually presented in some standard order. In Germany, the Federal Election Officer lists the sixteen states from North to South by their northern-most latitude. The states’ names are abbreviated by two-letter codes. See the top part of Table 2.2. Every party that stands in two or more states calls for a “sub-apportionment” (Unterzuteilung). Hence the 2009 election features five sub-apportionments; see Table 2.2. Column “Dir.” contains the number of direct seats a party gains in the state. Column “Second Votes” shows the statewide counts of second votes. The overall party-seats from the super-apportionment are apportioned again by means of the divisor method with standard rounding. The interim quotients for CDU are included in the top part of the table. For lack of space they are omitted from the SPD, FDP, LINKE and GRÜNE parts. The proportionality seats are exhibited in column “DivStd”. The final step relates direct seats to proportionality seats. If in a district a party’s number of direct seats exceeds its number of seats that is calculated on the basis of proportionality, then the direct seats persist and the proportionality seats become moot. In such a case additional overhang seats are brought to life and added to the notional house size of 598 seats. In all other cases the proportionality seats are dominant and persist. As an example consider the sub-apportionment to districts for CDU. The cases where the direct seats overrule the proportionality seats are marked in Table 2.2 with a dot (). In Schleswig-Holstein the CDU candidates win nine constituencies. The state-list’s share of the 173 countrywide CDU seats is but eight seats. This situation creates one overhang seat, as recorded in column “Overhang”. More overhang seats are brought into being in Mecklenburg-Vorpommern(2), Saxony (4), Thuringia (1), Rhineland-Palatinate (2), Baden-Württemberg (10), and Saarland (1). Thus the notional house size of the Bundestag increases by 21 CDU overhang seats. For the other four parties all direct seats are carried by the associated proportion- ality seats, even if barely so for SPD in the Free Hanseatic City of Bremen (HB: 2 direct seats and 2 proportionality seats) and in the State of Brandenburg (BB: 5 seats in either category). Nevertheless, the proportionality seats suffice to seat the constituency winners. The remaining seats are filled from the parties’ state-lists. In summary, the 17th Bundestag 2009 featured 24 overhang seats, three for CSU and 21 for CDU. They raised the Bundestag size beyond the notional size of 598 seats to an actual house size of 622 seats. They were shared between the six parties CDUWSPDWFDPWLINKEWGRÜNEWCSU in the ratios 194W146W93W76W68W45. 2.3 Per-Party Sub-Apportionments 2009 47

Table 2.2 Sub-apportionments of party-seats to districts: election to 17th Bundestag 2009 17BT2009 (continued) Dir. Second votes Quotient DivStd Overhang Sub-apportionment to districts: CDU SH Schleswig-Holstein 9 518,457 7.51 8 1 MV Mecklenburg-Vorpommern 6 287,481 4.2 4 2 HH Hamburg 3 246,667 3.6 4 0 NI Lower Saxony 16 1,471,530 21.3 21 0 HB Bremen 0 80,964 1.2 1 0 BB Brandenburg 1 327,454 4.7 5 0 ST Saxony-Anhalt 4 362,311 5.3 5 0 BE Berlin 5 393,180 5.7 6 0 NW North Rhine-Westphalia 37 3,111,478 45.1 45 0 SN Saxony 16 800,898 11.6 12 4 HE Hesse 15 1,022,822 14.8 15 0 TH Thuringia 7 383,778 5.6 6 1 RP Rhineland-Palatinate 13 767,487 11.1 11 2 BY Bavaria – – – – – BW Baden-Württemberg 37 1,874,481 27.2 27 10 SL Saarland 4 179,289 2.6 3 1 Sum (Divisor) 173 11,828,277 (69,000) 173 21 District Dir. Second votes DivStd Overhang Dir. Second votes DivStd Overhang Sub-apportionment to districts: SPD Sub-apportionment to districts: FDP SH 2 430;739 6 0 0 261;767 4 0 MV 0 143;607 2 0 0 85;203 1 0 HH 3 242;942 4 0 0 117;143 2 0 NI 14 1;297;940 19 0 0 588;401 9 0 HB 2 102;419 2 0 0 35;968 1 0 BB 5 348;216 5 0 0 129;642 2 0 ST 0 202;850 3 0 0 124;247 2 0 BE 2 348;082 5 0 0 198;516 3 0 NW 27 2;678;956 39 0 0 1;394;554 20 0 SN 0 328;753 5 0 0 299;135 4 0 HE 6 812;721 12 0 0 527;432 8 0 TH 0 216;593 3 0 0 120;635 2 0 RP 2 520;990 8 0 0 364;673 5 0 BY 0 1;120;018 16 0 0 976;379 14 0 BW 1 1;051;198 15 0 0 1;022;958 15 0 SL 0 144;464 2 0 0 69;427 1 0 Sum 64 9;990;488 146 0 0 6;316;080 93 0 (continued) 48 2 Imposing Constitutionality: The 2009 Bundestag Election

Table 2.2 (continued) District Dir. Second votes DivStd Overhang Dir. Second votes DivStd Overhang Sub-apportionment to districts: LINKE Sub-apportionment to districts: GRÜNE SH 0 127;203 2 0 0 203;782 3 0 MV 1 251;536 4 0 0 47;841 1 0 HH 0 99;096 1 0 0 138;454 2 0 NI 0 380;373 6 0 0 475;742 7 0 HB 0 48;369 1 0 0 52;283 1 0 BB 4 395;566 6 0 0 84;567 1 0 ST 5 389;456 6 0 0 61;734 1 0 BE 4 348;661 5 0 1 299;535 4 0 NW 0 789;814 11 0 0 945;831 14 0 SN 0 551;461 8 0 0 151;283 2 0 HE 0 271;455 4 0 0 381;948 6 0 TH 2 354;875 5 0 0 73;838 1 0 RP 0 205;180 3 0 0 211;971 3 0 BY 0 429;371 6 0 0 719;265 10 0 BW 0 389;637 6 0 0 755;648 11 0 SL 0 123;880 2 0 0 39;550 1 0 Sum 16 5;155;933 76 0 1 4;643;272 68 0 For CDU, the direct seats (column “Dir.”) overrule the proportionality seats (column “DivStd”) in seven states (marked ), thus giving rise to 21 overhang seats. For other states and other parties, the direct seats can be incorporated into the associated proportionality seats

The 1976 election to the 8th Bundestag was the last time when the notional house size did not have to be enlarged because of overhang seats. While Geyerhahn had envisioned the occurrence of overhang seats to be a rare eventuality, the rare eventuality is no longer an exception, but the rule.

2.4 Negative Voting Weights

Does the Bundestag seat apportionment fairly represent the whole people on the basis of the voters’ ballots on election day? As a matter of fact, the Federal Constitu- tional Court ruled on 3 July 2008 that the seat apportionment procedure, as described in the previous sections, violated the electoral principles of an equal and direct suffrage. Acknowledging that any amendment of the Federal Election Law poses a major challenge to the Bundestag, the Court allowed the subsequent 2009 election to be conducted according to the prevalent law despite its unconstitutionality. However, the Court ordered the Bundestag to restore the law’s constitutionality by 30 June 2011. The government majority tabled a proposal on 28 June 2011, and later voted it into law against a vehemently dissenting opposition. The amended law entered into force on 3 December 2011. The opposition minority and a group of dedicated 2.5 Direct and Universal Suffrage 49 citizens maintained that the amended law still violated the electoral principles of an equal and direct suffrage, and brought it before the Federal Constitutional Court. The Court ruled that the amended law indeed was incompatible with the electoral principles in the Basic Law, and declared it null and void. As of ten o’clock on 25 July 2012 when the court held a session to read its opinion, Germany had no applicable seat apportionment procedure specified in its Federal Election Law. The bone of contention is a bizarre effect called “negative voting weights”, or inverse voting weights. The name indicates a discordant behavior between votes and seats. A party may profit from fewer votes by getting more seats, other things being equal. In other words voters may support the party of their choice by not casting their votes for this party. Conversely, more votes may be detrimental because of entailing fewer seats. For example, CDU in 2009 might have profited from losing 18,000 second votes in Saxony (782,898 instead of 800,898), thereby winning an additional Bundestag seat (195 instead of 194). How? The loss of votes releases a proportionality seat of CDU in Saxony. The seat resurfaces in Lower Saxony. But CDU seats in Saxony are sealed as direct seats. Hence CDU in Saxony stays put, and CDU in Lower Saxony increases its seats by one. The final tally is more seats despite fewer votes. What seems a mere hypothetical construct turned into reality during the election to the 16th Bundestag in 2005. The main election took place on 18 September 2005. However, in the constituency Dresden I in Saxony, a candidate had died too suddenly for the party to nominate a substitute candidate for the main election. Thus a by-election had to be called; it took place two weeks later, on 2 October 2005. The real scenario was just like the hypothetical construct in the previous paragraph. The Dresden I by-election leaves no doubt that about ten thousand CDU supporters withheld their second votes from CDU in order not to harm the party of their choice. This incident proved to the Federal Constitutional Court that the Federal Election Law makes voters speculate on whether casting their votes for the party of their choice helps the party’s cause, or hinders it. The fact that casting a vote may prove detrimental, undermines the legitimizing function of democratic elections, and fools voters to a degree that is unacceptable. The Court ruled the law unconstitutional in so far as it allows negative vote weights to occur, and called upon the Bundestag to amend these provisions. The story continues in Chap. 13.

2.5 Direct and Universal Suffrage

The reasoning of the Federal Constitutional Court builds on the constitution, of course. Article 38, Section 1, of the German Basic Law specifies five electoral principles: Die Abgeordneten des Deutschen Bundestages werden in allgemeiner, unmittelbarer, freier, gleicher und geheimer Wahl gewählt. Sie sind Vertreter des ganzen Volkes, an Aufträge und Weisungen nicht gebunden und nur ihrem Gewissen unterworfen. 50 2 Imposing Constitutionality: The 2009 Bundestag Election

The Members of the German Bundestag are elected in a universal, direct, free, equal, and secret election. They are representatives of the whole people, not bound to orders nor instructions, and accountable solely to their conscience. The five principles of a direct and universal suffrage in a free, equal, and secret ballot also constitute Europe’s electoral heritage, according to the 2002 Code of Good Practice in Electoral Matters of the European Commission for Democracy Through Law (Venice Commission) of the Council of Europe. It is beyond the scope of this book to analyze the principles in full depth. We restrict ourselves to some brief comments to elucidate their meanings. The principle of direct suffrage demands that the translation of votes into seats is not impeded by any intervention. A prototype indirect election is the presidential election in the United States of America. The votes of the electorate are filtered by way of the Electoral College. The president is elected by the electors of the Electoral College, not directly by the people. Indirect, stratified electoral systems were common in the multi-layered stratified societies of the Middle Ages. In medieval times Augsburg, then a financial center of Europe, was a Free Imperial City in the Holy Roman Empire of the German Nation. The proceedings for the election of the Augsburg Mayor were somewhat circumstantial. Every male Augsburg citizen was a member of a guild, and his franchise was bound to the guild. In other words the electorate was subdivided into districts on the basis of social stratification, rather than geographical provenance. As a guild member, a man voted to elect his guild’s Council of Twelve (Zwölferrat). The Councils of Twelve of all Augsburg guilds elected the Great Council (Großer Rat). The Great Council elected the Governing Council (Kleiner Rat). The Governing Council elected the Mayor. Part of the layered structure of the society was the claim of the upper level to be the sounder part (sanior pars), the political board that knew better than the lower level. When in 1502 the Augsburg carpenter Marx Neumüller was elected to the Great Council and came to take his seat, he was sent home again (wider haim), because the council’s majority did not approve of his election. In contrast, the principle of a direct election leaves no leeway for any political body to claim to be the sounder part, nor to interfere otherwise in the translation of votes into seats. The principle of universal suffrage grants the franchise to every German. The meaning of “every German” has expanded over time. In imperial Germany it meant every male German of age 25 years and older, excluding the military, people living on welfare payments, and a few other groups of society. Nowadays the notion of every German embraces all male and female Germans who are at least 18 years old. Exclusions from the franchise are kept to a minimum. In the Middle Ages the concept of universal elections was unknown. All elections were limited to small electoral colleges. The archetype has always been the College of Cardinals to elect the Pope of the Roman Catholic Church. In addition many low level clerical institutions conducted elections to select their leaders. So did the secular world, again always limiting the electorate to an ensemble of privileged individuals. For example the King of the Holy Roman Empire of the German Nation 2.6 Free, Equal, and Secret Ballots 51 was elected by the Electoral College of the seven Prince Electors. The inception and enforcement of direct and universal elections are an achievement of modern history, growing out of the Era of Enlightenment, the founding of the United States of America, the French Revolution, and the times thereafter.

2.6 Free, Equal, and Secret Ballots

Yet even the Middle Ages had pondered already how electors could cast a free, equal, and secret ballot. In those days, the term “electors” meant the members of a small, distinguished, and well-defined electoral college. In 1299, the Catalan philosopher Ramon Llull (1232–1316) elaborated on the pros and cons of open versus secret votes. In 1433, the German clergyman Nicolaus Cusanus (1401– 1464), later promoted Cardinal of the Roman Curia, argued forcefully in favor of secret balloting to secure a free vote. It would keep electors from offering their votes for sale to the candidates, and it would prevent candidates from frightening the electors and exerting undue pressure on them. Moreover Cusanus held that a secret ballot was a necessity for all votes to acquire an equal impact on the final outcome. Historical sources prove that a free, equal, and secret ballot has become a prerequisite for the validity of an election. The principle of a free ballot means that voters are not subjected to any pressures about whom to vote for, that individuals may stand as candidates at their own discretion without anyone hindering them in doing so, and that parties may participate with a minimum of bureaucratic requirements. The principle of a secret ballot serves the same purpose today as it did in Cusanus’ time. It shields voters from being frightened or pressured by those who stand at the election, and it keeps them from selling their votes to candidates or parties. The principle of an equal ballot constitutes the essence of contemporary demo- cratic elections. The German Federal Constitutional Court has developed a com- prehensive jurisdiction concerning the issue of electoral equality in proportional representation systems. Of course, the Court’s rulings are binding only within Germany. We feel that they may radiate beyond, in view of their inner consistency as to what parliamentary elections are supposed to accomplish. Equality is a relation between many subjects, not a property of a single item. Equality depends on the reference set within which it applies. In Sects. 2.7–2.9 we examine three distinct reference sets: voters, Members of Parliament, and political parties. Within each set, equality may be captured by a precise numerical quantity. These quantities would be in a one-to-one relationship if seats were continuously divisible. This is not the case. Seats are discrete entities and come in whole numbers. For this reason it makes a difference whether equality aims at the voters, at the Members of Parliament, or at the parties, as emphasized in Sect. 2.10. 52 2 Imposing Constitutionality: The 2009 Bundestag Election

In the 2010 Treaty of Lisbon, the equality principle is missing from the parts that deal with the election of the EP. Section 3 of Article 14 reads: The members of the European Parliament shall be elected for a term of five years by direct universal suffrage in a free and secret ballot. Is the inclusion, or omission, of electoral equality at the discretion of the legislator? Every Member State of the European Union is among the 47 members of the Council of Europe and, as such, endorses the Venice Commission’s Code of Good Practice in Electoral Matters. However, it is a challenge to fill the abstract principle of electoral equality with a concrete meaning that would limit the margin of discretion of the legislator. Hence varied interpretations, not disagreement on its importance, may have prevented inclusion of the equality principle into the Treaty of Lisbon.

2.7 Equality of the Voters’ Success Values

The German Federal Constitutional Court uses the notion of “success value equality” (Erfolgswertgleichheit) of the voters’ votes to assess electoral equality in proportional representation systems. The court coined the notion right at the beginning of its functioning, in a decision of 5 April 1952 that refers to a similar decision of the Bavarian State Constitutional Court a month earlier. The subjects on whom this notion of electoral equality focuses are the voters: :::; alle Wähler sollen mit der Stimme, die sie abgeben, den gleichen Einfluss auf das Wahlergebnis haben. :::; all voters shall have the same influence on the result of the election with the vote they cast. Ever since, success value equality for all voters has been the central reference point for constitutional jurisdiction on proportional representation in Germany. Since the main protagonists at an election are the voters, it appears completely adequate to focus on them, and not on the candidates elected, nor on the parties mediating between those voting and those elected. In fact, this focus was empha- sized as early as in 1910. André Sainte-Laguë (Sect. 16.13 [1]) demands: Pour que l’égalité des bulletins de vote soit aussi complète que possible, chacun des électeurs doit avoir la même part d’influence. For the equality of the ballots to be as complete as possible, each voter must have an equal part in influence. Similarly George Pólya (Sect. 16.14 [1]) put voters ahead of political parties: Das Prinzip des gleichen Wahlrechts fordert die möglichst gleichmäßige Berücksichtigung der Wünsche aller Wähler, aber nicht der Parteien, als solcher. The principle of equal suffrage demands observing as evenly as possible the wishes of all voters, but not of the parties as such. 2.7 Equality of the Voters’ Success Values 53

A reference to Members of Parliament, or to parties, leads to manifestations of electoral equality distinct from voter-oriented equality, as explained in subsequent sections. Pursuant to the notion of “success value equality of the voters’ votes”, the Ger- man Federal Constitutional Court has developed a consistent body of jurisdiction on the principle of electoral equality. The jurisdiction is in text form, of course, but it is precise to a degree that it lends itself to a unique and compelling form of quantification. Electoral success manifests itself through the number of seats apportioned to the party of the voter’s choice, party P.AnyvoterofpartyP shares the individual success equally with the other voters who cast their vote for the same party. Hence the “success share” of a voter of party P is given by the ratio of seats relative to votes:

seat number of party P: vote count for party P

For example, in Table 2.1 (that is, without overhang seats) the success share of a CDU voter amounts to 173=11;828;277 D 0:000014626. Since the vote counts may range into the millions, a success share is a minute quantity. Besides being of an awkward order of magnitude numerically, success shares do not tell the full story. The impact of 173 seats is contingent on how many seats are available altogether. For instance, in a house of size 300 they constitute a straight majority, while in a house of size 598 they account for a bit more than a quarter of the seats. Similarly, the vote count of a party exhibits its true weight only when compared with the overall vote total. Thus the “success value of a voter’s vote‘” for party P is defined to be the ratio of seat shares relative to vote shares: = seat number of party P house size: vote count for party P = vote total

This definition turns the success values into manageable and meaningful quantities. Theoretically, if all votes enjoy the same success value, seat shares coincide with vote shares. All voters would have a success value unity, a 100% success. In practice, deviations from theoretical equality are unavoidable. For example in Table 2.1, the success value of a CDU voter turns out to be 0.997, while an SPD voter has success value 0.996. In other words, a CDU voter realizes a 99.7% success, an SPD voter a 99.6% success. The complete picture of the success values at the 2009 54 2 Imposing Constitutionality: The 2009 Bundestag Election

Bundestag election is as follows:

Vot er Success value (in %) Deviation from 100% CDU voter 99:7 0:3 SPD voter 99:6 0:4 FDP voter 100:4 0:4 LINKE voter 100:5 0:5 GRÜNE voter 99:8 0:2 CSU voter 101:2 1:2

Some voters stay short of a 100% success, others reach beyond. Immediately the question comes to mind as to whether the observed deviations from ideal equality can be reduced. We return to this question later, in Chap. 10. At present we have a more modest aim; simply to ask such questions. We may do so because the success value of a voter’s vote, while originally introduced as a normative, qualitative notion of constitutional law, may be identified with a quantitative, procedural formula that is amenable to a conceptual analysis.

2.8 Equality of Representative Weights

Another group of electoral protagonists are those elected, the Members of Parlia- ment. They enjoy the constitutional right of statutory equality. From their viewpoint the election is equal, provided every Member of Parliament represents the same number of voters. To this end we define the “representative weight” of a Member of Parliament of party P to be the average number of voters per seat:

vote count for party P : seat number of party P

Representative weights are party votes-per-party seat quotients. They result in fractional quantities which indicate the average number of voters represented by a party’s Member of Parliament. Representative weights are interpretable without further standardization. Ideally, if they were all equal to each other, they would coincide with the overall votes-to-seats ratio. For example, it is straightforward to evaluate the representative weight of a Mitglied des Bundestages (MdB, Member of Parliament) in Table 2.1. The weight of a CDU MdB amounts to 11;828;277=173 D 68;371:5 vote fractions. An SPD MdB carries a weight that is a bit heavier, 9;990;488=146 D 68;428 votes. The ideal representative weight, the votes-to-seats ratio, is 40;764;288=598 D 68;167:7 2.9 Satisfaction of the Parties’ Ideal Shares of Seats 55 vote fractions. Altogether the numbers emerging are as follows:

MdB Representative weight Deviation from 68,167.7 CDU MdB 68,371.5 203:8 SPD MdB 68,428.0 260:3 FDP MdB 67,914.8 252:9 LINKE MdB 67,841.2 326:5 GRÜNE MdB 68,283.4 115:7 CSU MdB 67,386.6 781:1

When looking for numerical evidence, the German Federal Constitutional Court calculates representative weights and rounds them to whole numbers. In this way the court evades the reference to fractional quantities, and replaces them by (whole) numbers of voters. While the court’s practice is convenient for communication purposes, it distracts from the technical difficulties of having to deal with interim quotients that are not whole numbers, in a context where only whole numbers make sense. In this book we stick to the original definition. Representative weights are votes-to-seats quotients and hence generally fractional numbers.

2.9 Satisfaction of the Parties’ Ideal Shares of Seats

Usually a parliament features just a handful or so of political parties, in contrast to hundreds of its members, and millions of its voters. In view of the size of the numbers it may seem that parties are simplest to deal with. They are not the most important group in an electoral system, though. From a constitutional viewpoint, parties rank only third in importance, behind voters, and behind Members of Parliament. In a proportional representation system a party may claim its “ideal share of seats”; that is, the share of seats corresponding to its share of votes:

vote count for party P house size: vote total For example, if a party gains 12.3% of the votes then it would claim 12.3% of the house seats. Since an arbitrary percentage of the house size usually yields a fractional number of seats, ideal shares are measured in seat fractions. For the 2009 Bundestag election in Table 2.1, the actual seats of a party, its ideal share of seats, and their difference, its seat excess, are as follows: 56 2 Imposing Constitutionality: The 2009 Bundestag Election

Party Actual seats Ideal share Seat excess CDU 173 173:5173 0:5173 SPD 146 146:5575 0:5575 FDP 93 92:6550 0:3450 LINKE 76 75:6360 0:3640 GRÜNE 68 68:1154 0:1154 CSU 42 41:5188 0:4812

Within each of the three groups—voters, candidates, and parties—the qualitative message is the same. For example, a CSU voter enjoys a success value that is larger than the ideal success value, by 1.2% points. A CSU MdB is better off with a representative weight that is lighter than the ideal representative weight, by 781.1 voter fractions. CSU as a party is allocated a seat number that exceeds the ideal share of seats, by 0.4812 seat fractions.

2.10 Continuous Fits Versus Discrete Apportionments

Let us hypothetically imagine an ideal world where Members of Parliament are divisible into continuous fractions. Then the ideal share of seats of a party P would constitute the solution. In terms of a formula, party P would be allocated an amount of seats given by the formula

vP yP D h; vC where vP denotes the vote count for party P, vC designates the total of all effective votes, and h signifies the house size. The outcome yP on the left-hand side is the amount of seat fractions sought. The change of lettering for fractional seats, yP, is meant to emphasize their different content as compared to the discrete seats xp eventually aimed at. The assignment yP complies perfectly well with all equality standards already mentioned. The success values of all voters turn out to be equal to unity and signal a uniform 100% success:

y =h P D 1: vP=vC

The representative weights of all Members of Parliament would become equal: v v P D C : yP h 2.10 Continuous Fits Versus Discrete Apportionments 57

The ideal share of seats of all parties coincides with the seat allocation, by the very definition of yP. The ideal world has no problems in coping with ideal equality. Alas, the world is real, not ideal. Members of Parliament are human beings who are entitled to be treated discretely, each in her or his own right. The problem is not to calculate continuous seat fractions. Rather, the discrete charm of each seat must be respected. The task is to determine a discrete apportionment, a procedure respecting the discrete character of the seats. Whatever continuous quantities are calculated during interim calculations, they must be rounded to whole numbers in the end. It seems easy enough to round fractional quotients to whole numbers. Rounding unfolds an enigmatic complexity, however, when it concerns parliamentary seats. This is no different from every-day life where we round in various ways to respect various sensitivities. When asked for age we round downwards until the very last minute when on our birthday we grow a year older. When paying a bill in a restaurant the appreciated rounding method is upward rounding, if only because of its implied expression of appreciation. When business partners negotiate contracts or convert currencies they use commercial rounding because experience has shown that it treats both partners in a fair and symmetric fashion. To cope with these issues in a systematic way, Chap. 3 develops a theory of rounding functions and rounding rules. Chapter 3 From Reals to Integers: Rounding Functions and Rounding Rules

Abstract A rounding function maps non-negative quantities into integers. Exam- ples are the floor function, the ceiling function, the commercial rounding function, and the even-number rounding function. A rounding rule maps non-negative quan- tities more lavishly into subsets of integers. Every rounding function or rounding rule induces a sequence of jumppoints, called signposts, where they advance from one integer to the next. Rounding rules map a signpost into the two-element set consisting of its neighboring integers, while non-signposts are mapped to singletons. Prominent examples are the rules of downward rounding, of standard rounding, and of upward rounding. The one-parameter families of stationary signposts and of power-mean signposts are of particular interest.

3.1 Rounding Functions

All seat apportionment methods need to map interim quotients of some sort into whole numbers. This is achieved by rounding functions, and by rounding rules. Rounding functions are discussed first. Let N WD f0; 1; 2; 3; : : :g denote the set of natural numbers. Definition A “rounding function” f W Œ0I1/ 7! N is a function from the non- negative half-axis into the set of natural numbers such that f is non-decreasing and satisfies f .n/ D n for all n 2 N. Consider the values f .t/ for arguments t in the “integer interval” Œn  1I n, with n  1.Att D n  1 we obtain f .n  1/ D n  1, while t D n yields f .n/ D n. Since f is non-decreasing, the interval Œn  1I n includes a jumppoint s.n/ at which the values of f jump from n  1 to n. The jumppoint s.n/ is mapped either to n  1, or else to n. Four instances deserve particular attention: the floor function, the ceiling func- tion, the commercial rounding function, and the even-number rounding function.

© Springer International Publishing AG 2017 59 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_3 60 3 From Reals to Integers: Rounding Functions and Rounding Rules

3.2 Floor Function

The mother of all rounding functions is the “floor function”, defined for all t  0 through

btcWDmax f n 2 N j n Ä t g :

From btcÄt < btcC1 we realize t  1

t DbtcC.t btc/:

Thus 12.34 has integral part b12:34cD12 and fractional part 12:34 b12:34cD 0:34. The floor function acts as the truncation operator that ignores all digits after the decimal point; this is why it is particularly suited for computing machinery. The function dates back to Gauss (1808) who denoted it by Œt; some authors refer to this notation as “Gauss brackets”. In this book brackets Œ will signify a general rounding function that is compatible with a general rounding rule, see Sect. 3.9.

3.3 Ties and the Need for Rules of Rounding

Seat apportionment problems involve the side condition that the sum of all seats must be equal to the preordained house size. The handling of these problems calls for a modification of rounding functions into rounding rules. We illustrate the insufficiency of rounding functions by means of the floor function. To this end we manipulate the data in the EP 2014 election in Austria in Table 1.3 so that they exhibit the cause of irritation, “ties”. Table 3.1 raises party A to 780,000 votes, and lowers party B to 650,000 votes. Thus both vote counts become integral multiples of the divisor 130,000. In a first attempt we stick to the divisor 130,000, and obtain the interim quotients in the fifth column of Table 3.1. The floor function rounds party A’s quotient 6:0 uniquely to 6 seats, and B’s quotient 5:0 to 5 seats. The other parties get 4W3W1 seats. The apportionment 6W5W4W3W1 deals out 19 seats, however. This is one seat too many in view of the Austrian allocation of 18 seats. Smaller divisors, such as 129,999, lead to the same complication. In a second attempt we increase the divisor to 130,001, so as to decrease the number of seats apportioned. Naturally, all interim quotients become smaller. Party A’s quotient 5.99995 is rounded downwards to 5 seats. Party B’s quotient 3.4 Rule of Downward Rounding 61

Table 3.1 Occurrence of ties (EP2014AT) Vot es Quotient DivDwn Quotient DivDwn Quotient DivDwn A 780;000 5:99995 5 6:0 6 6:00005 6 B 650;000 4:99996 4 5:0 4C 5:00004 5 C 556;835 4:3 4 4:3 4 4:3 4 D 410;089 3:2 3 3:2 3 3:2 3 E 229;781 1:8 1 1:8 1 1:8 1 Sum (Divisor) 2;626;705 (130,001) 17 (130,000) 18 (129,999) 19 The vote counts of the two strongest parties A and B in Table 1.3 are manipulated until they are tied. For house size 18 it is equally justified to allocate to them 6 and 4 seats, or 5 seats each. The two options are indicated by means of trailing plus and minus signs

4.99996 is rounded downwards to 4 seats. The resulting apportionment 5W4W4W3W1 accounts for 17 seats only, one seat too few. There are two options on how to proceed. Either we declare the problem of apportioning 18 seats to be unsolvable and turn away, with weird implications for the Austrian EP allocation. Or we surmise that, for the apportionment of the ten seats that are left to parties A and B, proper solutions lie in-between the first attempt that dealt out eleven seats between them, 6W5, and the second attempt that ended up with nine seats, 5W4.Thein- between solutions for ten seats are 6W4 or 5W5. Both solutions appear equally justified; a lot might be drawn to select one of them. It is not unusual, in a competition, that final results are tied, even though the event may occur only rarely. To save space we code a tie by a trailing plus sign or by a trailing minus sign. In Table 3.1, the notation 6 and 4C indicates two-element sets of feasible seat numbers, 6WDf6; 5g and 4CWDf4;5g. Every choice from these sets is allowed provided the sum of all seat numbers exhausts the preordained house size. With two parties tied there are two equally justified apportionments, 6W4W4W3W1 and 5W5W4W3W1. The number of tied apportionments is given by a binomial coefficient; see Sect. 4.7.

3.4 Rule of Downward Rounding

Rounding rules pave the way to handle ties in an efficient manner. They are set- valued mappings, admitting at their jumppoints the two-element set comprising the integer before the jump and the integer after the jump. We denote rounding rules by double brackets, as a reminder that they occasionally offer two feasible rounding values. The “rule of downward rounding” is defined for all t  0 through  fbtcg in case t ¤ 1;2;3;:::; bbtccWD ft  1; tg in case t D 1; 2; 3; : : : 62 3 From Reals to Integers: Rounding Functions and Rounding Rules

The expression n 2bbtcc readsas“n results from t via downward rounding”, or “t is rounded downwards to n”. For instance, in Table 3.1 the first three parties have     780;000 650;000 6 6; 5 ; 5 5; 4 ; 130;000 Dbb ccDf g 130;000 Dbb ccDf g   556;835 4:3 4 : 130;000 Dbb ccDf g

That is, downward rounding of the quotient of the first party yields 6 or 5, of the second party 5 or 4, of the third party uniquely 4. The information inherent in the rule of downward rounding is the same as that provided by the floor function in cases when t is not an integer. In these cases the floor function yields the value btc, while downward rounding is more circumstantial by packaging the same answer into a one-element set, fbtcg. However, in cases when t D 1; 2; 3; etc. is a positive integer the floor function results in a unique value, btcDt, whereas downward rounding yields a two-element set, bbtccDft  1; tg. Rounding rules are more laborious to deal with, but the added labor is worth the gain. The exposition provides persuasive evidence that rounding rules capture the occurrence of ties in a rather practical manner.

3.5 Ceiling Function and Rule of Upward Rounding

The counterpart of the floor function is the “ceiling function”, defined for all t  0 through

dteWDmin f n 2 N j n  t g :

The relation dte10)dte1. Only zero is mapped into zero, dteD0 ) t D 0. The “rule of upward rounding” that goes along with the ceiling function is  fdteg in case t ¤ 1;2;3;:::; t WD ft; t C 1g in case t D 1; 2; 3; : : :

Here two-valued rounding results are t Dft; t C 1g for positive integers t. For t D 0 the rounding result is unambiguously zero, 0 Df0g. The distinct orientation of upward rounding versus downward rounding becomes evident at the jumppoints t D 1; 2; 3; etc.: Upward rounding looks upwards, t Dft; t C 1g, while downward rounding turns downwards, bbtccDft  1; tg. 3.7 Rule of Standard Rounding 63

3.6 Commercial Rounding Function

A rounding function with a neutral orientation is the commercial rounding function (German: kaufmännische Rundung). Fractional parts are rounded downwards when smaller than one half, and upwards otherwise. The “commercial rounding function” is defined for all t  0 through  dte in case t btc0:5; htiWD btc in case t btc <0:5:

That is, values t in the half-open interval Œn  0:5I n C 0:5/, n  1, are rounded to n. Quantities t in Œ0I :5/ are rounded to zero. A mechanical prescription refers to the decimal representation of t. If its first digit after the decimal point is 0, 1, 2, 3 or 4, then the fractional part is considered to be a “minor fraction” and t is rounded downwards. If the first decimal place is 5, 6, 7, 8 or 9, then the fractional part is taken to be a “major fraction” and t is rounded upwards. In particular a fractional part exactly equal to one half is rounded upwards; this prescription leaves a residue of an upward orientation that lacks symmetry. The missing symmetry is restored by the “even-number rounding function”. By definition it coincides with the commercial rounding function except that numbers with a fractional part of one half are rounded to the nearest even integer:  1 1  dte in case t btc > 2 ; or t btcD 2 and dte even; hti WD < 1 ; 1 : btc in case t btc 2 or t btcD 2 and btc even

Hence the jumppoints 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, etc. are rounded to 0, 2, 2, 4, 4, 6, etc. The even-number rounding function is the default command implemented in software packages such as round in R, and Round in Mathematica. The reader may well ask: Why not odd? This is a good question, for which there is no persuasive answer.

3.7 Rule of Standard Rounding

The commercial and even-number rounding functions both jump at the midpoint n  1=2 of the integer interval Œn  1I n. For this reason they induce the same rounding rule, the “rule of standard rounding”. It is defined for all t  0 through  fhtig in case t ¤ 0:5;1:5;2:5;:::; hh tii W D ft  0:5; t C 0:5g in case t D 0:5; 1:5; 2:5; : : :

That is, if the fractional part of t is smaller than one half, then standard rounding rounds downwards. If the fractional part is larger than one half, then standard 64 3 From Reals to Integers: Rounding Functions and Rounding Rules

n+1 • •

n • •

n−1 • •

. .

2 • •

1 • •

• • ...... t −10 1 2 n−1 nn+1 • •

EXHIBIT 3.1 The rule of standard rounding. If the fractional part of t is equal to one half, standard rounding is undecided and may round either way, downwards and upwards. Otherwise t is rounded downwards if its fractional part is smaller than one half, and upwards if larger rounding rounds upwards. If the fractional part is exactly equal to one half, then standard rounding is undecided and turns both ways, downwards and upwards. The domain of applicability of standard rounding is extended to the negative half-axis by setting hh tii W D  hh  tii for t <0,wherefa; bgWDfa; bg. With this extension standard rounding also becomes well-behaved under translations by integers z 2 Z,

hh t C zii D hh tii C z for all t 2 R:

On the right-hand side the translation of sets is defined by fa; bgCz WD faCz; bCzg. See Exhibit 3.1. Standard rounding is the only rounding rule on all of the real line that conforms with a reflection around the origin as well as with a translation by an integer. Generally, the extension by reflection differs from the extension by translation. Therefore the domain of definition of general rounding rules remains restricted to the non-negative half-axis Œ0I1/. Only standard rounding has domain of definition R. 3.8 Signpost Sequences 65

3.8 Signpost Sequences

The three rules of downward rounding, standard rounding, and upward rounding provide but special instances of general rounding rules. Typically the integer interval Œn  1I n, n  1, is equipped with a jumppoint s.n/. The jumppoint itself is ambiguously rounded to the two-element set fn  1; ng. Otherwise the jumppoint determines the regions where to round downwards, or upwards. Below s.n/ a number t 2 Œn  1I n is rounded downwards to the singleton fn  1g. Above s.n/ the rounding of t is upwards to fng. Henceforward these jumppoints are called “signposts”. Sequences of signposts provide the expedient tool for the definition of rules of rounding. The present section introduces signposts sequences at a suitable level of generality, while Sect. 3.9 returns to the corresponding rounding rules. Specifically, the rule of downward rounding has signposts s.n/ D n, the rule of standard rounding s.n/ D n  1=2, and the rule of upward rounding s.n/ D n  1. The “localization property” s.n/ 2 Œn  1I n, n  1, is supplemented with the initialization s.0/ D 0, and with the “left-right disjunction” which features as part c of the following definition. Definition A “signpost sequence” s.0/; s.1/; s.2/; etc. is characterized by the three properties a, b,andc: a. (Initialization) The starting signpost is fixed at zero, s.0/ D 0. b. (Localization) All subsequent signposts belong to consecutive integer intervals:

s.n/ 2 Œn  1I n for all n D 1;2;::: c. (Left-right disjunction) If there exists a signpost hitting a positive left limit of its localization interval then all signposts are less than their right limits, and if there is a signpost hitting a right limit then all signposts, except possibly s.1/,are greater than their left limits:

s.m/ D m  1 for some m  2 H) s.n/m  1 for all m  2:

A signpost sequence is called “pervious” when the first signpost is positive: s.1/ > 0. It is called “impervious” when the first signpost vanishes: s.1/ D 0. The signpost sequences of the rounding rules already met in this chapter are:

Downward rounding; bbccW0; 1; 2; 3; : : : Standard rounding; hh  ii W 0; 0:5; 1:5; 2:5; : : : Upward rounding;  W 0; 0; 1; 2; : : :

Neglecting the initialization requirement s.0/ D 0, the remaining sequence s.1/, s.2/, s.3/; etc. is strictly increasing. Indeed, for n  1 non-strictness s.n/ D s.nC1/ 66 3 From Reals to Integers: Rounding Functions and Rounding Rules

...... s(0) = 0 123 n−1 nn+1 s(1) s(2) s(3) s(n) s(n+1)

EXHIBIT 3.2 Signpost sequences. Signpost sequences start with s.0/ D 0, and thereafter obey the localization property s.n/ 2 Œn  1I n, n  1. The exhibit sketches a signpost sequence with s.0/ D 0, s.1/ D 0:3, s.2/ D 1:6, s.3/ D 2:7, s.n/ D n  1=2,ands.n C 1/ D n C 1=2 entails s.n/ D n D s.n C 1/, because of the localization property. Thus s.n/ hits the right limit of its localization interval, and s.nC1/ hits its left limit. This constellation is ruled out by the left-right disjunction. The left-right disjunction makes another appearance when proving that the induced divisor method is exact; see Sect. 4.5. The name “signpost” is borrowed from Balinski and Young (1982, p. 62). It nicely alliterates with “Sprungstelle” in German, and with “seuil” in French. It is convenient to refer to s.n/ as the nth signpost. With the exception of the initial signpost s.0/ D 0, the labeling indicates that the first signpost s.1/ lies in the first integer interval Œ0I 1, the second signpost s.2/ in the second integer interval Œ1I 2, and so on. A signpost sequence decomposes the non-negative half-axis Œ0I1/ into suc- cessive intervals, Œ0I s.1//, Œs.1/I s.2//, Œs.2/I s.3//, and so on. For impervious sequences the initial interval Œ0I s.1// D Œ0I 0/ is empty and dispensable; it is retained for the sake of notational uniformity. The intervals are localized according to n 2 s.n/I s.n C 1/ . In other words, the interval Œs.n/I s.n C 1/ is the domain of attraction for rounding to the integer n. See Exhibit 3.2.

3.9 Rounding Rules

A rounding rule allows a signpost s.n/>0to be rounded ambiguously: downwards to n1 and, simultaneously, upwards to n. Thus it admits two values for the rounding of t D s.n/>0. Otherwise, it is single-valued. Definition A signpost sequence s.0/; s.1/; s.2/; etc. defines a “rounding rule” ŒŒ  by setting ŒŒ 0  WD f0g and, for all t >0and n 2 N,  fng in case t 2 .s.n/; s.n C 1// ; ŒŒ t WD fn  1; ng in case t D s.n/>0:

A rounding rule is called “pervious” when s.1/ > 0. It is called “impervious” when s.1/ D 0. The definition ensures that zero is always and unambiguously rounded to zero, ŒŒ 0  Df0g. Thus all rounding rules obey the “no input–no output law”. In the empirical examples, parties with vote count zero are not even mentioned as they 3.9 Rounding Rules 67 certainly do not get a seat. For impervious rules these are the only have-nots. For pervious rules, rounding annihilates small quantities: t 2 Œ0I s.1// ) ŒŒ t Df0g. The situation is similar to a sieve losing sand grains so small that they fall through and disappear. The attribute “pervious” (German: durchlässig) is meant to indicate that the rounding rule annihilates a positive input that is too small. In most instances a rounding rule returns a singleton, ŒŒ t Dfng. These instances apply whenever t lies between the nth jumppoint and its successor, s.n/0,forsomen  1. Then a rounding rule delivers the two- element set fn  1; ng. The rule considers it equally justified to round the input t D s.n/>0downwards to n  1, or upwards to n. The ambivalence that a tie may be rounded either way, downwards or upwards, is indispensable for realistically modeling practical electoral systems, as pointed out in Sect. 3.3. How do rounding rules relate to rounding functions? A rounding function Œ is said to be “compatible” with the rounding rule ŒŒ  when the rounding function maps to values that are feasible for the given rounding rule:

Œt 2 ŒŒ t for all t  0:

For example, the commercial rounding function is compatible with the rule of standard rounding, and so is the even-number rounding function. This shows that the relation of rounding rules to rounding functions is in general one-to-many. By definition rounding functions are non-decreasing (Sect. 3.1). Rounding rules share these properties in the sense of set-valued mappings. A rounding rule is “set- monotonic” in the sense of

t < T H) ŒŒ t Ä ŒŒ T ; where the right-hand side means that all n 2 ŒŒ t and all N 2 ŒŒ T satisfy n Ä N. Moreover, a rounding rule maps onto the set N inS the sense that the union of its ŒŒ  image sets is equal to the set of all natural numbers, t0 t D N. It is convenient to identify a rounding rule with its underlying signpost sequence without further ado. Hence we may start with a rounding rule ŒŒ  and instantly refer to its underlying signpost sequence s.n/, n  0. Or we name a specific signpost sequence and immediately apply the corresponding rounding rule. The identification is allowed due to the “fundamental relation”

n 2 ŒŒ t ” s.n/ Ä t Ä s.n C 1/ that holds for all t >0and for all n 2 N. The left-hand side refers to the rounding rule, the right-hand side to its signpost sequence. The fundamental relation is a direct consequence of the definition and is called upon in the sequel again and again. 68 3 From Reals to Integers: Rounding Functions and Rounding Rules

The class of all signpost sequences encompasses two subfamilies that deserve special attention because of their relevance to subsequent developments. The first is the family of stationary signposts; they are determined by a split parameter r 2 Œ0I 1. The second is the family of power-mean signposts; they are characterized by apowerparameterp 2 Œ1I 1.

3.10 Stationary Signposts

Definition The sequence of “stationary signposts with split” r 2 Œ0I 1 is defined through sr.0/ WD 0 and sr.n/ WD n  1 C r for all n  1. The stationary signpost sequence with split r is 0; r;1C r;2C r;3C r; etc. The position of the signposts sr.n/ D n  1 C r in the integer intervals Œn  1I n stays the same. The distance to the lower limit is r, to the upper limit, 1  r.The attribute “stationary” indicates this stability when passing from one integer interval to the next. For stationary signpost sequences the domains of attraction for rounding to a positive integer n  1 have the same length: sr.nC1/sr.n/ D .nCr/.n1Cr/ D 1. The length of the domain of attraction for rounding to zero is sr.1/  sr.0/ D r; it equals unity if and only if the split is unity: r D 1. All other splits r <1start with a first interval of length less than unity. This boundary effect has to be kept in mind. The family of stationary signposts starts from upward rounding (r D 0), passes through standard rounding (r D 0:5), and finishes with downward rounding (r D 1). It affords a smooth transition between these rounding rules. The family of power-mean signposts provides a similar embedding. Moreover it includes another two rules of traditional interest: harmonic rounding and geometric rounding.

3.11 Power-Mean Signposts

Definition The sequence of “power-mean signposts with power parameter” p 2 Œ1I 1 is defined throughesp.0/ WD 0 and, for all n  1, through

 Ã1= .n 1/p np p e . /  C ;0; ; sp n WD 2 for p ¤1 1 p es1.n/ WD n  1; es0.n/ WD .n  1/n; es1.n/ WD n:

The expression .n  1/p in the definition is nonsensical for n D 1 and p 2 .1I 0/. For these exceptional cases we setesp.1/ WD 0. The power-mean sequences with powers p 2 Œ1I 0 have first signpostesp.1/ D 0; they are impervious. Those with p 2 .0I1 satisfyesp.1/ > 0; they are pervious. 3.12 Simple Rounding Does Not Suffice! 69

The family of power-mean signposts includes the special rules of upward rounding (p D1), of standard rounding (p D 1), and of downward rounding 1; 0 (p D1). For p D it adjoins another two rules of traditional interest. The 1 1 1 signpostes1.n/ D .n  1/ =2 C n =2 is the harmonic mean of n  1 and n, whence the rule withp power parameter p D1 is called “harmonic rounding”. The signpostes0.n/ D .n  1/n designates the geometric mean of n  1 and n, whence the rule with power parameter p D 0 is called “geometric rounding”. In summary, the power-mean family embraces the following five “traditional rounding rules”:

Power p Split r Downward rounding W1 1 Standard rounding W 11=2 Geometric rounding W 0 – Harmonic rounding W1 – Upward rounding W10

The signpost esp.n/ is the power-mean with power parameter p of the limits n  1 and n of the integer interval Œn  1I n. A power-mean is a mean of powers of order p of two positive numbers:

 Ã1= ap C bp p ; a; b >0: 2 where

The exponents p ¤1;0;1 are immediately permissible. Since the expression is convergent as p tends to 1;0;1, the three exceptional cases fit in. Moreover, convergence takes place as a or b tend to zero. The limits give rise to the case distinctions in the definition of power-mean signposts. It is worth remembering that the parameterization is continuous: limq!pesq.n/ D esp.n/,foralln 2 N and all p 2 Œ1I 1.  Finally we note that calculus yields limn!1 esp.n/  .n  1=2/ D 0 whenever p 2 .1I 1/. Thus all power-mean rules for which the power parameter is finite converge as n !1to standard rounding. The diversity inherent in the power-mean family evaporates and reduces to three rules: upward rounding, standard rounding, and downward rounding. However, the transition to the limit n !1models a parliament with an infinite number of seats, whereas practical seat apportionment problems deal with smallish seat numbers n. Therefore both families merit attention, stationary signposts as well as power-mean signposts.

3.12 Simple Rounding Does Not Suffice!

Rounding rules are employed to map an individual quotient into a whole number, or into two whole numbers. The point is that usually quotients are rounded individually, one after the other, without respect to the other quotients that need 70 3 From Reals to Integers: Rounding Functions and Rounding Rules

Table 3.2 Insufficiency of simple rounding 1975 world population Population Proportion Percent Asia 2;295;000;000 0:57289 57 Europe 734;000;000 0:18323 18 Americas 540;000;000 0:13480 13 Africa 417;000;000 0:10409 10 Australia and Oceania 20;000;000 0:00499 0 Sum 4;006;000;000 1:00000 98 The percentages, obtained via commercial rounding of the proportions, sum to 98% only. The discrepancy of 2% of a world population of four billion accounts for more than eighty million people, a country of the size of Germany to be rounded, too. The procedural approach may be called “simple rounding”. A collective side condition such as exhausting a preordained house size cannot be accommodated. However, this condition forms an indispensable part of an apportionment problem. Since simple rounding makes no provisions to fulfill the side condition, its results generally fail to solve the apportionment problem. For example when quotients are rounded to percentages, the mere mention of the term “percentages” promises that the total is equal to 100. With simple rounding, however, it is rather likely that this promise does not come true. The sum possibly misses the target value 100 by a discrepancy of some percentage points, which may be too few or too many. Table 3.2 exhibits the 1975 world population; see Kopfermann (1991, p. 109). The percentage total equals 98, not 100, and leaves a discrepancy of 2% points. A missing 2% of a population of four billion accounts for more than eighty million people, a country of the size of Germany. Simple rounding annuls Germany by means of a rounding operation. Many statistical publications alert their readers by a disclaimer, hidden in the small print or somewhere else, that any percentages quoted may fail to sum to 100 “due to rounding effects”. However, when the 100 units signify parliamentary seats rather than percentage points, nobody dares to suggest that some seats disappear or others are brought into being solely because of “rounding effects”. The way-out of the dilemma is provided by apportionment methods. Chapter 4 is devoted to a class of apportionment methods called divisor methods. Chapter 5 deals with another class, quota methods. Both types of methods initialize their calculations by jumping into the vicinity of the preordained house size, but not necessarily hitting it exactly. The “discrepancy distribution”, the typical deviation of the initial seat total from the target house size, helps to analyze the complexity of the methods. Chapter 6 offers a detailed study of the discrepancy distribution. Chapter 4 Divisor Methods of Apportionment: Divide and Round

Abstract Apportionment methods are procedures for allocating a total number of seats proportionally to vote counts, census figures, or similar quantities. Apportion- ment methods must be anonymous, balanced, concordant, decent, and exact. Beyond these organizing principles the central issue is proportionality. This chapter focuses on the family of divisor methods; they follow the motto “Divide and round”. The properties of general divisor methods are elaborated in detail. Five divisor methods are of particular traditional interest: the divisor methods with downward rounding, with standard rounding, with geometric rounding, with harmonic rounding, and with upward rounding.

4.1 House Size, Vote Weights, and Seat Numbers

The seat apportionment problem presumes that there are h seats to be handed out. Thus the “house size” is a natural number: h 2 N. A single party would get all seats and the job would be done. Therefore the ensemble of parties 1;:::;` who are competing for the seats is assumed to comprise two or more parties. That is, the “size of the party system” is `  2. The apportionment is to be conducted in proportion to the parties’ vote weights v1;:::;v`. The weights often are vote counts, in which case they are positive integers. It is advantageous to allow them to vary continuously. A “vote weight” vj is defined to be any non-negative quantity, vj 2 Œ0I1/. We assemble the vote weights into a “vote vector” v and assume that some of its components are positive:

` v D .v1;:::;v`/ 2 Œ0I1/ ;v¤ 0:

The component sum of a vector is indicated by a subscript plus sign, vC WD v1 CCv` >0. Component sums of all sorts of vectors keep occurring in the sequel. For instance, if v1;:::;v` are vote counts, then the “vote share” of party j is given by wj WD vj=vC. Vote shares wj are fractional quantities between zero and one, with component sum equal to one: wC D 1. The vote share vector w D .w1;:::;w`/ qualifies as a permissible vote vector, as does the vote count vector v.

© Springer International Publishing AG 2017 71 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_4 72 4 Divisor Methods of Apportionment: Divide and Round

The output sought is a “seat vector” x D .x1;:::;x`/ for house size h;thatis, a vector whose components are natural numbers that sum to h. The component xj signifies the “seat number” of party j. Since seats are indivisible units, it is not possible to extend the range of seat numbers beyond the set of natural numbers. The solutions sought are discrete items, not quantities of a continuum. Similar apportionment problems are met when seats are allocated between several states in proportion to their census figures. Then the weight vj is the apportionment population of state j,andxj are the seats allocated. In other applications we may be given a set of measurements that need to be rounded to integral percentages xj 2f0; : : : ; 100g. Then the h D 100 apportionment units are percentage points, the weight vj is the score of the jth measurement, and xj is the percentage equivalent of vj. It is obvious how to adapt the apportionment results to these and other applications. Since the task of apportioning seats of a political body among political parties remains a core issue, we continue to orient the terminology towards this type of application. A concise notation for the set of all seat vectors for house size h is ˚ « ` ` N .h/ WD .x1;:::;x`/ 2 N j xC D h :

These vectors are familiar from combinatorial analysis. There they signify config- urations for placing h indistinguishable balls into ` cells. The total number of such hC`1 configurations is known to be given by the binomial coefficient `1 . Hence this coefficient determines the number of seat vectors that are feasible for house size h: Â Ã ` h C `  1 # N .h/ D : `  1

As an example we consider a situation as in Table 1.3  (EP2014AT). For the 22 7315 apportionment of 18 seats among five parties there are 4 D seat vectors. This is not an overly large number, yet it is too large for the options to be examined one by one. Or consider Table 2.1 (17BT2009) where 598 seats are apportioned 603 653;408;984;970 among six parties. There are 5 D feasible seat vectors, more than six hundred billion! Evidently a systematic approach to the apportionment problem is needed in order to identify a small subset of seat vectors that offers viable solutions. The approach starts out from apportionment rules (Sect. 4.2) which map a given house size and a given vote vector into a set of seat vectors in a general and unspecific manner. The general level is narrowed down by five organizing principles (Sect. 4.3) capturing basic requirements that are dictated by the apportionment problem. Apportionment rules which satisfy the organizing principles are called apportionment methods (Sect. 4.4); these are at the center of our interest. The most powerful apportionment methods are divisor methods. They are introduced in Sect. 4.5 and scrupulously studied in the remaining sections of the present chapter. 4.2 Apportionment Rules 73

4.2 Apportionment Rules

It would be nice if the apportionment problem could be solved by some function A mapping a house size h and a vote vector v D .v1;:::;v`/ into a unique seat vector A.hI v/ in N`.h/. However, asking for a unique answer is asking too much. For instance if three seats must be apportioned among two parties which are equally strong, then the two seat vectors .2; 1/ and .1; 2/ are the only feasible solutions accepted by common sense. It looks capricious to single out one of them as the unique answer. A lasting definition must handle ties in a fashion more satisfactory than arbitrarily selecting one of them, as indicated in Sect. 3.3. Hence the mapping A cannot be a single-valued function. Instead, it is taken to be set-valued. The shift of viewpoint induces a shift of terminology. The mappings to be studied are set-valued “rules”, not single-valued “functions”. Thus an apportionment rule A yields a subset of seat vectors: A.hI v/  N`.h/. All seat vectors x; y;:::;z in the set A.hI v/ are legitimate solutions; they are mathematically equally justified. In fact, in virtually all practical applications (Chap. 1) the resulting set is a singleton: A.hI v/ Dfxg. In these cases the solution to the problem is unique after all, even though it appears in unusual packaging. But the exceptional occurrence of ties can now be handled in a satisfactory fashion. The set A.hI v/ simply enumerates the tied solutions. For instance the solution set of three seats among two equally strong parties simply documents the unavoidable ambiguity: A .3I .v1;v1// D f.2; 1/; .1; 2/g. Yet another peculiarity needs to be addressed. If the formal notion of an apportionment rule is to comply with practical applications of the rule, then its definition must not involve a reference to the size of the party system to which it is applied. Nobody distinguishes the “divisor method with downward rounding for five parties” in the Austrian EP election (Table 1.3) from the “divisor method with downward rounding for thirty-nine parties” in the Spanish EP election (Table 1.12). The rule is called the “divisor method with downward rounding”, without any reference to the size of the party system. Hence the definition of apportionment rules must not refer to the number of participating parties, `. Instead of invoking the precise number of parties we content ourselves with the fact that there are just finitely many of them. We simply append an infinite number of zeros and thereby introduce a “vote sequence” v D .v1;:::;v`;0;0;:::/.The explicit reference to ` is circumvented by demanding that only finitely many terms of the vote sequence v are positive. All other terms are zero. A sequence of real numbers is called “finitary” (German: finitär) when only finitely many of its terms are non-zero. All conceivable vote sequences are assembled into the set V: ˚ « V WD v 2 Œ0I1/N j v finitary;v ¤ 0 :

Whenever we look at a particular vote sequence v 2 V, we know that it has a last term that is positive: v` >0say. Hence the whole sequence v D .v1;:::;v`;0;0;:::/ conveys the same information as does its initial section 74 4 Divisor Methods of Apportionment: Divide and Round

.v1;:::;v`/. These preparations place us in a position to define apportionment rules in a formal manner. Definition An “apportionment rule” A maps every house size h 2 N and every vote sequence v 2 V into a “solution set” consisting of integer sequences, A.hI v/ Â NN, where the solution set A.hI v/ fulfills the three properties a, b,andc: a. (Non-emptiness) The solution set A.hI v/ is non-empty. b. (Inheritance of zeros) Every sequence x 2 A.hI v/ inherits all zeros of v:

vj D 0 H) xj D 0 for all j 2 N: c. (House size feasibility) Every sequence x 2 A.hI v/ has component sum h. Property a guarantees that an apportionment rule never finishes empty-handed. For every problem there is one solution at least, and possibly two or more. Since the input v is finitary so is the output x, by Property b. More precisely, if the input sequence v vanishes beyond its `th component, so does every associated output sequence x. Hence in Property c the component sum xC D x1 C x2 C  D x1 CCx` reduces to a sum of finitely many terms and is well-defined. Having defined apportionment rules in a formal way we hasten to rid ourselves of pretentious formalism. In all applications we must handle a vote sequence v that is known to vanish beyond its `th component. Therefore only the first ` components of a solution sequence matter. This is expressed by either of the following two displays: ˚ « ` A .hI .v1;:::;v`;0;0;::://Â .x1;:::;x`;0;0;:::/j .x1;:::;x`/ 2 N .h/ ; ` A .hI .v1;:::;v`// Â N .h/:

The first display, though formally correct, looks clumsy and is hard to grasp. The omission of the dazzling zeros makes the second display slightly incorrect. But it looks benign and is easy to comprehend. Since the parsimonious second display is unlikely to invite misinterpretations, we prefer it over the first and use it in the sequel. A solution set A.hI v/ is called “tie-free” when it is a singleton, A.hI v/ Dfxg. In this case it is unambiguous to identify the solution set fxg with its element x, and to say that the apportionment rule yields the unique solution x. The notation x 2 A.hI v/ states that the seat vector x is an apportionment of h seats according to thevotevectorv, but there may be further “tied solutions” y;:::;z 2 A.hI v/. For an abstract apportionment rule to prove practically useful it must satisfy five general principles. 4.3 The Five Organizing Principles 75

4.3 The Five Organizing Principles

A reasonable apportionment rule must be anonymous, balanced, concordant, decent, and exact. We discuss the five principles one after the other. Anonymity An apportionment rule A is called “anonymous” when every rear- rangement of the vote weights induces the same rearrangement of the seat numbers. Whether a party is listed first or last has no impact on its seat number. Chapters 1 and 2 make use of anonymity by ranking parties by decreasing vote counts, v1    v`. Thus the first party j D 1 is the votewise strongest party, the second party j D 2 is second strongest, and so on until the last party j D ` which is weakest. However, districts usually are listed following a fixed geographical order. Whichever order applies, the resulting seat numbers are carried along. Balancedness An apportionment rule A is called “balanced” when the seat num- bers of equally strong parties differ by at most one seat. That is, all seat vectors .x1;:::;x`/ 2 A .hI .v1;:::;v`// and all parties i; j D 1;:::;`satisfy ˇ ˇ ˇ ˇ vi D vj H) xi  xj Ä 1:

It would be tempting to insist that equally strong parties gain the same number of seats. But when two parties with identical vote counts must share an odd number of seats, a one-seat imbalance is unavoidable. Balancedness ascertains that in tied instances the spread does not grow beyond the inevitable minimum, one seat. Concordance An apportionment rule A is called “concordant” when, of any two parties, the stronger party gains at least as many seats as the weaker party. That is, all seat vectors .x1;:::;x`/ 2 A .hI .v1;:::;v`// and all parties i; j D 1;:::;`satisfy

vi >vj H) xi  xj:

Concordance is easy to check visually, provided parties are listed by decreasing vote counts. Then the corresponding seat numbers must be non-increasing too. When vote counts are ordered otherwise, as with districts, a visual check may need to be supplemented by a computer check. One would tend to believe that a “discordant result”, that is a non-concordant result, is an academic artifact and does not occur in practice. This is not so. Discordant apportionments emerge in electoral systems that involve a succession of several computational stages. The formation of alliances gives rise to many examples of discordant seat apportionments, see Sect. 7.11. Decency An apportionment rule A is called “decent”, or positively homogeneous of degree zero, when its solutions for vote weights .v1=c;:::;v`=c/ remain the same 76 4 Divisor Methods of Apportionment: Divide and Round whatever the scaling constant c.Thatis,forallc >0we have  . v/ 1 v : A hI D A hI c

In particular, scaling with the vote total vC turns the raw vote counts vj into the corresponding vote shares wj D vj=vC. Decency guarantees that the apportionment that is based on vote counts coincides with the apportionment based on vote shares. The transition to vote shares gives the apportionment problem a probabilistic twist. Since vote shares sum to one, wC D 1, they may be interpreted as probabilities. Because of the usually large size of the denominator vC, vote shares are virtually continuous quantities. In contrast, the output signifies probabilities x1=h;:::;x`=h with discrete numerators, xj 2f0;1;:::;h  1; hg. The stochastic view turns the task into an approximation problem. A probability distribution with continuous weights is to be approximated by a probability distribution with discrete weights. Exactness The last principle links the continuum property of the input to the discrete nature of the output. It looks a bit technical, yet the underlying idea is simple. A fitting input cannot but reproduce itself as the unique output: A.hI x/ D fxg for all x 2 N`.h/. The idea extends to sequences of vote vectors v.n/ that converge to x. Since the set N`.h/ is finite the induced sequence of solutions y.n/, n  1, is convergent if and only if it is eventually equal to some seat vector y. Hence without loss of generality we may assume that the sequence is constant throughout: y.n/ D y for all n  1. An apportionment rule A is called “exact” when all seat vectors y 2 N`.h/ that are obtainable from vote vectors v.n/ close to a given seat vector x 2 N`.h/ actually coincide with x. That is, if a seat vector y is common to all solution sets A .hI v.n//, then it is equal to x:

y 2 A .hI v.n// for all n  1 H) y D x; (4.1) whenever the vote vectors v.n/, n  1,convergetox and their weights satisfy vj.n/ D 0 when xj D 0. The definition of exactness simplifies in two situations in which the zeros of x are of no concern. Firstly, if all components of the limiting seat vector x are positive, as practically happens more often than not, then (4.1) simply says that all sequences of vote vectors which tend to a seat vector x eventually produce the solution x. Secondly, if the sequence of vote vectors is constant, v.n/ D x for all n  1, then (4.1) says that every seat vector x 2 N`.h/, when viewed as an input vote vector, is reproduced as the unique solution:

A.hI x/ Dfxg: (4.2)

Indeed, (4.1) entails y 2 A.hI x/ ) y D x;thatis,(4.2). The message of (4.2)is pleasing. The result of an exact apportionment rule A cannot be changed, let alone be improved, by repeatedly applying A to its solutions. 4.5 Divisor Methods 77

Definition (4.1) links the continuum character of the vote vector quadrant Œ0I1/` to the discrete nature of the seat vector grid N`.h/ in a manner more sophisticated than (4.2). The full force of (4.1) is needed in part IV of the proof of the Coherence Theorem in Sect. 9.8.

4.4 Apportionment Methods

Definition An apportionment rule A that is anonymous, balanced, concordant, decent, and exact is called an “apportionment method”. The five organizing principles say nothing about the main issue we are aiming at, the preservation of proportionality. Perfect proportionality would require the existence of a proportionality constant D >0satisfying xj D vj=D for all j Ä `. On the left-hand side the seat numbers xj are markedly discrete items. On the right-hand side the quotients vj=D are practically continuous quantities. Therefore perfect proportionality is generally beyond reach. We must be satisfied with some sort of approximate proportionality, xj vj=D. This is where the multitude of apportionment methods comes into play. There are plenty of methods purporting to achieve approximate proportionality in a satisfactory manner. A family of apportionment methods amenable to a fairly comprehensive analysis are divisor methods. They follow the motto “Divide and round”. The essential ingredient for generating a divisor method is a rounding rule ŒŒ  with its underlying signpost sequence s.0/; s.1/; s.2/; etc., see Sects. 3.8 and 3.9.

4.5 Divisor Methods

Definition The “divisor method A with rounding rule ŒŒ  ” maps a house size h 2 N ` and a vote vector .v1;:::v`/ 2 Œ0I1/ into the solution set n ˇ hh ii hh ii o ` ˇ v1 v` A.hI v/ WD .x1;:::;x`/ 2 N .h/ ˇ x1 2 ;:::;x` 2 for some D >0 : D D A divisor method is called “pervious” when the underlying rounding rule is pervious; it is called “impervious” when the rounding rule is impervious. In other words the seat numbers xj are obtained by applying the rounding rule ŒŒ  to the quotients of the vote weight vj and some common divisor D >0.The divisor D is determined so that the seat numbers exhaust the given house size: xC D h. Often the input quantities are not vote counts vj but vote shares wj D vj=vC. Since vote shares sum to one, wC D 1, they must be enlarged to reach the house size h. It is then conducive to talk of a “multiplier method” rather than a divisor method. Divisors D for vj, and multipliers  for wj are related through v v v j D C w D w I that is,  D C : D D j j D 78 4 Divisor Methods of Apportionment: Divide and Round

Both views could be subsumed under the neutral heading of “scaling method”. However, the term “divisor method” is firmly established in the literature. If seats were divisible items then the particular divisor vC=h, the votes-to-seats ratio, would provide perfectly proportional solutions, the “ideal shares” of seats .vj=vC/h. However, seats are indivisible units. A final rounding step is unavoidable. The inevitable rounding effects are outweighed by an appropriate adjustment of the divisor. Therefore the definition admits flexible divisors and only demands that “some” divisor D >0will do the job. Since the rounding effects remain small even when aggregated, it is safe to predict that a feasible divisor D lies in the vicinity of the votes-to-seats ratio vC=h. Or, what amounts to the same thing: a feasible multiplier  for the vote shares wj will be close to the house size h. The class of divisor methods comprises five traditional divisor methods of past and present prominence. They warrant individual identifiers instead of the generic symbol A to help memorize their relationship with the five traditional rounding rules (Sect. 3.11). We propose the following six-letter identifiers:

Identifier Name of method Traditional divisor methods DivDwn Divisor method with downward rounding DivStd Divisor method with standard rounding DivGeo Divisor method with geometric rounding DivHar Divisor method with harmonic rounding DivUpw Divisor method with upward rounding Families of divisor methods

DivPwrp Divisor method with power-mean rounding, p 2 Œ1I 1

DivStar Divisor method with stationary rounding, r 2 Œ0I 1

The divisor methods with downward and standard rounding are pervious; those with geometric, harmonic, and upward rounding are impervious. Divisor methods are apportionment methods as introduced in Sect. 4.4. Verifica- tion of the required properties is straightforward, though a bit lengthy. Skipping these details now incurs no lasting loss later. The arguments used resurface in various disguises again and again. Lemma Every divisor method is an apportionment rule that is anonymous, bal- anced, concordant, decent, and exact. Proof Let A be the divisor method with rounding rule ŒŒ  and underlying signposts s.n/, n  0. Part I establishes that A is an apportionment rule. Part II verifies the five organizing principles.

I. We claim that for every vote vector v D .v1;:::;v`/ the solution set A.hI v/ is non-empty. If some vote weight is zero, the associated seat number is also zero: vj D 0 ) xj D 0, due to the no input–no output–law. Hence we may assume that all components of the vote vector are positive, v 2 .0I1/`. The proof of 4.5 Divisor Methods 79

the non-emptiness of A.hI v/ is by induction on h. First we assume that A is pervious. Then we turn to impervious methods. In the pervious case the first signpost of the rounding rule is positive, s.1/ > 0. Starting with house size h D 0,leti be a party whose vote weight is strongest: vi D maxjÄ` vj >0. We select the divisor D WD 2vi=s.1/ > 0. Then all parties j Ä ` satisfy vj=D D .vj=vi/s.1/=2 Ä s.1/=2 < s.1/. Hence all quotients vj=D are rounded downwards to zero, and yield xj D 0. Thus h D 0 starts out with a solution set that is non-empty, .0;:::;0/2 A.0I v/. For the induction step we assume the solution set for house size h to be non- empty. Let .x1;:::;x`/ 2 A.hI v/. The definition of divisor methods guarantees the existence of some divisor D >0such that for all j Ä ` we have xj 2 ŒŒ v j=D ; that is, s.xj/ Ä vj=D Ä s.xj C 1/. This yields a lower bound for the divisor: D  maxjÄ` vj=s.xj C 1/ > 0.Leti be a party for which the lower bound is attained, and select the new divisor d WD vi=s.xi C 1/ > 0. Now the quotient vi=d D s.xi C 1/ may be rounded downwards to xi, or upwards to xi C 1.For progressing from house size h to house size h C 1, we round upwards to obtain the seat numbers yi WD xi C 1,andyj WD xj for j ¤ i. The seat vector .y1;:::;y`/ is a member of the solution set A.h C 1I v/. Hence the set is non-empty. The induction proof for the pervious case is complete. The impervious case has s.1/ D 0 0. Assuming all weights to be positive, the quotients 0vk ) ŒŒ v j=D  ŒŒ v k=D ) xj  xk (Sect. 3.9). (D) Decency is immediate since a scaling of the vote weights vj into vj=c is matched by scaling the divisor D into D=c. 80 4 Divisor Methods of Apportionment: Divide and Round

(E) Exactness is proved in two steps. The first step claims the weaker property that a divisor method A reproduces every seat vector x 2 N`.h/ uniquely: A.hI x/ Dfxg;seedisplay(4.2) in Sect. 4.3. The localization property s.xj/ Ä xj Ä s.xj C1/ (Sect. 3.8b) shows that xj 2 ŒŒ xj . With divisor D.x/ D 1 the vector x is seen to be a member of its solution set: x 2 A.hI x/. The assumption that there exists a solution y 2 A.hI x/ that is distinct from x leads to a contradiction. Indeed, if y has divisor D.y/<1D D.x/ then ŒŒ v j=D.y/  ŒŒ v j=D.x/ implies yj  xj (Sect. 3.9). Since y and x share the component sum h, they would have to be identical which they are not. A similar argument excludes D.y/>1. Hence we have D.y/ D 1 D D.x/,and xj; yj 2 ŒŒ xj for all j Ä `. Because of the common component sum h there exist two parties i ¤ k with xi < yi and xk > yk.Nowxi; yi 2 ŒŒ xi entails xi C 1 D yi D s.xi C 1/ > 0, while xk; yk 2 ŒŒ xk yields xk  1 D yk D s.xk/>0. Setting n WD xi C 1  1 and m WD xk  2, the joint fulfillment of s.n/ D n and s.m/ D m  1 violates the left-right disjunction (Sect. 3.8c). Hence the solution set is a singleton, A.hI x/ Dfxg, and the claim is established. The second step verifies definition (4.1) of Sect. 4.3. For a given seat vector x 2 N`.h/,letv.n/ 2 Œ0I1/`, n  1, be vote vectors that converge to x and that satisfy vj.n/ D 0 when xj D 0. Assume that the seat vector y is common to all solution sets A.hI v.n//, n  1. We need to show that y D x.Fromy 2 A.hI v.n// we know that there exists a divisor D.n/ satisfying yj 2 ŒŒvj.n/=D.n/ for all j Ä `.LetD be an accumulation point of the sequence D.n/, n  1. A passage to the limit n !1in s. yj/ Ä vj.n/=D.n/ Ä s. yj C 1/ yields s. yj/ Ä xj.D Ä s. yj C 1/,forallj Ä `.This means y 2 A.hI x/. The first step now secures y D x and thus verifies exactness. ut

4.6 Max-Min Inequality

The following result offers an easy check on whether a seat vector that is feasible for house size h belongs to the solution set of the divisor method under consideration. The check circumvents an explicit appeal to divisors. We restrict attention to positive ` vote vectors, v 2 .0I1/ , and as usual set vj=0 D1for vj >0. Theorem Let the divisor method A be induced by the rounding rule with signpost sequence s.0/; s.1/; s.2/, etc. Then a seat vector x 2 N`.h/ belongs to the solution set of a vote vector v 2 .0I1/`,

x 2 A.hI v/; if and only if v v max j Ä min j : jÄ` s.xj C 1/ jÄ` s.xj/ 4.6 Max-Min Inequality 81

Proof We have x 2 A.hI v/ if and only if there exists a divisor D >0satisfying xj 2 ŒŒ v j=D for all j Ä `. The fundamental relation s.xj/ Ä vj=D Ä s.xj C 1/ yields vj=s.xj C 1/ Ä D Ä vj=s.xj/. Hence the existence of D entails the max-min inequality. Conversely, every number D between the left maximum and the right minimum is a feasible divisor. ut As already mentioned we occasionally interpret a divisor method for vote counts vj as a multiplier method for vote shares wj D vj=vC. Since divisors and multipliers are inversely related, the multiplier version of the max-min inequality reads

s.x / s.x C 1/ x 2 A.hI w/ ” max j Ä min j : jÄ` wj jÄ` wj

In the remainder of this book we refer to either version as “the” max-min inequality. It will be clear from the context whether the reference is to the divisor version, as in the theorem, or to the multiplier version, as in the current paragraph. The max-min inequality is extremely fruitful. It designates three sets of pertinent importance: the delimited interval, the set of parties for which the lower interval limit is attained, and the set of parties for which the upper interval limit is attained: Ä  v v D.v; x/ WD max j I min j ; jÄ` s.xj C 1/ jÄ` s.xj/  ˇ  ˇ v v ˇ i j I.v; x/ WD i Ä ` ˇ D max ; s.xi C 1/ jÄ` s.xj C 1/  ˇ  ˇ v v ˇ k j K.v; x/ WD k Ä ` ˇ D min : s.xk/ jÄ` s.xj/

The “divisor interval” D.v; x/ consists of the divisors feasible for house size h and vote vector v. The “set of increment options” I.v; x/ assembles the parties i eligible to receive the .h C 1/st seat as soon as the divisor falls below its smallest feasible value. The “set of decrement options” K.v; x/ consists of the candidate parties k that give up the hth seat when the divisor increases beyond the largest feasible value and only h1 seats are available. Hence a seat vector x 2 A.hI v/ for house size h allows an immediate passage to seat vectors for house sizes h C 1 and h  1:

.x1;:::;xi1; xi C 1; xiC1;:::;x`/ 2 A.h C 1I v/ for all i 2 I.v; x/;

.x1;:::;xk1; xk  1; xkC1;:::;x`/ 2 A.h  1I v/ for all k 2 K.v; x/:

These relations encapsulate a rather efficient way to determine feasible divisors. 82 4 Divisor Methods of Apportionment: Divide and Round

4.7 Jump-and-Step Procedure and the Select Divisor

Given a house size h and a positive vote vector v 2 .0I1/` we now turn to calculating the solution set A.hI v/. Unfortunately no closed formula is available. Yet a simple “jump-and-step procedure” does the job. As with all algorithms, initialization is crucial. With a good initialization, the jump-and-step procedure quickly yields the solution. With a bad initialization, the procedure takes longer to find the same answer. Feasible divisors are likely to be close to the votes-to-seats ratio (Sect. 4.4). Trusting that vC=h is a promising divisor initialization we jump to an initial seat vector y with seat numbers yj 2 ŒŒ . v j=vC/h . One of following cases a, b, or c occurs:

(a) The component sum of y exhausts the house size: yC D h. In this case the initial seat vector is a solution: x D y. The procedure terminates. (b) The component sum of y stays below the house size: yC < h. In this case a further seat is handed out to some increment option i 2 I.v; y/. The increment step is repeated until the incremented seat vector x exhausts the house size: xC D h. (c) The component sum of y exceeds the house size: yC > h. In this case a seat is retracted from some decrement option k 2 K.v; y/. The decrement step is repeated until the decremented seat vector x exhausts the house size: xC D h. The jump-and-step procedure terminates with a seat vector x in the solution set A.hI v/. Calculations conclude by selecting a user-friendly divisor D encouraging users to verify that the seat numbers xj result from rounding the interim quotients vj=D. If the divisor interval is degenerate, D.v; x/ D ŒDI D, there is no choice other than D. If the divisor interval is non-degenerate, D.v; x/ D ŒaI b with a < b,every number in the interval is feasible. Our “select divisor” (German: Zitierdivisor) is obtained by commercially rounding the interval’s midpoint .a C b/=2 to as few significant digits as possible while remaining in the interval’s interior. The solution may be succinctly paraphrased as: “Every D votes justify roughly one seat.” The adverb “roughly” reminds us that the seat numbers x1;:::;x` undergo a final rounding step. The select divisor D is also a rounded quantity. However, the two rounding operations have distinct meanings. For seat numbers, rounding is imperative; they cannot be but whole numbers. For divisors, rounding is optional; the select divisor is user-friendly and eases communication. The exact limits a and b of the divisor interval are also rounded before printing. Here the rounding makes sure that the printed numbers stay inside the interval. The left limit a is rounded upwards to six significant digits (or more, if needed), and the right limit b is rounded downwards. Usually the select divisor D is an integer which is a multiple of a power of ten. On rare occasions it is a decimal number. For example, in the 1965 Bundestag election the apportionment method used was the divisor method with downward rounding. Its divisor interval is Œ63;119:2I 63;198:7. The midpoint 63,158.9 leads to the select divisor 63,160. In contrast, the divisor method with standard rounding yields the 4.8 Uniqueness, Multiplicities, and Ties 83

Table 4.1 A narrow divisor interval 5BT1965 Second votes Quotient DivDwn Quotient DivStd SPD 12;813;186 202.9 202 202.2 202 CDU 12;387;562 196.1 196 195.4997 195 CSU 3;136;506 49.7 49 49.5001 50 FDP 3;096;739 49.03 49 48.9 49 Sum (Divisor) 31;433;993 (63,160) 496 (63,363.6) 496 Divisor interval Œ63;119:2I 63;198:7 Œ63;363:5I 63;363:7 The divisor method with downward rounding has a divisor interval large enough to contain the select divisor 63,160. The divisor interval of the divisor method with standard rounding is so narrow that all divisors are fractional, including the select divisor 63,363.6 divisor interval Œ63;363:5I 63;363:7. The interval is so narrow that the select divisor requires one digit after the decimal point: 63,363.6. See Table 4.1.

4.8 Uniqueness, Multiplicities, and Ties

Uniqueness of a solution vector x and uniqueness of a feasible divisor D are complementary events. The solution is unique if and only if the divisor is not unique. That is, the solution is not unique if and only if the divisor is unique. These equivalences are a corollary of the max-min inequality. Corollary Every seat vector x 2 N`.h/ and every vote vector v 2 .0I1/` satisfy v v fxgDA.hI v/ ” max j < min j ; jÄ` s.xj C 1/ jÄ` s.xj/ v v fxg ¤ A.hI v/ ” max j D min j : jÄ` s.xj C 1/ jÄ` s.xj/

Proof The two statements are negations of each other. We prove the second. For the direct part assume that x; y 2 A.hI v/ are two distinct solutions: x ¤ y. Let D.x/ be a divisor for x,andD.y/ beadivisorfory.IfD.x/>D.y/, monotonicity entails xj Ä yj for all j Ä `. Because of equal component sums, xC D h D yC,the two vectors coincide: x D y, contradicting the assumption that they are distinct. A similar argument excludes D.x/ yk.Butxi; yi 2 ŒŒ v i=D implies xi C 1 D yi and the equality vi=D D s.xi C 1/. Similarly xk; yk 2 ŒŒ v k=D entails xk  1 D yk and the equality vk=D D s.xk/. Altogether we have v v v v D D i Ä max j Ä min j Ä k D D: s.xi C 1/ jÄ` s.xj C 1/ jÄ` s.xj/ s.xk/

This establishes equality in the max-min inequality. The direct part is proved. 84 4 Divisor Methods of Apportionment: Divide and Round

For the converse part we assume that the two sides of the max-min inequality share the same value, D. The increment options i 2 I.v; x/ satisfy the equality vi=D D s.xi C1/; this yields ŒŒ v i=D Dfxi; xi C1g. Similarly the decrement options k 2 K.v; x/ satisfy the equality vk=D D s.xk/; this leads to ŒŒ v k=D Dfxk  1; xkg. In the current case the option sets are disjoint: I.v; x/ \ K.v; x/ D;.Otherwise some component j, with vj=s.xj C 1/ D D D vj=s.xj/, would violate the fact that all signpost sequences are increasing. Hence we may fix two distinct components, i 2 I.v; x/ and k 2 K.v; x/, and define the seat vector y through yi WD xi C 1, yk WD xk 1,andyj WD xj for all j ¤ i; k.Theny is distinct from x, but still a member of the solution set A.hI v/. This proves the converse part. ut The proof of the Corollary also reveals how an identified member x relates to the other seat vectors in the solution set A.hI v/.Leta WD #I.v; x/ be the number of increment options of x,andletb WD #K.v; x/ be the number of decrement options. The seat vector x has a C b parties tied to signposts. When all parties are rounded upwards, the resulting component sum is h C a. This would exceed the target house size h by a seats. The house size fits when a of the aCb ties are rounded downwards, and the remaining b ties are rounded upwards. Thus the cardinality of the solution set is given by the binomial coefficient:  à  à vj vj a C b a C b max D min H) #A.hI v/ D D  2: jÄ` s.xj C 1/ jÄ` s.xj/ a b

For a compact notation we introduce trailing plus and minus signs. A natural number n is followed by a trailing plus sign when n C 1 is also feasible: nCWD fn; n C 1g. A trailing minus sign indicates that n as well as n  1 are allowed: nWD fn; n  1g.Table4.2 is a manufactured example pushing ties to an extreme. Every interim quotient hits a signpost. With two increment options  and three decrement v 5 10 options, the solution set DivDwn(18; ) consists of the 2 D equally justified seat vectors. The ten solutions are enumerated in the table. Each seat vector with

Table 4.2 Manufactured ties Ten equally justified DivDwn solutions Vot es Quotient #1 #2 #3 #4 #5 #6 #7 #8 #9 #10 A 780;000 6 6 6 6 5C 6 6 5C 6 5C 5C B 650;000 5 5 5 4C 5 5 4C 5 4C 5 4C C 520;000 4 4 3C 4 4 3C 4 4 3C 3C 4 D 390;000 3 2C 3 3 3 2C 2C 2C 3 3 3 E 260;000 2 1C 1C 1C 1C 2 2 2 2 2 2 Sum (Div.) 2;600;000 (130,000) 18 18 18 18 18 18 18 18 18 18 With divisor 130,000 all interim quotients hit some signpost, s.n/ D n, and may be rounded to n, or to n  1. Seat numbers may be decreased by one seat when flagged with a trailing minus sign. 3C2 10 With a trailing plus sign they may be increased by one seat. There are 2 D equally justified apportionments 4.9 Tie Resolution Provisions 85 its trailing plus or minus signs is sufficiently informative to recover the other nine solutions.

4.9 Tie Resolution Provisions

In popular elections vote counts reach into the thousands or millions, and ties occur only rarely. Nevertheless many electoral laws include a provision on how to resolve ties. The provision conveys a message beyond addressing an oddity of numbers. It demonstrates the legislator’s firm intention to design an electoral system that is definitive and unambiguous. The tie resolution provision practiced in Spain and elsewhere follows the motto “Stronger parties first”. For instance, when three seats are to be shared by five tied parties, the three strongest parties receive one seat each. In Table 4.2 this rule determines seat vector no. 1 as the final result. Ties persist only in the event that two (or more) vote counts are precisely equal. Another common tie resolution provision is to draw lots. The lots may extend over the equally justified seat vectors (in Table 4.2: ten). Equivalently, simple random sampling without replacement may be used to choose those parties in the tie group that gain a seat (in Table 4.2: three out of five). There is a variety of other tie resolution provisions, particularly when a single seat is tied between two parties. The seat may go to the stronger party (as mentioned above); or to the weaker party; or to the party that registered earlier; or to the oldest candidate of the tied parties; or to the candidate with the most preference votes; or to the candidate with the most children. Other provisions call for a run-off election. Or they grant the chairperson a decisive vote. In other instances it is the particulars at the election that entail a particular tie resolution decision. For example a tie emerged during the 2006 election of the 36-seat city council of the City of Uster in the Swiss Canton of Zurich. The pertinent electoral law governs all elections in the Canton of Zurich, including some with multiple districts. The number of seats to be filled in a district is known as the district magnitude. In each district, a voter may mark as many candidates on the ballot sheet as given by the preordained district magnitude. Summing raw vote counts over several districts provides an equal treatment of ballot marks, not of voters. In order to measure the support of a party with reference to voters, a party’s vote count is converted into the party’s “voter number” (German: Wählerzahl). It is defined to be the commercially rounded quotient of the vote count of a party over the district magnitude:   vote count voter number D : district magnitude

The formula does not take into account that some voters may not exhaust all the votes they have, or that other voters spread their votes over several lists (panachage, 86 4 Divisor Methods of Apportionment: Divide and Round

Table 4.3 Uster, 2006 city council election 2006UsterZhCH Voter number Quotient DivStd Vote count Quotient DivStd 01 SP 1996 10.7 11 71;847 10.7 11 02 SVP 1798 9.7 10 64;728 9.6 10 03 FDP 1017 5.47 5 36;613 5.4 5 04 EVP 378 2.0 2 13;600 2.0 2 05 CVP 271 1.46 1 9756 1.45 1 06 SD 160 0.9 1 5745 0.9 1 07 EDU 93 0.5 1 3353 0.499 0 08 SEDU 104 0.6 1 3752 0.6 1 09 GP 371 2.0 2 13;369 2.0 2 10 GLP 458 2.46 2 16;476 2.45 2 11 JEDU 93 0.5 0C 3365 0.501 1 Sum (Divisor) 6739 (186) 36 242;604 (6720) 36 With reference to voter numbers, EDU and JEDU are tied with common interim quotient 0.5. With reference to vote counts, JEDU outperforms the EDU by a margin of twelve votes. Therefore the tie is resolved in favor of JEDU cross voting). Judging from past elections in Zurich, there is no evidence that these two aspects entail a bias of any significance. The definition of voter numbers seems to serve all practical needs. For example, in Table 4.3 the EDU voter number is h3353=36iDh93:1iD93, while the JEDU voter number is h3365=36iD h93:47iD93. The law decrees the conversion of vote counts into voter numbers even when there is just a single district, as is the case in Uster. It so happens that a seat is tied between EDU and JEDU, in terms of voter numbers. Both lists have voter number 93. The seats are apportioned using the divisor method with standard rounding. With divisor 186, both lists have interim quotient 0.5. One of the two quotients has to be rounded upwards, the other one downwards. In the 2006 election, the election officer cast a lot and gave the seat to EDU. An attentive voter complained by pointing out that, in terms of vote counts, JEDU ranked above EDU by a margin of 12 votes. The electoral authorities overruled the lot decision, and awarded the tied seat to JEDU.

4.10 Primal Algorithms and Dual Algorithms

The remainder of this chapter is devoted to an in-depth discussion of the jump- and-step procedure. The jump-and-step procedure falls into the category of dual algorithms. Generally there is a distinction between primal algorithms and dual algorithms. A primal algorithm never leaves the set of feasible seat vectors, N`.h/. Given an initial seat vector whose component sum is h it examines the max-min inequality 4.11 Adjusted Initialization for Stationary Divisor Methods 87 that belongs to the divisor method to be applied. If the maximum exceeds the minimum, a seat is transferred between any two parties that attain the maximum and the minimum. The direction of the seat transfer is chosen so as to reduce the excess. Eventually the max-min inequality holds true, and then the current seat vector is a solution. With a good initialization, a primal algorithm reaches a solution with just a few seat transfers. With a bad initialization, it takes longer to finish the job. A good initialization is the seat vector that is provided by the Hare-quota method with residual fit by greatest remainders. A bad initialization would be the seat vector that gives all seats to the strongest party and no seat to the others. A dual algorithm proceeds along a sequence of seat vectors all of which result from the underlying divisor method. They may fail to exhaust the correct house size though. Hence seats are adjoined or retracted until the house size is met. The jump-and-step procedure is a dual algorithm. As with all iterative procedures, the crucial question is to find a good starting point. It is a misconception to believe that different initializations breed different procedures. Nor is someone who starts from a bad initialization and works diligently towards the result more serious about the problem than someone who uses a clever initialization and gets the job finished sooner. We present three initializations for the jump-and-step procedure: an initializa- tion adjusted to stationary divisor methods (Sect. 4.11), a universal initialization (Sect. 4.12), and a bad initialization (Sect. 4.13). Alas, the bad initialization is the favorite of law-making divisions and political scientists. Its tiring calculations are often obscured by jargon, by focusing on “highest comparative scores” (Sect. 4.14).

4.11 Adjusted Initialization for Stationary Divisor Methods

The effects of choosing different starting points are illustrated with the data from the Austrian EP election 2014 (Table 1.3). Austria uses the divisor method with downward rounding; its signposts are s.n/ D n. The initialization that is adjusted to stationary divisor methods promptly hits the solution. With the universal initialization, three steps are needed to reach the final result (Sect. 4.12). The bad initialization labors through eighteen steps, one for each available seat (Sect. 4.13). Each instance concludes with a few lines to calculate the select divisor. For the divisor method with stationary rounding, Sect. 6.1 recommends adjusting the initial divisor to the split parameter by way of D.r/ WD vC=.h C `.r  1=2//. The family includes the divisor method with downward rounding, with split r D 1. Hence the adjusted divisor is D.1/ D vC=.hC`=2/. For the Austrian data we obtain D.1/ D 2;638;781=.18 C 5=2/ D 128;721:02. The interim quotients vj=D.1/ are rounded downwards, and yield the initial seat vector: y D .5;5;4;3;1/. Since its component sum fits the house size, yC D 18 D h, the initial seat vector is the solution, x D y. No increment or decrement steps are needed, in this case. Every 128,721.02 vote fractions justify roughly one seat. 88 4 Divisor Methods of Apportionment: Divide and Round

D.1/ D vC=.h C `=2/ ÖVP SPÖ FPÖ GRÜNE NEOS Sum (Divisor)

Vote count vj 761,896 680,180 556,835 410,089 229,781 2,638,781 Quotient 5.9 5.3 4.3 3.2 1.8 (128,721.02)

Final seats xj 5 5 4 3 1 18 Calculation of the select divisor vj=s.xj/ 152,379.2 136,036Ž 139,208.8 136,696.3 229,781 (Ž D min) vj=s.xj C 1/ 126,982.7 113,363.3 111,367 102,522.3 114,890.5 ( D max) Divisor interval D Œ126;982:7I 136;036, Midpoint D 131;509:4, Select divisor D 130;000

For the readers who find vote fractions too awkward a format for a divisor, a few lines are adjoined to calculate the more charming select divisor (Sect. 4.7). In the line labeled “vj=s.xj/” the lowest score is marked (Ž). This is the critical divisor above which the house size decreases by one seat, from 18 down to 17. In line “vj=s.xjC1/” the highest score is marked (). This is the critical divisor below which the house size increases by one seat, from 18 up to 19. The marked quantities delimit the divisor interval: Œ126;982:7I 136;036. Its midpoint, 131,509.4, is reduced to as few significant digits as is possible while staying in the interval’s interior. The resulting select divisor is D D 130;000. Every 130,000 votes justify roughly one seat.

4.12 Universal Initialization

Without reference to the specifics of the underlying rounding rule, the votes-to-seats ratio provides a universal initialization that proves to perform quite satisfactorily. The Austrian data yield Duniv D 2;638;781=18 D 146;598:9. Division of the counts in the “Votes” line leads to the quotients, as shown below. Rounding them downwards, the procedure jumps to the “Initial seats” vector: y D .5;4;3;2;1/. It stays below the house size by three seats: yC D 15 < 18. Three seats must be incremented. According to Sect. 4.5 the increment options I.v; y/ are identified by means of the scores vj=s.yj C 1/. The signposts s.yj C 1/ are shown in the upper half of each “Seat” line, in small print. The scores vj=s.yj C 1/ are to be compared. The highest comparative score is marked with a dot (). It tells us to accord the sixteenth seat to FPÖ. Incrementation from the sixteenth to the seventeenth seat follows the same recipe, but is less laborious. Undotted comparative scores are copied from the “Seat 16” line into the “Seat 17” line. Only the dotted FPÖ score needs to be recalculated. The highest comparative score in the line “Seat 17”, again dotted, allocates the seventeenth seat to GRÜNE. In the same way the eighteenth seat is given to SPÖ. At this juncture the seat vector .5;5;4;3;1/ is obtained. Since it matches the house size, xC D 18 D h, it is the final result sought. 4.13 Bad Initialization 89

Duniv D vC=h ÖVP SPÖ FPÖ GRÜNE NEOS Sum (Divisor)

Vote count vj 761,896 680,180 556,835 410,089 229,781 2,638,781 Quotient 5.2 4.6 3.8 2.8 1.6 (146,598.9)

Initial seats yj 5 4 3 2 1 15 Seat 16 v1=.5 C 1/ D v2=.4 C 1/ D v3=.3 C 1/ D v4=.2 C 1/ D v5=.1 C 1/ D Increment: 126,982.7 136,036 139,208.8 136,696.3 114,890.5 FPÖ Seat 17 126,982.7 136,036 v3=.4 C 1/ D 136,696.3 114,890.5 GRÜNE 111,367 Seat 18 126,982.7 136,036 111,367 v4=.3 C 1/ D 114,890.5 SPÖ 102,522.3

Final seats xj 5 5 4 3 1 18 Calculation of the select divisor vj=s.xj/ 152,379.2 136,036Ž 139,208.8 136,696.3 229,781 (Ž D min) vj=s.xj C 1/ 126,982.7 113,363.3 111,367 102,522.3 114,890.5 ( D max) Divisor interval D Œ126;982:7I 136;036, Midpoint D 131;509:4,SelectdivisorD 130;000

For the readers who wish to report the result with a phrase like “Every D votes justify roughly one seat”, the last few lines calculate the select divisor D D 130;000. Every 130,000 votes justify roughly one seat.

4.13 Bad Initialization

The bad initialization starts with a huge initial divisor Dbad D1.What- ever the seat numbers, the induced interim quotients are rounded downwards to zero. To begin with, nobody gets anything. Thereafter seats are handed out one after the other. The incrementation strategy is the same as described in the previous section. The labor is diligently carried on until all available seats are allocated. This example has 18 seats and covers about one page. The 96 seats of the German EP allocation (Table 1.8) would fill five pages; the 631 seats in the German Bundestag (Table 2.1) thirty. With the bad initialization the complexity of the jump-and-step procedure grows linearly with the house size h. In contrast, the house size is irrelevant when the procedure starts with the adjusted initialization, or with the universal initialization. Then the number of steps needed is bounded in terms of the number of participating parties; see Sects. 6.2 and 6.3. 90 4 Divisor Methods of Apportionment: Divide and Round

Dbad D1 ÖVP SPÖ FPÖ GRÜNE NEOS Sum

Vote count vj 761,896 680,180 556,835 410,089 229,781 2,638,781 Seat 1 v1=.0 C 1/ D v2=.0 C 1/ D v3=.0 C 1/ D v4=.0 C 1/ D v5=.0 C 1/ D Increment: 761,896 680,180 556,835 410,089 229,781 ÖVP Seat 2 v1=.1 C 1/ D 680,180 556,835 410,089 229,781 SPÖ 380,948 Seat 3 380,948 v2=.1 C 1/ D 556,835 410,089 229,781 FPÖ 340,090 Seat 4 380,948 340,090 v3=.1 C 1/ D 410,089 229,781 GRÜNE 278,417.5 Seat 5 380,948 340,090 278,417.5 v4=.1 C 1/ D 229,781 ÖVP 205,044.5 Seat 6 v1=.2 C 1/ D 340,090 278,417.5 205,044.5 229,781 SPÖ 253,965.3 Seat 7 253,965.3 v2=.2 C 1/ D 278,417.5 205,044.5 229,781 FPÖ 226,726.7 Seat 8 253,965.3 226,726.7 v3=.2 C 1/ D 205,044.5 229,781 ÖVP 185,611.7 Seat 9 v1=.3 C 1/ D 226,726.7 185,611.7 205,044.5 229,781 NEOS 190,474 Seat 10 190,474 226,726.7 185,611.7 205,044.5 v5=.1 C 1/ D SPÖ 114,890.5 Seat 11 190,474 v2=.3 C 1/ D 185,611.7 205,044.5 114,890.5 GRÜNE 170,045 Seat 12 190,474 170,045 185,611.7 v4=.2 C 1/ D 114,890.5 ÖVP 136,696.3 Seat 13 v1=.4 C 1/ D 170,045 185,611.7 136,696.3 114,890.5 FPÖ 152,379.2 Seat 14 152,379.2 170,045 v3=.4 C 1/ D 136,696.3 114,890.5 SPÖ 139,208.8 Seat 15 152,379.2 v2=.4 C 1/ D 139,208.8 136,696.3 114,890.5 ÖVP 136,036 Seat 16 v1=.5 C 1/ D 136,036 139,208.8 136,696.3 114,890.5 FPÖ 126,982.7 Seat 17 126,982.7 136,036 v3=.4 C 1/ D 136,696.3 114,890.5 GRÜNE 111,367 Seat 18 126,982.7 136,036 111,367 v4=.3 C 1/ D 114,890.5 SPÖ 102,522.3

Final seats xj 5 5 4 3 1 18 Calculation of the select divisor

vj=s.xj/ 152,379.2 136,036Ž 139,208.8 136,696.3 229,781 (Ž D min)

vj=s.xj C 1/ 126,982.7 113,363.3 111,367 102,522.3 114,890.5 ( D max) Divisor interval D [126,982.7; 136,036], Midpoint D 131; 509:4, Select divisor D 130;000 4.14 Highest Comparative Scores 91

4.14 Highest Comparative Scores

The excessive space needed by the bad initialization may be abridged by arranging the comparative scores vj=s.n/ into a “table of comparative scores”. The table is assembled seat by seat. Our table indicates the succession in which the seats are apportioned by trailing numbers in italics: -1, :::,-18. For the first seat, vote counts are divided by the signpost s.1/ and entered into the initial row. The party with the highest comparative score gains the seat (ÖVP). For the next seat, the just successful party (ÖVP) activates its next comparative score v = .2/ ( ÖVP s ) and appends it to its column. The party with the highest value among the now active comparative scores gains the seat (second seat: SPÖ). The process continues until all available seats are dealt with:

ÖVP SPÖ FPÖ GRÜNE NEOS Vote count vj 761,896 680,180 556,835 410,089 229,781 vj=s.1/ 761,896-1 680,180-2 556,835-3 410,089-4 229,781-9 vj=s.2/ 380,948-5 340,090-6 278,417.5-7 205,044.5-11 114,890.5 vj=s.3/ 253,965.3-8 226,726.7-10 185,611.7-13 136,696.3-17 vj=s.4/ 190,474-12 170,045-14 139,208.8-16 102,522.3 vj=s.5/ 152,379.2-15 136,036-18 111,367 vj=s.6/ 126,982.7

Final seats xj 5 5 4 3 1

A table of comparative scores abridges the longer table in Sect. 4.13, but its growth still is linear in the house size h. The table needs about w1h rows, where w1 is the vote share of the strongest party. With a large house size h, such as 631 seats in the German Bundestag or 751 seats in the European Parliament, tables of comparative scores become inefficient and unwieldy. They provide no reasonable alternative to the jump-and-step procedure of Sect. 4.7. Moreover a table of comparative scores is devoid of the transparency that makes divisor methods so attractive. Some of the attractiveness can be rescued in the particular instance of the divisor method with downward rounding. With signposts s.n/ D n, comparative scores may be interpreted as votes-to-seats averages: vj=n. Highest comparative scores thus turn into “highest averages” (German: Höchstzahlen) that sound a bit friendlier to an uninitiated novice. However, the term “highest average” is to the point only for the divisor method with downward rounding. It makes no sense at all with any other divisor method. It is deplorable jargon to refer to the scores vj=s.n/ as averages for any method other than the divisor method with downward rounding. Yet highest-average aficionados managed to also force the divisor method with standard rounding into their ill-conceived terminology. With signposts 1=2; 3=2; 5=2 etc., the parties with highest values among the comparative scores vj=.1=2/; vj=.3=2/; vj=.5=2/ etc. evidently are the same as the ones with highest values among the “averages” vj=1; vj=3; vj=5 etc. With a view towards this obscure 92 4 Divisor Methods of Apportionment: Divide and Round cross relation the divisor method with standard rounding is occasionally called the “odd-number method”. This raises the curious question whether there also is something like an “even- number method”. Yes, there is. Clearly the “averages” vj=2; vj=4; vj=6 etc. induce the same sequencing of parties as do the comparative scores vj=1; vj=2; vj=3 etc. The latter belong to the signposts s.n/ D n. To cut a long story short: the even-number method is nothing but the divisor method with downward rounding.

4.15 Authorities

Experts often prefer a somewhat cryptic jargon by naming a seat apportionment method after an authority who fought for it. Unfortunately there exists no interna- tional agreement which person deserves the honor most. For example the divisor method with downward rounding is called the “Jefferson method” in the USA, the “D’Hondt method” in most of Europe, or the “Hagenbach-Bischoff method” in Switzerland. Here is a list of celebrities associated with the five traditional divisor methods: DivDwn Thomas Jefferson (Sect. 16.1), principal author of the US Declaration of Independence, third US President 1801–1809 Victor D’Hondt (Sect. 16.7), Professor of Law, Ghent University, co- founder of the Belgian Association réformiste pour l’adoption de la Représentation Proportionnelle 1881 Eduard Hagenbach-Bischoff (Sect. 16.6), Professor of Physics, Univer- sity of Basel, and cantonal politician DivStd Daniel Webster (Sect. 16.3), US statesman, Senator from Massachusetts, US Secretary of State Jean-André Sainte-Laguë (Sect. 16.13), Professor of Mathematics, Con- servatoire national des arts et métiers, Paris Hans Schepers (b. 1928), Physicist, Head of the Data Processing Unit, Scientific Services of the German Bundestag DivGeo Joseph Adna Hill (Sect. 16.8), Statistician, Assistant Director of the Census, US Bureau of the Census Edward Vermilye Huntington (Sect. 16.11), Professor of Mathematics, Harvard University, Cambridge, Massachusetts DivHar James Dean (1776–1849), Professor of Astronomy and Mathematics, University of Vermont, Burlington, Vermont DivUpw John Quincy Adams (1767–1848), US diplomat and statesman, sixth US President 1825–1829 4.15 Authorities 93

Chapter 16 quotes from the work of these protagonists in greater detail, in order to pay homage to their individual contributions to apportionment methodology. The next Chap. 5 introduces another important family of apportionment methods, quota methods. Quota methods are procedures complementary to divisor methods. Divisor methods fix the rounding rule and adjust the divisor. Quota methods fix the divisor and adjust the rounding rule. Chapter 5 Quota Methods of Apportionment: Divide and Rank

Abstract Quota methods constitute another important family of apportionment methods. They rely on a fixed divisor of some intrinsic persuasiveness, called quota, and follow the motto “Divide and rank”. The most prominent member of the family, the Hare-quota method with residual fit by greatest remainders, is discussed in the first part of the chapter. The second part addresses various variants of the quota, and various variants of the residual apportionment step. As a whole, the family of quota methods offers a more eclectic approach to apportionment problems than the family of divisor methods.

5.1 Quota Methods

Quota methods solve the same seat apportionment problem to which divisor methods are applied: h seats are to be apportioned among parties j Ä ` in proportion to their vote counts vj  0 with vC >0. Again a divisor is used to downscale the vote counts into the vicinity of the final seat numbers xj. The point is that quota methods consider the divisor to be fixed, vindicated by its intrinsic persuasiveness. To emphasize this point the fixed divisor is called “quota” and designated by the letter Q. A quota that appears compelling to some may appear debatable to others. Section 5.8 lists a variety of quotas that have found their way into electoral laws in former and present times. Quota methods are two-step procedures. The first step is called the “main apportionment”. It calculates an interim quotient vj=Q, and apportions its integral part yj Dbvj=Qc to party j.Letm WD yC denote the number of seats assigned by the main apportionment. The quota Q is supposed to be such that it assigns no more seats than are available: m Ä h; and that it leaves no more seats unassigned than there are parties: h  m Ä `. The second step, the “residual fit”, allocates the h  m 2f0;:::;`g residual seats left. The most popular procedure is the residual fit “by greatest remainders”. It ranks the fractional parts .vj=Q/bvj=Qc by decreasing size, and allocates one seat to each of the h  m greatest remainders. Thus quota methods are captured by the motto “Divide and rank”.

© Springer International Publishing AG 2017 95 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_5 96 5 Quota Methods of Apportionment: Divide and Rank

5.2 Hare-Quota Method with Residual Fit by Greatest Remainders

The most popular quota method is the “Hare-quota method with residual fit by greatest remainders”. We abbreviate it by HaQgrR. The quota used is the votes-to- seats ratio, vC=h, called “Hare-quota” in the context of quota methods. The induced quotients with the vote counts are the ideal shares of seats: vj=.vC=h/ D .vj=vC/h. The main apportionment allocates to party j Ä ` the integral part of its ideal share of seats: yj WD b.vj=vC/hc. Hence the number of seats accounted for in the main apportionment stays below the house size:     v1 v` v1 v` m D h CC h Ä h CC h D h: vC vC vC vC

The main apportionment exhausts the house size, m D h, if and only if all vote counts vj are integer multiples of the Hare-quota vC=h. As this is highly unlikely to happen, the method usually enters the second step, the fit of h  m residual seats. As all fractional parts are less than unity, .vj=vC/hb.vj=vC/hc <1, the number of residual seats is smaller than the number of parties: Â  Ã Â  Ã v1 v1 v` v` h  m D h  h CC h  h <`: vC vC vC vC

Because of integer constraints the inequality h  m <`tightens to h  m Ä `  1. That is, the main apportionment takes care of at least hC1` seats. In most practical applications the house size is much larger than the number of parties, whence the vast majority of seats is dealt with. For example in the German Bundestag with h D 598 seats and ` D 6 parties (Table 2.1), the main apportionment step would settle at least 593 seats. It would leave at most five seats for the residual fit. In the case of the Bulgarian EP election the method is applied with h D 17 and ` D 5 (Table 1.5). The main apportionment step is guaranteed to assign at least thirteen seats, and to leave at most four residual seats. With the 2014 data it actually allocates fifteen seats in the main apportionment and leaves two seats for the residual fit. The second step, the residual fit by greatest remainders, ranks the ideal shares’ fractional parts, and adds for the h  m largest of them one seat to the preliminary seat numbers of the main apportionment. Thus the final seat numbers are xi D yi C1 for the parties i with a larger remainder, and xk D yk for the parties k with a smaller remainder. Ties emerge if and only if several parties share the same remainder and this particular remainder is the split which separates larger remainders from smaller remainders. The Hare-quota method with residual fit by greatest remainders is anonymous, balanced, concordant, decent, and exact, as is easily verified. Hence it qualifies as an apportionment method in the sense of Sect. 4.3. 5.4 Shift-Quota Methods 97

5.3 Greatest Remainders and the Select Split

We illustrate the pertinent calculations with the Bulgarian 2014 EP election. The main apportionment rests on the ideal shares of seats listed in the column “Quotient” in Table 1.5. Their integral parts provide the preliminary seat numbers yj:

yGERB D 6; yBSP D 3; yDPS D 3; yBWC D 2; yRB D 1:

Thus m D 15 of the h D 17 seats are allocated in the main apportionment. Two seats are left for the residual fit. After ranking the ideal shares’ fractional parts f . j/ WD .vj=vC/h b.vj=vC/hc in decreasing order, the two parties with the largest remainders have their preliminary seat numbers increased by one seat, while the three parties with smaller remainders stay as is:

f .BSP/ D 0:845; f .DPS/ D 0:507 ) xi D yi C 1;

f .RB/ D 0:311; f .GERB/ D 0:174; f .BWC/ D 0:164 ) xk D yk:

This verifies the seat numbers in the last column of Table 1.5:

xGERB D 6; xBSP D 4; xDPS D 4; xBWC D 2; xRB D 1:

The same result is obtained using the divisor method with standard rounding. In fact, every split from the interval

Œ f .RB/I f .DPS/ D Œ0:311I 0:507 separates the fractional parts of the ideal shares of the parties that do not profit from the residual fit from those that do. Table 1.5 quotes the particular split of 0:5.The strategy for determining the select split r is the same as the strategy for working out the select divisor D (Sect. 4.7). The midpoint of the interval is rounded to as few significant digits as the interval’s interior permits. There is one extra rule. If the interval happens to contain the value 0:5, then we select r D 0:5. The extra rule indicates when standard rounding would do the job. The example shows that the results of the Hare-quota method with residual fit by greatest remainders can be replicated by means of some stationary divisor method. The replication approach extends to the larger class of quotas called shift-quotas. They perturb the Hare-quota so little that the second apportionment step can be conducted via a residual fit by greatest remainders.

5.4 Shift-Quota Methods

The “shift-quota” with “shift” s 2 Œ1I 1/, denoted by Q.s/,isdefinedby v Q.s/ WD C : h C s 98 5 Quota Methods of Apportionment: Divide and Rank

The “shift-quota method” with residual fit by greatest remainders and with shift s is abbreviated as shQgrRs.Theshifts D 0 retrieves the Hare-quota method with residual fit by greatest remainders: shQgrR0 =HaQgrR. Although there is no practical interest in negative shifts, s <0, all shift-quotas Q.s/, s 2 Œ1I 1/, allow the main apportionment to be paired with a residual fit by greatest remainders. Indeed, the main allocation for party j is     vj vj yj D D .h C s/ : Q.s/ vC

Hence the main apportionment has the upper bound yC Ä h C s < h C 1 seats. Because of integer constraints the inequality tightens to yC Ä h. The lower bound yC > h C s  `  h  `  1 tightens to yC  h  `. The range h  ` Ä yC Ä h confirms feasibility of a residual fit by greatest remainders. The next theorem presents a max-min inequality similar to that for divisor methods (Sect. 4.5). The theorem’s charm lies in avoiding a direct appeal to remainders.

5.5 Max-Min Inequality

Theorem Consider a shift-quota method shQgrRs with shift s 2 Œ1I 1/.Thena seat vector x 2 N`.h/ belongs to the solution set of a vote vector v 2 .0I1/`,

. v/; x 2 shQgrRs hI if and only if  à  à vj vj max .h C s/  xj Ä min .h C s/ C 1  xj : jÄ` vC jÄ` vC

Proof It is convenient to switch to the vote shares wj D vj=vC. The shift-quota Q.s/ D vC=.h C s/ then yields interim quotients vj=Q.s/ D wj.h C s/. For the proof of the direct part let x be a solution vector in shQgrRs.hI v/.The parties i that receive no residual seat are assembled in the set I. The complement K WD I0 includes the parties that are awarded one of the residual seats:

I WD f i Ä ` jxi Dbwi.h C s/c g ; K WD f k Ä ` jxk Dbwk.h C s/cC1 g :

In the left maximum of the max-min inequality parties i 2 I have a non-negative difference, wi.h C s/  xi D wi.h C s/ bwi.h C s/cDWf .i/  0. Note that f .i/ is 5.5 Max-Min Inequality 99 the fractional part of the interim quotient wi.h C s/. Parties k 2 K have a negative difference and drop out. We have  max wj.h C s/  xj D max .wi.h C s/  xi/ D max f .i/: jÄ` i2I i2I  The right minimum equals minjÄ` wj.h C s/ C 1  xj D mink2K f .k/. The residual fit by greatest remainders entails that the maximum stays below the minimum: maxi2I f .i/ Ä mink2K f .k/. For the proof of the converse part the max-min inequality is expressed as

wi.h C s/  xi Ä wk.h C s/ C 1  xk for all i; k Ä `:

A first summation over k ¤ i gives .`  1/ .wi.h C s/  xi/ Ä .1  wi/.h Cs/ C.`  1/  .h  xi/,foralli Ä `. This simplifies to wi.h C s/  xi Ä .`  1 C s/=` < 1.A second summation over i ¤ k yields an inequality leading to 0 Ä wk.hCs/C1xk, for all k Ä `. Together we obtain

wj.h C s/  1

The inequality string leaves just two possibilities, xj Dbwj.h C s/c or xj Dbwj.h C 0 s/cC1. Let the set I assemble the parties i with xi Dbwi.h C s/c,andK WD I the others. The first part of the proof shows that the max-min inequality takes the form maxi2I f .i/ Ä mink2K f .k/. Hence the parties i 2 I that are limited to their seats from the main apportionment have an interim quotient with a fractional part less than or equal to that of the parties k 2 K that are awarded one of the residual seats. This . v/ establishes x 2 shQgrRs hI . ut The max-min inequality gives rise to the “split interval” Ä Â Ã Â Ã vj vj R.v; x/ WD max .h C s/  xj I min .h C s/ C 1  xj : jÄ` vC jÄ` vC

The preceding proof reveals that R.v; x/ consists of the splits r 2 Œ0I 1 that separate the fractional part f .i/ of the interim quotient of a party i that has to make do with the main apportionment, xi Dbvi=Q.s/c, from the fractional part f .k/ of the interim quotient of a party k which is awarded one of the residual seats, xk Dbvk=Q.s/cC1. This feature is complementary to that of divisor methods. Divisor methods fix the rounding rule and adjust the divisor. Quota methods fix the divisor and adjust the rounding rule. 100 5 Quota Methods of Apportionment: Divide and Rank

5.6 Shift-Quota Methods and Stationary Divisor Methods

Œ 1 1/ Corollary Consider a shift-quota method shQgrRs with shift s 2  I . For all ` house sizes h 2 N and for all vote vectors .v1;:::;v`/ 2 .0I1/ there is a split r 2 Œ0I 1, which may depend on v (and on h and s), such that the shift-quota method with shift s and the stationary divisor method with split r have the same solution sets:

. v/  . v/: shQgrRs hI D DivStar hI

Proof The proof of the max-min inequality 5.5 shows that, given an arbitrary seat . v/  .v; / vector x 2 shQgrRs hI , every split r in the split interval R x establishes the assertion. ut The corollary mitigates the motto “Divide and rank”. The ranking of parties by fractional parts of their interim quotients actually is a transient step. The decisive step is the partitioning of parties into a group with smaller fractional parts where quotients are rounded downwards, and a complementary group with larger fractional parts where quotients are rounded upwards. The partitioning is succinctly specified by publishing a split r.Oncer is made known it is no longer needed to establish the ranking of the fractional parts. The motto “Divide and split” would subsume the functioning of quota methods more pertinently than “Divide and rank”.

5.7 Authorities

The Hare-quota method with residual fit by greatest remainders carries the name of Thomas Hare (Sect. 16.4), an English barrister and proponent of proportional representation systems. In his writings Hare repeatedly referred to the votes-to-seats ratio; hence this ratio may rightly be called the Hare-quota. However, the system that Hare fought for was a single transferable vote (STV) scheme. In terms of achieving proportionality the STV scheme is close to the apportionment methods discussed in this book. Nevertheless, the philosophy underlying STV schemes is rather different from quota methods, as are ballot structure and vote counting. Hence crediting the Hare-quota method with residual fit by greatest remainders to Hare is a misnomer of sorts. In the United States of America the Hare-quota method with residual fit by great- est remainders is called the Hamilton method, after Alexander Hamilton (Sect. 16.2). Hamilton successfully proposed the method to the House of Representatives, only to then see it vetoed by President George Washington. In Germany the method is associated with the name of mathematician Horst F. Niemeyer (Sect. 16.15). The Hare-quota method with residual fit by greatest remainders often comes under the alternative name of “largest remainder (LR) method”. The name emphasizes the second step of the method, the residual fit by greatest remainders. 5.8 Quota Variants 101

It entirely neglects the main apportionment step although this is where most of the seats are apportioned. For this reason we maintain acronyms of the type HaQgrR, even though they are somewhat bulky. Proponents of the short name LR method would presumably argue that there is no need to explicitly highlight the main apportionment step because it is natural and self-evident. History teaches otherwise.

5.8 Quota Variants

The term “quota”, as used by Hare and other authors, signifies a number of voters—to wit: of human beings—who are sufficient for a candidate to become their representative. Accordingly the original Hare-quota is a whole number: not the unrounded votes-to-seats ratio, but its integral part. With a hopefully pardonable inversion of the historical roots we refer to the integral part of the votes-to-seats ratio as a variant of the Hare-quota, the “Hare-quota variant-1” HQ1. Another variant rounds the votes-to-seats ratio upwards, the “Hare-quota variant-2” HQ2, or downwards and adds unity, the “Hare-quota variant-3” HQ3. In summary the unrounded Hare-quota, HaQ, is accompanied by three variants: j k l m j k vC vC vC vC HaQ D ; HQ1 D ; HQ2 D ; HQ3 D C 1: h h h h Another set of quotas is associated with the name of Henry Richmond Droop (Sect. 16.5). He proposed what came to be called the “Droop-quota”, DrQ WD bvC=.h C 1/cC1. For all practical purposes the Droop-quota is smaller than the Hare-quota. Therefore it increases the number of seats allocated in the main apportionment, whence the main apportionment reinforces its persuasive power. Fewer seats are passed on to the residual fit. Four Droop-quota variants may be met in theory and practice. Variant-1 omits the addition of unity, variant-2 revives it almost surely by rounding upwards instead of downwards, variant-3 uses standard rounding, and variant-4 remains unrounded:       v v v DrQ D C C 1; DQ1 D C ; DQ2 D C ; h C 1 h C 1 h C 1   v v DQ3 D C ; DQ4 D C : h C 1 h C 1

Variant 1 of the Hare-quota, and variants 1 and 3 of the Droop-quota may theoretically become zero. In such a case, when they could not be used as a denominator, they are redefined to be equal to unity (Sect. 1.6). 102 5 Quota Methods of Apportionment: Divide and Rank

The first seven quota variants are practically relevant. They are in current use, or have been used in the past:

HaQ Bulgaria, EP 2014 election (Table 1.5) HQ1 Italy, EP 2014 election (Table 1.18) HQ2 Lithuania, EP 2014 election (Table 1.19) HQ3 Austria, parliamentary elections, since 1992

DrQ Ireland, EP 2014 election (Table 1.17) DQ1 Solothurn, cantonal elections 1896–1977 DQ2 Solothurn, cantonal elections 1981–1993 DQ3 Slovakia, EP 2014 election (Table 1.29)

The Droop-quota variant-4 may be viewed as an unrounded approximation to the integral Droop-quota DrQ and its variants 1–3. Moreover it is the closure of the shift-quota family, lims!1 Q.s/ D DQ4.

5.9 Residual Fit Variants

At least three alternatives are available to substitute for the residual fit by greatest remainders, as mentioned already in Sect. 1.7. The variant gR1 relies on the greatest remainders of the parties’ interim quotients, but includes only those parties that receive at least one seat in the main apportionment. We refer to gR1 as the “full-seat restricted residual fit by greatest remainders”. The variant WTA follows the adage “winner take all” by awarding all residual seats to the strongest party. The variant -EL is peculiar to Greece (Sect. 1.4). All variants are or were used:

grR Bulgaria, EP 2014 election (Table 1.5) gR1 Lithuania, EP 2014 election (Table 1.19) WTA Solothurn, cantonal elections 1896–1917 -EL Greece, EP 2014 election (Table 1.11)

While the Greek version is one of a kind, the other three variants reflect the transition from plurality voting systems to proportional representation systems. In 1896 the Swiss Canton of Solothurn moved from plurality to proportionality. It implemented DQ1WTA, the Droop-quota variant-1 method with residual fit by winner-take-all. While from today’s viewpoint the winner-take-all imperative appears unacceptably biased, at the time its bias may have appeared negligible. In the abolished plurality system the winner-take-all rule applied to all seats while, in the novel proportional system, it applied to no more than a few residual seats. 5.10 Quota Method Variants 103

In 1917 Solothurn adopted the residual variant gR1. By eliminating parties that receive no seat in the main apportionment, the variant implements a kind of polar adage “loser-get-nil”. Naturally it is a widespread game for stronger parties to devise stumbling blocks for weaker parties to enter parliament. However, it is hard to defend such hurdles from a conceptual viewpoint. Quota methods presuppose that the quota represents voters, and so do quota remainders. They are human beings who are treated by different standards, not just fractional parts of interim quotients. For example Sect. 5.3 displays the fractional parts of the interim quotients for the Bulgarian 2014 EP election. These fractional numbers are a measure of human beings. The Hare-quota signifies batches of 110,280.1 voters. The six GERB main apportionment seats thus take care of 6 110;280:1 D 661;680:4 “voters”. This leaves 680;838  661;680:4 D 19;157:6 GERB “voters” as yet unconsidered. They are taken care of in the residual fit. In the example the “unused voting power”, u. j/, that is left after the main apportionment is

u.BSP/ D 93;196:8; u.DPS/ D 55;884:8; u.RB/ D 34;251:9; u.GERB/ D 19;157:6; u.BWC/ D 18;068:9:

Since the residual fit commands two seats, the two strongest of the six groups are able to achieve representation. Regrettably no seats are left for the three weaker groups of unconsidered voter. The full-seat restricted variant gR1 would look at these groups conditional on whether their fellow voters have secured representation or not. The unused voting power of party j and the fractional part f . j/ of the party’s interim quotient are related through u. j/ D f . j/Q,whereQ is the applicable quota. The notion of unused voting power points more readily towards human beings than the notion of fractional parts. On the other hand fractional parts are more convenient in technical calculations.

5.10 Quota Method Variants

The present book keeps the focus on the unabridged residual fit by greatest remainders, grR. Still we encounter difficulties. The reason is that most of the quota variants in Sect. 5.7 produce a whole number, and hence involve a rounding step. Therefore the ensuing quota methods with residual fit by greatest remainders, HQ1grR and so on, fail to be decent in the sense of Sect. 4.3. By abuse of termi- nology we continue to speak of “apportionment methods”. This usage admittedly means a stretching of the term’s meaning beyond its proper definition. We claim that only the unrounded Hare-quota HaQ, its variants HQ1, HQ2, and HQ3, and the genuine Droop-quota DrQ are such that their main apportionment always can be paired with a residual fit by greatest remainders. Furthermore we claim that the four Droop-quota variants DQ1–4 are so small that the main 104 5 Quota Methods of Apportionment: Divide and Rank apportionment occasionally allocates more seats than are available. Hence the residual apportionment would have to retract seats, not to hand out yet more seats. To verify the two claims we sort Hare- and Droop-quotas by decreasing magnitude: jv k lv m v jv k HQ3 D C C 1  HQ2 D C  HaQ D C  HQ1 D C ; h h h h         v v v v DrQ D C C 1  DQ2 D C  DQ3 D C  DQ1 D C : h C 1 h C 1 h C 1 h C 1

The unrounded Droop-quota variant-4 lies between DQ2 and DQ1, as does DQ3. In typical applications the vote total vC dwarfs the house size h by fulfilling

vC  h.h C 1/: (*)

Assuming (*) it is easy to see that the Hare-quota variant-1 remains above the Droop-quota, HQ1  DrQ. Thus the first inequality string runs into the second:  DQ3  DQ1 HQ3  HQ2  HaQ  HQ1  DrQ  DQ2  DQ4  DQ1

We now invoke the shift-quotas of Sect. 5.4. Assumption (*) yields Q.1/ D vC=.h  1/  vC=h C 1 bvC=hcC1 D HQ3. On the other hand the Droop-quota satisfies DrQ >vC=.h C 1/ D Q.1/. Therefore there exists an admissible shift s <1with DrQ D Q.s/. Hence the first section of the inequality string is framed by shift-quotas:

Q.1/  HQ3  HQ2  HaQ  HQ1  DrQ D Q.s/:

Let yj.Q/ Dbvj=Qc denote the seats allocated to party j in the main apportionment that uses quota Q. The bounds of the shift-quota totals (Sect. 5.4) imply that the totals yC.Q/ satisfy

h  ` Ä yC.HQ3/ Ä yC.HQ2/ Ä yC.HaQ/ Ä yC.HQ1/ Ä yC.DrQ/ Ä h:

This proves the claim that the four larger quotas are such that it is always possible to complete their main apportionments with a residual fit by greatest remainders. Moreover, all nine quotas respect the lower feasibility bound h  `. The quotas never overcharge the residual fit with more than ` seats. Furthermore we show that the four smaller quotas can lead to seat totals beyond the upper bound h. To see this consider a vote total of the form vC D n.h C 1/ for some n 2 N. Then the Droop-quota variant-2 equals DQ2 DdvC=.h C 1/eDn.If 5.10 Quota Method Variants 105 all vote counts vj happen to be multiples of n, then the main apportionment hands out one seat too many: j k j k v1 v` v1 v` y .DQ2/ D CC D CC D h C 1: C n n n n

Illustrative figures are easily contrived, such as h D 9 and v D .40; 30; 20; 10/. Hence the four Droop-quota variants DQ1–4 are infeasible. Additional instructions would be needed to clarify situations when the main apportionment deals out too many seats. Such instructions are at odds with the philosophy of quota methods. A quota of votes is perceived as a sacrosanct measure codifying how many voters must come together to be guaranteed a Member of Parliament to represent them. Suddenly a deficiency in the electoral system revokes the guarantee. Some seat is declared to be one too many and is retracted. The only way-out is to soften the sacrosanct status of the quota, and to inject some dose of flexibility. However, if a sensible invocation of flexibility is what is asked for, then the answer is divisor methods, not quota methods. Chapter 6 Targeting the House Size: Discrepancy Distribution

Abstract Technical aspects are discussed that are common to all seat apportion- ment methods. Typical calculations start with an initial apportionment that, while aiming at the target house size, misses it by some discrepancy. The range of variation of the discrepancy is analyzed. For stationary divisor methods an initialization with an appropriately adjusted divisor is recommended. The discrepancy distribution is determined in two models. One model assumes that the vote shares are uniformly distributed, and allows the house size to be finite. The other model assumes that the distribution of the vote shares is absolutely continuous, and lets the house size grow to infinity. An invariance principle shows that the associated discrepancy distributions converge to a limit that is a convolution of uniformly distributed rounding residuals.

6.1 Seat Total and Discrepancy

Apportionment methods operate in two steps. The initial step jumps to some reasonable seat vector y D .y1;:::;y`/ without guaranteeing that its component sum yC is equal to the house size h. The finalizing step advances from y to a final ` seat vector x D .x1;:::;x`/ 2 N .h/ by adjoining or removing individual seats until the house size is met. The present chapter investigates the properties of the initial seat assignment y. Divisor methods choose some initial divisor D, divide it into the vote counts vj, and obtain the interim quotients vj=D. The underlying rounding rule ŒŒ  then yields the initial seat numbers hh v ii y .D/ 2 j for all j Ä `: j D

The resulting seat vector is denoted by y.D/ D .y1.D/;:::;y`.D//. The question is how the “seat total” yC.D/ WD y1.D/ CCy`.D/ compares to the target house size h.

Definition The difference yC.D/  h is called “discrepancy”. Quota methods constitute a special case. The divisor D is replaced by the quota Q, and the rounding rule applied is downward rounding, yj.Q/ 2bbvj=Qcc. To maintain a high level of generality the development focuses on divisor methods.

© Springer International Publishing AG 2017 107 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_6 108 6 Targeting the House Size: Discrepancy Distribution

In the case of a vanishing discrepancy, yC.D/  h D 0, we are in the lucky situation that the initial seat vector represents a final solution: x D y.D/ 2 N`.h/. Unfortunately there is no divisor D that fits all vote vectors v D .v1;:::;v`/.For every divisor D there exists a vote vector v whose discrepancy is non-zero. Yet some initial divisors perform better than others. A bad initialization is the choice D D1, with initial seat vector y.1/ D 0. For a start nobody gets anything. The discrepancy yC.1/  h Dh means that there remains a deficiency of h seats. The finalizing step then passes through a great many rounds to allocate one seat after the other until all h seats are apportioned. This marathon effort is the favorite approach of many legislators and political scientists. Section 4.13 exemplifies its inefficiency. A good, universal initialization is the votes-to-seats ratio: D D vC=h.Forastart every party is allocated its ideal share of seats except for some rounding inaccuracy. The ensuing discrepancy yC.vC=h/h is bounded by ˙` (Sect. 6.2). Hence at most ` seats remain to be handled. The gain in efficiency is spectacular, from h seats down to at most ` seats. Section 4.12 illustrates the pertinent calculations. A yet more efficient divisor initialization emerges for stationary divisor methods. Given a split r 2 Œ0I 1 the “adjusted divisor” D.r/ is defined by  à vC 1 D.r/ WD ; where hr WD h C ` r  : hr 2

The term hr is called the “adjusted house size”. In Sect. 4.11 we have already met the Austrian example where the adjusted divisor D.1/ instantly leads to the final solution. The adjusted divisor D.r/ resembles the shift-quota Q.s/ D vC=.h C s/ from Sect. 5.4. In Sect. 6.3 we show that the discrepancy yC .D.r//  h is bounded by ˙`=2. Hence the removel of the discrepancy retracts or adds at most b`=2c seats. This range is the smallest possible, the probability of a zero discrepancy is largest, and the discrepancy values ˙z become rapidly less likely as z moves away from zero; see Sects. 6.7 and 6.11.

6.2 Universal Divisor Initialization

Consider a general divisor method with its underlying rounding rule ŒŒ  .LetD >0 be an arbitrary divisor. The initial seat numbers yj.D/ 2 ŒŒ v j=D and the interim quotients vj=D satisfy 1 Ä yj.D/  vj=D Ä 1. Summation yields v ` Ä y .D/  C Ä `: C D 6.3 Recommended Divisor Initialization 109

With universal divisor D D vC=h the difference yC.vC=h/  h coincides with the discrepancy. The inequalities tell us that the discrepancy attains an integer between ` and `: v Á y C  h 2 f`;:::;`g : C h

The initial seat assignment to party j is its rounded ideal share: yj.vC=h/ 2 ŒŒ . v j=vC/h .

6.3 Recommended Divisor Initialization

A stationary rounding rule ŒŒ  r with split r 2 Œ0I 1 bounds the difference of initial seat numbers yj.D/ and interim quotients vj=D in an asymmetric fashion, r Ä yj.D/  vj=D Ä 1  r. Summation and symmetrization yield

v ` v ` ` `r Ä y .D/  C Ä `.1  r/I that is,  Ä y .D/  C C `r  Ä : C D 2 C D 2 2

Insertion of the adjusted divisor D.r/ D vC=hr shows that the discrepancy yC .D.r//  h attains an integer value between `=2 and `=2. Due to integer constraints the range tightens:      ` ` y .D.r// h ;:::; : C  2  2 2

The discrepancy range of the adjusted divisor D.r/ is half as large as the discrepancy range of the universal divisor vC=h. This is the narrowest discrepancy range generally possible. For example, Table 3.2 assigns h D 100 percentage points to ` D 5 continents. The tabled percentages yj.vC=h/ Dh.vj=vC/hi leave a discrepancy of 98  100 D2 percentage points. Thus the lower bound of the discrepancy range is attained, b5=2cD2. Table 6.1 presents another example from Happacher (1996, p. 5). The commer- cial rounding function is applied to obtain whole percentages, for seven fractional percentages with an accuracy of a hundredth of a percent. Presumably most people would spot no exciting differences when glancing over the numbers in sets A and B. All first and last digits are the same. The second digit differs only for item 3, but both numbers are properly rounded to the common value 14. Could anything go astray? When rounded to whole percentages, set A forfeits three percentage points and exhibits the smallest possible discrepancy, b7=2cD3. Set B fabricates three percentage points in excess and features the largest possible discrepancy: b7=2cD3. 110 6 Targeting the House Size: Discrepancy Distribution

Table 6.1 Discrepancy range Item Proportion A Percentage A Proportion B Percentage B Item 1 28:39 28 28:59 29 Item 2 20:42 20 20:52 21 Item 3 14:47 14 13:67 14 Item 4 12:48 12 12:58 13 Item 5 11:38 11 11:58 12 Item 6 7:41 7 7:51 8 Item 7 5:45 5 5:55 6 Sum 100:00 97 100:00 103 Proportions are rounded to integer percentages using standard rounding and its adjusted divisor 100:00=100 D 1. The discrepancy of b7=2cD3 percentage points for set A is the smallest possible. The discrepancy b7=2cD3 for set B is the largest possible

For standard rounding .r D 1=2/ the universal divisor and the adjusted divisor coincide and agree with the votes-to-seats ratio: D.1=2/ D vC=h. The coincidence is indicative of the distinguished role played by standard rounding. For upward rounding .r D 0/ the adjusted divisor is D.0/ D vC=.h  `=2/. It is larger than the votes-to-seats ratio. Hence the interim quotients vj=D.0/ D .vj=vC/.h`=2/ are smaller than the ideal shares of seats .vj=vC/h. This downward shift anticipates the upward trend that is inherent in the final step of upward rounding. For downward rounding .r D 1/ the adjusted divisor is D.1/ D vC=.h C `=2/. It is smaller than the votes-to-seats ratio. Hence its interim quotients are larger than the ideal shares of seats. This upward shift outweighs the final downward rounding. The divisor D.1/ was noted already more than a century ago by the Swiss propor- tional representation activist Gfeller (1890): :::on obtiendra le diviseur au moyen de la division du total des suffrages par le nombre des candidats plus la moitié du nombre des listes. :::one obtains the divisor by means of the division of the total of the votes by the number of parties plus half of the number of parties. Gfeller’s note fell into oblivion among political scientists and constitutional lawyers even though the divisor method with downward rounding is one of their favorite procedures. Next we determine the distribution of the discrepancy yC.D/  h. Without extra effort the results are derived for the seat totals yC.D/ with an arbitrary divisor D >0. Section 6.7 returns to the adjusted divisors D.r/ as a special case. 6.4 Distributional Assumptions 111

6.4 Distributional Assumptions

How likely is it that the seat total attains a given value m? The probability of the event yC.D/ D m depends on the distributional assumptions for the vote counts. The set of all vote counts is the quadrant Œ0I1/`, an unbounded set. It is advantageous to normalize the vote counts vj into the vote shares wj D vj=vC. The set of all weight vectors w D .w1;:::;w`/ constitutes the “probability simplex” ˚ ˇ « ` ˇ ` WD .w1;:::;w`/ 2 Œ0I 1 wC D 1 :

This is a compact set in the affine subspace of vectors with component sum unity. Let  .` 1/  `1 denote  -dimensional Lebesgue measurep over the probability simplex `. The total mass is well-known to be `1.`/ D `=.`  1/Š; the exposition makes no use of this formula though. Our subsequent results assume the distribution of the weight vectors to be (a) uniform or, more generally, (b) absolutely continuous. Definition a. The weight vectors are said to follow the “uniform distribution” when the probability of a Borel subset of weights B Â ` is given by normalized Lebesgue measure:

` 1.B/ P.B/ D  : `1.`/ b. The weight vectors are said to follow an “absolutely continuous distribution” when the probability of a Borel subset of weights B Â ` is given by a Lebesgue density f : Z

P.B/ D fd`1: B

The uniform distribution is the particular absolutely continuous distribution that has a constant density function: f .w/ D 1=`1.`/. An important consequence of absolutely continuous distributions is that the occurrence of ties disappears in null-sets. Given a multiplier >0, a scaled weight wj is a tie if and only if it hits a positive signpost: wj D s.n/>0. Hence the vectors .w1;:::;w`/ 2 ` with a fixed component wj D s.n/= lie in a lower-dimensional subspace and have Lebesgue measure zero. It is safe to neglect ties in the remainder of the present chapter. In the absence of ties the set-valued rounding rule ŒŒ  is substituted by a compatible rounding function Œ W Œ0I1/ 7! N (Sect. 3.9). The set-oriented notation yj.D/ 2 ŒŒ v j=D is replaced by the number-oriented notation yj.D/ D Œvj=D. 112 6 Targeting the House Size: Discrepancy Distribution

We recall that divisors for vote counts vj and multipliers for vote shares wj D vj=vC satisfy

vj vC vj vC D D wj; where  D : D D vC D P P . / Œv =  Œ  This relation turns the seat total into yC D D jÄ` j D D jÄ` wj . It visibly displays the pertinent variables: `, ,andwj. The event that the seat total attains a given value m is

B.m/ WD f .w1;:::;w`/ 2 ` jŒw1 CCŒw` D m g :

The probabilities of the events B.m/ for m 2 N constitute the seat total distribution. From now on we restrict attention to a stationary divisor method with split r 2 Œ0I 1; the associated rounding function is denoted by Œr. We assume that the weight vectors follow a uniform distribution and that the house size h is fixed. With these Passumptions the Lemma in Sect. 6.5 determines the distribution of the seat total Œ  >0 jÄ` wj r for arbitrary multipliersP . The Theorem in Sect. 6.7 derives the Œ  distribution of the discrepancy jÄ` hrwj r  h where the adjusted house size hr plays the role of the “adjusted multiplier” that is induced by the adjusted divisor D.r/. The discrepancy probabilities (Sect. 6.7) prove dramatically practical. They apply to apportionment methods in electoral systems, to the transformation into percentages or the like, to the fitting of contingency tables, as well as to a wealth of other rounding problems. Their practicality comes as a surprise since a weight distribution that is uniform would appear to be a bold model to capture the pecu- liarities of this fan of applications. The reason for the wide applicability is that the distributions depend on their input only through the rounding residuals (Sect. 6.9). When the house size grows to infinity, the rounding residuals become exchangeable and uniformly distributed under the rather general and mild assumption that the weights follow an arbitrary absolutely continuous distribution (Sect. 6.10). A last piece of notation needed is the “positive part” tpos of a real number t 2 R:

t Cjtj t t;0 : pos WD 2 D maxf g

. /n n Its powers tpos are abbreviated as tpos. The following lemma calculates the seat  v = ` . / total distribution for a given multiplier . The boundsP C D  r Ä yC D Ä v = `.1 /  ` Œ   `.1 / C D C  r from Sect. 6.3 translate into  r Ä jÄ` wj r Ä C  r . We assume that the multiplier is not too close to zero, >`r. The assumption guarantees that the seat total is positive, and leapfrogs irrelevant boundary effects in the vicinity of zero. 6.5 Seat-Total Distributions 113

6.5 Seat-Total Distributions

Lemma Consider the stationary divisor method with split r 2 Œ0I 1, and a multiplier >`r. If the weight vectors are uniformly distributed then the seat total attains the values m Dd  `re;:::;b C `  `rc with probabilities ˚ ˇ « ˇ P .w1;:::;w`/ 2 ` Œw1r CCŒw`r D m ! ! ! X` k X`k .1/ ` `  k m C k  1 `1 D .  m C i.1  r/  kr/ : `1 k i i C k  1 pos kD0 iD0

Proof P  ` Œ   `.1 / I. The seat total is bounded by  r Ä jÄ` wj r Ä C  r (Sect. 6.3). Hence its values m range between d  `re and b C `  `rc. The assumption >`r secures m  1, and leapfrogs the trivial singleton N`.0/ Df0g. The domain of attraction A.y/ of a seat vector y 2 N`.m/ is defined to consist of the weight vectors w such that w is rounded to y:

A.y/ WD f .w1;:::;w`/ 2 ` jŒw1r D y1;:::;Œw`r D y` g :

The intersection of two such domains consists of ties and hence is a null-set. Therefore the probability sought is the sum of the probabilities of the domains of attraction: 08 91 < ˇ X = X @ . ;:::; /  ˇ Œ  A . . // P : w1 w` 2 ` wj r D m ; D P A y jÄ` y2N`.m/ X ` 1 .A.y// D  : `1.`/ y2N`.m/

` II. We fix a vector y D .y1;:::;y`/ 2 N .m/. A component yj attracts all values from an interval ŒajI bj. Its limits are aj WD yj  1 C r if yj  1,andaj WD 0 if yj D 0,andbj WD yj C r. The rectangle ŒaI b WD Œa1I b1  Œa`I b`, with south-west corner a D .a1;:::;a`/ and north-east corner b D .b1;:::;b`/, encompasses the vectors v 2 Œ0I1/` that are rounded to y. Introducing for a set ` B  R the scaled set B WD fv j v 2 Bg we obtain A.y/ D .`/ \ ŒaI b. The set of corner vectors of the rectangle ŒaI b is partitioned into subsets Ek.y/,fork D 0;:::;`.ThesetEk.y/ is defined to contain the corner vectors c such that `k components of c are copied from a,andk components are copied from b. Every corner vector c induces a quadrant Q.c/ WD Œc1I1/  Œc`I1/ 114 6 Targeting the House Size: Discrepancy Distribution

that recedes unboundedly towards infinity. Since the volume of the set .`/ \ ŒaI b is the alternating sum of the volumes of the sets .`/ \ Q.c/ we have

1 1 X` X k `1 .A.y// D `1 .A.y// D .1/ `1 ..`/ \ Q.c// : `1 `1 kD0 c2Ek .y/

In the case   cC the intersection is .`/ \ Q.c/ D c C .  cC/`. It originates from the probability simplex ` through a translation by the vector c and through a scaling by the factor   cC. Lebesgue measure `1 is invariant under translation and scales with power `  1. In the case 

X  . . // X` . 1/k X X `1 A y  . /`1 : D `1  cC pos `1.`/  y2N`.m/ kD0 y2N`.m/ c2Ek.y/

k ` III. It remains to evaluate the component sums cC,forc 2 E .y/ and y 2 N .m/. For i D 0;:::;` k we define the “.i; k/-generator” to be the generic corner vector which has

i initial components from a where yj  1, they are equal to yj.1r/; `  k  i middle components from a where yj D 0, they are equal to 0; k terminal components from b, they are equal to yj C r D yj C 1  .1  r/. The .i;k/-generator has component sum m  i.1  r/ C kr. The sum is common ` `k . ; / to k i corner vectors. There are as many i k -generators as there are ways for yC C k D m C k indistinguishable objects to occupy i C k cells leaving none of the mCk1 cells empty. This occupancy problem is well-known to have iCk1 solutions. The proof is complete. ut The double sum lends itself to rapid machine computation. For values m outside the range d  `re;:::;b C `  `rc the double sum can be shown to be zero.

6.6 Hagenbach-Bischoff Initialization

We briefly digress to appreciate the contributions of Eduard Hagenbach-Bischoff, a physics professor at Basel University (Sect. 16.6). As the leading proportional representation proponent in the Great Council of the Canton of Basel in Switzerland he fought for the introduction of the divisor method with downward rounding, and succeeded. His writings on the topic are still today a fruitful and reliable source. The 28-page pamphlet Hagenbach-Bischoff (Sect. 16.6 [6, pp. 15, 26]) is the first to calculate discrepancy probabilities. 6.6 Hagenbach-Bischoff Initialization 115

Hagenbach-Bischoff remarks that the divisor method with downward rounding executes faster when the initial divisor is chosen somewhat smaller than the votes- to-seats ratio vC=h. His choice is bvC=.h C 1/cC1. This divisor coincides with the Droop-quota. (In view of this coincidence some authors erroneously file Hagenbach-Bischoff ’s work under the heading of quota methods.) The feasibility bounds h  ` Ä yC.DrQ/ Ä h of Sect. 5.10 mean that the discrepancy is non- positive: ` Ä yC.DrQ/  h Ä 0. How likely is it that the discrepancy is z D1 and one seat must be adjoined, or that it is z D2 and two seats are needed, and so on? For the calculation of these probabilities Hagenbach-Bischoff makes two vital assumptions which he specifies in the appendix “Mathematische Ergänzungen”. First, the initial divisor is simplified to vC=.h C 1/ without any rounding. With this simplification, the divisor induces the multiplier  D h C s with shift s D 1. Second, he interprets the discrepancies via rounding residuals, as we do in Lemma in Sect. 6.9. In our terms this means that the house size h grows beyond limits. Hence Hagenbach-Bischoff ’s results follow from Lemma in Sect. 6.5 when r D 1, m D h C z,  D h C s,andh !1. The limit of the inner sum involves only the last term i D `  k. As shown in the upcoming proof of Theorem a in Sect. 6.7 we areleftwithasinglesum: 08 0 1 91 ˇ ! < ˇ X  ˘ = X` . 1/k ` @  ˇ @ . / A A  . /`1 : lim P w 2 ` ˇ h C s wj  h D z D s  z  k pos h!1 : ; .`  1/Š k jÄ` kD0

With s D 1, the formula yields the probabilities of Hagenbach-Bischoff (Sect. 16.6 [6]):

If two parties are present, then the probability is 1 that the discrepancy equals zero and the initial seat vector is final. If three parties are present, then the probability is 1=2 that the discrepancy equals zero and the initial seat vector is final, 1=2 that the discrepancy equals 1 and is removed by adding one seat. If four parties are present, then the probability is 1=6 that the discrepancy equals zero and the initial seat vector is final, 2=3 that the discrepancy equals 1 and is removed by adding one seat, 1=6 that the discrepancy equals 2 and is removed by adding two seats. If five parties are present, then the probability is 1=24 that the discrepancy equals zero and the initial seat vector is final, 11=24 that the discrepancy equals 1 and is removed by adding one seat, 11=24 that the discrepancy equals 2 and is removed by adding two seats, 1=24 that the discrepancy equals 3 and is removed by adding three seats.

Gfeller (1890) recommended the adjusted divisor vC=.h C `=2/. The induced multiplier is  D h C `=2.Theshifts D `=2 responds to the size of the party system, in contrast to Hagenbach-Bischoff ’s constant choice s D 1. The multiplier 116 6 Targeting the House Size: Discrepancy Distribution that is associated with the adjusted divisor is the adjusted house size h C `=2.An application of the above formula with s D `=2,for` D 3; 4; 5, yields discrepancy distributions which are much more favorable than those of Hagenbach-Bischoff :

If two parties are present, then the probability is 1 that the discrepancy equals zero and the initial seat vector is final. If three parties are present, then the probability is 3=4 that the discrepancy equals zero and the initial seat vector is final, 1=8 that the discrepancy equals 1 and is removed by adding one seat, 1=8 that the discrepancy equals C1 and is removed by retracting one seat. If four parties are present, then the probability is 2=3 that the discrepancy equals zero and the initial seat vector is final, 1=6 that the discrepancy equals 1 and is removed by adding on seat, 1=6 that the discrepancy equals C1 and is removed by retracting one seat. If five parties are present, then the probability is 115=192 that the discrepancy equals zero and the initial seat vector is final, 19=96 that the discrepancy equals 1 and is removed by adding one seat, 19=96 that the discrepancy equals C1 and is removed by retracting one seat, 1=384 that the discrepancy equals 2 and is removed by adding two seats, 1=384 that the discrepancy equals C2 and is removed by retracting two seats.

The replacement of the shift parameter s D 1 by s D `=2 may appear negligible, but actually is significant. With s D `=2 a vanishing discrepancy is the most probable event. The probability of a nonvanishing discrepancy value z decreases rapidly as z moves away from zero. The general formulas that govern this behavior are as follows.

6.7 Discrepancy Probabilities: Formulas

Theorem Consider the stationary divisor method with split r 2 Œ0I 1, and a house size h >`=2. With adjusted house size hr D h C `.r  1=2/ let

p`;r;h.z/ WD P .f .w1;:::;w`/ 2 ` j Œhrw1r CCŒhrw`r  h D z g/ denote the probability that the discrepancy attains the values z Db`=2c;:::; b`=2c. a. (Double-Sum Formula) If the weight vectors are uniformly distributed then ! ! ! ` `k  Ã`1 X .1/k ` X `  k z C h C k  1 ` p`; ; .z/ D  z  k  .`  k  i/.1r/ : r h h`1 k i i C k  1 2 kD0 r iD0 pos 6.7 Discrepancy Probabilities: Formulas 117 b. (Single-Sum Formula) For large house sizes the probabilities p`;r;h.z/ have a limit: ! `  Ã` 1 X .1/k ` `  lim p`;r;h.z/ D  z  k DW g`.z/: h!1 .`  1/Š k 2 kD0 pos c. (Approximation Formula) For large numbers of parties the limits g`.z/ converge: r r ! r ` ` 1 6 z2=2 6z2=` lim g` z D p e I that is; g`.z/ e : `!1 12 12 2 `

Proof a. The double-sum formula is the Lemma in Sect.ı6.5 with  D hr and m D z C h. zChCk1 `1 iCk` b. In the double-sum formula, the ratio iCk1 hr is of order h . Hence the inner sum has all ratios with i <` k tending to zero. (The terms originate from corner vectors with `  k  i >0zeros.) The last ratio with i D `  k converges to 1=.`  1/Š. (These are the corner vectors with no zeros.) Hence the double-sum formula has limit g`.z/. A reference to the split parameter is no longer needed because the limit is the same for all r. c. Let the random variables V1;:::;V` be stochastically independent and uniformly distributed over the interval Œ1=2I 1=2.ThesumVC has a density that for t 2 Œ`=2I `=2 is given by

Z Z 1 tC 2 g`.t/ D g`1.u/g1.t  u/ du D g`1.u/ du 1 t 2 ` Â ÃÂ Ã` 1 X .1/k ` `  D  t  k : .`  1/Š k 2 kD0 pos

The last identity is proved by induction, see Fellerp (1971p, p. 28). TheÁ sum VC has expectation zero and variance `=12. Hence `=12 g `=12 z is the density of the standardized sum. The Central Limit Theorem states that for ` !1the 2 p limit is ez =2= 2. An Edgeworth expansion details the speed of convergence, see Happacher (1996, pp. 90, 95): ˇ Â Ã r ˇ ˇ 3 18 2 36 4 6 ˇ ˇ z z 6z2=` ˇ 5=2 ˇ g`.z/  1  C  e ˇ Ä ` : ut ˇ 20` 5`2 5`3 ` ˇ

The discrepancy probabilities g`.z/ in part b remain relevant even when the weight vectors follow some absolutely continuous distribution, rather than being restricted to the uniform distribution as in part a. This is proved in the Theorem in 118 6 Targeting the House Size: Discrepancy Distribution

Sect. 6.11 under the assumption that the house size grows beyond limits: h !1. Yet we pause before accumulating more theory, and contemplate some practical aspects.

6.8 Discrepancy Probabilities: Practice

Discrepancy examples abound. In December 1999 a public German television station conducted a telephone poll to rank five pop music groups. The top group was fortunate to win by the narrow margin of one percentage point. However, the five percentages that flashed over the television screen summed to 101 and not to 100:

29 C 28 C 25 C 13 C 6 D 101:

The excess percentage point 101  100 D 1 hit the press. The tabloid Bild am Sonntag made a mockery of the television people’s inability to count to one hundred. In a press release the television company pointed out that the excess point had not been instrumental in determining the winner. In our parlance they said that simple rounding, though deficient, still is concordant. Simple rounding is all too likely to miss the target house size (Sect. 3.12). Sect. 6.7 assesses the likelihood numerically. The probabilities that troubled the television people are given by the double-sum formula (` D 5; r D 1=2; h D 100):

Discrepancy z 2 1 0 1 2 Sum Double-Sum Formula [in %] 0.3 20.2 59.8 19.4 0.3 100.0 The discrepancy vanishes with probability 0.598. That is, the discrepancy is zero in about sixty of one hundred instances. The television station risked a sizable chance of 40% of encountering a non-zero discrepancy and of ridiculing themselves. For the example in Table 6.1 (` D 7; r D 1=2; h D 100) Theorem in Sect. 6.7 yields the following probabilities [in %]:

Discrepancy z 3 2 1 0 1 2 3 Sum a. Double-Sum 0:002 1:652 23:591 51:093 22:176 1:484 0:002 100:000 b. Single-Sum 0:002 1:567 22:880 51:102 22:880 1:567 0:002 100:000 c. Approximation 0:023 1:694 22:166 52:234 22:166 1:694 0:023 100:000

The discrepancies 3 or 3 each occur just twice per a hundred thousand instances. The example in Table 6.1 is an artful construction, not a typical constellation. We note that the double-sum formula yields a skewed distribution, with slightly more weight on negative discrepancies than on their positive counterparts. The skewness is caused by the scaled simplex in the proof in Sect. 6.5, `, that cuts the non-negative quadrant Œ0I1/` into two pieces. The piece towards the origin 6.8 Discrepancy Probabilities: Practice 119 is a compact pyramid. The piece away from the origin is a truncated pyramid receding to infinity. On lower-dimensional faces (in Sect. 6.5 for i <` k,when corner vectors have components that are zero) the orientation matters; negative discrepancies (towards the origin) are more likely than positive discrepancies (away from the origin); see Happacher (1996, p. 55). Within the relative interior (i D `k) the two directions look locally alike, whence the limit single-sum distribution is symmetric; see Sect. 6.11. The discrepancy distributions accord rather well with observed data. Accordance is illustrated with the 1996 presidential elections in the United States of America and in the Russian Federation. In both cases the vote counts are rounded to a tenth of a percent (h D 1000) using standard rounding (r D 1=2). For the 1996 US election the International Herald Tribune of 7 November 1996 reported the vote counts for the three leading candidates (` D 3) in the 50 states, the District of Columbia, and the whole country. This provides a sample of size N D 52. The expected values 52 p3;1=2;1000.z/—which are reals—are converted into predicted counts—which are integers—using the divisor method with standard rounding:

Discrepancy z 1 0 1 N Expected values 6:51 39:00 6:49 52:00 Predicted counts 7 39 6 52 Observed counts 5 39 8 52

The predicted counts are in good accordance with the observed counts. The 1996 Russian presidential election data were reported in the Rossijskaja Gazeta of 1 July 1996. The electorate could vote for one of ten candidates, or against all of them (` D 11). The 89 Constitutional Subjects of the Russian Federation, the votes abroad, and the countrywide totals provide a sample of size N D 91. Discrepancy values as extreme as ˙5 and ˙4 did not materialize.

Discrepancy z 3 2 1 0 1 2 3 N Expected values 0:23 4:41 22:32 37:40 22:10 4:32 0:22 91:00 Predicted counts 0 4 23 38 22 4 0 91 Observed counts 0 9 18 37 20 6 1 91

Again the predicted counts accord well with those observed. The most desired event is that the discrepancy is zero; that is, the target house size is met. For the American data (` D 3) the observed discrepancy vanishes in 39 out of 52 instances; that is, the observed probability is 39=52 D 0:75. With the Russian data (` D 11/ this occurs in 37 out of 91 instances; that is, the observed probability is 37=91 D 0:41. Theorem c in Sect. 6.7 determines the probability of 120 6 Targeting the House Size: Discrepancy Distribution meeting the house size to be 1:4 P .f .w1;:::;w`/ 2 ` jŒh w1 CCŒh w` D h g/ g`.0/ D p : r r r r ` p The last term decreases to zero as ` increases. Rather generously we equate 1:4= ` to one half for all one-digit system sizes ` Ä 9. Thus, as a rule of thumb, simple rounding meets the target house size in about half of all instances. The other half misses the target by one seat or, more rarely, by two or more seats. See Table 6.2. Whether non-zero discrepancies are vexing or not is context dependent. When apportioning one hundred percentage points, or one thousand units of a tenth of a percent or the like, non-zero discrepancies are explained by rounding effects and generally appear tolerable. Nevertheless they may occasionally be a cause for ridicule, as experienced by the television company. However, a non-zero discrepancy is unacceptable when apportioning parliamentary seats. The public would not accept seats to remain vacant or to be brought to life on the grounds of rounding effects. Sometimes people take pains to secure impeccable totals. For example, the Augsburger Allgemeine newspaper used to convert vote shares into tenths of a percent (h D 1000). These numbers persistently used to sum to 100.0. Never did they total 99.9 nor 100.1. Persistent fits cannot originate from simple rounding. So what did the journalists do to balance the field? They fitted the strongest party last. All other parties are subjected to simple rounding, nj WD h1000 vj=vCi for all j D 2;:::;`P. Finally the strongest party must even out a possible imbalance: 1000 ` n1 WD  jD2 nj. The “journalists’ apportionment method” may be classified as a “Hare-quota method with standard rounding and with a residual fit by winner-take-all”. Indeed, it relies on the quota Q D vC=1000 to obtain interim quotients vj=Q.Then

Table 6.2 Discrepancy Discrepancy z 0 ˙1 ˙2 ˙3 ˙f4;::: g Sum distributions for increasing ` 2 100 0 100 system sizes D 3 76 12 100 4 66 17 0 100 5 60 20 0 100 10 44 24 4 0 0 100 20 30 23 10 2 0 100 30 26 21 11 4 1 100 40 22 19 12 6 2 100 50 20 17 12 7 4 100 100 14 13 11 8 11 100 The single-sum probabilities in Sect. 6.7.b are listed after being symmetrized around zero [in %]. The distributions spread out ` and flatten as grows. Thep probability of a vanishing discrep- ancy tends to zero like 1:4= ` 6.9 Discrepancy Representation by Means of Rounding Residuals 121 standard roundingP is applied. The residual fit follows the winner-take-all rule: 1000 ` n1 WD  jD2 nj. Thus a few tenths of a percent are added or subtracted according to whether the discrepancy is negative or positive.P An obvious variant to 1000 ensure a fitting total is the loser-take-all rule, n` WD  j<` nj. The following lemma builds on a similar strategy, to temporarily deal with all but one party, and to eventually revive the omitted party to complete the field. The lemma expresses the discrepancy in terms of rounding residuals uj.h/. Out of all ` parties, `  1 are dealt with and one is omitted.

6.9 Discrepancy Representation by Means of Rounding Residuals

Lemma Consider the stationary divisor method with split r 2 Œ0I 1.Lethr D hC`.r1=2/ denote the adjusted house size. For weight vectors w D .w1;:::;w`/ 2 ` such that hrw is tie-free, the rounding residual of party j Ä ` is designated by   Â Ã Ä  1 1 1 1 u .h/ h w r h w r : j WD r j  C 2  r j  C 2 2 2I 2

Then, omitting some party j, the discrepancy satisfies 0 1 * + X X @ A Œhrwir  h D uC.h/ D ui.h/ : iÄ` i¤j

Proof The fundamental relation for stationary rounding with shift r is n  1 C r Ä t Ä n C r. It translates into the fundamental relation for standard rounding: n  1=2 Ä t  r C 1=2 Ä n C 1=2. Thus stationary rounding is expressed through standard rounding: ŒŒ t r Dhht  r C 1=2ii .Sincehrw is assumed to be tie-free we revert to rounding functions. We obtain Œhrwir Dhhrwi  r C 1=2iDWyi,and ui.h/ D yi  .hrw  r C 1=2/. Summation over all parties i Ä ` yields uC.h/ D yC  hr C `.r  1=2/ D yC  h which is the first identity. As for the second identity we note that the rounding residuals ui.h/ are continuous quantities while the discrepancy yC h is a discrete variable. The breach of measurement levels is easily mended. Setting z WD yC  h 2 Z we select some party j and separate its rounding residual from the other rounding residuals: Ä  * + X 1 1 X u .h/ z u .h/ z z z u .h/ : i D  j 2  2I C 2 I that is, D i i¤j i¤j

The argument makes use of the fact that standard rounding is defined on all of R (Sect. 3.7). ut 122 6 Targeting the House Size: Discrepancy Distribution

The representation of the discrepancy as a function of the rounding residuals casts new light on the problem. The weight vector determines the discrepancy not directly, but only indirectly through the rounding residuals. The weight vectors do not need to follow a uniform distribution, as is assumed in Theorem a in Sect. 6.7. What is needed is that the induced rounding residuals are uniformly distributed. It is a folklore belief that rounding residuals tend to be uniformly distributed. The belief is substantiated by the invariance principle below, under rather mild assumptions. Theorem a in Sect. 6.7 assumes the weight vectors to be uniformly distributed and the house size to be fixed. The present assumptions are complementary. Now the weight vectors are permitted to follow any distribution as long as it is absolutely continuous. Loosely speaking the result is invariant with regard to the underlying distribution. The price for the invariance is that the house sizes ought to be large: h !1. The invariance principle states that the rounding residuals eventually look like random variables u1;:::;u` that have the following properties: every single variable uj is uniformly distributed over the range Œ1=2I 1=2; jointly they are stochastically independent of the weight vectors, uncorrelated, and exchangeable. Exchangeability means that the joint distribution is the same irrespective of the order in which the residuals are assembled into a vector.

6.10 Invariance Principle for Rounding Residuals

Theorem Consider the stationary divisor method with split r 2 Œ0I 1.Lethr D h C `.r  1=2/ denote the adjusted house size. If the weights w1;:::;w` follow an absolutely continuous distribution over the probability simplex ` then the rounding residuals uj.h/ WD hhrwj  r C 1=2i .hrwj  r C 1=2/ 2 Œ1=2I 1=2 and the weights wj,forjÄ `, jointly converge in distribution:

in distribution .u1.h/;:::;u`.h/; w1;:::;w`/ ! .u1;:::;u`; w1;:::;w`/: h!1

Here any single variable uj is uniformly distributed over the interval Œ1=2I 1=2. Jointly the variables u1;:::;u` are stochastically independent of w1;:::;w`, and exchangeable. Omitting any variable uj the remaining `  1 variables ui,i¤ j, are stochastically independent. If `  3 then the variables u1;:::;u` are uncorrelated. Proof I. In the first part of the proof we omit the last component ` and set k WD `  1. The other components are assembled into the vectors u.h/ D .u1.h/;:::;uk.h// and w D .w1;:::;wk/. By assumption w admits a Lebesgue density f on the k- k dimensional simplex f .w1;:::;wk/ 2 Œ0I 1 j w1CCw` <1g. Convergence in distribution is verified via the Lévy continuity theorem for characteristic 6.10 Invariance Principle for Rounding Residuals 123

functions, as inP Janson (2012). Ap prime signifies the scalar product of two 0 1 vectors, s t WD jÄk sjtj,andiWD  is the imaginary unit. The random variables 2uj.h/ fall into the interval ŒI . Since the trigonometric system over the unit circle is complete, integer coefficients k suffice. With integer vector s D .s1;:::;sk/ 2 Z and real vector t D k .t1;:::;tk/ 2 R we introduce the characteristic functions  Á 2i s0u.h/Ci t0w 'h.s; t/ WD E e :

We claim that they converge to the characteristic function of the limit distribu- tion: Y Á  Á  0 2isjuj i t w lim 'h.s; t/ D E e E e : h!1 jÄk

Evidently the claim is true when s D 0. Otherwise some component sj is non- 2 zero and the factor E e isjuj D 0 annihilates the right-hand side. Hence the claim simplifies to

s ¤ 0 H) lim 'h.s; t/ D 0: h!1

The periodicity e2iz D 1 obliterates the integer hh w  r C 1=2i,  P r j Á .2 0 . // 2 . 1=2/ and leaves exp i s u h D exp  i jÄk sj hrwj  r C D 0 exp .2ihr s w C 2i.r  1=2/sC/. Thus 'h.s; t/ is determined by the Fourier transformbf of the density f of w:  Á 0 i .t2hr s/ w 2i.r1=2/sC b 2i.r1=2/sC 'h.s; t/ D E e e D f .t  2hr s/ e :

Now the Riemann/Lebesgue Lemma is invoked, see Bauer (1991, p. 196). It states that Fourier transforms of Lebesgue densities vanish at infinity: b lim jjt  2hr sjj D 1 H) lim f .t  2hr s/ D 0: h!1 h!1

This proves the convergence of .u.h/; w/ to .u; w/ where the components of u WD .u1;:::;uk/ are stochastically independent of each other and of w. II. The second part revives theP omitted component `. Independence extends to the omitted weight w` D 1  <` w . Sect. 6.9 shows that the omitted rounding DP j E j P . / . / . / residual is u` h D j<` uj h  j<` uj h . The transformation is almost surely continuous whence the Continuous Mapping TheoremP applies. ThusP the . / rounding residuals u` h converge in distribution to u` WD h j<` uji j<` uj. Clearly u` is stochastically independent of w1;:::;w`. III. In the third part of the proof we repeat the previous arguments while omitting a component j other than the last: j <`. Then the convergence is to random 124 6 Targeting the House Size: Discrepancy Distribution

variables uQ , k ¤ j, and the missing component to be evaluated in is uQ WD DP E k P j e k¤j uk  k¤j uQk. Since the limit distribution is unique, we conclude that the joint distribution of u1;:::;u` is characterized by two properties:

(A) Every .`  1/-element subset of u1;:::;u` is such that its components are stochastically independent and identically distributed according to the uniform distribution over Œ1=2I 1=2. (B) The total component sum is a whole number, uC 2 Z. As both properties (A) and (B) are invariant under permutations, the random variables u1;:::;u` are exchangeable. IV. Finally we establish uncorrelatedness of u1;:::;u` in the case `  3. With three or more residuals, any two residuals can be embedded in an .`  1/-element subset. As just shown the subset has independent components. Hence any two residuals are uncorrelated. ut The invariance principle is rather powerful. It applies invariably whatever density governs the weight vectors. It overcomes diverting technicalities owed to finiteness of the house size. It shows that the limit behavior is the same for all stationary divisor methods when invoking the adjusted house size hr. As a first application we establish the universal validity of the discrepancy probabilities g`.z/ introduced in Theorem b in Sect. 6.7. We refer to this distribution as the “discrepancy distribution” although, strictly speaking, it is the limit of the discrepancy distributions as the house size increases towards infinity. More applications of the invariance principle are encountered in the proof of the seat bias formulas in Sects. 7.10 and 7.12.

6.11 Discrepancy Distribution

Theorem Consider the stationary divisor method with split r 2 Œ0I 1.Lethr D h C `.r  1=2/ denote the adjusted house size. If the weights w1;:::;w` follow an absolutely continuous distribution over the probability simplex ` then the discrepancy probabilities are convergent as h !1: ˚ ˇ « ˇ lim P .w1;:::;w`/ 2 ` Œhrw1r CCŒhrw`r  h D z D g`.z/; h!1 for all z Db`=2c;:::;b`=2c, with limit probabilities as given in Theorem b in Sect. 6.7: ! `  Ã` 1 X .1/k ` `  g`.z/ D  z  k : .`  1/Š k 2 kD0 pos 6.11 Discrepancy Distribution 125

If ` D 2 then the discrepancy distribution is the one-point distribution at zero. If `  3 then the discrepancy distribution is symmetric and has expectation zero and variance `=12. Proof The Lemma in Sect. 6.9 and the invariance principle in Sect. 6.10 imply that the discrepancy converges in distribution to uC, with random variables ;:::; . / u1 ˚P u` as specified in the invariance« R principle. Hence follows P fuC D zg D Œ 1=2 1=2 zC1=2 . / . / P i<` ui 2 z  I z C D z1=2 g`1 t dt D g` z . If ` D 2 then all mass concentrates at zero, g2.0/ D 1.If`  3 then induction proves the distribution to be symmetric. Indeed, g`1.t/ D g`1.t/ implies Z Z Z zC1=2 zC1=2 zC1=2 g`.z/ D g`1.t/ dt D g`1.t/ dt D g`1.t/ dt D g`.z/: z1=2 z1=2 z1=2

. / 0 1=12 The expectation is zero, E uC D . TheP uniform variablesP uj all have variance . . / . / . ; / `=12 Uncorrelatedness entails Var uC D jÄ` Var uj C i¤j Cov ui uj D . ut This concludes the technical excursion. We now return to the study of what apportionment methods can achieve and how they respond to practical needs. The next chapter investigates whether a method produces seat numbers that are unbiased or biased; that is, whether parties receive on average their ideal shares of seats or not. Chapter 7 Favoring Some at the Expense of Others: Seat Biases

Abstract Seat biases are quantitative measures assessing the deviation of the number of seats apportioned to a party from the party’s ideal share of seats. Seat bias formulas for stationary divisor methods are calculated when parties are ordered by their vote strengths. The formulas are rather telling, and entail manifold consequences. The divisor method with standard rounding emerges as the unique stationary divisor method treating all parties in an unbiased fashion. In the presence of party alliances the seat bias formulas prove puzzling. It is virtually impossible to predict whether it is advantageous or disadvantageous for a party to join an alliance.

7.1 Seat Excess of a Party

Electoral laws often stay the same over long periods of time. Does a seat apportion- ment method entail systematic effects of practical interest when applied repeatedly? For example the divisor method with downward rounding has a long tradition. Its proponents acknowledged early on that the method is biased. It favors stronger parties at the expense of weaker parties. More generally this chapter encompasses all divisor methods that are stationary. The chapter offers quantitative formulas that allow assessment of seat biases ahead of time. Seat biases are predictable because they capture the average behavior of an apportionment method. Before analyzing averages, however, we need to address a single instance. Again let h denote the house size, and wj thevoteshareandxj the seat number of party j. Definition The “seat excess of party j” is defined to be the difference between the number of seats apportioned to the party and its ideal share of seats, xj  wjh. As seat numbers are integers and ideal shares are reals, their difference is unlikely to vanish. If a seat excess is positive then the party is better off than demanded by ideal proportionality. If negative, the party is worse off. In every single instance the sum of all seat excesses is zero: xC  wCh D h  h D 0. It is inevitable that, in each particular apportionment, some parties are privileged to enjoy positive seat excesses, while unfortunately other parties must cope with negative seat excesses. What about the average behavior over many instances? Perhaps seat excesses again and again exhibit the same trend allowing to predict who will be the lucky

© Springer International Publishing AG 2017 127 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_7 128 7 Favoring Some at the Expense of Others: Seat Biases winners, who the unlucky losers. Or perhaps good luck and bad luck are balanced and even out. A quantitative orientation is provided by the seat bias formula (Sect. 7.4). The formula is contingent on the rank-order of the parties, as manifested by their rank scores.

7.2 Rank-Order of Parties by Vote Shares

The qualifiers “strong” and “weak” that we attribute to parties solely refer to voter support in terms of vote counts or, equivalently, in terms of vote shares. Other aspects are neglected, such as party budget, number of party members, campaign contributions, media presence, or the like. We list parties by their vote shares from strongest to weakest:

w1 w`:

As a visual reminder when dealing with rank-ordered parties we replace the otherwise favored party subscript j by the letter k. Thus the subscript k identifies the party’s rank score: k D 1 is the strongest party, and k D 2 is the second-strongest party. The weakest party comes last, k D `. Naturally the parties’ rank-order can be ascertained only retrospectively, when all vote counts are available. Moreover, a party’s rank score generally varies from one election to the next. It may not be the same party that remains strongest in one election and a subsequent election.

7.3 Vote Share Thresholds

A last detail needs to be mentioned. Many electoral systems impose a threshold which the vote share of a party must meet or exceed in order for the party to participate in the apportionment process. A threshold t requires the vote shares to satisfy

wj  t for all j Ä `:

When ` parties participate in the apportionment process the threshold t D 1=` forces all vote shares to be equal: wj D 1=` for all j Ä `. Threshold values greater than 1=` are impossible. Hence thresholds are required to lie in the right-open interval Œ0I 1=`/. A threshold often takes the form of a percentage hurdle. For example, Table 1.32 mentions countries with no threshold, and with thresholds of 1.8%, or 3%, 4%, or 5%. The percentages are sometimes calculated relative to the set of votes cast. On other occasions the base is the set of valid votes. There is yet another 7.4 Seat Bias Formula 129 conceivable reference set: the effective votes. These are the valid votes that enter into the apportionment calculations. With regard to the seat bias formula we do not distinguish between the three reference sets. We trust that the difference among them is practically negligible. A threshold excludes not just some parties from participating in the seat apportionment process. The effect from the voters’ point of view is more dramatic. A threshold inhibits a part of the electorate from obtaining parliamentary representa- tion. These voters cast votes which, though valid, are wasted. Chapter 1 takes pains to document the number of voters who fall victim to the presence of a threshold.

7.4 Seat Bias Formula

These preparations enable us to discuss the seat bias formula. We consider this formula the most important result of this chapter, if not of this treatise. .t/ The “seat bias of the kth-strongest party”, denoted by Br .k/, is an average of the party’s seat excess values xk  wkh. The averaging operation includes all seat excess values that are obtainable from rank-ordered vote shares above the threshold: w1    w`  t. The operation assumes all feasible constellations to be equally likely. The house size h is taken to be large: h !1. A formal definition of seat biases is deferred to Sect. 7.10. Theorem Assume that the vote share vector w follows the uniform distribution over the probability simplex ` and that t 2 Œ0I 1=`/ is a preordained threshold. For the stationary divisor method with split r 2 Œ0I 1, the seat bias of the kth-strongest party is given by  à 1  Á B.t/.k/ r H` 1 .1 `t/; r D  2 k   P ` ` .1= / where Hk WD nDk n is a partial sum of the harmonic series. Proof The proof is deferred to Sect. 7.10. ut The formula consists of three factors that neatly correlate with the ingredients of the problem. The first factor is the “method factor”: it reflects the influence of the apportionment method under investigation. The second factor is the “party factor”: it captures the impact of the party’s position within the entire party system. The third factor is the “threshold factor”: it describes the impact of the vote share threshold. The divisor method with standard rounding (r D 1=2) has method factor zero. .t/ Hence every party has seat bias zero, B1=2.k/ D 0 for all k Ä `. Every party may expect its due share of seats. On average no party is advantaged, nor is any party disadvantaged. The divisor method with standard rounding is the unique member among all stationary divisor methods that is “unbiased”. 130 7 Favoring Some at the Expense of Others: Seat Biases

The method factor satisfies r  1=2 D..1  r/  1=2/. Hence the stationary divisor method with split 1  r has seat biases of opposite sign:

.t/. / .t/ . /: Br k DB1r k

While the methods with a split larger than one half, r >1=2, favor stronger parties at the expense of weaker parties, the methods with split r <1=2favor weaker parties at the expense of stronger parties. ` 1 The party factor Hk  is positive for stronger parties, and negative for weaker parties. This is worked out in greater detail in Sect. 7.8. The threshold factor 1  `t involves the threshold parameter t. It discounts the seat biases of all parties in a like manner. Practical thresholds are such that the contribution of the term `t is negligible. In the sequel we therefore concentrate on the seat biases with no threshold (t D 0): Â Ã 1  B.0/.k/ r H` 1 : r D  2 k 

7.5 Biasedness Versus Unbiasedness

A prominent pair of biased apportionment methods are the divisor method with downward rounding (r D 1) and the divisor method with upward rounding (r D 0). The noticeable seat bias of the divisor method with downward rounding, in favor of stronger participants and at the expense of weaker participants, has been acknowledged ever since the inception of the method in the eighteenth century in the United States of America and in the nineteenth century in continental Europe. The monograph Balinski and Young (1982) tells the tale of the New World, how arguments swayed back and forth to promote a preferred apportionment method over its competitors. Among the plot’s protagonists we meet illustrious figures such as Thomas Jefferson, Alexander Hamilton, Daniel Webster,tonamebutafew. In the Old World the divisor method with downward rounding was resolutely advocated by the Belgian activist Victor D’Hondt and his Swiss contemporary Eduard Hagenbach-Bischoff. Once the method got firmly rooted in the minds of proportional representation protagonists it was accorded with all sorts of fanciful claims: that it is the only apportioning method existing, that it is the simplest method, that it is the method most widely used. Of these claims, the last is correct. The frequency in Table 1.32 is indicative of the method’s predominance. Seventeen of the 28 Member States of the 2014 European Union use it for their EP elections. Pólya (Sect. 16.14 [1]) was first to investigate seat biases in a formal manner. For the divisor method with downward rounding (r D 1/ in a three-party system he 7.5 Biasedness Versus Unbiasedness 131 calculated the seat biases of the strongest party, the median party, and the weakest party to be

.0/ 5 .0/ 1 .0/ 4 B .1/ ; B .2/ ; B .3/ : 1 D 12 1 D12 1 D12

That is, the strongest party may look forward to five bonus seats per every twelve elections. The median party must bear a loss of one seat and the weakest party the loss of four seats. Pólya considered the quantitative assessment of the seat biases as das wichtigste Resultat dieser Abhandlung. the most important result of this treatise. He emphasized that the divisor method with standard rounding is unbiased (see above), as is the Hare-quota method with residual fit by greatest remainders (see Sect. 7.12). Unbiased apportionment methods are often preferred in the second stage of a two-stage systems, as is evidenced in the Netherlands and in Poland (Sect. 1.8). In the Netherlands, sub-apportionments become necessary when parties join into an alliance. An unbiased sub-apportionment makes it easier for stronger parties to persuade weaker parties to join into a prospective alliance. At second glance though the weaker partners amplify the strength of the stronger partner. Irrationalities of alliances are the theme of Sect. 7.9. In Poland the countrywide seats of a party are forwarded to its various district lists. Within a party there is no reason for the sub-apportionment to be biased by favoring more populous districts at the expense of less populous districts. Hence sub-apportionments tend to employ unbiased apportionment methods. Altogether the between-party seat apportionment is biased and favors stronger parties at the expense of weaker parties, but the within-party seat apportionment is unbiased. More variation is met when the seats of a house are apportioned among regional districts in proportion to census figures. The seats of the French Assemblée nationale are allocated to the 101 Départements using the divisor method with upward rounding; see Balinski (2004, p. 92). The method is biased in favor of less populous (rural) districts at the expense of more populous (urban) districts. The bias has a straightforward explanation in terms of power politics. All that is needed is that the parliamentary majority finds their voter support in the countryside outnumbering that in the cities. Then overrepresentation of less populous districts and underrepresentation of more populous districts becomes the method of choice. Unbiasedness is not a virtue by itself. The central issue is its status in the country’s constitution. The constitution sets the limits how much non-zero bias is constitutionally tolerable. While the tasks of apportioning seats among political parties, or of apportioning seats among geographical districts are formally the same, their constitutional appraisal may differ significantly. Many countries feature some less populous districts that, under standard rounding or downward rounding or any other pervious rounding rule, would end up with no parliamentary representation 132 7 Favoring Some at the Expense of Others: Seat Biases at all. For this reason constitutions often inject a dose of biasedness in favor less populous districts by guaranteeing a minimum representation. For example the Constitution of the United States of America decrees that “each state shall have at least one representative”. Indeed the 435 seats of the House of Representatives are allocated among the fifty states using the divisor method with geometric rounding. Being impervious the method guarantees every state at least one seat, as stipulated by the Constitution. The clause may be interpreted as a constitutional mandate to favor less populous states at the expense of more populous states. For infinite house sizes the diversity of the power-mean family reduces to three stationary divisor methods: with upward rounding, standard rounding, and downward rounding (Sect. 3.11). Moreover, since the seat bias formula is asymptotic (h !1), it does not distinguish between the divisor method with geometric rounding and the divisor method with standard rounding. Asymptotically both methods are unbiased. But here application is to finite house sizes, see Sect. 7.7. This raises the question of which house sizes are large enough for the seat bias formula to become meaningful.

7.6 House Size Recommendation

In this section we argue in support of the following house size recommendation: The seat bias formula is applicable for all practical purposes provided the house size meets or exceeds twice the number of parties: h  2`. Empirical evidence substantiates the house size recommendation. Schuster et al. (2003, p. 657) evaluate data with three-party seat excesses calculated from N D 49 parliamentary elections in the German State of Bavaria. During 1966–1998 Bavaria conducted nine elections of its diet. In seven elections just three parties pass the 5% threshold and qualify for the seat apportionment. Since Bavaria is subdivided into seven electoral districts the data provide a sample of size N D 7 7 D 49. House sizes varied between 19 and 65 seats. The voting attitudes of the electorate may be considered sufficiently stable to allow the assumption that the seat excesses .t/ originate from the same distribution. The theoretical seat biases Br .k/ are shown to conform rather satisfactorily with the empirical averages. Another data set provided by those authors is from the Swiss Canton of Solothurn. During the period 1896– 1997, the ten Solothurn districts provide a sample of N D 143 three-party elections, with house sizes ranging from 7 to 29 seats. Again the predicted seat biases agree exceedingly well with the observed seat biases. The recommendation does not extend to house sizes less than 2`. With too small a house size the values of the seat excesses xk  hwk vary erratically. Their averages do not allow identification of a systematic trend. The lamento of Riker (1982, p. 260) of what happens when ten seats are apportioned among twelve parties is vacuous. 7.7 Cumulative Seat Biases by Subdivisions into Districts 133

In the sequel we apply the seat bias formula uniformly for all house sizes h  2`. Every time the apportionment method is used the kth-strongest party faces a seat .t/ bias as predicted by Br .k/. Whatever the house size h, each deployment of the apportionment method contributes a bias of the same size.

7.7 Cumulative Seat Biases by Subdivisions into Districts

Multiple applications of an apportionment method arise regularly as soon as an electoral region is subdivided into districts. If the districts are evaluated separately then the seat biases accumulate. The cumulative seat biases can reach an impressive level. This sheds new light on a phrase as in Sect. 7.5: “The strongest party may look forward to five bonus seats per every twelve elections”. One scenario would be to realize twelve elections consecutively in time. Assuming that the twelve elections take place every 4 years the accumulation period would span 48 years. A property that needs half a century to materialize could easily be denounced as a theoretical artifact of no practical relevance. The subdivision into districts is a different scenario. Assuming that the electoral region is subdivided into twelve districts bonus seats may accumulate to become of utmost importance. We give two examples: the 1990 election to the Bavarian state diet, and the 2011 election to the Spanish national parliament. For the election of the 204 members of the 1990 Bavarian state diet the state was subdivided into seven electoral districts. The districts are separately evaluated using the divisor method with downward rounding. In 1990 the district magnitudes were 65, 20, 19, 20, 28, 23, and 29 seats. A 5% threshold applied; four parties gained seats. In all districts the strongest party was CSU. Hence CSU could expect .0:05/ a cumulative bias of 7 B1 .1/ 3 bonus seats. In 1990 facts excelled fiction. CSU actually garnered 127 seats, while its ideal share was but 121.1 seat fractions. In this instance the seat excess of nearly six seats doubled expectations. It was a Pyrrhic victory. The opposition parties challenged the election in the Bavarian State Constitutional Court. The Constitution would guarantee proportional representation. Proportionality was blatantly missed by awarding a single party six bonus seats. The constitution could be served better by other procedures. For instance the Hare-quota method with residual fit by greatest remainders would entitle CSU to 121 seats. In a 1992 landmark decision the Court barred the divisor method with downward rounding from future use in state diet elections. The Court urged the state diet to switch to the Hare-quota method with residual fit by greatest remainders. Soon thereafter the Bavarian state diet amended the electoral provisions accordingly. For the election of the 350 deputies to the Spanish national parliament the country is divided into 52 districts. In 2011 there were two single-seat districts where candidates were elected by plurality vote. The other fifty districts were subject to 134 7 Favoring Some at the Expense of Others: Seat Biases proportional representation by way of the divisor method with downward rounding. In 2011 twenty-five district magnitudes ranged from 2 to 5 seats, and eighteen from 6 to 8 seats. The seven district magnitudes remaining were 10, 10, 12, 12, 16, 31, and 36 seats. The dominant party was PP, finishing strongest in forty-three districts and second-, third- and forth-strongest in the other seven districts. With a countrywide vote share of 46%, PP’s ideal share of seats came to roughly 160 seats. In reality PP garnered 186 seats. The spectacular excess of 26 bonus seats secured PP a straight majority of 53% of the 350 house seats. The cause for this striking unproportionality is the concurrence of two short- comings: a biased apportionment method and small district magnitudes. Unbiased methods by themselves would not remedy the situation. The divisor method with standard rounding would allot PP 176 seats, thus leaving a notable excess of 16 seats. The Hare-quota method with residual fit by largest remainders would apportion PP 167 seats, still entailing an excess of 7 seats. The excess residue that appears inextinguishable is due to the small district magnitudes. With two seats or only a few more, proportionality is hard to achieve and impossible to guarantee. A small district magnitude entails a high natural threshold. High thresholds make it a foregone illusion for weak parties to gain seats; see Sect. 11.8. If proportionality is to be achieved, one option might be to merge districts until district magnitudes are sufficiently raised. Another option is to adopt a double proportional method as discussed in Chap. 14.

7.8 Total Positive Bias: The Stronger Third, the Weaker Two-Thirds P ` 1 ` ` .1= / The party factor Hk  ,whereHk D nDk n , is decisive for determining the .t/ sign of the seat biases Br .k/. If the party factor is positive then the kth-strongest party is advantaged for splits r >1=2. It is disadvantaged for splits r <1=2.Ifthe party factor is negative, advantages and disadvantages are reversed. The party factor switches sign at the “biasfree rank score” k` for which the party ` 1 0 factor vanishes: Hk`  D . In view of the logarithmic approximation Z X` 1 ` 1 H` D dx D log `  log k k n x nDk k we solve log.`=k`/ D 1 to obtain k` Dh`=eiDh0:37`i. Roughly speaking the stronger third of the parties has a positive party factor. The weaker two-thirds have a negative factor. When r >1=2the stronger third enjoys positive seat biases, while the weaker two-thirds must endure negative seat biases. The situation is familiar from day-to-day life: a minority of well-to-do enjoy a surplus, at the expense of a majority of not-so-well-to-do who must be satisfied with less than their fair share. 7.9 Alliances of Lists 135

The logarithmic approximation of the party factor shows that it decreases logarithmically as a function of the rank score k. On the other hand we may be interested in the front runner, k D 1, when the number of weaker parties that contest the election is increasing. Then ` increases and the party factor grows like log `. A measure for the aggregate seat bias of a party system of size ` is the “total positive bias”, TPB, defined to be the sum of all positive seat biases when using a stationary divisor method with split r 2 Œ0I 1. It satisfies  à  Ã à 1 X  1 ` TPB.`; r/ WD r  H`  1 r   1 : 2 k 2 e `>1 kWHk

The approximate equality is established under the assumption `  5, whence ` ` 2 1 0 ` `= `= k  . With Hk`  and by replacing k Dh ei by e we obtain

X  Xk`  Á kX`1 Xn 1 ` ` k`1 ` H  1 D H C H  1 D k`  1  1: k k k` n e `>1 kD1 nD1 kD1 kWHk

In summary the total positive bias TPB.`; r/ is obtained from a tripartition of the party system. The tripartition has already been employed for discriminatory purposes by Balinski and Young (1982, pp. 75, 126).

7.9 Alliances of Lists

The advocates of the divisor method with downward rounding recognized early on that the method is compromised by being biased in favor of stronger parties at the expense of weaker parties. To assuage weaker parties, Hagenbach-Bischoff (Sect. 16.6 [5]) devised the construct of an “alliance of lists” (German: Listen- verbindung, French: apparentement des listes, also referred to as coalition or cartel). His intention was to allow weaker parties to gather strength by joining their lists into an alliance, and thereby to evade or at least to lessen threatening seat biases. In electoral systems with multiple districts it may well happen that in one district parties A and B join into a first alliance and parties C and D into a second alliance, while in another district parties A and C register an alliance and parties B and D stand alone, and in a third district yet another partition is realized. It is more pertinent to speak of alliances of lists (per district) rather than of alliances of parties (countrywide). Many electoral laws permit parties to register alliances, in particular when the law stipulates the notoriously biased divisor method with downward rounding. The stipulations persist even when an unbiased apportionment method is introduced. It is a safe bet that such laws have precursors that formerly used the divisor method with downward rounding, and that the unbiased method was adopted only later on. The legislators who prepared the amendment lacked insight and forgot to ban alliances. 136 7 Favoring Some at the Expense of Others: Seat Biases

Party officials are fond of the chance to forge an alliance. They view it to be a smart lever for diverting the electoral outcome into a direction more desirable to them. From the voters’ viewpoint alliances interfere with the principle of a direct election because of the intervention of obscuring calculations. The functioning of alliances is not at all transparent. An alliance entails a two-stage seat apportionment process. In the super- apportionment (Sect. 1.4) the seats are apportioned among the alliances registered and the lists that stand alone. Every alliance then calls for a second stage of calculations. This sub-apportionment distributes the alliance’s joint seats among its partners. The 2008 local election in the Bavarian City of Friedberg provides an intriguing example. Six lists contested the thirty seats in the city council. As usual we rank- order them from strongest (1) to weakest (6). The second-, third-, and fifth-strongest lists registered an alliance, denoted by f2;3;5g. The other lists stood for themselves. In formal terms the other lists constituted a singleton alliance each: f1g, f4g,andf6g. It so happened that the aggregate votes for the alliance f2;3;5g surpassed the vote count for the strongest list 1. The four alliances finished in the rank-order f2;3;5g, f1g, f4g, f6g. The resulting seat apportionment is disturbing. See Table 7.1. Firstly, the second-strongest list 2 emerged as the ultimate winner. In the super- apportionment the alliance f2;3;5g garnered 16 seats. Without an alliance the three lists would have accumulated 15 seats. The within-partner sub-apportionment accorded the bonus seat to the strongest of the three partners, list 2. List 2 owed its last seat to the alliance, not to its voters. The result is a discordant seat apportionment. List 2 finished with more seats than list 1 despite having obtained fewer votes. Secondly, Hagenbach-Bischoff nurtured the view that the alliance would aid lists 3 and 5. True, alliance f2;3;5g finished strongest. But it made no difference

Table 7.1 Discordance BY2008Friedberg Vot es Quotient DivDwn victory via an alliance, f2; 3; 5g 194;141 16 divisor method with Alliance 16.04 downward rounding List 1 150;615 12.4 12 List 4 28;428 2.4 2 List 6 12;010 0.99 0 Sum (Divisor) 385;194 (12,100) 30 Partners Vot es Quotient DivDwn Sub-apportionment within Alliance f2; 3; 5g List 2 145;292 13.2 13 List 3 30;558 2.8 2 List 5 18;291 1.7 1 Sum (Divisor) 194;141 (11,000) 16 The alliance gains 16 seats; 13 of them go to list 2. The end result is discordant: List 2 has fewer votes but more seats than list 1. Without alliance the divisor is 12,000, and list 2 is entitled to 12 seats 7.9 Alliances of Lists 137 to lists 3 and 5. They gained as many seats with the alliance as they would have gained without. Thirdly, there is the desolate experience of the weakest list 6. Without an alliance, list 6 would have won a seat. Alas, the alliance pocketed its seat. List 6 did nothing but look on, yet had to pay the bill. Surprise? Not really. The presence of the alliance f2;3;5g upsets the seat biases:

Total positive List (rank-ordered) 1 2 3 4 5 6 bias Without alliances 0:725 0:225 0:025 0:192 0:317 0:416 0:950 With alliance f2; 3; 5g 0:317 0:507 0:160 0:295 0:245 0:444 0:984

Without alliances the strongest list is advantaged most, with seat bias 0.725. With alliance f2;3;5g its bias reduces to 0.317, and the second-strongest list 2 with seat bias 0.507 takes the lead; the numbers are obtained from the formula below. The example records an instance where the forecast of the average behavior of seat biases actually materialized. Meanwhile the Bavarian State Diet amended the legal provisions for local elections. In 2010 the divisor method with downward rounding was replaced by the Hare-quota method with residual fit by greatest remainders. In 2017 the permission to register list alliances was canceled. An alliance Q is a subset of parties: Q Âf1;:::;`g. An alliance with q partners is called “proper” when 1

p q  1 X a. p; Q/ WD C ; H` WD H`: ` ` where Q i HQ i2Q

Seat Bias Formula for List Alliances Suppose the party system of size ` is partitioned into p alliances. If the kth-strongest party of the ` parties is among the q partners of the alliance Q, then super- and sub-apportionment jointly entail a seat bias of party k that is approximated by  à 1  B.0/ .k p; Q / r a. p; Q/H` 1 : r j D  2 k 

Proof The proof is deferred to Sect. 7.11. ut 138 7 Favoring Some at the Expense of Others: Seat Biases

The extreme cases p D 1 and p D ` are degenerate, yet instructive. In the case p D 1 a single alliance comprises all parties (q D `). There is no need for ` ` a super-apportionment. In view of H1 CCH` D ` the alliance coefficient is .0/ a.1; f1;:::;`g/ D 1. All seat biases remain unaffected: Br . k j 1; f1;:::;`g / D .0/ Br .k/. In the case p D ` all alliances are singletons. There is no need for a sub- apportionment. In view of q D 1 the alliance coefficients equal unity: a.`; fkg/ D 1. .0/ .0/ Again seat biases are unaffected: Br . k j `;fkg / D Br .k/. The cases of interest are those with 1

1partners, the other `q parties are singleton alliances. Altogether there are p D `  q C 1 alliances. The alliance coefficient for the parties k 2 Q is larger than unity: ! 1 1 a .` .q 1/; Q/ 1 .q 1/ >1:   D C  `  ` HQ

.0/ .0/ It follows that B1 . k j `  q C 1; Q />B1 .k/. A positive seat bias grows more positive, a negative seat bias becomes less negative. For example, in the 2008 local elections in Bavaria proper alliances were registered in 668 communities. Two thirds of the communities (456) had a sole proper alliance. The partners of these alliances could look forward to bonus seats simply because their competitors were napping. Rule 2: If you decide to stand alone while other parties form proper alliances, you lose. The presence of at least one proper alliance entails p <`. Hence the alliance coefficient of every singleton alliance fkg is smaller than unity: p a . p; k / <1: f g D `

.0/ .0/ Hence we realize B1 . k j p; fkg/

7.10 Proof of the Seat Bias Formula

The remainder of this chapter deals with the more technical aspects of calculating seat biases. The current section presents a proof of the seat bias formula in Sect. 7.4. We assume throughout that the underlying apportionment method is the stationary divisor method with split r 2 Œ0I 1. Lemma A decomposes the seat excess xj wjh into three terms. Conceptually the three terms may be interpreted as systematic effect, discrepancy effect, and rounding effect. Technically each term is later analyzed individually when averaging over all vote share vectors w D .w1;:::;w`/ and passing to the limit h !1. Lemma A (Seat Excess Decomposition) For every tie-free seat vector x 2 DivStar.hI w/ the seat excess of party j decomposes as follows:  à 1   x w h r `w 1 x y u .h/; j  j D  2 j  C j  j C j where the adjusted multiplier hr WD hC`.r 1=2/ is used to obtain the seat number yj WD Œhrwjr and its rounding residual uj.h/ WD yj  .hrwj  r C 1=2/.

Proof The seat number yj D Œhrwjr that uses the adjusted multiplier hr can be obtained through standard rounding via yj Dhhrwj  r C 1=2i; see Sect. 6.9. Obviously it satisfies  à  à 1 1  y w h h w r w h u .h/ r `w 1 u .h/: j  j D r j  C 2  j C j D  2 j  C j

This leads to the seat excess decomposition as claimed. ut The first term will soon transpire to be the contribution to the seat excess that persists systematically. The second term .xj  yj/ captures the number of adjustment seats needed to progress from the initial seat number yj to the final seat number xj via the steps that are prescribed by the jump-and-step procedure (Sect. 4.7). We aim to show that the adjustment seats average out to zero. To this end Lemma B expresses the term .xj yj/ by means of rounding residuals. Since in this chapter we focus on vote shares wj the max-min inequality is employed in its multiplier version that is based on the signpost-to-vote share ratios s.n/=wj (Sect. 4.6). Therefore we look for lowest comparative scores, rather than for highest comparative scores that are pertinent to vote counts vj and associated vote count-to- signpost scores vj=s.n/ (Sect. 4.14). We discriminate between incrementation and decrementation using the sign function sgn.t/ which attains the values 1,0,or1 according as t 2 R is negative, zero, or positive.

Lemma B (Jump-and-Step Procedure Revisited) The modulus jxj  yjj is the count of how often party j appears among the juC.h/j lowest values of the numbers   n  1=2  sgn .uC.h// ui.h/ ` ani WD ; where n D 1;:::; ; i D 1;:::;`: wi 2 140 7 Favoring Some at the Expense of Others: Seat Biases

Proof The Lemma in Sect. 6.9 states that uC.h/ is an integer that is equal to the discrepancy yC  h. Section 6.3 delimits its values to the range extending from b`=2c to b`=2c. In the case uC.h/ 2fb`=2c;:::;1g the discrepancy is negative. Its removal increments party j by xj  yj Djxj  yjj seats. As the multipliers increase above hr, party j is entitled to as many additional seats as the party contributes to the uC.h/ lowest values of the numbers   s . y C n/ ` r i ; where n D 1;:::; ; i D 1;:::;`: wi 2

The shift hr D.hrwi/=wi leaves the order of these ratios as is. The numerators transform to  à 1 1 1 y n 1 r h w n y h w r n u .h/: i C  C  r i D  2 C i  r i  C 2 D  2 C i

In view of sgn .uC.h// D1 we look for the juC.h/j lowest values of the numbers ani. In the case uC.h/ D 0 the discrepancy is zero and the initial seat number is final, jxj  yjjD0. In the case uC.h/ 2f1;:::;b`=2cg the discrepancy is positive. Its removal reduces the seats of party j by yj  xj Djxj  yjj seats. As the multipliers decrease below hr party j has to give up as many seats as it contributes to the uC.h/ highest values of the numbers   s . y  .n  1// ` r i ; where n D 1;:::; ; i D 1;:::;`: wi 2

Thesameshifthr D.hrwi/=wi as above unifies the format of the numerators: Â Ã Â Ã Â Ã 1 1 1 y n r h w n y h w r n u .h/: i  C  r i D  2 C i  r i  C 2 D  2 C i

Multiplication by 1 shows that we look for the juC.h/j lowest values of the numbers ani. ut Now we turn to the calculation of seat excess averages when the vote share vector w D .w1;:::;w`/ varies through the probability simplex ` Dfw 2 ` .0I 1/ j wC D 1 g. It is a helpful custom in probability theory to emphasize by way of capital letters that all conceivable values of the random vector W are under investigation and not just a single particular value. In the same vein the final seat number Xj turns into a random variable, as does the initial seat number Yj. Lemma C shows that for large house sizes the seat excess reduces to the systematic term .r  1=2/.`wj  1/. The adjustment seats Zj.h/ WD Xj  Yj that are caused by the discrepancy removal average out to zero, as do the rounding residuals Uj.h/. 7.10 Proof of the Seat Bias Formula 141

Lemma C (Systematic Seat Excess) If the vote share vector W follows an absolutely continuous distribution then the conditional expectations of the seat excess Xj  hWj of party j given the vote shares W D w converge in distribution according to  à  in distribution 1  E Xj  hWj j W1 D w1;:::;W` D w` ! r  `wj  1 : h!1 2

Proof In view of the absolutely continuous distribution of the vote shares all seat vectors are almost surely tie-free. Lemma A yields the decomposition  à 1  X hW r `W 1 Z .h/ U .h/: j  j D  2 j  C j C j

The initial term is decisive, but least problematic. It does not even depend on h: ÂÂ Ã Ã Â Ã 1  ˇ 1  r `W 1 ˇ W w r `w 1 : E  2 j  D D  2 j 

The last term is the rounding residual Uj.h/. The invariance principle 6.10 states that its limit Uj is stochastically independent of W and uniformly distributed over the interval Œ1=2I 1=2. Hence the limiting conditional expectation is the unconditional expectation of Uj which is zero:

 in distribution  E Uj.h/ jW D w ! E Uj jW D w D E.Uj/ D 0: h!1 It remains to discuss the middle term. We claim that ˇ  ˇ in distribution E Zj.h/ W D w ! 0: h!1

The term Zj.h/ says how many seats are adjoined or retracted from the Yj.h/ initial seats of party j. Lemma B states that Zj.h/ depends on Xj.h/ and Yj.h/ only through Wj and the rounding residuals U1.h/;:::;U`.h/. The relevant limiting distributions are provided by the invariance principle 6.10 through the random vectors U WD .U1;:::;U`/ and W. Thus the terms Zj.h/ converge in distribution to Zj.U; W/ WD sgn.UC/Nj.U; W/, with Nj.U; W/ counting how often party j appears among the jUCj smallest values of the numbers   n  1=2  sgn.UC/Ui ` ani WD ; where n D 1;:::; ; i D 1;:::;`: Wi 2

Since U and W are stochastically independent the conditional expectations of Zj.h/ given W D w converges to the unconditional expectation of Zj.U; w/: ˇ    ˇ in distribution E Zj.h/ W D w ! E Zj.U; W/ jW D w D E Zj.U; w/ : h!1 142 7 Favoring Some at the Expense of Others: Seat Biases

The random variables Z1.U; w/;:::;Z`.U; w/ inherit exchangeability  from U1;:::;U`. Hence they share the same expectation. From ` E Zj.U; w/ D E .ZC.U; w// DE.UC/ D 0 we finally confirm E Zj.U; w/ D 0. The proof is complete. ut The systematic seat excesses .r  1=2/.`wj  1/ of all parties j Ä ` evidently sum to zero. That is, if some parties are advantaged by positive seat excesses, others are disadvantaged by negative seat excesses. Conversely, if some parties are disadvantaged then others are advantaged. One man’s meat is another man’s poison. Numerical results that are practically relevant emerge for parties with a given rank score. From now on let us assume that parties are ordered from strongest to weakest and that all of them pass a preordained threshold t 2 Œ0I 1=`/:

W1 W`  t:

Hence Wk signifies the vote shares of the party that is kth-strongest. This is the setting tailored to the discussion of seat biases. Definition The “seat bias of the kth-strongest party” is defined to be the limit for large house sizes, h !1, of the conditional expectation of the seat excesses of party k given that the parties are rank-ordered and obey the preordained threshold t:

.t/. / . /: Br k WD lim E Xk  hWk jW1 W`  t h!1

.t/ .t/ The seat biases of all parties sum to zero, Br .1/ CCBr .`/ D 0.Thisis immediate from XC D h and WC D 1. Hence either all seat biases are zero; in this case we call the apportionment method “unbiased”. Otherwise the method is “biased”: some parties enjoy a positive seat bias while others suffer from a negative seat bias. Favoritism of some parties spawns distress for others. Our wording that a method “is biased in favor of some parties at the expense of other parties” always acknowledges the existence of both groups, those advantaged and those disadvantaged. Now we can present the proof of the seat bias formula of Sect. 7.4. The result is established under the narrower assumption that the vote share vector W is uniformly distributed over the probability simplex `. Actually the uniformity assumption may be relaxed. It suffices that it applies to the ordered and truncated wedge f W1    W`  t g. In the wedge the vote share of the kth-strongest party is delimited by its two neighbors, Wk 2 ŒWk1I WkC1. The smaller range of variation promises an expectation that is fairly stable with respect to a deviation from the uniform distribution. This explains the surprisingly vast domain of validity of the seat bias formula. 7.10 Proof of the Seat Bias Formula 143

Theorem (Seat Bias Formula) Assume that the vote share vector W follows the uniform distribution over the probability simplex ` and that t 2 Œ0I 1=`/ is a preordained threshold. For the stationary divisor method with split r 2 Œ0I 1 the seat bias of the kth-strongest party is given by  à 1  B.t/.k/ r H` 1 .1 `t/; r D  2 k   P ` ` .1= / where Hk WD nDk n is a partial sum of the harmonic series. Proof Lemma C takes care of the passage to the limit h !1.Wehave  à 1 ˇ  .t/ ˇ B .k/ r .`E 1/ ; E W W1 W` t : r D  2  where WD E k  

The key role is played by the term `E  1,whereE is the expectation of Wk given the conditioning event f W1 W`  t g that is constrained by the threshold t. To dispose of the threshold restriction we introduce the shifted and scaled variables Vj WD .Wj  t/=.1  `t/ for j Ä `. The vector V WD .V1;:::;V`/ has non- negative components summing to unity. Thus it takes its values in the probability simplex `. Generated by translation and scaling, V inherits the uniform distribution of W. The conditioning event satisfies f W1   W`  t gDfV1   V`  0 g. The substitution Wk D t C .1  `t/Vk now yields

E D t C .1  `t/E . Vk jV1 V`  0/:

It remains to find the expectation of Vk given the conditioning event V1    V`  0 which disposes of a threshold restriction. The conditional expectation of the vector V is the arithmetic mean of the vertices of its range: 0 0 1 1 0 1 0 1 0 1 0 1 V1 1 1=2 1=3 1=` B B C C B C B C B C B C B B C C B 0 C B 1=2 C B 1=3 C B 1=` C B B V2 C C B C B C B C B C B B C C 1 B 0 C 1 B 0 C 1 B 1=3 C 1 B 1=` C B B V3 C V1 V` 0 C B C B C B C B C : E B B C j   C D ` B CC ` B CC ` B CCC` B C B B : C C B : C B : C B : C B : C @ @ : A A @ : A @ : A @ : A @ : A V` 0 0 0 1=`

. 0/ `=` ` 1 With kth component E Vk j V1    V`  D Hk we obtain E  D ` 1 .1 ` / Hk   t . The proof is complete. ut 144 7 Favoring Some at the Expense of Others: Seat Biases

7.11 Proof of the Seat Bias Formula for List Alliances

The seat bias formula for list alliances also assumes that the vote share vector W follows the uniform distribution over the probability simplex `.However,the threshold is set to be zero. We recall the alliance coefficient from Sect. 7.9 which depends on the number of alliances, p, and on the specific alliance, Q, that embraces the party under investigation as one of its q partners:

p q  1 X a. p; Q/ WD C ; where H` WD H`: ` H` Q i Q i2Q

Theorem (Seat Bias Formula for List Alliances) Suppose the party system of size ` is partitioned into p alliances. If the kth-strongest party of the ` parties is among the q partners of the alliance Q then super- and sub-apportionment jointly entail a seat bias of party k that is approximated by  à 1  B.0/ . k p; Q / r a. p; Q/ H` 1 : r j D  2 k 

Proof The argument focuses on the term `E  1 which plays the key role in the proof of the seat bias formula (Sect. 7.10). The term is evaluated first for the super-apportionment and then for the sub-apportionment. Finally the two stages are aggregated. We save space by omitting the factor .r  1=2/ that is common to all intermediate results. The conditional expectation of a random variable Z given the event fW1 W`  0g is indicated by a superscript .0/:

.0/ E .Z/ WD E . Z jW1 W`  0/:

All conditional expectations are determined given the event fW1   W`  0g. Thereby we pass over separate rank-orderings of the p alliances, and of the q partners in the alliance Q. P In the super-apportionment alliance Q features the cumulative vote share WQ WD i2Q Wi.Astherearep alliances the key term for the seat bias of alliance Q is .0/ approximated by pE .WQ/  1. The seat bias of alliance Q is shared among its partners. The share for party k is  à  .0/ Wk .0/ E pE .WQ/  1 : WQ

In the sub-apportionment party k acquires the relative vote share Wk=WQ.As there are q partners the key term for the seat bias of party k is  à W qE.0/ k  1: WQ 7.12 Seat Biases of Shift-Quota Methods 145

Finally the two stages are aggregated. Under the uniformity assumption for the vote share vector W, the weight WQ of the alliance Q relative to all p alliances and the weight Wk=WQ of party k relative to all q partners in its alliance transpire to be stochastically dependent to such a low degree that the dependence may be neglected. Approximate independence entails  à  à  .0/ . / .0/ Wk .0/ .0/ . / .0/ Wk E Wk : E E WQ E Wk I that is, E .0/ WQ WQ E WQ

Aggregation of the key terms from the super- and sub-apportionments now yields  à  à  à  1 .0/ Wk .0/. / 1 .0/ Wk 1 q  .0/. / 1: E pE WQ  CqE  p C .0/ E Wk  WQ WQ E .WQ/

.0/. / `=` .0/. / ` =` Inserting E Wk D Hk and E WQ D HQ we obtain the desired factor . ; / ` 1 a p Q Hk  . The proof is complete. ut

7.12 Seat Biases of Shift-Quota Methods

Already Pólya (Sect. 16.14 [1]) pointed out that the Hare-quota method with residual fit by greatest remainders is unbiased. The approach gains in lucidity by considering Œ 1 1/ the larger family of shift-quota methods, shQgrRs, with shift s 2  I and shift- quota Q.s/ D vC=.hCs/ (Sect. 5.4). When using the shift-quota method with shift s in the presence of a preordained threshold t the seat bias of the kth-strongest party is denoted by

.t/. / . /: Bs k D lim E Xk  hWk jW1 W`  t h!1

The Hare-quota method with residual fit by greatest remainders (s D 0) turns out to be the only shift-quota method that is unbiased. Theorem Assume that the vote share vector W follows the uniform distribution over the probability simplex ` and that t 2 Œ0I 1=`/ is a preordained threshold. For the shift-quota method with shift s 2 Œ1I 1/ the seat bias of the kth-strongest party is given by s  B.t/.k/ H` 1 .1 `t/; s D ` k   P ` ` .1= / where Hk WD nDk n is a partial sum of the harmonic series. 146 7 Favoring Some at the Expense of Others: Seat Biases

Proof The arguments run parallel to those for divisor methods in Sect. 7.10. . / A. A tie-free solution x 2 shQgrRs hI w has seat excess decomposition  às  s 1 x w h `w 1 x y u .h/; j  j D ` j  C j  j C `  2 C j

where yj WD b.hCs/wjcDh.hCs/wj 1=2 i is the seat number of party j Ä ` in the main apportionment and uj.h/ WD yj  .h C s/wj  1=2 2 Œ1=2I 1=2 is the associated rounding residual. The decomposition instantly follows from the identity

s  s 1 y w h `w 1 u .h/: j  j D ` j  C `  2 C j

B. Next the residual fit is expressed by means of rounding residuals. The main apportionment leaves fractional part .h C s/wj  yj D 1=2  uj.h/ for party j. The number of residual seats is given by the negative discrepancy: h  yC D `=2  s  uC.h/. A party gains a residual seat if and only if its fractional part is among the `=2  s  uC.h/ greatest remainders 1=2  u1.h/;:::;1=2 u`.h/. C. Turning to probabilistic arguments we again use capital letters to indicate random variables. Assuming that the vote share vector W follows an absolutely continuous distribution we claim that the conditional expectations of the seat excess Xj  hWj of party j given the vote shares W D w converge in distribution:

 in distribution s  E Xj  hWj jW1 D w1;:::;W` D w` ! `wj  1 : h!1 `

Indeed let Zj.h/ WD Xj  Yj be the variable indicating whether party j gains a residual seat or not. The invariance principle 6.10 permits replacing the rounding residuals Uj.h/ by their limits Uj. Thus the conditional expectations E. Zj.h/ j W1 D w1;:::;W` D w` / converge to random variables Zj.U; w/,forallj Ä `. They inherit exchangeability from U1;:::;U`. The sum of their expectations is `E Zj.U; w/ D E .ZC.U; w// D E.`=2  s  UC/ D `=2  s. This implies E Zj.U; w/ D 1=2  s=`. Hence the seat excess decomposition has second and third terms whose limiting conditional expectations vanish. The first term verifies the claim. .t/ The seat bias of the kth-strongest party now becomes Bs .k/ D .s=`/.`E1/. ` 1 ` 1 .1 ` / The equality E D Hk   t is established in the proof of the theorem in Sect. 7.10. ut In summary the family of shift-quota methods includes a unique member that is unbiased, s D 0: the Hare-quota method with residual fit by greatest remainders. The limiting extreme s D 1 yields the bias of the Droop-quota method with residual fit by greatest remainders. It favors stronger parties at the expense of weaker parties. 7.12 Seat Biases of Shift-Quota Methods 147

Since its coefficient 1=` is smaller than the coefficient 1=2 of the seat biases of the divisor method with downward rounding, the Droop seat biases are less pronounced. So far we have investigated how a fixed apportionment method behaves under repeated applications. In a nutshell: one method, many vote share vectors. The next chapter reverts the emphasis: many methods, one vote share vector. The comparison invokes the majorization order, a relation with fruitful applications in many fields of the natural and behavioral sciences. Chapter 8 Preferring Stronger Parties to Weaker Parties: Majorization

Abstract Majorization is an order relation for the comparison of two apportion- ment methods. Some method A is majorized by some other method B when every group of stronger parties garners under A at most as many seats as it garners under B. Specifically, a divisor method is majorized by another divisor method if and only if their signpost ratios are increasing. The family of stationary divisor methods is monotonic with regard to majorization, as is the family of power-mean divisor methods.

8.1 Bipartitions by Vote Strengths

Majorization is a well-established partial order for the comparison of integer vectors that share a common sum. We first review the classical concept of vector majorization. Let x D .x1;:::;x`/ and y D .y1;:::;y`/ be two seat vectors when h seats are apportioned among ` parties. A comparison always involves at least two items. Choosing the items to be single parties does not lead to a suitable comparison of two seat vectors x and y.Ifthe vectors are equal, the comparison is trivial. If unequal, they still share the component sum h. Hence there always exist some parties i and j such that the passage from x to y is advantageous for one party and disadvantageous for the other: xi < yi and xj > yj. Two-party comparisons have little informative value. Moreover, consideration of just two parties disregards the other `  2 parties. Therefore the items to be compared are groups of parties, not single parties. Majorization compares complementary subsets of parties: I Âf1;:::;`g and I0 WD f1;:::;`gnI. Every such bipartition is exhaustive of the entire system; no party is lost from sight. Each party features in one of the sets, either in I or in I0. Interest is in bipartitions that contrast stronger parties with weaker parties. As always in this book, the strength of party j is reflected by its vote count vj (or equivalently, by its vote share wj D vj=vC). A “group of stronger parties” is a subset of parties, I,such that all of its members are at least as strong as the parties in the complementary set I0:

0 vi  vj for all i 2 I; j 2 I :

© Springer International Publishing AG 2017 149 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_8 150 8 Preferring Stronger Parties to Weaker Parties: Majorization

Majorization compares an arbitrary group of stronger parties with its complement, the group of weaker parties that remain. This leaves `  1 bipartitions to look at: the strongest party versus the `  1 weaker parties, the two strongest parties versus the `  2 weaker parties, and so on until the `  1 strongest parties versus the weakest party. Bookkeeping turns easy when we rank-order the parties from strongest to weakest: v1 v` (Sect. 7.2). Then the initial section f1;:::;kg constitutes the group of the k strongest parties. Its complement fk C 1;:::;`g is the tail section of the `  k weakest parties. Let A be a general apportionment method (Sect. 4.4). Since the method is anonymous, a re-arrangement of parties by rank-order does no harm. Since the method is concordant, a strict rank-order of the parties leads to ordered seat numbers: v1 >  >v` ) x1    x`. The seat vectors with decreasingly ordered components are assembled in the set ˚ ˇ « ` . / . ;:::; / `. / ˇ : N h WD x1 x` 2 N h x1 x`

The group of the k strongest parties garners a total of x1 CCxk seats.

8.2 Majorization of Two Seat Vectors

` . / ` . / Definition A vector x 2 N h is said to be “majorized” by a vector y 2 N h , denoted by x  y, when the sum of the k largest components of x is less than or equal to the sum of the k largest components of y:

x1 CCxk Ä y1 CCyk for all k <`:

When passing from x to y every group of stronger parties garners more seats or maintains its level. Every group of weaker parties finishes with fewer seats if any. In other words the passage from x to y is beneficial for groups of stronger parties and disadvantageous for groups of weaker parties, or it is neutral for the two groups. The definition excludes the sum of all components since they are equal anyway: ` . / xC D yC D h. Vector majorization equips the set N h with a partial order. That is, the relation is reflexive (x  x), transitive (x  y and y  z ) x  z), and antisymmetric (x  y and y  x ) x D y). A transitivity example for house size h D 100 is provided by the seat vectors

x D .40; 30; 20; 10/  y D .41; 29; 20; 10/  z D .41; 29; 22; 8/:

The example attests to the superiority of the majorization order over pairwise comparisons. The passage from x to y involves the two strongest parties only: .40; 30/  .41; 29/. The second-strongest party must give up a seat to the strongest party. The passage from y to z has the fourth-strongest party giving up two seats to the third-strongest party: .20; 10/  .22; 8/. So far, so good. But the passage from 8.4 Majorization of Two Apportionment Methods 151

4 x to z affects all four parties and invites 2 D 6 pairwise comparisons. Sometimes a weaker party has to give up some seats to a stronger party, but not always. Take a look at the second- and third-strongest parties, with .30; 20/ seats in x,and.29; 22/ seats in z. The stronger party loses seats and the weaker party gains seats. The “pairwise give-up relation from weaker to stronger parties” fails to be transitive. Nevertheless pairwise comparisons provide a sufficient condition for majorization, as follows.

8.3 A Sufficient Condition via Pairwise Comparisons

; ` . / Lemma Let the vectors x y 2 N h be given. If, in all pairwise comparisons of seat numbers, the stronger party does not decrease or the weaker party does not increase, then x is majorized by y:

.xi Ä yi or xk  yk for all i < k/ H) x  y:

Proof The proof is indirect. Assuming that x is not majorized by y we show that there are two parties i < k such that xi > yi and xk < yk.Ifx is not majorized by y then some party i is first to violate the defining inequalities; that is, x1 CCxk Ä y1 CCyk for all k < i,butx1 CCxi > y1 CCyi.Theith terms must satisfy xi > yi. Due to equal component sums, some later party k > i has xk < yk. Thus the two parties i < k fulfill xi > yi and xk < yk. ut Next we apply the concept of majorization to apportionment methods. We demand that all seat vectors of one method are majorized by all seat vectors of the other method whenever the rank-order of the parties is strict. For two sets of seat vectors the relation A.hI v/  B.hI v/ is defined to mean x  y for all x in A.hI v/ and y in B.hI v/. That is, all seat vectors in one solution set are majorized by all seat vectors in the other solution set. Reflexivity is lost since A.hI v/  A.hI v/ entails both, x  y and y  x. This implies x D y and would force every solution set A.hI v/ to be a singleton. Yet a strict notion of majorization is sensible.

8.4 Majorization of Two Apportionment Methods

Definition An apportionment method A is said to be “majorized” by an apportion- ment method B, denoted by A  B, when the two methods are distinct and satisfy

v1 > >v` H) A.hI v/  B.hI v/;

` for all system sizes `, vote vectors .v1;:::;v`/ 2 .0I1/ , and house sizes h. 152 8 Preferring Stronger Parties to Weaker Parties: Majorization

The following theorem provides a majorization check when the two methods under consideration are divisor methods. Recall that a divisor method is charac- terized by a signpost sequence s.0/; s.1/; s.2/; : : : (Sect. 3.8). Since every signpost sequence starts with s.0/ D 0, the significant terms are s.n/ with n  1.Since impervious methods have s.1/ D 0 we use the conventions 0=0 D 0 and c=0 D1 for c >0.

8.5 Majorization of Divisor Methods

Theorem Let A be a divisor method with signpost sequence s.n/,n 0, and let B be a divisor method with signpost sequence t.n/,n 0. Then method A is majorized by method B if and only if the two methods satisfy signpost ratio monotonicity:

s.n/ s.n C 1/ A  B ” < for all n  1: t.n/ t.n C 1/

Proof The proof of the direct part is indirect. Assuming that some N  1 satisfies

s.N/ s.N C 1/ s.N C 1/ t.N C 1/  ; that is, Ä ; t.N/ t.N C 1/ s.N/ t.N/ we show that A is not majorized by B. The assumption implies s.N/>0. Setting a WD s.N C 1/=s.N/ we obtain 1 v2 WD 1=.aC1/. We claim that x WD .N C1; N 1/ is contained in the solution set A .2NI .v1;v2//. The applicable max-min inequality is     s.N C 1/ s.N  1/ s.N C 2/ s.N/ max ; Ä min ; : v1 v2 v1 v2

The inequality is easily verified by comparing each term on the left-hand side with each term on the right-hand side. A similar verification of the max-min inequality     t.N/ t.N/ t.N C 1/ t.N C 1/ max ; Ä min ; v1 v2 v1 v2 shows that y WD .N; N/ lies in B .2NI .v1;v2//. Vector x is not majorized by vector y, and so method A is not majorized by method B. The proof of the direct part is complete. For the proof of the converse part we consider a house size h, ordered weights v1 >  >v`, and arbitrary seat vectors x in A.hI v/ and y in B.hI v/. The desired majorization relation x  y follows from the Lemma in Sect. 8.3. We show that signpost ratio monotonicity prevents any two parties i < k from fulfilling xi > yi 8.6 Majorization-Increasing Parameterizations 153 and xk < yk. That is, if signpost ratio monotonicity holds true then so does the premise of the Lemma in Sect. 8.3. Its conclusion ascertains x  y. Assume that there are two parties i < k fulfilling xi > yi and xk < yk. Then they also satisfy xi 1  yi and xk C1 Ä yk. Due to concordance vi >vk implies yi  yk. We realize xk C 1 Ä xi  1

v v t.y C 1/ t.x / s.x / v v i D i Ä i Ä i < i Ä i D i : vk vk t.yk/ t.xk C 1/ s.xk C 1/ vk vk

This inequality string is a contradiction. The proof of the converse part is complete. ut The localization property s.n/ 2 Œn  1I n implies .n  1/=n Ä s.n/=t.n/ Ä n=.n  1/. Hence a sequence of signpost ratios always converges to unity: limn!1 s.n/=t.n/ D 1. Monotone convergence entails s.n/

8.6 Majorization-Increasing Parameterizations

Theorem a. (Stationary divisor methods) A stationary divisor method DivStar is majorized by another stationary divisor method DivStaR if and only if r < R. b. (Power-mean divisor methods) A power-mean divisor method DivPwrp is majorized by another power-mean divisor method DivPwrP if and only if p < P. Proof a. With stationary signposts sr.n/ D n  1 Cr, signpost ratio monotonicity .n  1 C r/=.n  1 C R/<.n C r/=.n C R/ instantly reduces to r < R. 154 8 Preferring Stronger Parties to Weaker Parties: Majorization b. The power-mean signposts sQp.n/ are the mean of order p of the interval endpoints n  1 and n (Sect. 3.12). Verification of signpost ratio monotonicity reduces to proving that the function

 Ã1= .n  1/p C np p g. p/ WD np C .n C 1/p

is increasing, from g.1/ D .n  1/=n to g.1/ D n=.n C1/. The proof follows from Proposition 5.B.3 in Marshall and Olkin (1979, p. 130). ut Thus the five traditional divisor methods, which have power parameters 1, 1, 0, 1, and 1, are seen to be ordered by majorization:

DivUpw  DivHar  DivGeo  DivStd  DivDwn:

From left to right the methods are more and more beneficial to groups of stronger parties and detrimental to groups of weaker parties.

8.7 Majorization Paths

The family of stationary divisor methods generates a sequence of seat vectors, and so does the family of power-mean divisor methods. In these sequences every seat vector is majorized by its successor. Although the parameter increases continuously, the evolving sequence of seat vectors is finite because of the inherent discretization step. Both paths start with DivUpw .D DivSta0 D DivPwr1/, pass through DivStd .D DivSta:5 D DivPwr1/, and finish with DivDwn .D DivSta1 D DivPwr1/. Table 8.1 presents an example. The top portion displays the path in the stationary family. The bottom part shows the path in the power-mean family. Both paths happen to feature eight seat vectors each. They have six seat vectors in common, while the third and the seventh are peculiar to just one path. Every seat vector is majorized by its successor. From left to right, groups of stronger parties accumulate more seats or maintain what they have. Groups of weaker parties do worse or stay as is. The fate of a single party is less deterministic. The strongest party never loses a seat since it is a singleton group. The weakest party never wins a seat. Parties in-between may oscillate, as does the fifth-strongest party. A seat transfer to a party i from a party k can occur only when the two are tied:

s.x C 1/ s.x / i D k : vi vk 8.7 Majorization Paths 155 5 2 1 1 1 5 2 1 1 1 13 11 13 11 11 10 11 10 45 45 1 1 100 100 1 1 1 1   C C 2 6 5 2 1 1 2 1 1 1 97 : 13 11 13 11 11 10 11 10 44 44 50 0 100 100 1 1 1 1   C C 6 2 2 6 2 2 1 1 1 1 9 9 8 13 11 11 13 11 11 : 44 44 18 0 100 100 1 1 1 1   C C 6 2 6 2 2 1 1 2 1 1 5 13 11 10 13 11 10 11 11 : 43 43 0 100 1 100 ) 1 1 1 1 1   C C C 6 3 2 6 3 2 9 1 1 9 1 1 47 6 13 11 13 11 11 11 : 43 43 : 0 100 0 100 p 1 1 1 1   C C r ) to a stronger party ( 1 6 3 2 2 6 3 2 2 9 1 9 1  44 4 13 11 13 11 10 : 42 11 43 : 0 100 100 0 1 1 1 1   C C 6 3 2 2 2 6 3 2 2 2 9 9 3 13 11 10 13 11 10 42 42 : 0 100 0 100 1 1 1 1   C C 0 6 3 2 2 6 3 2 2 2 2 D1 D 10 10 41 13 11 10 41 13 11 10 Majorization path of stationary divisor methods, DivSta Majorization path of power-mean divisor methods, DivPwr r p 100 100 9547 5708 2502 1898 1461 1457 9547 5708 2502 1898 1461 1457 42;919 13;048 10;879 10;581 42;919 13;048 10;879 10;581 Majorization paths 100;000 100;000 Votes Votes Rank 1 2 3 4 5 6 7 8 9 10 Sum Rank 1 2 3 4 5 6 7 8 9 10 Sum Table 8.1 Top: Stationary divisor methods.contiguous seat Bottom: columns, Power-mean a divisor seat is methods. given Both up from paths a start weaker party with ( DivUpw, pass through DivStd, and finish with DivDwn. In 156 8 Preferring Stronger Parties to Weaker Parties: Majorization

The tie equation is instrumental in verifying that the paths in Table 8.1 are complete. For the stationary family, the tie equation and its solution are

xi C r xk  1 C r xivk  .xk  1/vi D ; that is, r1.i; k/ WD : vi vk vi  vk

In the presence of majorization we know that the receiving party i must be stronger than the donor k;thatis,i < k and vi >vk. As the split point r increases from r0 D 0 onwards, the smallest of the solutions r1.i; k/ is encountered first: 1289 r1 WD min r1.i; k/ D r1.1; 5/ D 0:155: i

At the change-point r1, a seat is transferred from the fifth strongest party to the strongest party, thus giving rise to the second seat vector. Beyond r1 calculations start afresh. The second seat vector persists until the next change-point r2 is reached: 3989 r2 WD min r2.i; k/ D r2.4; 10/ D 0:437: i

The interval of constancy is Œ0:155I 0:437;Table8.1 quotes the select split 0:3. A traditional parameter value is preferred whenever possible. The select splits r3;:::;r7 are obtained in the same manner. For the family of power-mean divisor methods the tie equation for the power p is

x p C .x C 1/p vp i i D i : . 1/p p vp xk  C xk k

No explicit solution for p is available, but a numerical solution is easily obtained by machine calculation. The select powers p1;:::;p7 yield the path shown in Table 8.1. Finally we show that the parameterization of the shift-quota methods (Sect. 5.4) is majorization-increasing too. Furthermore these methods lie between the divisor method with upward rounding and the divisor method with downward rounding.

8.8 Majorization of Shift-Quota Methods

Theorem Two shift-quota methods with shifts s < S satisfy

: DivUpw  shQgrRs  shQgrRS  DivDwn

Proof For house size h and vote vector v 2 .0I1/` fixed, the Corollary in Sect. 5.6 provides splits r and R that reproduce the shift-quota solution sets, . v/  . v/ . v/  . v/ shQgrRs hI D DivStar hI and shQgrRS hI D DivStaR hI .Letx be 8.8 Majorization of Shift-Quota Methods 157 a seat vector in the first set, y, in the second. Unless the two vectors are equal,  there exists a party j with seat numbers xj > yj.Fromxj  1 C r Ä vj=Q.s/ and  vj=Q.S/ Ä yj C R we obtain v v j C 1  R Ä y C 1 Ä x Ä j C 1  r: Q.S/ j j Q.s/

Insertion of the shift-quotas Q.s/ D vC=.hCs/ yields 0<.Ss/vj=vC D vj=Q.S/     vj=Q.s/ Ä R  r . Hence the splits are ordered, r < R . Theorem a in Sect. 8.6. yields DivUpw.hI v/  DivStar .hI v/  DivStaR .hI v/  DivDwn.hI v/. ut Seat biases in favor of some parties at the expense of others, and the preferential treatment of groups of stronger parties as compared to groups of weaker parties are important characteristics of an apportionment method. However, there are yet other features that may become essential. An apportionment method should translate vote counts into seat numbers in a manner that is consistent and does not invite dispute. The next chapter judges the performance of a method from a holistic point of view. The resulting seat vectors ought to be such that the whole vector and its parts fit together in a coherent way that is devoid of irritating paradoxes. Chapter 9 Securing System Consistency: Coherence and Paradoxes

Abstract This chapter assesses apportionment methods from an overall viewpoint as to whether vote weights and seat numbers always correspond in a fair manner. A new organizing principle turns out to be decisive: coherence. It demands that every solution for a general apportionment problem agrees with the solutions for all embedded subproblems. The Coherence Theorem states that an apportionment method is coherent if and only if it is compatible with a divisor method. The ground for the proof is prepared by showing that coherent methods are house size monotone and vote ratio monotone. In contrast, quota methods may produce non-monotonic results of a seemingly paradoxical nature.

9.1 The Whole and Its Parts

Apportionment problems involve quite a few variables: the house size h,the size ` of the party system, and the vote weights v1;:::;v`. The five organizing principles aim to secure a reasonable interrelation between the variables. An abstract apportionment rule must be anonymous, balanced, concordant, decent, and exact in order to be elevated to a viable apportionment method (Sects. 4.2–4.4). However the five principles suffer from a common weakness. They are insen- sitive to variations of the house size and of the system size. With house size and system size fixed, each principle focuses solely on the variation of the vote weights v1;:::;v`. Anonymity permutes the weights, balancedness and concordance com- pare them by pairs, decency rescales them, and exactness addresses the special case when they are integers. The house size h and the system size ` are treated as if they were constants. As a consequence the notion of an apportionment method still admits procedures that are rather silly. For instance let A be the apportionment method that for even system sizes ` coincides with the divisor method with standard rounding, while for odd system sizes ` it concurs with the Hare-quota method with residual fit by greatest remainders. Even more weird let B be the method that coincides with the stationary divisor method with split r D 1=h in the case ` is even, and with the shift-quota method with shift s D 1=h in the case ` is odd. Both procedures A and B are apportionment methods: they are anonymous, balanced, concordant, decent, and exact. Yet nobody, in their right minds, would recommend these methods for practical use.

© Springer International Publishing AG 2017 159 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_9 160 9 Securing System Consistency: Coherence and Paradoxes

The weakness is remedied by introducing a sixth organizing principle that views the apportionment problem in a more comprehensive manner than the first five. Balinski and Young (1982, p. 141) put it this way: “An inherent principle of any fair division is that every part of a fair division should be fair.” The ensuing principle might be called fairness; the name would nicely conform with the alphabetic order of the first five principles. Alas, the sixth principle comes under the heading of coherence.

9.2 The Sixth Organizing Principle: Coherence

The notion of coherence captures the idea that the whole and its parts must fit together in a coherent way. An apportionment method is coherent when every overall seat vector is such that any partial vector is a valid solution for the associated partial problem. While this informal description is persuasive, the formal definition of coherence is cumbersome notationally. As usual xC D x1 CCx` designates the component sum of the grand seat vector x D .x1;:::;x`/.LetI f1;:::;`g denote a subset of parties, and let 0 I Df1;:::;`gnI be the complementary subset.P We indicateP the cumulative seats 0 0 garnered by the parties in I and I by xI WD i2I xi and xI WD k2I0 xk, respectively. Now suppose that the overall seat vector .x1;:::;x`/ is a solution for the apportionment of h seats in a large system of ` parties. Coherence demands (a) that every party subsystem I f1;:::;`g admits the subvector .xi/i2I as a solution for the apportionment of xI seats among the parties in I, and (b) that, if the subproblem admits another solution . yi/i2I, then substituting it for .xi/i2I yields another overall solution. For notational convenience we replace the house size h by the component sum xC. Definition An apportionment method A is called “coherent” when, for all system sizes `  2 and all vote vectors .v1;:::;v`/, every seat vector x 2 A.xCI v/ and all party subsets I f1;:::;`g fulfill properties a and b: a. (Coherence of partial problems) The partial vector .xi/i2I solves the associated partial apportionment problem; that is, .xi/i2I 2 A .xII .vi/i2I /. b. (Coherence of substituted solutions) When tied partial solutions are substituted for partial vectors in x, the resulting vector is an overall solution as is x;that is, all seat vectors . yi/i2I 2 A .xII .vi/i2I/ and .zk/k2I0 2 A .xI0 I .vk/k2I0 / satisfy .. yi/i2I ;.zk/k2I0 / 2 A .xCI ..vi/i2I ;.vk/k2I0 //. Coherence of partial problems (a) is a top-down requirement. It demands that every partial vector that is extracted from an overall seat vector is a valid apportionment solution for the associated partial problem. Coherence of substituted solutions (b) is a bottom-up demand. Tied solutions for partial problems that are related to a grand solution x through no more than the partial sums xI and xI0 ,when substituted into the overall solution x, must yield tied overall solutions. 9.3 Coherence and Completeness of Divisor Methods 161

The definition makes sense only in the presence of anonymity. Without anonymity we would have to observe the order in which the parties are presented, which would force us to consider subsequences. With anonymity the order is negligible, and consideration of subsets suffices. If helpful we are allowed to rearrange the system so that the parties in the subset I come first, and those in I0 last.

9.3 Coherence and Completeness of Divisor Methods

Divisor methods are coherent, as shown next. Another property is adjoined: completeness. While the related notion of exactness stresses the discrete role of the seat vectors (Sect. 4.3), completeness emphasizes the continuum character of thevotevectors. An apportionment method A is called “complete” when all seat vectors that are obtainable from weights close to a given vote vector v 2 Œ0I1/` actually belong to the solution set of v. That is, if a seat vector y is common to all sets A .hI v.n// then it lies in the solution set A.hI v/:

y 2 A .hI v.n// for all n  1 H) y 2 A.hI v/; whenever the vote vectors v.n/, n  1,convergetov and their weights satisfy vj.n/ D 0 when vj D 0. Theorem Every divisor method is coherent and complete. Proof Consider a solution x 2 A.hI v/ of a divisor method A. Coherence of partial solutions holds true since every divisor D feasible for x is also feasible for all partial solutions .xi/i2I. Coherence of substituted solutions is evident in the case when .xi/i2I and .xk/k2I0 both are unique solutions of their respective partial problems. Otherwise the divisor is unique (Sect. 4.8), both in the partial problems as well as in the grand problem. Hence the divisor is feasible for all other partial solutions as well. This establishes coherence. Now we tackle completeness. Since the zeros of v are preserved we may assume all vote weights to be positive: vj >0.LetD.n/, n  1, be feasible divisors for a common solution y 2 A .hI v.n//.Fromyj 2 ŒŒ v j.n/=D.n/ we obtain vj.n/=s. yj C 1/ Ä D.n/ Ä vj.n/=s. yj/. Hence the divisors D.n/ are convergent, possibly along some subsequence, to a limit D.Fromvj=s. yj C 1/ Ä D Ä vj=s. yj/ we conclude y 2 A.hI v/. This establishes completeness. ut Completeness is a feature that is not endorsed by electoral practice. On the contrary there are electoral laws which start from a divisor method D,butthen modify it into another method A by imposing a particular tie resolution rule. For example in Spain ties are resolved by the motto “Stronger parties first” (Sect. 4.9). While failing completeness, the modification A still is compatible with D,inthe following sense. 162 9 Securing System Consistency: Coherence and Paradoxes

An apportionment method A is called “compatible with the apportionment meth- od D” when every solution set of A is included in the corresponding solution set of D. That is, the methods satisfy

A.hI v/ Â D.hI v/ for all house sizes h and vote vectors v. If the solution set D.hI v/ contains two or more tied solutions, some of them may be lacking in the set A.hI v/. If the solution set D.hI v/ is a singleton, the two methods yield the same solution. The momentous consequence of coherence is to reduce the vast class of all apportionment methods to those methods that are compatible with divisor methods.

9.4 Coherence Theorem

Theorem Every coherent apportionment method is compatible with a divisor method. Proof The proof is deferred to Sect. 9.8. ut An immediate consequence is a strengthening of the Theorem in Sect. 9.3. Corollary An apportionment method is coherent and complete if and only if it is a divisor method. Proof The direct part follows from the Coherence Theorem. Every coherent appor- tionment method A is compatible with a divisor method D. Because of completeness every solution set of A includes all ties. So do the solution sets of the divisor method D, by definition. Hence all solution sets coincide; that is, A D D.The converse part is the Theorem in Sect. 9.3. ut The proof of the Coherence Theorem is accomplished by first establishing addi- tional structural properties. Section 9.5 shows that a coherent apportionment method is house size monotone. Section 9.6 adds vote ratio monotonicity. Section 9.7 contemplates how to retrieve a signpost sequence whose existence is granted. Section 9.8 turns to the proof proper. Divisor methods, being coherent, are house size monotone and vote ratio monotone. The choice is further narrowed down by imposing additional properties. Transfer-proofness leads to the family of stationary divisor methods (Sect. 9.9). Compatibility with natural two-party apportionments leaves just a single procedure: the divisor method with standard rounding (Sect. 9.10). Apportionment jargon paraphrases a violation of principles or a breach of mono- tonicity as a “paradox”. The new states paradox indicates a failure of coherence (Sect. 9.11), the Alabama paradox a failure of house size monotonicity (Sect. 9.12), and the population paradox a failure of vote ratio monotonicity (Sect. 9.13). Prime examples are quota methods. They are neither coherent, nor house size monotone, nor vote ratio monotone. 9.5 House Size Monotonicity 163

9.5 House Size Monotonicity

House size monotonicity ensures that an increase of the house size never goes along with the loss of a seat. If x is a seat vector for house size h and y is a seat vector for the larger house size h C 1 then some party i does better and no other party ` does worse: yi D xi C 1 and yk D xk for all k ¤ i. For two vectors x; y 2 N the componentwise ordering xj Ä yj, j Ä `, is abbreviated by x Ä y and by y  x. Definition An apportionment method A is called “house size monotone” when for all house sizes h < H and vote vectors v 2 Œ0I1/` it fulfills properties a and b: a. .` D 2/ In two-party systems every seat vector .x1; x2/ 2 A .hI .v1;v2// and every seat vector . y1; y2/ 2 A .HI .v1;v2// satisfy .x1; x2/ Ä . y1; y2/. b. .`  3/ Systems with three or more parties are such that for every seat vector x 2 A.hI v/ there exists a seat vector y 2 A.HI v/ satisfying x Ä y, and that for every seat vector y 2 A.HI v/ there exists a seat vector x 2 A.hI v/ satisfying y  x. Property a offers a symmetric handling of the two situations when stepping up from a small house size h to a larger house size H, and when going down from a larger house size H to a smaller house size h. Either way all seat vectors .x1; x2/ for house size h and . y1; y2/ for house size H satisfy .x1; x2/ Ä . y1; y2/;thatis, . y1; y2/  .x1; x2/. Property b handles the two situations separately because in the first situation the seat vector y depends on x, while in the second x depends on y. A symmetric dependence is too much to ask for. It is not hard to see why. The example in Table 4.2 displays ten tied seat vectors x for house size 18. A vector y that dominates all of them must satisfy y  .6;5;4;3;2/and have component sum yC  20. House size 19 would be missed out. For this reason property b is more modest: For every seat vector x 2 A.18I v/ there exists a solution y 2 A.19I v/ such that x Ä y,andfor every seat vector y 2 A.19I v/ there exists a solution x 2 A.18I v/ satisfying y  x. All divisor methods are house size monotone. In fact, the monotone progression from house size h with seat vector x to house size h C 1 with seat vector y  x is part of the argument that divisor methods are well-defined; see Sect. 4.5. In the presence of coherence house size monotonicity comes for free. Theorem Every coherent apportionment method is house size monotone. Proof The proof is presented in two parts. Part I treats systems with two parties, part II systems with three or more parties. Let A denote a coherent apportionment method. I. Part I investigates two-party systems. It suffices to establish the assertion for house sizes h and H D h C 1. Given a vote vector v D .v1;v2/ we set t D v1=v2 and pass to the scaled weights v=v2 D .t;1/, which is allowed by decency. For two arbitrary seat vectors, . y1; y2/ 2 A .hI .t;1// and .z1; z2/ 2 A .h C 1I .t;1// we need to show that . y1; y2/ Ä .z1; z2/. 164 9 Securing System Consistency: Coherence and Paradoxes

Consider the apportionment of 2hC1 seats among four parties with respective weights t, 1, t,and1. We claim that in every solution .x1; x2; x3; x4/ the first two components sum to h or h C 1:

.x1; x2; x3; x4/ 2 A .2h C 1I .t;1;t;1// H) x1 C x2 2fh; h C 1g:

The claim is proved as follows. Since x1 and x3 are tied via the weight t,andx2 and x4 are tied via the weight 1, balancedness restricts the ranges of x3 and x4 to x1  1 Ä x3 Ä x1 C 1 and x2  1 Ä x4 Ä x2 C 1. If the sum of the first two components is h  1 seats or less or if it is h C 2 seats or more, then the sum of all four components cannot match the house size 2h C 1:

x1 C x2 Ä h  1 H) xC Ä .h  1/ C .x1 C 1/ C .x2 C 1/ Ä 2.h  1/ C 2 D 2h;

x1 C x2  h C 2 H) xC  .h C 2/ C .x1  1/ C .x2  1/  2.h C 2/  2 D 2h C 2:

The only possibilities left are x1 C x2 D h or x1 C x2 D h C 1. The claim is proved. The set A .2h C 1I .t;1;t;1// also includes the solution .x3; x4; x1; x2/ where the first two components are interchanged with the last two components, because of anonymity. If x1 C x2 D h then x3 C x4 D h C 1, while if x1 C x2 D h C 1 then x3 C x4 D h. We continue with the first case, x1 C x2 D h. The second case is dealt with in a similar way. On the one hand coherence of partial problems secures that .x1; x2/ lies in A .hI .t;1//, as does . y1; y2/. Coherence of substituted solutions permits replacing the first two components in the seat vector .x1; x2; x3; x4/ by . y1; y2/:

. y1; y2; x3; x4/ 2 A .2h C 1I .t;1;t;1//:

On the other hand coherence of partial problems ascertains that .x3; x4/ lies in A .h C 1I .t;1//, as does .z1; z2/. Coherence of substituted solutions allows replacing the last two components in the seat vector . y1; y2; x3; x4/ by .z1; z2/:

. y1; y2; z1; z2/ 2 A .2h C 1I .t;1;t;1//:

Here y1 and z1 are tied via the weight t,andy2 and z2 are tied via the weight 1. Balancedness restricts the ranges of z1 and z2 to y1  1 Ä z1 Ä y1 C 1 and y2  1 Ä z2 Ä y2 C 1. The option z1 D y1  1 fails since it implies z1 C z2 Ä . y1 1/C. y2 C1/ Ä h, which contradicts z1 Cz2 D hC1. The option z2 D y2 1 fails for the same reason. The remaining options entail y1 Ä z1 and y2 Ä z2.The proof of part I is complete. II. Part II turns to systems with `  3 parties. The proof of monotonicity when stepping up from house size h to H D h C 1 is presented in detail. We assume that the apportionment method A is house size monotone for all systems up to size `  1, but violates house size monotonicity in an `-party 9.6 Vote Ratio Monotonicity 165

system. That is, for some vote vector v D .v1;:::;v`/ there exists a seat vector .x1;:::;x`/ 2 A.hI v/ such that all seat vectors z D .z1;:::;z`/ 2 A.h C 1I v/ embrace a party i losing seats: xi > zi. We fix such a vector z. Since the house size grows there is another party k ¤ i doing better than before, xk < zk. Anonymity permits moving the two parties to positions i D 1 and k D 2,givingx1 > z1 and x2 < z2. Coherence of partial problems yields .x1; x2/ 2 A .x1 C x2I .v1;v2// and .z1; z2/ 2 A .z1 C z2I .v1;v2//. In the case x1 C x2 < z1 C z2 part I implies x1 Ä z1 which contradicts x1 > z1. In the case x1 C x2 > z1 C z2 part I implies x2  z2 which contradicts x2 < z2. The remaining case is x1 C x2 D z1 C z2. Setting n D h  .x1 C x2/ D h  .z1 C z2/, coherence of partial problems secures

.x3;:::;x`/ 2 A .nI .v3;:::;v`// ; .z3;:::;z`/ 2 A .n C 1I .v3;:::;v`// :

By assumption there is a seat vector . y3;:::;y`/ 2 A .n C 1I .v3;:::;v`// satisfying .x3;:::;x`/ Ä . y3;:::;y`/. Coherence of substituted solutions allows replacing, in z, the first two components by .x1; x2/ and the last `2 components by . y3;:::;y`/. The two substitutions yield a vector y D .x1; x2; y3;:::;y`/ 2 A.h C 1I v/ satisfying x Ä y. This contradicts the assumption that method A violates house size monotonicity for the `-party system with vote vector v. Monotonicity for decreasing house sizes is handled similarly. The proof of part II is complete. ut

9.6 Vote Ratio Monotonicity

Vote ratio monotonicity compares two elections that share the same house size h and the same system size `. Let the vote vectors v D .v1;:::;v`/ and w D .w1;:::;w`/ yield the seat vectors x 2 A.hI v/ and y 2 A.hI w/. How would two parties assess their performances from one election to the other? Due to decency we may standardize the weight of the second party to be unity. The standardized vote vectors are  à  à 1 v1 v3 v` 1 w1 w3 w` v D ;1; ;:::; ; w D ;1; ;:::; : v2 v2 v2 v2 w2 w2 w2 w2

Which seat constellations appear adequate when the standardized weight of the first party increases, v1=v2 < w1=w2, while the second party is stuck with its weight of unity? Common sense would frown upon an outcome where the first party loses seats, x1 > y1, while at the same time the second party gains seats, x2 < y2.The opposite should occur. Being relatively more successful the first party may gain seats: x1 Ä y1. Or the second party with its lack of advance may lose seats: x2  y2. These considerations motivate the notion of vote ratio monotonicity. 166 9 Securing System Consistency: Coherence and Paradoxes

Definition An apportionment method A is called “vote ratio monotone” when all seat vectors x 2 A.hI v/ and y 2 A.hI w/ and all parties i; k Ä ` satisfy

vi wi < H) xi Ä yi or xk  yk: vk wk

All divisor methods are vote ratio monotone. Indeed, the case xi Ä yi visibly verifies the definition. In the case xi > yi the scaled weights satisfy vi=D  wi=E, with feasible divisors D for x and E for y.ThisgivesE=D  wi=vi > wk=vk.Now vk=D > wk=E establishes xk  yk. Again the definition is verified. Coherence guarantees vote ratio monotonicity. Theorem Every coherent apportionment method is vote ratio monotone. Proof The proof comes in two parts. Part I treats systems with two parties, part II systems with three or more parties. Let A denote a coherent apportionment method.

I. Part I handles two-party systems. For seat vectors .z1; z2/ 2 A .hI .v1;v2// and . y1; y2/ 2 A .hI .w1; w2// we show that v1=v2 < w1=w2 implies z1 Ä y1 or z2  y2. With s D v1=v2 and t D w1=w2 decency simplifies the weights: . y1; y2/ 2 A .hI .t;1//and .z1; z2/ 2 A .hI .s;1//. Consider the apportionment of 2h seats among four parties with respective vote weights t, 1, s,and1,wheret > s. We claim that in every solution .x1; x2; x3; x4/ the sum of the first two components is greater than or equal to h:

.x1; x2; x3; x4/ 2 A .2hI .t;1;s;1// H) x1 C x2  h:

The claim is proved by establishing the inequality x1 C x2  x3 C x4.The inequality instantly gives 2.x1 C x2/  xC D 2h, whence x1 C x2  h.In view of t > s concordance yields x1  x3. In the case x2  x4 the inequality x1 C x2  x3 C x4 is immediate. In the case x2 < x4 balancedness entails x4 D x2 C 1. Hence x2 C x4 D 2x2 C 1 is odd. Since xC D 2h is even, x1 C x3 is odd, too. It follows that x1 ¤ x3, whence x1 > x3;thatis,x1  x3 C 1.Now x1 C x2  .x3 C 1/ C .x4  1/ D x3 C x4 establishes the inequality in the case x2 < x4. The claim is proved. The set A .2hI .t;1;s;1//also includes the solution obtained from interchang- ing in x the second and fourth components: xQ WD .x1; x4; x3; x2/, by anonymity. If x2 > x4 we continue the argument with xQ in place of x. Therefore there is no loss of generality in assuming x2 Ä x4. The conclusion x1 C x2  h is split into the case x1 C x2 D h and the case x1 C x2 > h. The case x1 C x2 D h implies x3 C x4 D h. Coherence of substituted solutions allows substitution of .x1; x2/ by . y1; y2/, and of .x3; x4/ by .z1; z2/. In the seat vector . y1; y2; z1; z2/ 2 A .2hI .t;1;s;1//concordance ascertains y1  z1 as was to be shown. The case x1 C x2 > h is more delicate. On the one hand coherence of partial problems secures .x1; x2/ 2 A .x1 C x2I .t;1//. Compared with . y1; y2/ 2 A .hI .t;1//house size monotonicity implies x2  y2. 9.7 Retrieval of an Underlying Signpost Sequence 167

On the other hand coherence of partial problems ascertains .x3; x4/ 2 A .x3 C x4I .s;1//,wherex3 C x4 D 2h  .x1 C x2/ y1 and z2 < y2. With s D v1=v2 and t D w1=w2 coherence of partial problems yields .z1; z2/ 2 A .z1 C z2I .s;1// and . y1; y2/ 2 A .y1 C y2I .t;1//,wheres < t as well as z1 > y1 and z2 < y2. Contradictions evolve, as follows. In the case z1 C z2 D y1 C y2 part I yields z1 Ä y1, whence z2  y2.This contradicts both, z1 > y1 as well as z2 < y2. In the case z1 C z2 < y1 C y2 we consider a solution .x1; x2/ 2 A .y1 C y2I .s;1//. For the passage from .z1; z2/ to .x1; x2/ house size monotonicity yields z1 Ä x1 and z2 Ä x2. For the passage from .x1; x2/ to . y1; y2/ part I gives x1 Ä y1, whence x2  y2. Together these imply z1 Ä y1, which contradicts z1 > y1. In the case z1 C z2 > y1 C y2 we consider a solution .x1; x2/ 2 A .z1 C z2I .t;1//. For the passage from .z1; z2/ to .x1; x2/ part I gives z1 Ä x1, whence z2  x2. For the passage from .x1; x2/ to . y1; y2/ house size monotonicity yields x1  y1 and x2  y2. Together these imply z2  y2, which contradicts z2 < y2. The proof of part II is complete. ut

9.7 Retrieval of an Underlying Signpost Sequence

Normally a divisor method D is given by a signpost sequence s.0/; s.1/; s.2/; : : : and the accompanying rounding rule ŒŒ  . Then a solution set D.hI v/ needs a feasible divisor to verify its definition. As a preparation for the proof of the Coherence Theorem in Sect. 9.8 we briefly digress and study the inverse problem. Given all solution sets D.hI v/, how do we retrieve the signpost sequence that defines D? The clue is to start from a sequence that is proportional to the signpost sequence, but that has one of its members scaled to be equal to unity. Scaling by the first signpost s.1/ 2 Œ0I 1 is problematic since it may be zero. The second signpost is certainly positive, s.2/  1. Therefore we define the scaled sequence

s.1/ s.3/ s.n/ t.1/ WD ; t.2/ WD 1; t.3/ WD ; :::; t.n/ WD ; ::: s.2/ s.2/ s.2/ 168 9 Securing System Consistency: Coherence and Paradoxes

Two-party systems suffice for the retrieval process. For n  1 we consider the apportionment of n C 1 seats among a party with weight t.n/ and another party with weight unity. Decency permits multiplying the weights by s.2/, whence we realize D .n C 1I .t.n/; 1// D D .n C 1I .s.n/; s.2/// D f.n;1/;.n  1; 2/g. Indeed, the signpost s.n/ may be rounded upwards to n or downwards to n  1, while s.2/ may be rounded upwards to 2 or downwards to 1. Since the house size is n C 1, one signpost is rounded upwards and the other downwards. The two solutions are .n;1/ and .n  1; 2/. The seat vector .n;1/ ceases to be a solution if the weight of the first party becomes smaller than t.n/ while the second party maintains its weight of unity. This reveals the value t.n/ to be the infimum weight for a party to win n seats in a house of size n C 1 when competing against another party whose weight is unity:

t.n/ D inf f t >0j .n;1/2 D .n C 1I .t;1//g :

The infimum is taken over a set containing the value t D n, by exactness. This entails t.n/ Ä n and t.n/=n Ä 1. The quotients t.n/=n converge to a limit L:

t.n/ 1 s.n/ 1 L D lim D lim D : n!1 n s.2/ n!1 n s.2/

Indeed, the localization property s.n/ 2 Œn  1I n implies limn!1 s.n/=n D 1. A division by L retrieves the original signpost sequence: t.n/=L D s.n/. In summary the retrieval of the signpost sequence proceeds in three steps. The first step uses the relations .n;1/2 D .n C 1I .t;1//to define the sequence t.n/.The second step calculates the limit L of the quotients t.n/=n. The third step retrieves the signposts via the scaling t.n/=L D s.n/. These steps reappear in the proof of the Coherence Theorem. The proof is more demanding since it does not presuppose the existence of a signpost sequence, but needs to construct one from scratch.

9.8 Proof of the Coherence Theorem in Sect. 9.4

Theorem Every coherent apportionment method is compatible with a divisor method. Proof Let A be a coherent apportionment method. Parts I–III of the proof investigate solution sets A.hI v/ in two-party systems. Part I considers equal weights v D .1; 1/, part II special weights v D .t;1/, and part III arbitrary weights v D .v1;v2/.PartIV constructs the signpost sequence that induces the desired divisor method D.PartV verifies that A is compatible with D.

I. Part I begins with two parties with equal vote weights, v1 D v2. Decency permits a passage to the vote vector .1; 1/. Here the solution sets are fully determined through the organizing principles. If the house size is even, 9.8 Proof of the Coherence Theorem in Sect. 9.4 169

balancedness forces the two parties to split the seats evenly. If the house size is odd, balancedness permits a party to be one seat ahead of the other party. Anonymity implies that the solution set also includes the permuted solution, with the other party ahead of the first. In summary, 8 Â Ã ˆ n C 1 n C 1 < ; n 1 ; 2 2 in case C is even A .n C 1I .1; 1// D n Á  Áo :ˆ n n n n 1; ; ; 1 n 1 2 C 2 2 2 C in case C is odd.

II. Part II studies two-party systems where the second party has weight unity. Hence the vote vector is .t;1/.Forn  1 let Tn be the set consisting of all t >0so that in a house of size n C 1 the first party with weight t gains n seats and the second party with weight unity gains one seat: ˚ ˇ « ˇ Tn D t >0 .n;1/2 A .n C 1I .t;1// :

Exactness entails n 2 Tn. Vote ratio monotonicity implies that Tn is an interval, but this fact is not needed in the sequel. Central quantities for the proof are the infima of the sets Tn:

t.n/ D inf Tn 2 Œ0I n:

The infimum weights fulfill t.2/ D 1 Ä t.n/ for all n >2. Indeed, part I says .2; 1/ 2 A .3I .1; 1//, whence 1 2 T2; thus t.2/ Ä 1.Fort <1concordance yields .2; 1/ 62 A .3I .t;1//, whence t 62 T2 and t.2/  1. Thus we have t.2/ D 1. For n >2we have t 62 Tn for t <1, whence 1 Ä t.n/. Weights t > t.n/ imply a lower bound for the seats of the first party or an upper bound for the seats of the second party, uniformly for all house sizes h and for all seat vectors .x1; x2/ 2 A .hI .t;1//:

t > t.n/ H) x1  n or x2 Ä 1: (9.1)

Indeed, vote ratio monotonicity implies that all solutions . y1; y2/ 2 A .n C 1I .t;1//satisfy y1  n, whence y2 Ä 1. For house sizes h ¤ n C 1 and arbitrary solutions .x1; x2/ 2 A .hI .t;1// we apply house size monotonicity: either h > n C 1 and x1  y1  n,orh < n C 1 and x2 Ä y2 Ä 1.Thisproves (9.1). Weights 0

s < t.m/ H) x1 Ä m  1 or x2  2: (9.2) 170 9 Securing System Consistency: Coherence and Paradoxes

Indeed, vote ratio monotonicity implies that all solutions . y1; y2/ 2 A .m C 1I .s;1//fulfill y1 Ä m  1, whence y2  2. For house sizes h ¤ m C 1 and arbitrary solutions .x1; x2/ 2 A .hI .s;1// we again call upon house size monotonicity: either h < m C 1 and x1 Ä y1 Ä m  1,orh > m C 1 and x2  y2  2. This proves (9.2). III. Part III admits arbitrary vote vectors .v1;v2/. Fixing some indices n; m  1 with t.m/>0we claim that for all house sizes h every seat vector . y1; y2/ 2 A .hI .v1;v2// satisfies

v1 t.n/ > H) y1  n or y2 Ä m  1: (9.3) v2 t.m/

Indeed, the premise yields t.n/=v1 < t.m/=v2. With average a D .t.n/=v1 C t.m/=v2/=2 the weights are scaled into t D av1 and s D av2. The scaled weights satisfy t > t.n/ and s < t.m/, thus preparing for an invocation of (9.1)and(9.2). To this end we apportion n C m seats among three parties with respective weights t, s,and1. We verify that in every solution .x1; x2; x3/ the component x1 is greater than or equal to n while x2 is less than or equal to m  1:

.x1; x2; x3/ 2 A .n C mI .t; s;1// H) x1  n and x2 Ä m  1:

Verification splits into the cases x3  2 and x3 Ä 1. In the first case coherence of partial problems yields .x1; x3/ 2 A .x1 C x3I .t;1//.From(9.1) we obtain x1  n or x3 Ä 1. The latter inequality is ruled out by assumption. Hence x1  n holds true. We cannot have x2 > m  2 since it implies xC > n C.m  2/C2 D n C m D xC, a contradiction. Hence x2 Ä m  1 holds true, too. The second case has x3 Ä 1. Coherence of partial problems yields .x2; x3/ 2 A .x2 C x3I .s;1//.From(9.2) we obtain x2 Ä m  1 or x3  2. The latter inequality is ruled out by assumption. Hence x2 Ä m  1 holds true. We cannot have x1 < n since it implies xC < n C .m  1/ C 1 D n C m D xC,a contradiction. Hence x1  n holds true, too. Verification is complete. Coherence of partial problems secures .x1; x2/ 2 A .x1 C x2I .t; s//, with x1  n and x2 Ä m  1. Returning to (9.3) we consider an arbitrary seat vector . y1; y2/ in A .hI .v1;v2// D A .hI .t; s//. In the case h D x1 C x2 coherence of substituted solutions yields . y1; y2; x3/ 2 A .n C mI .t; s;1//, whence y1  n and y2 Ä m  1. Otherwise we employ house size monotonicity: either h > x1 C x2 and y1  x1  n,orh < x1 C x2 and y2 Ä x2 Ä m  1.This proves (9.3). IV. Part IV establishes and explores the inequalities

n t.n/ > for all n  1 and for all m  2: (9.4) m  1 t.m/ 9.8 Proof of the Coherence Theorem in Sect. 9.4 171

The argument is indirect. Let us assume that there are indices n  1 and m  2 satisfying n=.m  1/ Ä t.n/=t.m/. The inequality necessitates t.n/>0.Wemovetowards(9.3) by passing to reciprocals: .m  1/=n  t.m/=t.n/. When compared with (9.3) the interchange of the indices n and m is negligible. However, we cannot neglect that the inequality in the premise of (9.3) is strict. Strict inequality is achieved by the weights v1.k/ WD m  1 C 1=k and v2.k/ WD n with k  1:

v1.k/ m  1 t.m/ >  : v2.k/ n t.n/

With v.k/ WD .v1.k/; v2.k// every solution .y1.k/; y2.k// 2 A .m  1 C nI v.k// satisfies y1.k/  y1.kC1/  m, by vote ratio monotonicity and (9.3). Therefore the integer sequence y1.k/, k  1, eventually takes on a constant integer value y1  m. Hence the seat vector y D . y1; m  1 C n  y1/ is common to all solution sets A .m  1 C nI v.k// eventually for all k. Since the vote vectors v.k/ converge to x WD .m  1; n/, exactness requires the solutions y and x to be equal (Sect. 4.3). But they are distinct: y1  m > m  1 D x1. Our assumption is untenable. This proves (9.4). Exploration of (9.4) uses the rearrangement t.n/=n < t.m/=.m1/ to obtain a string of inequalities: t.n/ t.n/ t.m/ t.m/ t.n/ lim sup Ä sup Ä inf Ä lim inf D lim inf : 2 n!1 n n1 n m m  1 m!1 m  1 n!1 n

Thus equality holds throughout. Therefore the quotients t.n/=n converge to a limit, L say, which is equal to the supremum of t.n/=n and to the infimum of t.m/=.m  1/. Hence we obtain t.n/ t.m/ Ä L Ä for all n  1 and for all m  2: n m  1 This puts us in a position to define numbers s.0/ D 0 and s.n/ D t.n/=L for n  1.Forn D 1 we obtain s.1/ 2 Œ0I 1 from part II. For n D m  2 the last display yields s.n/=n Ä 1 Ä s.n/=.n  1/. This is just another way of expressing the localization property s.n/ 2 Œn  1I n.Furthermore(9.4) turns into s.n/=n < s.m/=.m  1/ for all n  1 and m  2. This is just another way of expressing the left-right disjunction. Hence s.0/; s.1/; s.2/; : : : is a signpost sequence. V. Part V verifies that the apportionment method A is compatible with the divisor method D that is induced by the signpost sequence just constructed. We claim that every seat vector x 2 A.hI v/ satisfies the max-min inequality that belongs to D: v v max j Ä min j : jÄ` s.xj C 1/ jÄ` s.xj/ 172 9 Securing System Consistency: Coherence and Paradoxes

Indeed, for every two parties i ¤ k we have .xi; xk/ 2 A .xi C xkI .vi;vk//,by coherence of partial problems. Let us assume that some such pair with xk  1 violates the max-min inequality:

v s.x C 1/ t.n/ i > i D ; vk s.xk/ t.m/

where n D xi C 1 and m D xk.From(9.3) we obtain xi  n D xi C 1 or xk Ä m  1 D xk  1, a contradiction. Our assumption is untenable. With v = . 1/ v = . / v = . 1/ v = . / i s xi C Ä k s xk we obtain maxiÄ` i s xi C Ä minkWxk1 k s xk . The convention vk=0 D1permits adjoining the cases xk D 0. The claim is established. The proof of the Coherence Theorem is complete. ut Our presentation emphasizes house size monotonicity and vote ratio monotonic- ity as a means to prove the Coherence Theorem. The conclusion that they serve this single purpose is rather misleading however. Monotonicity requirements, in the specific forms as they apply to the variables of apportionment problems, are quite natural. Indeed they are so natural that situations from which they are absent are felt so upsetting to be termed paradoxes. Paradoxical instances are reviewed in Sect. 9.11–9.14. Prior to a parade of paradoxes we complete the constructive part of the structural discussion. The Coherence Theorem reduces the vast class of all apportionment methods to the subclass of divisor methods. Still there is an abundance of divisor methods. Section 9.9 combines transfers of votes and seats in a dynamic way that leads to the much smaller family of stationary divisor methods, a one-parameter subclass. Section 9.10 coherently extends the natural solutions for two parties to larger systems. This extension is characteristic of the divisor method with standard rounding.

9.9 Coherence and Stationary Divisor Methods

This section discusses a criterion that leads to stationary divisor methods through well-behavior of successive apportionment solutions. The criterion asks that, as voters swing more and more from one party to another party, at some point the receiving party is apportioned one more seat, the shrinking party one less seat, and the seats of the remaining parties are as they were. If the ensemble of swing voters continues to grow, the receiving party is assigned eventually two more seats, the diminishing party two seats less, and the other parties remain as they were; etc. An apportionment method A is termed “transfer-proof” when for all solutions x 2 A.hI v/ every transfer of z seats from a party k to a party i is realized while transferring votes from party k to party i,forallz D 1;:::;xk  1. To be precise let y.z/ denote the seat vector which transfers z seats to party i from party k without affecting the other parties; its components are yi.z/ WD xi C z, 9.9 Coherence and Stationary Divisor Methods 173 yk.z/ WD xk  z,andyj.z/ WD xj for all j ¤ i; k. Let the vote vector w.u/ mirror the defection of u swing voters from party k to party i; its weights are wi.u/ WD vi C u, wk.u/ WD vk  u,andwj.u/ WD vj for all j ¤ i; k. The property of transfer-proofness demands that every seat vector y.z/ is a solution that belongs to some vote vector w.u/;thatis,y.z/ 2 A .hI w.u//. Three subtleties are worth noting. Firstly, the range of variation of z is given by 1;:::;xk  1. Therefore a shrinking party k is of interest only if it starts with two or more seats: xk  2. The definition neglects single-seat parties since for them the range is empty. Thus pervious and impervious divisor methods can be treated simultaneously. Secondly, a swing loss u does not have to be an integer even though we refer to it as a set of voters. Technically it may be any value in the interval Œ0I vk. Thirdly, transfer-proofness does not force the path of solutions which belong to the vote vectors w.u/, u 2 Œ0I vk, to consist exclusively of the seat vectors y.1/;:::;y.xk  1/. Perturbations by u may give rise to additional seat vectors in- between. Transfer-proofness only insists that the path visits all seat vectors y.z/ and does not omit any one of them. In the case that u voters defect from party k to party i and all other vote counts remain constant, the seat vectors change from x 2 A.hI v/ to y 2 A .hI w.u// say. Vote ratio monotonicity states that the seat numbers satisfy xi Ä yi or xk  yk. However, in the presence of a bilateral voter swing the disjunction “or” is superseded by the conjunction “and”:

xi Ä yi and xk  yk:

Indeed, let us assume to the contrary that party i loses seats: xi > yi. Then vote ratio monotonicity implies not only xk  yk,butalsoxj  yj for all j ¤ i; k.In particular the component sum of x is larger than that of y. This is impossible as both are equal to the house size h. Our assumption being untenable, the conjunction “and” is vindicated. Thus a voter swing to party i from party k implies that the growing party i cannot lose seats and the shrinking party k cannot gain seats. The fate of the other parties j ¤ i; k is indeterminate. Transfer-proofness demands that, as u varies through Œ0I vk, the swing process realizes all of the specific seat vectors y.z/ which transfer z seats to party i from party k while leaving the other parties unaffected. Theorem An apportionment method is coherent, complete and transfer-proof if and only if it is a stationary divisor method. Proof In view of the Corollary in Sect. 9.4 it suffices to prove the simpler statement: A divisor method A is transfer-proof if and only if A is stationary. The direct part is proved indirectly. We show that a non-stationary divisor method A is not transfer-proof. For a stationary divisor method the spacing of signposts is constant: sr.n C 1/  sr.n/ D 1 for all n  1 (Sect. 3.10). Since A is assumed non- stationary there are signposts fulfilling s.nC1/s.n/ ¤ s.nC2/s.nC1/ DW b >0, for some n  1. The proof elaborates the case s.n C 1/  s.n/>b; the case s.n C 1/  s.n/0is smaller than s.n C 3/  s.n C 2/ and s.n C 1/ .1  s.n C 3/=s.n C 4//. With these terms we obtain s.n C 1/  s.n/ D ma C b. Consider three parties with weights v1 D s.n/ C .m  3/a, v2 D s.n C 2/ C a, and v3 D s.n C3/.Weverifythatx D .n; n C2;n C2/ is the unique solution for the apportionment of h D 3n C 4 seats. The divisor D D 1 proves feasible. Because of v1=D > s.n/ and v1=D < s.n/ C ma C b D s.n C 1/ the quotient v1=D is rounded to n. Because of v2=D > s.n C 2/ and v2=D D s.n C 2/ C a < s.n C 3/ the quotient v2=D is rounded to n C 2. The quotient v3=D D s.n C 3/ can be rounded to n C 2 or to n C 3. With house size 3n C 4 only the first option is viable. Verification is complete. A transfer of u 2 Œ0I v2 votes from the second party to the first party induces the weights w1.u/ D s.n/C.m3/aCu, w2.u/ D s.nC2/Cau,andw3.u/ D s.nC3/. The seat vectors y D . y1; y2; y3/ in the solution sets A .3n C 4I w.u// are as follows:

u 2 Œ0I a/ H) y D .n; n C 2;n C 2/; (9.5) u D a H) y D .n; n C 2;n C 2/ or y D .n; n C 1; n C 3/; (9.6) u 2 .aI a C b H) y D .n; n C 1; n C 3/; (9.7) u 2 .a C bI 2a C b/ H) y D .n; n C 1; n C 3/; (9.8) u D 2a C b H) y D .n; n C 1; n C 3/ or y D .n C 1; n; n C 3/; (9.9)

u 2 .2a C bI v2 H) y1  n C 1; y2 Ä n; y3 D 3n C 4  y1  y2: (9.10)

The seat vector y.1/ D .n C 1; n C 1; n C 2/ which in x transfers one seat from the second party to the first party does not materialize. Hence the divisor method A is not transfer-proof. It remains to establish statements (9.5)–(9.10). In lines (9.5)–(9.7) the divisor D D 1 proves feasible. The arguments to determine y run parallel to the verification of x above. In lines (9.8)and(9.9) the swing parameter takes the form u D aCbCc with c 2 .0I a. A feasible divisor is D D .s.n C 1/  c/=s.n C 1/ < 1. In line (9.8) the first quotient w1.a C b C c/=D D s.n C 1/  2.a  c/=D is rounded to n. The second quotient is equal to s.n C 1/ and may be rounded to n or n C 1.By choice of a the third quotient lies between s.n C 3/ and s.n C 4/ and is rounded to n C 3. Hence the unique solution is y D .n; n C 1; n C 3/. In line (9.9)thefirst quotient also becomes equal to s.n C 1/ and thus gives rise to the second solution y D .n C 1; n; n C 3/.Line(9.10) follows from the conjunction that is discussed prior to the theorem. The proof of the direct part is complete. The converse part shows that a stationary divisor method A D DivStar is transfer- proof. Let x 2 A.hI v/ be a seat vector which apportions to some party k at least two seats: xk  2. A feasible divisor D for x fulfills xi  1 C r D sr.xi/ Ä vi=D Ä 9.10 Coherence and the Divisor Method with Standard Rounding 175 sr.xi C 1/ D xi C r, i Ä `.Fromxk  2 we obtain sr.xk/  xk  1 and .xk  1/D Ä sr.xk/D Ä vk. This allows a transfer of zD 2 Œ0I vk votes from party k to apartyi ¤ k,forz D 1;:::;xk  1, leading to the vote vector w.zD/. We claim that the seat vector y.z/ which transfers z seats to party i from party k belongs to w.zD/;thatis,y.z/ 2 A .hI w.zD//. The divisor D is feasible also for y.z/.The fundamental relations are verified simply via sr .yi.z// D sr.xi C z/ D sr.xi/ C z and wi.zD/=D D vi=D C z. ut

9.10 Coherence and the Divisor Method with Standard Rounding

Two-party problems always admit seat vectors x D .x1; x2/ that most people would agree to accept as the “natural two-party apportionment” of h seats. They emerge from standard rounding of the ideal shares of seats with regard to the two parties:     v1 v2 x1 2 h ; x2 2 h : v1 C v2 v1 C v2

The same solutions are delivered by the divisor method with standard rounding and by the Hare-quota method with residual fit by greatest remainders. In fact, for two- party systems the two methods coincide:

DivStd .hI .v1;v2// D HaQgrR .hI .v1;v2// ;

2 for all house sizes h 2 N and for all vote pairs .v1;v2/ 2 Œ0I1/ . The natural two-party apportionments can be coherently extended to arbitrary party systems. The unique extension is the divisor method with standard rounding. Theorem The unique coherent extension of the natural two-party apportionments to party systems of arbitrary size is the divisor method with standard rounding. Proof Evidently DivStd is a coherent extension of the natural two-party apportionments. It remains to prove uniqueness. Let A be a coherent extension of the natural two- party apportionments. We claim that

A.hI v/ D DivStd.hI v/; for all house sizes h 2 N, all system sizes `  2, and all vote vectors v 2 Œ0; 1/`. The direct inclusion A.hI v/  DivStd.hI v/ is established indirectly. We assume that some seat vector z 2 A.hI v/ is not contained in DivStd.hI v/. For arbitrary seat vectors y 2 DivStd.hI v/ let c. y/ be the count of components j Ä ` with yj ¤ zj. Since the solution set DivStd.hI v/ is finite there is a vector x 2 DivStd.hI v/ such that c.x/ is minimum. By assumption we have c.x/  2. There exist two parties i 176 9 Securing System Consistency: Coherence and Paradoxes and k satisfying xi < zi and xk > zk. Without loss of generality we continue with the case p WD xi C xk  zi C zk DW q. With regard to the two parties i and k we have xi 2hhpvi=.vi C vk/ii and zi 2hhqvi=.vi C vk/ii . Thus pvi=.vi C vk/  qvi=.vi C vk/, but xi < zi. Because of set-monotonicity (Sect. 3.9) this is impossible unless p D q. Since xi and zi are two distinct rounding values of the two-party ideal share, the fractional remainder of pvi=.viCvk/ must equal one half. Then so does the fractional remainder of pvk=.vi C vk/. Thus .xi; xk/ D .zi  1; zk C 1/ and .zi; zk/ are tied solutions for apportioning p seats proportionally to the two-party vote vector .vi;vk/. Coherence of substituted solutions states that the vector

y WD .x1;:::;xi1; zi; xiC1;:::;xk1; zk; xkC1;:::;x`/ lies in DivStd.hI v/. Clearly y differs from z in fewer components than x.This contradicts the minimality of c.x/. Thus the direct inclusion A.hI v/ Â DivStd.hI v/ is established. A similar argument verifies the converse inclusion. The proof of uniqueness is complete. ut As an example Table 9.1 reconsiders the 2009 Bundestag election data of Table 2.1. SPD and CSU garner 188 seats. With regard to the two-party total of 12,820,726 votes, SPD wins 77.9245% of the total, CSU 22.0755%. The two-party ideal share is 146.498 seat fractions for SPD, and 41.502 seat fractions for CSU. The natural two-party apportionment is 146 seats for SPD, 42 seats for CSU. The table shows that the divisor method with standard rounding complies with the natural two-party apportionment. In contrast, the Hare-quota method with residual fit by greatest remainders yields a grand solution that does not embrace the natural two- party apportionment. The grand solution accords 147 seats to SPD, and 41 seats to CSU (marked ). The example also demonstrates that the Hare-quota method with residual fit by greatest remainders is not coherent. Its lack of coherence is scrutinized in the remaining sections of this chapter. For, if a method is not coherent, it might still be house size monotone or vote ratio monotone. Alas, quota methods violate not only coherence, but also house size monotonicity and vote ratio monotonicity.

Table 9.1 Coherence of DivStd, and incoherence of HaQgrR (17BT2009) Vot es Quotient DivStd Quotient HaQgrR CDU 11;828;277 173.4 173 73.517 173 SPD 9;990;488 146.497 146 146.558 147 FDP 6;316;080 92.6 93 92.655 93 LINKE 5;155;933 75.6 76 75.636 76 GRÜNE 4;643;272 68.1 68 68.115 68 CSU 2;830;238 41.502 42 41.519 41 Sum (divisor, split) 40;764;288 (68,196) 598 (0.54) 598 The natural two-party apportionment of 188 seats to SPD and CSU is 146W42. It is coherent with the grand DivStd apportionment, but distinct from the grand HaQgrR apportionment which assigns 147W41 seats to SPD and CSU 9.11 Violation of Coherence: New States Paradox 177

9.11 Violation of Coherence: New States Paradox

Table 9.2 evaluates the 2014 EP election in Germany with the Hare-quota method with residual fit by greatest remainders. The legal procedure is the divisor method with standard rounding (Table 1.8) which is coherent. The alternative evaluation with the Hare-quota method with residual fit by greatest remainders produces instances of incoherence. The fourth column of Table 9.2 shows the over-all seat apportionment. From this column we extract the partial solution for SPD, GRÜNE, FREIE WÄHLER, and PIRATEN to be 26W10W2W2. When the cumulative forty seats are apportioned solely between the four parties, the Hare-quota method with residual fit by largest remainders leads to the solution 27W11W1W1. The two solutions differ. The whole and its part do not fit together. This is a violation of coherence. More incoherence can be detected. Consider the subset of first fourteen parties: I WD fCDU;:::;DIE PARTEIg. These are the parties gaining one or more seats. The complement are the “11 Others” with no seat. Do the losers matter when distributing the seats among the winners? For the Hare-quota method with residual fit by greatest remainders the answer is: Yes, losers matter. With losers the method produces the solution in the fourth column. Without losers the method yields the solution in the sixth column. This constitutes another violation of coherence.

Table 9.2 Incoherence of the Hare-quota method with residual fit by greatest remainders Including 11 Others Excluding 11 Others (EP2014DE) Vot es Quotient HaQgrR Quotient HaQgrR CDU 8;812;653 28.820 29 29.332 29 SPD 8;003;628 26.174 26 26.639 27 GRÜNE 3;139;274 10.266 10 10.449 11 DIE LINKE 2;168;455 7.092 7 7.217 7 AFD 2;070;014 6.770 7 6.890 7 CSU 1;567;448 5.126 5 5.217 5 FDP 986;841 3.227 3 3.285 3 FREIE WÄHLER 428;800 1.402 2 1.427 1 PIRATEN 425;044 1.390 2 1.415 1 TIERSCHUTZPARTEI 366;598 1.199 1 1.220 1 NPD 301;139 0.985 1 1.002 1 FAMILIE 202;803 0.663 1 0.675 1 ÖDP 185;244 0.606 1 0.617 1 DIE PARTEI 184 709 0.604 1 0.615 1 11 Others 512;442 – 0 Sum (Split) 29;355;092 (0.37) 96 (0.44) 96 Hare-quota 305,782.2 300,444.3 Seats are swapped between parties () depending on whether the 11 Others are included in the apportionment, or excluded. Of the 11 other parties, each has fewer than 110,000 votes and a remainder below 0.37 178 9 Securing System Consistency: Coherence and Paradoxes

Table 9.2 shows what happens numerically. In the grand system the Hare-quota is 29;355;092=96 D 305;782:2. The partial system of the winning fourteen has a smaller vote total: 28,842,560. Hence the Hare-quota is smaller: 300,444.3. The decrease of the quota increases the interim quotients. The increase is proportional to party strength. The quotients of stronger parties increase more significantly than those of weaker parties. The increase of the remainders causes SPD and GRÜNE to overtake FREIE WÄHLER and PIRATEN. A vexing seat transfer troubled the 1993 Council of the Free and Hanseatic City of Hamburg. The Council comprised four political groups of sizes 58W36W20W5 and two indeps. The task was to allocate the fifteen seats of the Committee of Advisory Deputies. The apportionment method used was the Hare-quota method with residual fit by greatest remainders. By including the two indeps into the calculation the seat allocation turned out to be 7W4W3W1W0W0. Without the indeps it would have been 7W5W2W1. Thus inclusion or exclusion of the indeps proved decisive for the second- strongest and third-strongest to divide their seven seats as 4W3,oras5W2. The problem is not bound to parties receiving no seats. We chose have-nots for a start only because they facilitate the exposition. The phenomenon also arises when new parties join and are accompanied by new seats to proportionally enlarge the prevailing house size. Such instances do occur. For example, the German Democratic Republic acceded to the Federal Republic of Germany in 1990 while the Bundestag’s legislative period was in full swing. The apportionment method then in use was the Hare-quota method with residual fit by greatest remainders. The Bundestag decided to enlarge its house size and the sizes of its committees by proportionally adding new seats. Had the apportionments been recalculated from scratch, the old parties would have had to swap some seats for no other reason than the purportedly proportional enlargement. The issue also arises when allocating seats between geographical districts. The expansion of the United States of America from formerly 15 to currently 50 states led to apportionment problems with a growing number of participants. The lack of coherence of the Hare-quota method with greatest remainders when new states accede the Union is termed “new states paradox” by Balinski and Young (1982, p. 44). They illustrate the problem with the accession of Oklahoma in 1907. Giving Oklahoma its ideal share of five seats, New York and Maine would have to swap a seat if the apportionment were calculated from scratch. The new states paradox is avoided by methods whose grand solutions conform with the embedded partial solutions. The whole and its parts must fit together: this is the content of coherence. Lack of coherence of an apportionment method raises suspicions as to whether the method harbors more weaknesses. 9.12 Violation of House Size Monotonicity: Alabama Paradox 179

9.12 Violation of House Size Monotonicity: Alabama Paradox

A non-monotone apportionment method A is prone to a disconcerting behavior. If for house sizes h and h C 1 two seat vectors x 2 A.hI v/ and y 2 A.h C 1I v/ violate the componentwise growth, x — y, then there exists a party that finishes with fewer seats than before even though there are more seats to go around. Here is a perplexing example. In a five-party system with vote vector v D .280; 275; 270; 90; 85/, the Hare-quota method with residual fit by greatest remain- ders produces seat vector x D .1;1;1;1;1/for house size five, but y D .2;2;2;0;0/ for house size six. The extra seat works wonders. Three parties double their representation, and two parties disappear from the scene. Evidently the Hare-quota method with residual fit by greatest remainders violates house size monotonicity. The effect first came to light in the United States of America. The Constitution decrees to conduct a census every ten years and to apportion the seats in the House of Representatives among the States of the Union accordingly. However, it does not specify the apportionment method to be used. In the aftermath of every census fierce debates erupted in the House of Representatives regarding which method to use. The discussions produced a wealth of political arguments and statistical ramifications to advertise specific methods and to criticize counter-proposals. Balinski and Young (1982) tell the tale in a vivid and enlightening manner. The size of the House of Representatives has been fixed at 435 members by statute when Arizona and New Mexico became states in 1912. Until then the determination of the house size was part of the agenda of the decennial apportionment legislation. The Bureau of the Census would prepare tables for different house sizes for the purpose of aiding the decision-making process. After the 1880 census Charles Wesley Seaton, chief clerk of the Census Office, computed apportionments for all house sizes between 275 and 350 using the Hare-quota method with residual fit by greatest remainders. As reported by Balinski and Young (1982, p. 38), Seaton noted a startling quirk that he described in a letter to Congress dated 25 October 1881: While making these calculations I met with the so-called “Alabama” paradox where Alabama was allotted 8 Representatives out of a total of 299, receiving but 7 when the total became 300. Since then the label “Alabama paradox” has commanded a firm place in the field, as a synonym for a breach of house size monotonicity. The data of the 1880 census are shown in Table 9.3. On the way from house size 299 to 300 all interim quotients grow larger of course. Since growth is proportional to size, the more populous states Illinois and Texas advance substantially and pass the split point 0:6715. The progression of less populous Alabama is much smaller and lets it fall below the split point. Thus Alabama loses the residual seat won earlier, while Illinois and Texas pick up an extra seat each. 180 9 Securing System Consistency: Coherence and Paradoxes

Table 9.3 Alabama paradox, US census 1880 US1880Census Population Quotient HaQgrR Quotient HaQgrR New York 5;082;871 30.783 31 30.886 31 Pennsylvania 4;282;891 25.938 26 26.025 26 Ohio 3;198;062 19.368 19 19.433 19 Illinois 3;077;871 18.640 18 18.702 19 Missouri 2;168;380 13.132 13 13.176 13 Indiana 1;978;301 11.981 12 12.021 12 Massachusetts 1;783;085 10.799 11 10.835 11 Kentucky 1;648;690 9.985 10 10.018 10 Michigan 1;636;937 9.914 10 9.947 10 Iowa 1;624;615 9.839 10 9.872 10 Texas 1;591;749 9.640 9 9.672 10 Tennessee 1;542;359 9.341 9 9.372 9 Georgia 1;542;180 9.340 9 9.371 9 Virginia 1;512;565 9.160 9 9.191 9 North Carolina 1;399;750 8.477 8 8.505 8 Wisconsin 1;315;497 7.967 8 7.993 8 Alabama 1;262;505 7.646 8 7.671 7 Mississippi 1;131;597 6.853 7 6.876 7 New Jersey 1;131;116 6.850 7 6.873 7 Kansas 996;096 6.033 6 6.053 6 South Carolina 995;577 6.029 6 6.050 6 Louisiana 939;946 5.692 6 5.711 6 Maryland 934;943 5.662 6 5.681 6 California 864;694 5.237 5 5.254 5 Arkansas 802;525 4.860 5 4.876 5 Minnesota 780;773 4.728 5 4.744 5 Maine 648;936 3.930 4 3.943 4 Connecticut 622;700 3.771 4 3.784 4 West Virginia 618;457 3.745 4 3.758 4 Nebraska 452;402 2.740 3 2.749 3 New Hampshire 346;991 2.101 2 2.108 2 Vermont 332;286 2.012 2 2.019 2 Rhode Island 276;531 1.675 2 1.680 2 Florida 269;493 1.632 1 1.638 1 Colorado 194;327 1.177 1 1.181 1 Oregon 174;768 1.058 1 1.062 1 Delaware 146;608 0.888 1 0.891 1 Nevada 62;266 0.377 1 0.378 1 Sum (Split) 49;371;340 (0.643) 299 (0.6715) 300 Hare-quota 165,121.5 164,571.1 An increase of the house size from 299 to 300 increases all interim quotients, for the larger states Illinois and Texas more so than for Alabama (). With its remainder falling below the split point 0:6715, Alabama loses its residual seat from before 9.13 Violation of Vote Ratio Monotonicity: Population Paradox 181

Another detail of Table 9.3 is worth mentioning. The bottom line features Nevada, a state so small receiving no seat in the main apportionment nor in the residual fit. Yet the table lists a seat for Nevada. The seat is due to the constitutional warranty that “each state shall have at least one representative”, mentioned in Sect. 7.5. The constitutional order is executed as follows. The lacking seats are filled-in from the residual seats before entering the residual apportionment stage. For house sizes 299 and 300 the main apportionment provides all states with at least one representative except Delaware and Nevada. Hence from the residual seats Delaware and Nevada receive one seat each, and only the remaining residual seats are fed into the residual fit by greatest remainders. Chapter 12 has more to say on how to handle seat restrictions.

9.13 Violation of Vote Ratio Monotonicity: Population Paradox

So far our parade of paradoxes focused on the Hare-quota method with residual fit by greatest remainders. This may invite speculations that it is the residual fit by greatest remainders that causes the malfunctioning of the method. This is not so, as is testified by Table 9.4: Every Hare-quota method with a residual fit that obeys concordance violates vote ratio monotonicity. As a consequence these methods fail coherence too. Table 9.4 aims to apportion seven seats among four parties, once with the vote vector v D .717; 96; 94; 93/ and once with vote vector w D .570; 285; 72; 73/. In either situation the Hare-quota is Q D 1000=7. In view of the quotients v=Q, party A is eligible for six or five seats, party B one or zero seats, etc. Concordance imposes the requirement that no party can have more final seats than a weaker party. This allows two seat vectors, x.1/ D .6;1;0;0/ and x.2/ D .5;1;1;0/. As for vote vector w, note that parties C and D swap ranks. Otherwise the same argument produces the seat vectors y.1/ D .4;2;0;1/, y.2/ D .4;1;1;1/,and y.3/ D .3;2;1;1/. Whichever pairing the method realizes, party A loses one seat

Table 9.4 Violation of vote ratio monotonicity of Hare-quota methods with a concordant resid- ual fit Party Vot es Quotient x.1/ x.2/ Vot es Quotient y.1/ y.2/ y.3/ A 717 5.019 6 5 570 3.990 4 4 3 B 96 0.672 1 1 285 1.995 2 1 2 C 94 0.658 0 1 72 0.504 0 1 1 D 93 0.651 0 0 73 0.511 1 1 1 Sum (Split) 1000 (0.655) 7 7 1000 (0.51) 7 7 7 Party A’s vote ratio relative to D increases from 717=93 D 7:7 (left) to 570=73 D 7:8 (right). Yet A loses a seat and D gains a seat, in every pairing of one of the seat vectors x.1/ and x.2/ (left) with one of the seat vectors y.1/, y.2/,ory.3/ (right) 182 9 Securing System Consistency: Coherence and Paradoxes at least, and party D gains a seat. This is incompatible with vote ratio monotonicity. Party A’s vote weight relative to party D is v1=v4 D 717=93 D 7:7 in the first instance, but w1=w4 D 570=73 D 7:8 in the second. Hence party A is relatively more successful than party D in terms of votes, yet A loses a seat to party D.This behavior is inhibited by vote ratio monotonicity. Vote ratio monotonicity admits a kind of dynamic interpretation when the inequality with which it starts is rearranged in another way: v w w w i < i ” k < i : vk wk vk vi

This rearrangement is informative in situations when v is a vote vector that emerged earlier in time than w. The inequality wk=vk < wi=vi means that party k grows slower than party i. Vote ratio monotonicity rules out that with regard to seats party k does better and party i does worse. The interpretation is attractive when allocating seats between geographical districts. For example, w may assemble current census figures, and v previous census figures. If the ratio wi=vi is larger than the ratio of another state k, then the population of state i grows at a faster rate than state k. The conclusion demands that state i does potentially better by meeting or exceeding its former seat number, or that state k does worse or stays as is. In this setting vote ratio monotonicity turns into “population monotonicity”. Population monotonicity plays a pivotal role in the exposition of Balinski and Young (1982, p. 108). Lack of monotonicity is what those authors term a “population paradox”. That is, state i grows faster than state k and yet state i loses seats to state k. Vote ratio monotonicity and population monotonicity pose problems that are identical as far as the numbers are concerned. However, political perceptions and practical consequences of the concepts differ. While the translation of votes into seats remains the prerogative of the political legislator, the representation of geographical districts involves considerable administrative expertise. All modern states entertain statistical offices updating census figures and monitoring population mobility meticulously and laboriously. It would seem counterproductive to belittle these efforts by eventually employing an apportionment method that is flawed by a population paradox. This is the reason why the German Bundestag disposed of the Hare-quota method with residual fit by greatest remainders. The lack of population monotonicity was considered irritating when apportioning the 299 constituencies to the sixteen German States. In 2008 the Bundestag adopted the divisor method with standard rounding for the allocation of the 299 constituencies. In order to avoid the use of several apportionment methods in the Federal Election Law, the Bundestag decided to introduce the divisor method with standard rounding also for the apportionment of the notional 598 Bundestag seats among the political parties. 9.14 Violation of Voter Monotonicity: No-Show Paradox 183

9.14 Violation of Voter Monotonicity: No-Show Paradox

An apportionment method A is called “voter monotone” when all seat vectors x 2 A.hI v/ and y 2 A.hI w/ with vi < wi and vj D wj for all j ¤ i satisfy xi Ä yi. That is, if party i wins more votes while all other vote weights remain the same then party i cannot lose seats. More votes never result in fewer seats when the other vote counts are unchanged. Every apportionment method that is vote ratio monotone is automatically voter monotone. The opposite ordering xi > yi is ruled out. Because of a common house size h there would have to be a party k ¤ i with vk D wk and xk < yk. But vote ratio monotonicity does not allow a situation where vi=vk < wi=wk occurs together with xi > yi and xk < yk. Hence voter monotonicity comes for free whenever vote ratio monotonicity holds true. We already know that quota methods may violate vote ratio monotonicty. With an integer quota, they may even fail voter monotonicity. Table 9.5 exhibits a party that attracts another voter and thereupon loses a seat. This is called a “no-show paradox”: The voter would serve her or his party best by not showing up at the election. In Sect. 2.4 such instances are discussed under the heading of negative voting weights. The behavior in Table 9.5 is caused by the fact that the underlying Droop-quota variant-3 (DQ3) is required to be an integer. While the addition of a single vote is minute, the induced rounding effect for the quota is notable. In summary coherence captures an intrinsic aspect of apportionment methods. The notion may not appear readily accessible, but entails remarkably strong consequences. Coherence restricts the vast class of all apportionment methods to the smaller class of divisor methods. Stationary divisor methods are distinguished by being transfer-proof. Conformity with the natural two-party apportionments leaves just a single procedure, the divisor method with standard rounding.

Table 9.5 Violation of voter monotonicity by DQ3grR Party Vot es Quotient DQ3grR Vot es Quotient DQ3grR A 683,622 76.00022 76 683,623 75.99189 75 B 269,844 29.99933 29 269,844 29.99600 30 C 162,000 18.01001 18 162,000 18.00800 18 D 89,945 9.99944 10 89,945 9.99833 10 E 80,950 8.99944 9 80,950 8.99844 9 F 71,959 7.99989 8 71,959 7.99900 8 Sum (Split) 1,358,320 (0.9994) 150 1,358,321 (0.994) 150 Quota 8995 8996 With 683,622 votes party A is apportioned 76 seats. With one vote more, party A is apportioned one seat less Chapter 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

Abstract Perfect electoral equality is a conceptual ideal defying reality. A certain degree of inequality must be tolerated in practice. Ways and means are discussed on how to numerically evaluate the inevitable residue of disproportionality. Three approaches are explored. A first approach investigates all-embracing goodness-of-fit criteria; that is, functions that measure a system’s deviation from ideal equality by a single real number. Different criteria are seen to lead to different methods. A second approach applies disparity functions to pairwise comparisons. The aim is to reduce a pending imbalance by transferring a seat from a party that is advantaged to another party that is disadvantaged. A third approach examines whether the parties’ realized shares of seats and ideal shares of seats are as close to each other as could be.

10.1 Optimization of Goodness-of-Fit Criteria

It is an established approach of all sciences to explore a complex system by means of real functions. Naturally a single number cannot possibly mirror the full complexity of a high-dimensional system. Yet valuable information may be obtained when the criterion function is geared to the essentials of the system. Apportionment methods aim at near proportionality between input vote counts ` ` .v1;:::;v`/ 2 .0I1/ and output seat numbers .x1;:::;x`/ 2 N .h/. Usually the vote total vC is distinct from the seat total xC D h. Therefore the vectors are stan- dardized so that their component sum share the same value: unity. Goodness-of-fit criteria then measure how the output, the seat share vector x=h D .x1=h;:::;x`=h/, conforms to the input, the vote share vector v=vC D .v1=vC;:::;v`=vC/. Perfect proportionality would require the two vectors to be equal. However, because of the discreteness of the seat numbers, equality is almost always beyond reach and some residue of inequality must be tolerated. The question is: Which inequality? Inequality between whom? An election features at least three groups of protagonists who may claim a constitutional right to equality among their group members. The largest group is composed by the vC voters. A middle-size group is formed by the h Members of Parliament elected. The smallest group is constituted by the ` parties that mediate between the first group, those voting, and the second group, those elected. For each of these groups there exists an adequate criterion function for mea- suring within-group inequity. Minimization of different criteria points to different

© Springer International Publishing AG 2017 185 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_10 186 10 Appraising Electoral Equality: Goodness-of-Fit Criteria apportionment methods, though. The voter-oriented criterion leads to the divisor method with standard rounding (Sect. 10.2). The criterion that minimizes parlia- mentary inequity yields the divisor method with geometric rounding (Sect. 10.5). Minimization of inequity among parties leads to the Hare-quota method with residual fit by greatest remainders (Sect. 10.6). Generally there is no agreement about the relative merits of one criterion function versus the other. There is no obligation to dispose of other apportionment methods in favor of the one that optimizes some specific criterion. Every criterion mirrors but a partial view of the whole system. On the other hand, if an apportionment method is optimal with respect to some criterion, the method evidently harmonizes well with the aspects that are captured by the given criterion.

10.2 Voter Orientation: DivStd

Arguably the most important protagonists at an election are the voters. How do we appraise disproportionality from the voters’ viewpoint? ` Suppose we are given an arbitrary vote vector v D .v1;:::;v`/ 2 .0I1/ and an ` arbitrary seat vector x D .x1;:::;x`/ 2 N .h/. A voter casting a vote for party j is but one of vj supporters of party j.Allvj voters secure the party’s success of xj seats. A single voter contributes the success share xj=vj. Since usually the numerator is much smaller than the denominator they are suitably standardized (Sect. 2.7). Standardization leads to the success value that is ascribed to an individual voter: = xj h : vj=vC

If perfect proportionality were possible, seat shares and vote shares would coincide. The ideal success values would equal unity. However, in reality success values deviate from the ideal value unity almost always. If the ratio is larger than unity, voters enjoy a bit of good luck and are more successful than promised by perfect proportionality. If smaller, voters suffer from a bit of bad luck. In order to neutralize the direction of the deviation, the difference between the realized success value of a voter of party j and the ideal success value unity is squared:

 Ã2 x =h j  1 : vj=vC

This per-voter index is counted once for each of the vj voters of party j, yielding

 Ã2 xj=h vj  1 : vj=vC 10.2 Voter Orientation: DivStd 187

An aggregation of these terms over all groups of voters gives rise to a criterion embracing every voter in the entire electorate:

X Â Ã2 xj=h fh;v.x/ WD vj  1 : vj=vC jÄ`

A seat vector that minimizes this goodness-of-fit criterion minimizes electoral inequality from the voters’ viewpoint. Theorem A seat vector x 2 N`.h/ minimizes the above goodness-of-fit criterion if and only if x belongs to the divisor method with standard rounding:

` fh;v .x/ Ä fh;v. y/ for all y 2 N .h/ ” x 2 DivStd.hI v/:

Proof Upon setting w D v =v and developing the squares the criterion function j j C P Á 2 2 turns into f ;v .x/ D v =h x =w  v . Hence the function to be h P C jÄ` j j C . / 2= minimized is F x WD jÄ` xj wj. For the proof of the direct part let x be a minimum point: fh;v .x/ Ä fh;v. y/ for ` all y 2 N .h/. A seat transfer to a party i from a party k ¤ i with xk  1 yields a new seat vector y with components yi WD xi C 1, yk WD xk  1,andyj WD xj for ; . / . / 2= all j ¤ i k. Minimality secures F x Ä F y . This inequality reduces to xi wi C 2= . 1/2= . 1/2= . 1=2/= . 1=2/= xk wk Ä xi C wi C xk  wk;thatis, xk  wk Ä xi C wi. A passage to the maximum (left) and the minimum (right) verifies the max-min inequality 4.6 of the divisor method with standard rounding. This establishes x 2 DivStd.hI w/ D DivStd.hI v/. The proof of the converse part starts with a vector x 2 DivStd.hI w/. For a seat vector y 2 N`.h/ other than x, y ¤ x, we assemble the parties with incremented seat numbers in the set I WD fi Ä ` j yi > xigDfi Ä ` j yi  xi C 1g, and those with smaller components in the set K WD fk Ä ` j yk < xkgDfk Ä ` j yk Ä xk  1g.For i 2 I and k 2 K the max-min inequality yields

.x C y /=2 x  1=2 x C 1=2 .x C y /=2 k k Ä k Ä i Ä i i : wk wk wi wi . /= . /= It follows that the difference a WDP mini2I xi C yi wi  maxk2K xk C yk wk is 0 . / nonnegative: a  .LetS WD i2I yi  xi denote the aggregated surpluses. SinceP surpluses are balanced by deficits, S also signifies the aggregated deficits: . / S D k2K xk  yk . We obtain 0 1 0 1 ! ! X y2 X x2 X X @ j A @ j A xi C yi xk C yk  D . yi  xi/  .xk  yk/  aS  0: wj wj wi wk jÄ` jÄ` i2I k2K

This establishes F. y/  F.x/. ut 188 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

With vote shares wj D vj=vC the criterion function takes the form  X 2 vC xj  wjh fh;v.x/ D : h wjh jÄ`

The sum is the familiar chi-square statistic. However, an allusion to the chi-square distribution is inappropriate. The prevalent assumption is that the vote share vector w follows anP absolutely continuous distribution (Sect. 6.4). Then the distributions of . /2=. / the sums jÄ` xj  wjh wjh converge to a limit distribution (as h !1, then ` !1, with appropriate standardizations). But the limit is a Lévy-stable distribution, not the chi-square distribution. Yet the persuasive rationale for the current criterion function remains the same as with the chi-square statistic. It is illustrated best with a toy example. Suppose the house size is h D 100 and the electorate divides into three parties with vote counts v D .45; 35; 20/. Because of exactness every apportionment method yields the same answer: the seat vector .45; 35; 20/. Being exact, the solution is devoid of any disproportionality: fh;v.45; 35; 20/ D 0. How about seat vectors that suffer from a malapportionment of two seats for the strongest party? The two seats may be allocated (10.1) to the middle party, or (10.2) to the middle and the weakest parties, or (10.3) to the weakest party. The three alternatives incur the following disproportionality indices:

fh;v.45  2;35 C 2;20 C 0/ D 0:089 C 0:114 C 0 D 0:203; (10.1)

fh;v .45  2;35 C 1; 20 C 1/ D 0:089 C 0:029 C 0:05 D 0:168; (10.2)

fh;v .45  2;35 C 0; 20 C 2/ D 0:089 C 0 C 0:2 D 0:289: (10.3)

The index is 0.203 when the two diverted seats benefit the middle party (10.1). It is aggravated to 0.289 when they boost the weakest party (10.3). The increase from 0.203 to 0.289 appears reasonable since a two-seat surplus weighs heaviest with the weakest party. Disproportionality is least, 0.168, when the two seats are shared by the middle and weakest party (10.2). This is appealing since a kind of local equity is realized in a state missing exactness. As Gauss (1821, p. 97) argued when advertising the method of least squares: The squared-error function weighs deviations in a quite natural manner indem man sich gewiss lieber den einfachen Fehler zweimal als den doppelten einmal gefallen läßt. as one certainly bears the simple error twice more willingly than the double error once. By minimizing the sum of the squared deviations of the voters’ realized success values from the ideal success value unity, the divisor method with standard rounding harmonizes exceedingly well with equality among the individuals in the electorate. 10.3 Curtailment of Overrepresentation: DivDwn 189

10.3 Curtailment of Overrepresentation: DivDwn

Success values larger than unity are indicative of overrepresenting the voters of a party. The extreme overshoot is determined by the success value that is largest:

xj=h fh;v.x/ WD max : jÄ` vj=vC

Minimization of the largest success value curtails worst-case overrepresentation. The criterion is closely related to the multiplier version of the max-min inequality of the divisor method with downward rounding (Sect. 4.6):

x x C 1 max j Ä min j : jÄ` vj jÄ` vj

The following theorem involves not only the objective function itself, but also the “number of decrement options” b.x/ for the seat vector x:  ˇ  ˇ ˇ xk xj b.x/ WD #K.v; x/; where K.v; x/ D k Ä ` ˇ D max : vk jÄ` vj

In the presence of ties the number of increment options and the number of decrement options determine the number of tied solutions (Sect. 4.8). Theorem A seat vector x 2 N`.h/ minimizes the above goodness-of-fit criterion with the smallest number of decrement options if and only if x belongs to the divisor method with downward rounding:

` . fh;v .x/ Ä fh;v. y/ and fh;v .x/ D fh;v . y/ ) b.x/ Ä b. y// for all y 2 N .h/ ” x 2 DivDwn.hI v/:

Proof It suffices to simplify the objective function to F.x/ WD maxjÄ` xj=vj.The direct part is proved by negation. We show that a seat vector x 62 DivDwn.hI v/ does not minimize the objective criterion or does not have a smallest number of decrement options. Since x violates the max-min inequality for DivDwn there is a party i fulfilling .xi C 1/=vi < F.x/. In the case b.x/ D 1 a unique party k ¤ i attains the maximum: xk=vk D F.x/. That is, all other parties j ¤ k have xj=vj < F.x/. Now the seat vector y with components yi WD xi C1, yk WD xk 1,andyj WD xj for j ¤ i; k satisfies F. y/

objective value: F. y/ D xk2 =vk2 D F.x/. But it features fewer decrement options: b. y/ D b.x/  1

x x C 1 x C 1 y y F.x/ D max j Ä min j Ä i Ä i Ä max j D F. y/: (10.4) jÄ` vj jÄ` vj vi vi jÄ` vj

Indeed, the first inequality is the max-min inequality for DivDwn, the second restricts attention to an arbitrary party i 2 I, the third follows from the definition of I, and the fourth passes to the maximum over all parties. Finally we show that the case F.x/ D F. y/ implies b.x/ Ä b. y/. Equality in (10.4) says that all parties with more seats under y than under x gain exactly one seat each: yi D xi C 1 for i 2 I. Because of xi=vi <.xi C 1/=vi D yi=vi D F.x/, these parties are decrement options for y: I Â K.v; y/; but not for x: K.v; x/ Â I0. Denoting by n the number of parties in I, the passage from x to y thus gives rise to n new decrement options. How many old decrement options of x may get lost on the way? Either n parties in K.v; x/ lose one seat each and cease to be decrement options for y; then the tally for y yields b.x/ D b. y/. Or some parties in K.v; x/ give up more than one seat or other parties in I0 contribute too; this yields b.x/

DivDwn .10I .3;2;1//D .6;3C;1C/ D f.6;3;1/;.5;4;1/;.5;3;2/g:

The three solution vectors have a single decrement option each: the first component, the second, and the third, respectively. However, the function F.x/ D maxjÄ` xj=vj is minimized also by .6;4;0/, .6;2;2/,and.4;4;2/. These additional minimizers are no valid solutions. They have two decrement options each: the first and second components, the first and third, and the second and third, respectively. To appreciate the workings of the criterion we evaluate the criterion function for the example of Sect. 10.2:

fh;v.45  2;35 C 2;20 C 0/ D maxf0:956; 1:057; 1gD1:057; (10.5)

fh;v .45  2;35 C 1; 20 C 1/ D maxf0:956; 1:029; 1:05gD1:05; (10.6)

fh;v .45  2;35 C 0; 20 C 2/ D maxf0:956; 1; 1:1gD1:1: (10.7)

The criterion tends to spotlight weaker parties more than stronger parties. The reason is that the success values of weaker parties have small denominators vj=vC. Weaker parties respond to changes in the numerator with bigger leaps than stronger 10.4 Alleviation of Underrepresentation: DivUpw 191 parties. In the example the stepsize of the strongest party is 1=45 D 0:02;ofthe middle party: 1=35 D 0:03; of the weakest party: 1=20 D 0:05. For this reason the minimization of the criterion is more likely to push the quotients of weaker parties down to unity or even below. The consequence is that the divisor method with downward rounding is biased in favor of stronger parties at the expense of weaker parties (Chap. 7). For the same reason the method is more preferential towards stronger party groups than towards weaker party groups in terms of majorization (Chap. 8).

10.4 Alleviation of Underrepresentation: DivUpw

Voters with a success value smaller than unity are subject to underrepresentation. The extreme deficiency is given by the success value that is smallest:

xj=h fh;v.x/ WD min : jÄ` vj=vC

Maximization of the smallest success value alleviates worst-case underrepresen- tation. The motivation is familiar from welfare economics when maximizing the income of the poorest person in order to equalize the distribution of wealth in society. The criterion is closely related to the multiplier version of the max-min inequality of the divisor method with upward rounding (Sect. 4.6):

x  1 x max j Ä min j : jÄ` vj jÄ` vj

The following characterization involves not only the objective function itself, but also the “number of increment options” a.x/ for the seat vector x:  ˇ  ˇ ˇ xi xj a.x/ WD #I.v; x/; where I.v; x/ D i Ä ` ˇ D min : vi jÄ` vj

In the presence of ties the number of increment options and the number of decrement options determine the number of tied solutions (Sect. 4.8). Theorem A seat vector x 2 N`.h/ maximizes the above goodness-of-fit criterion with the smallest number of increment options if and only if x belongs to the divisor method with upward rounding:

` . fh;v .x/  fh;v. y/ and fh;v .x/ D fh;v . y/ ) a.x/ Ä a. y// for all y 2 N .h/ ” x 2 DivUpw.hI v/: 192 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

Proof It suffices to simplify the objective function to F.x/ WD minjÄ` xj=vj.The direct part is proved by negation. We show that a seat vector x 62 DivUpw.hI v/ does not maximize the objective criterion or does not have a smallest number of increment options. Since x violates the max-min inequality for DivUpw there is a party k fulfilling .xk  1/=vk > F.x/. In the case a.x/ D 1 a unique party i ¤ k attains the minimum: xi=vi D F.x/. That is, all other parties j ¤ i have xj=vj > F.x/. Now the seat vector y with components yi WD xi C1, yk WD xk 1,andyj WD xj for j ¤ i; k satisfies F. y/>F.x/. Hence F.x/ is not a maximum. In the case a.x/  2 there are parties i1; i2;:::;ia.x/ other than k which attain the =v =v =v . / minimum: xi1 i1 D xi2 i2 DDxia.x/ ia.x/ D F x . Here the seat vector y with components yi1 WD xi1 C 1, yk WD xk  1,andyj WD xj for j ¤ i1; k has the same objective value: F. y/ D xi2 =vi2 D F.x/. But it features fewer increment options: a. y/ D a.x/  1

x x  1 x  1 y y F.x/ D min j  max j  k  k  min j D F. y/: (10.8) jÄ` vj jÄ` vj vk vk jÄ` vj

Indeed, the first inequality is the max-min inequality for DivUpw, the second restricts attention to an arbitrary party k 2 K, the third follows from the definition of K, and the fourth passes to the minimum over all parties. Finally we show that the case F.x/ D F. y/ implies a.x/ Ä a. y/. Equality in (10.8) says that all parties with fewer seats in y than in x lose exactly one seat each: yk D xk 1 for k 2 K. Because of xk=vk >.xk 1/=vk D yk=vk D F.x/ these parties are increment options for y: K Â I.v; y/; but not for x: I.v; x/ Â K0. Denoting by n the number of parties in K, the passage from x to y thus creates n new increment options for y. How many old increment options of x may get lost on the way? Either n parties in I.v; x/ gain one seat each and thereby cease to be increment options for y; then the tally for y yields a.x/ D a. y/. Or some parties in I.v; x/ gain more than one seat or other parties in K0 profit too; then we have a.x/

DivUpw .8I .3;2;1//D .4;3;1C/ D f.4;3;1/;.4;2;2/;.3;3;2/g :

Each of the three solution vectors x has a single increment option: the third, second, or first component, respectively. However, the objective function is maximized also by .5;2;1/, .3;4;1/,and.3;2;3/. Each of these additional maximizers y has two increment options: the second and third, first and third, and first and second components, respectively. We have fh;v .x/ D fh;v. y/,buta.x/

10.5 Parliamentary Orientation: DivGeo

A second group of electoral protagonists are those elected. How would Members of Parliament assess any disproportionality between a vote vector v and a seat vector x? The representative weight of each of the xj Members of Parliament of party j is the average number of voters he or she represents (Sect. 2.8):

vj

xj:

In the ideal case that all Members of Parliament enjoy the same representative weight, it must be equal to the votes-to-seats ratio: vj=xj D vC=h for all j Ä `.In practice deviations from the votes-to-seats ratio are unavoidable. Again the sign of a deviation is neutralized by squaring: Â Ã v v 2 j  C : xj h

This per-seat index is counted once for each of the xj representatives of party j:

 Ã2 vj vC xj  : xj h

The sum of these terms over all groups of representatives results in a criterion incorporating every Member of Parliament:

X Â Ã2 vj vC fh;v.x/ WD xj  : xj h jÄ`

A seat vector that minimizes this goodness-of-fit criterion minimizes electoral inequality from the viewpoint of the Members of Parliament. Theorem A seat vector x 2 N`.h/ minimizes the above goodness-of-fit criterion if and only if x belongs to the divisor method with geometric rounding:

` fh;v.x/ Ä fh;v. y/ for all y 2 N .h/ ” x 2 DivGeo.hI v/: P Á 2 2 Proof Simple algebra yields f ;v.x/ D v =x  v =h. Hence the function to P h jÄ` j j C . / v2= v =0 v >0 be minimized is F x WD jÄ` j xj. Recall the convention j D1for j . 194 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

For the proof of the direct part let x be a minimum point. Let the seat vector y be obtained from x by a seat transfer to a party i from some party k ¤ i with xk  1. v2= v2= v2=. 1/ v2=. 1/ It followsp from minimality thatp i xi C k xk Ä i xi C C k xk  .This yields vi= xi.xi C 1/ Ä vk= .xk  1/xk. Now the max-min inequality 4.6 for the divisor method with geometric rounding establishes x 2 DivGeo.hI v/. For the proof of the converse part we assume x 2 DivGeo.hI v/. Given a vector ` y 2 N .h/, y ¤ x, we again use the sets I WD fi Ä ` j yi  xi C 1g and K WD fk Ä ` j yk Ä xk  1g.Foralli 2 I and k 2 K the max-min inequality yields v v v v q i Ä q i Ä q k Ä q k : xiyi xi.xi C 1/ .xk  1/xk xkyk

v2=. / v2=. / It follows that the differenceP a WD mink2PK k xkyk  maxi2I i xiyi is nonnega- 0 . / . />0 tive: a  . With S WD i2I yi  xi D k2K xk  yk we obtain 0 1 0 1 ! ! X v2 X v2 X v2 X v2 @ j A @ j A k i  D .xk  yk/  . yi  xi/  aS  0: yj xj xkyk xiyi jÄ` jÄ` k2K i2I

This establishes F. y/  F.x/. ut With the vote shares wj D vj=vC the criterion takes the form

 Á2 X 2 vC .xj  wjh/ fh;v.x/ D : h xj jÄ`

The sum is a modification of the chi-square statistic where the denominator is the realized seat number xj, not the ideal share of seats wjh. As an illustration we reconsider the example from Sect. 10.2. The transfer of two seats from the strongest party to the others leads to the following criterion values:

fh;v .45  2;35 C 2;20 C 0/ D 0:093 C 0:108 C 0 D 0:201; (10.9)

fh;v.45  2;35 C 1; 20 C 1/ D 0:093 C 0:028 C 0:048 D 0:169; (10.10)

fh;v.45  2;35 C 0; 20 C 2/ D 0:093 C 0 C 0:182 D 0:275: (10.11)

The numbers send the same message as before. The new denominator does not hamper the rationale for the criterion’s usefulness. The divisor method with geometric rounding secures electoral equality among the Members of Parliament. It does so by minimizing the sum of the squared deviations of the deputies’ realized representative weights from the votes-to-seats ratio. 10.6 Party Orientation: HaQgrR 195

10.6 Party Orientation: HaQgrR

The last relevant group at an election system assembles the parties. They are the institutions mediating between the people voting and the representatives elected. The success of party j is expressed through the number of seats apportioned to it: xj. Perfect proportionality would promise the party its ideal share of seats wjh,where wj signifies party j’s vote share. Since in practice the seat number xj deviates from the ideal share wjh, a non-zero seat excess xj  wjh must be tolerated. It may be positive, or negative. The direction of the deviation is again neutralized by squaring:

2 xj  wjh :

Aggregation of the squared seat excesses of all parties gives rise to the criterion function X 2 xj  wjh : jÄ`

2 Minimization of the squared-error criterion t ,wheret D xj  wjh, generalizes considerably. The square t2 may be substituted by a more general function b.t/: X  fh;v;b.t/.x/ WD b xj  wjh ; jÄ` wj D vj=vC. The “score function” b.t/ is assumed to have a “slope function”

b.t/  b.t  1/ c.t/ WD D b.t/  b.t  1/; t 2 R; t  .t  1/ that is non-decreasing on R and increasing on Œ0I 1. For instance the square b.t/ D t2 is a feasible score function, as is the modulus b.t/ Djtj. Every convex function with a unique minimum at zero is feasible. Moreover every strictly convex function on R is feasible, such as b.t/ D et or b.t/ D et. Theorem Let b.t/ be a score function as specified above. A seat vector x 2 N`.h/ minimizes the above goodness-of-fit criterion if and only if x belongs to the Hare-quota method with residual fit by greatest remainders:

` fh;v;b.t/.x/ Ä fh;v;b.t/. y/ for all y 2 N .h/ ” x 2 HaQgrR.hI v/: 196 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

. / Proof In the objectiveP function the terms b xj  wjh extend viaP the slope function . / xj . / . / . / intoP b wjh C  nD1 c n  wjh . This yields fh;v;b.t/ x D jÄ` b wjh C xj . / nD1 c n  wjh . The function to be minimized is seen to be

X Xxj F.x/ WD c.n  wjh/: jÄ` nD1

The proof of the direct part is in two steps. The first step delivers bounds for the entries of a seat vector x 2 N`.h/ that is assumed to be a minimum:

xj bwjhc for all j Ä `; (10.12)

xj ÄbwjhcC1 for all j Ä `: (10.13)

We demonstrate (10.12), by indirect argument. Assuming that there is a component i with xi < bwihc we show that x is not a minimum point. It is impossible that all other components k ¤Pi fulfill xk Äbwkhc as this would lead to the contradiction < h D xC bwihcC k¤ibwkhcÄh. Hence some component k ¤ i satisfies xk > bwkhc. By transferring a seat to i from k we generate a rival seat vector y with entries yi WD xi C 1, yk WD xk  1,andyj WD xj for all j ¤ i; k. We obtain

F.x/  F. y/ D c.xk  wkh/  c.xi C 1  wih/: (10.14)

But xi < bwihc entails xi C 1  wih Ä 0, while xk > bwkhc implies xk  wkh >0. The monotonicity behavior of c yields c.xi C 1  wih/ Ä c.0/ < c.xk  wkh/.It follows that F.x/>F. y/, whence x is not a minimum point. A similar reasoning demonstrates (10.13). The second step verifies the max-min inequality (Sect. 5.5) with shift s D 0 that pertains to HaQgrR. Let x continue to be a minimum point. We fix an arbitrary component k with xk  1, and again generate a rival vector y by transferring a seat from k to another component i ¤ k. Minimality implies F.x/ Ä F. y/. We claim that the following implication holds true:

F.x/ Ä F. y/ H) xk  wkh Ä xi C 1  wih: (10.15)

The claim is shown indirectly. Assuming that xk  wkh > xi C 1  wih, the bounds (10.12)and(10.13)imply0 ÄbwihcC1  wih Ä xi C 1  wih < xk  wkh Ä bwkhcC1  wkh Ä 1. Thus the arguments lie in the interval Œ0I 1 where the slope function is increasing, c.xi C 1  wih/F. y/. The claim is proved. The inequality in (10.15) yields wih  xi Ä wkh C 1  xk,foralli and for all k ¤ i with xk  1. Clearly inequality (10.15) holds true also for k D i and, in view of (10.12), for the components k that have xk D 0. This verifies the max-min inequality (Sect. 5.5), and establishes x 2 HaQgrR.hI v/. 10.6 Party Orientation: HaQgrR 197

The proof of the converse part follows the pattern of the previous proofs. Consider a seat vector x 2 HaQgrR.hI v/, a competitor y 2 N`.h/, y ¤ x,and ` > the sets of subscripts where the two vectorsP differ: I WDP fi Ä j yi xig and ` < . / . />0 K WD fk Ä j yk xkg.WehaveS WD i2I yi  xi D k2K xk  yk . Monotonicity of the slope function and the max-min inequality imply

a WD min c.xi C 1  wih/  max c.xk  wkh/  0: i2I k2K

We obtain 0 1 0 1 X Xyi X Xxk @ A @ A F. y/  F.x/ D c.n  wih/  c.n  wkh/

i2I nDxiC1 k2K nDykC1 ! ! X X  . yi  xi/c.xi C 1  wih/  .xk  yk/c.xk  wkh/  aS  0: i2I k2K

Minimality of F.x/ is established: F. y/  F.x/. ut Despite the theorem’s ostensible generality, of admitting a large class of score functions b.t/, the feasible criteria are impaired by a lack of sensitivity. For the example of Sect. 10.2 we obtain

fh;v;b.t/.45  2;35 C 2;20 C 0/ D b.2/ C b.2/ C b.0/; (10.16)

fh;v;b.t/.45  2;35 C 1; 20 C 1/ D b.2/ C b.1/ C b.1/; (10.17)

fh;v;b.t/.45  2;35 C 0; 20 C 2/ D b.2/ C b.0/ C b.2/: (10.18)

Whatever the score function b.t/, the criterion function provides no guideline whether the two seats cause less inequality when transferred to the middle party (10.16) rather than to the weakest party (10.18). A popular score function is the modulus b.t/ Djtj. It induces the goodness-of-fit criterion ˇ ˇ X ˇ v ˇ ˇ j ˇ fh;v;jtj.x/ D ˇxj  hˇ : vC jÄ`

This criterion assigns the common weight 4 to all three cases of the example. It is rather insensitive. Its lack of discriminating power is apparent also in Sects. 10.2 and 10.5. There the objective functions would become identical (up to the constant vC=h) when the square is replaced by the modulus: ˇ ˇ ˇ ˇ ˇ ˇ X ˇ = ˇ X ˇv v ˇ v X ˇ v ˇ ˇ xj h ˇ ˇ j C ˇ C ˇ j ˇ vj ˇ  1ˇ D xj ˇ  ˇ D ˇxj  hˇ : vj=vC xj h h vC jÄ` jÄ` jÄ` 198 10 Appraising Electoral Equality: Goodness-of-Fit Criteria

It makes no difference whether voters, Members of Parliament, or parties constitute the reference set in which to assess inequality. The discriminative power of the modulus score function b.t/ Djtj is too weak. Squared-error criteria are more informative. The preceding sections exhibit, and delimit, the scope of an optimization approach to apportionment problems. The plethora of goodness-of-fit criteria divulge an impression of arbitrariness. All are attractive, none is mandatory. Despite these reservations the optimization approach involves arguments inviting further scrutiny. In the apportionment problem the value in dispute is one seat at the least, whether it stays where it is or whether it is transferred to another party. It takes two to quarrel. The idea suggests a path of improvement by way of pairwise comparisons: Does the transfer of a seat promise an improvement? If so, carry it out, and continue investigating further transfers.

10.7 Stabilization of Disparity Functions

Can the seat apportionment be improved by transferring a seat from one party to another? Or does the apportionment prove to be stable? The answer depends on the precise meaning of the notion of “stability”. From the voters’ viewpoint (and from the viewpoint of the German Federal Constitutional Court; see Sect. 2.7) the focus ought to be on the voters’ success values. The concept of pairwise comparisons is exemplified with the 2009 Bundes- tag election data (Table 2.1). The success value .xj=h/=.vj=vC/ of a voter of the weakest party, CSU, is found to be

42=598 0:070234 101:2 : 2;830;238=40;764;288 D 0:069429 D %

The success values of voters of the other parties are: 99.7% for voters of CDU, 99.6 for voters of SPD, 100.4 for voters of FDP, 100.5 for voters of LINKE, and 99.8 for voters of GRÜNE. The 146 SPD seats mean a success of only 99.6%. The 42 CSU seats secure a 101.2% success. At this election the voters of SPD are least successful, the voters of CSU are most successful. The disparity between the success values of a voter of SPD and a voter of CSU amounts to

gh;vISPD,CSU.146; 42/ Dj99:62  101:16jD1:54 percentage points:

The disparity is the difference, in absolute terms, of the individual success values of a voter of SPD and a voter of CSU. It is tempting to try and decrease the disparity by transferring a seat to the underrepresented SPD from the overrepresented CSU. However, the test is negative:

gh;vISPD,CSU.147; 41/ Dj100:30  98:75jD1:55 percentage points: 10.8 Success-Value Stability: DivStd 199

When the seat is transferred the disparity of the success values is not decreasing, 6 but increasing. There are 2 D 15 pairings that can be drawn from the six parties. Whichever seat transfer is examined the disparity is seen to increase and deteriorate, rather than to decrease and improve. In this sense the specific apportionment in Table 2.1 is seen to be stable. With regard to general disparity functions gh;vIi;k the notion of stability is as follows. Definition A seat vector x 2 N`.h/ is said to “stabilize” the disparity functions gh;v;i;k.xi; xk/ when, for all parties i; k Ä ` with i ¤ k and xk  1, it satisfies

gh;v;i;k.xi; xk/ Ä gh;v;i;k.xi C 1; xk  1/:

The remainder of the chapter shows that success-value stability distinguishes the divisor method with standard rounding (Sect. 10.8). Of course other disparity criteria are conceivable. Representative-weight stability characterizes the divisor method with harmonic rounding (Sect. 10.9). Some disparity criteria are unworkable. All relative disparity criteria lead to the divisor method with geometric rounding (Sect. 10.10).

10.8 Success-Value Stability: DivStd

For a general seat vector x 2 N`.h/, the “success-value disparity” of a voter of party i andavoterofpartyk is given by ˇ ˇ ˇ = = ˇ ˇ xi h xk h ˇ gh;v;i;k.xi; xk/ WD ˇ  ˇ : vi=vC vk=vC

A stabilizing seat vector x inhibits any desire to allocate one of its seats differently because any such transfer impairs the disparity function. Theorem A seat vector x 2 N`.h/ stabilizes the success-value disparity if and only if x belongs to the divisor method with standard rounding: gh;v;i;k.xi; xk/ Ä gh;v;i;k.xi C1; xk  1/ for all i; k Ä ` ” x 2 DivStd.hI v/:

Proof It is notationally convenient to switch to the vote shares wj D vj=vC.The stability definition refers to parties i; k Ä ` with i ¤ k and xk  1, while the theorem also admits the cases i D k and xk D 0. No harm is done. The case i D k always satisfies gh;v;i;i.xi; xi/ D 0<2=.wih/ D gh;v;i;i.xiC1; xi1/. The case xk D 0 always fulfills gh;v;i;k.xi;0/D xi=.wih/<.xi C 1/=.wih/ C 1=.wkh/ D gh;v;i;k.xi C 1; 1/. For the direct part of the proof we assume x to be stable. Any two parties i and k fulfill xi=wi  xk=wk or xi=wi Ä xk=wk. In the first instance the inequality .xk  1=2/=wk Ä .xi C 1=2/=wi is immediate. In the second instance stability yields 200 10 Appraising Electoral Equality: Goodness-of-Fit Criteria gh;v;i;k.xi; xk/ D xk=.wkh/  xi=.wih/ Ä gh;v;i;k.xi C 1; xk  1/. The latter cannot have value .xk  1/=.wkh/  .xi C 1/=.wih/, due to the ensuing contradiction: 0 Ä 1=.wkh/1=.wih/.Thevaluegh;v;i;k.xiC1; xk1/ D .xiC1/=.wih/.xk1/=.wkh/ leads to the inequality .xk 1=2/=wk Ä .xi C1=2/=wi again. Now the pertinent max- min inequality establishes x 2 DivStd.hI v/. For the converse part we assume x 2 DivStd.hI v/ and fix two parties i and k. If xi=wi  xk=wk then we obtain gh;v;i;k.xi; xk/ D xi=.wih/  xk=.wkh/ Ä .xi C 1/=.wih/  .xk  1/=.wkh/ D gh;v;i;k.xi C 1; xk  1/.Ifxi=wi Ä xk=wk then we invoke the inequality .xk  1=2/=wk Ä .xi C 1=2/=wi which is part of the max-min inequality. The latter means xk=wk  xi=wi Ä .xi C 1/=wi  .xk  1/=wk. Hence we conclude gh;v;i;k.xi; xk/ D xk=.wkh/  xi=.wih/ Ä gh;v;i;k.xi C 1; xk  1/.This establishes stability. ut Thus the divisor method with standard rounding is the unique method that is success-value stable. This provides yet another vindication of the method, beyond the optimization of the success-value oriented goodness-of-fit criterion in Sect. 10.2.

10.9 Representative-Weight Stability: DivHar

Stability from the parliamentary viewpoint leads to the divisor method with harmonic rounding, not the divisor method with geometric rounding (Sect. 10.5). For a general seat vector x 2 N`.h/ the “representative-weight disparity” for a representative of party i and a representative of party k is given by ˇ ˇ ˇv v ˇ ˇ i k ˇ gh;v;i;k.xi; xk/ WD ˇ  ˇ : xi xk

Representative-weight stability conforms to electoral equality from the viewpoint of the Members of Parliament. Theorem A seat vector x 2 N`.h/ stabilizes the representative-weight disparity if and only if x belongs to the divisor method with harmonic rounding: gh;v;i;k.xi; xk/ Ä gh;v;i;k.xi C1; xk 1/ for all i; k Ä ` ” x 2 DivHar.hI v/:

.0/ .1/ 0 . / Proof The harmonic signposts are sQ1 DQs1 D ,andsQ1 n D 1 2 .n  1/1 C n1 for n  2 (Sect. 3.11). That is, for n  2 we have 1=.n  1/ C 1=n D 2=sQ1.n/. For the proof of the direct part we assume x to be stable. Any two parties i and k satisfy vi=xi Ä vk=xk or vi=xi  vk=xk. The first instance implies vi=.xi C 1/ Ä vk=.xk  1/. Summation of the two inequalities yields vi=sQ1.xi C 1/ Ä vk=sQ1.xk/. The second instance has gh;v;i;k.xi; xk/ D vi=xi  vk=xk Ä gh;v;i;k.xi C 1; xk  1/ D vk=.xk 1/vi=.xi C1/. Again we obtain vi=sQ1.xi C1/ Ä vk=sQ1.xk/. The pertinent max-min inequality establishes x 2 DivHar.hI v/. 10.10 Unworkable Disparity Functions 201

For the proof of the converse part we assume x 2 DivHar.hI v/. Any parties i and k with vi=xi Ä vk=xk fulfill gh;v;i;k.xi; xk/ D vk=xk  vi=xi Ä vk=.xk  1/  vi=.xi C 1/ D gh;v;i;k.xi C 1; xk  1/. The other instance vi=xi  vk=xk uses the max-min inequality to establish stability. ut In summary, parliament-oriented equality is such that distinct specifications result in distinct methods. Representative-weight stability fosters the divisor method with harmonic rounding. Optimization of the goodness-of-fit criterion advances the divisor method with geometric rounding (Sect. 10.5). It seems only natural that different questions have different answers. The rare occurrence of identical answers is what is truly remarkable. Voter- oriented equality points to one and the same procedure: the divisor method with standard rounding (Sects. 10.2 and 10.8).

10.10 Unworkable Disparity Functions

Other disparity functions for pairwise comparisons are readily invented. However, some prove to be unworkable. They lead into a never ending circle of improvements. The constitutional courts of the German states of Bavaria (1961) and Lower Sax- ony (1978) demand, incidentally, that electoral equality requires the seat numbers of two parties to be in the same ratio as their vote counts. The courts’ proposal sounds persuasive, but fails practically. The wording points to the disparity function ˇ ˇ ˇ v ˇ ˇ xi i ˇ gh;v;i;k.xi; xk/ WD ˇ  ˇ : xk vk

Here is an example of a vicious circle: the apportionment of sixteen seats among three parties with vote counts v D .729; 534; 337/. With votes-to-seats ratio 1600=16 D 100, the ideal shares of seats are v=100 D .7:29; 5:34; 3:37/.The ideal shares point to three seat vectors of interest, x D .8;5;3/, y D .7;6;3/,and z D .7;5;4/. The search for a stable vector loops endlessly from x to y to z,tox to y to z,tox to y to z, and so on: ˇ ˇ ˇ ˇ ˇ 8 729 ˇ ˇ 7 729 ˇ gh;vI1;2.x1; x2/ D 5  534 D 0:24 > 0:20 D 6  534 D gh;vI1;2. y1; y2/; ˇ ˇ ˇ ˇ ˇ 6 534 ˇ ˇ 5 534 ˇ gh;vI2;3. y2; y3/ D 3  337 D 0:42 > 0:34 D 4  337 D gh;vI2;3.z2; z3/; ˇ ˇ ˇ ˇ ˇ 4 337 ˇ ˇ 3 337 ˇ gh;vI3;1.z3; z1/ D 7  729 D 0:11 > 0:09 D 8  729 D gh;vI3;1.x3; x1/:

No solution is stable, yet each has its merits. The vector x belongs to the divisor method with standard rounding, y to the divisor method with geometric rounding, and z to the divisor method with harmonic rounding. The example is taken from Huntington (Sect. 16.11 [2]). The author champions comparison tests; that is, pairwise comparisons by means of disparity functions. A 202 10 Appraising Electoral Equality: Goodness-of-Fit Criteria plethora of thirty-two criteria is presented. Twelve of them turn out to be “unwork- able”: they may lead to endless circles of improvement. The other twenty disparity functions are shown to lead to the five traditional divisor methods (Sect. 4.4). Huntington dismisses the twenty workable disparity functions as “undesirable” because of leading to “a confusion of miscellaneous results”. Huntington disposes of these disparity functions for the persuasive reason that he has something better in the offering. He creates “relative disparity functions” by dividing a disparity function by the smaller of its terms. In this way every (absolute) disparity gets matched with a relative companion version. For instance, the relative success-value disparity and the relative representative-weight disparity are ˇ ˇ ˇ ˇ ˇ ˇ ˇ = = ˇ ˇ ˇ ˇv v ˇ ˇ xi h xk h ˇ ˇ xi xk ˇ ˇ i k ˇ ˇ  ˇ ˇ  ˇ ˇ  ˇ v =v v =v v v x x i C k C  D i k  ; i k  : x =h = x x v v min i ; xk h min i ; k min i ; k vk=vC vi=vC vi vk xi xk

Sparing his readers the sight of such bulky formulas throughout his exposition, Huntington nevertheless assures them of his firm belief that “in the present problem it is clearly the relative or percentage difference, rather than the mere absolute difference, which is significant”. The main result of the author states that the relative variants of the thirty-two disparity functions lead to but one and the same procedure: the divisor method with geometric rounding. This is Huntington’s favorite method, successfully promoted under the winning label “method of equal proportions”. We fear that Huntington’s relative disparity functions run the risk of being unconstitutional, due to indirect scaling. A vote for party i is scaled with some weight when compared to a vote for a second party j, and with another weight when compared to a vote for a third party k. Repeated weightings run counter to such principles as electoral equality. Thus we may complete Huntington’s rating: Twelve disparity functions are unworkable, twenty are undesirable, and thirty-two are unconstitutional. The crux is that any claim to exclusiveness is futile. Electoral systems are com- plex systems. They have plenty of facets, and admit plenty of criteria highlighting one aspect or another.

10.11 Ideal-Share Stability: DivStd

The stability concept extends to the parties’ ideal shares of seats. Consider a situation where party i’s seat number misses the ideal share of seats by more than half a seat: xi < wih  1=2. Let us assume that there is a party k whose seat number exceeds the ideal share of seats by more than half a seat: xk > wkh C 1=2.The transfer of a seat to party i from party k generates a vector y with components yi D xi C 1, yk D xk  1,andyj D xj for all j ¤ i; k, such that parties i and k 10.12 Ideal Share of Seats Versus Exact Quota of Seats 203 both move closer to their ideal shares of seats. The seats of the other parties remain as they were. For a seat vector to be enduring it ought to be immune against this kind of componentwise improvement. ` Given a house size h and a vote share vector .w1;:::;w`/, a seat vector x 2 N .h/ is said to be “ideal-share stable” when  à  à 1 1 x w h j ` x w h j ` : j  j  2 for all Ä or j Ä j C 2 for all Ä

A seat vector x fails to be ideal-share stable if and only if some parties i and k satisfy xi < wih  1=2 and xk > wkh C 1=2, as discussed in the previous paragraph. Ideal-share stability is a trait of the divisor method with standard rounding. Theorem The divisor method with standard rounding is the only divisor method for which all solution vectors are ideal-share stable. Proof Let x 2 DivStd.hI w/ be a solution vector of the divisor method with standard rounding. With a suitable multiplier  all parties j satisfy .xj  1=2/=wj Ä  Ä .xj C 1=2/=wj;thatis,xj  wj  1=2 and xj Ä wj C 1=2.If  h then the first stability inequality holds true. If  Ä h then the second stability inequality applies. This proves ideal-share stability of DivStd. Let A ¤ DivStd be another divisor method. There is a two-party problem where the two methods differ because otherwise the two divisor methods would rely on the same signpost sequence and hence be identical (Sect. 9.7). Therefore, for some house size h and some vote share vector w D .w1; w2/, we may select a seat vector x belonging to A but not to DivStd: x 2 A.hI w/ and x 62 DivStd.hI w/. The negation of the max-min inequality for DivStd is .x1 C1=2/=w1 <.x2 1=2/=w2. This yields x1 D w1.x1 C 1=2/ C w2.x1 C 1=2/  1=2 < w1.x1 C 1=2/ C w1.x2  1=2/  1=2 D w1h  1=2. In a similar manner we obtain x2 > w2h C 1=2. Hence x 2 A.hI w/ is not ideal-share stable. ut

10.12 Ideal Share of Seats Versus Exact Quota of Seats

In the final section of this chapter we philosophize on the merits of nomenclature. This book calls the seat fractions wjh the “ideal share of seats” of party j.Other authors speak of the “exact quota of seats” of party j. What difference does it make? Our substitutions of “ideal” for “exact” and of “share” for “quota” are intentional. The attribute “exact” subsists as one of the organizing principles of apportionment methods (Sect. 4.3). If for all parties the ideal shares of seats are whole numbers, xj D wjh 2 N for all j Ä `, then they are the only acceptable solution to the apportionment problem and the attribute “exact” is singularly appropriate. Otherwise the seat fractions wjh are idealized quantities that are in no way exact. The notion of a “quota” has its place in electoral parlance, too. There is the Hare-quota and the Droop-quota and many more quota variants (Sect. 5.8). A quota 204 10 Appraising Electoral Equality: Goodness-of-Fit Criteria of votes signifies a bunch of votes to justify a seat. A typical phrase reads “The candidate is 99 votes short of the quota”. The phrase indicates that there is a decreed quota which is missed by 99 votes. The notions of fishing quota, sales quota, production quota, women’s quota, and the like are used in the same manner. The term “quota of seats” would impose an obligatory entitlement to a certain number of seats. This is not what is really meant. In contrast, the term “share of seats” offers more leeway. Owners of shares in a company are aware that the monetary value of their shares becomes known only when sold or traded. Reference to a “share of seats” indicates the seat number only vaguely. An apportionment method is needed to turn the broad meaning of seat shares into a seat number that is definitive. The ideal share of seats wjh is a fractional number of seats indicating a theoretical entitlement that a party would claim if fractional seats could be realized. But seats are indivisible. The fractional quantity wjh proves practical only when rounded to a whole number. An apportionment method A is said to “stay within the ideal region” when all seat vectors x 2 A.hI w/ have components coinciding with the floor or the ceiling of the ideal shares of seats: ˚ « xj 2 bwjhc; dwjhe :

Thus xj is restricted to be one of two consecutive integers. The integers collapse to a singleton in the rare instances when the ideal share happens to be a whole number. Most examples with a method A not staying within the ideal region appear less offensive when scrutinized. Typical instances feature a single strong party together with many weak parties. The contrived data in Table 10.1 use the divisor method with standard rounding. For house size 95 the strongest party A may claim an ideal

Table 10.1 Staying within the ideal region Ideal share Divisor method with standard rounding Ideal share Party Vot es of h D 95 for house sizes from 95 to 104 of h D 104 A 5023 47:7185 50 50 50 50 50 50 50 50 50 50 52:2392 B 557 5:2915 5 6 6 6 6 6 6 6 6 6 5:7928 C 556 5:2820 5 5 6 6 6 6 6 6 6 6 5:7824 D 555 5:2725 5 5 5 6 6 6 6 6 6 6 5:7720 E 554 5:2630 5 5 5 5 6 6 6 6 6 6 5:7616 F 553 5:2535 5 5 5 5 5 6 6 6 6 6 5:7512 G 552 5:2440 5 5 5 5 5 5 6 6 6 6 5:7408 H 551 5:2345 5 5 5 5 5 5 5 6 6 6 5:7304 I 550 5:2250 5 5 5 5 5 5 5 5 6 6 5:7200 K 549 5:2155 5 5 5 5 5 5 5 5 5 6 5:7096 Sum 10 000 95:0000 95 96 97 98 99 100 101 102 103 104 104:0000 Party A fails to stay within its ideal region of seats, by C2 seats for house size 95 and by 2 seats for house size 104. The intermediate seats are apportioned to the equally weak nine parties B–K to promptly raise their seat numbers by one seat 10.12 Ideal Share of Seats Versus Exact Quota of Seats 205 share of 47.7 seat fractions but is awarded an excess of two seats: 50. For house size 104 the party is entitled to 50 seats, two seats below its ideal share of 52.2 seat fractions. These effects are counterbalanced by attending to the nine mini-parties B– K. Most reasonably, they are each promptly raised from five seats to six. In practice the divisor method with standard rounding fails to stay within the ideal region only very rarely. None of the empirical data in this book offers a practical illustration. Staying within the ideal region of seats does not guarantee flawless apportion- ments. The Hare-quota method with residual fit by greatest remainders always stays within the ideal region. Yet it suffers from paradoxical weaknesses (Sects. 9.11– 9.14). In conclusion the term “exact quota” is misconceived. The promised rigor fails to materialize with real data. The notions of “ideal share of seats” and “ideal region of seats” are wider and, more appropriately, closer to practical needs. Optimization of goodness-of-fit criteria or stabilization of disparity functions aim at embracing the whole system. A complementary approach is to investigate the variables of the system separately and to highlight properties that are more isolated. This is the topic of the next chapter. Chapter 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

Abstract Various sorts of vote thresholds are studied in detail. The minimum vote share and the maximum vote share that are compatible with a given number of seats are determined. Particular cases are the threshold of representation and the threshold of exclusion. Amendments are presented ensuring that a party with a straight majority of votes, however narrow, is guaranteed a straight majority of seats. A specific amendment, the majority-minority partition clause, applies to a majority of votes of a single party as well as to a majority of aggregated votes of a coalition of parties.

11.1 Vote Share Variation for a Given Seat Number

The apportionment task is to determine a seat vector x D .x1;:::;x`/ given the vote shares wj D vj=vC of the parties j Ä `. A converse task is to find all values of the vote share wj that are compatible with a given seat number xj. If the value is too small, it entails fewer than xj seats. If too large, it produces too many seats. Hence the values of wj that comply with precisely xj seats form an interval: Œa.xj/I b.xj/ say. The lower endpoint a.xj/ is termed the “minimum vote share given xj seats”. The upper endpoint b.xj/ is the “maximum vote share given xj seats”. Two instances are of special interest. The minimum vote share given one seat, a.1/, is the lowest vote share for a party to gain a seat and to achieve parliamentary representation. For this reason a.1/ is called the “threshold of representation”. The maximum vote share given no seat, b.0/, is the highest vote share for a party to stay out of parliament because of too few votes. Hence b.0/ is termed “threshold of exclusion”. Our aim is to determine a.xj/ and b.xj/, the extreme vote shares given xj seats. The minimum vote share given xj seats, a.xj/, is met when a single vote less for party j implies the loss of a seat. That is, a seat is transferred from party j to another party i ¤ j. The maximum vote share given xj seats, b.xj/, is attained when a single vote more for party j secures an additional seat. That is, a seat is transferred from another party i ¤ j to party j. A decrease to xj  1 seats or an increase to xj C 1 seats can occur only in situations that feature appropriate ties. These situations characterize the attainment of the extreme vote shares, as detailed in the following sections.

© Springer International Publishing AG 2017 207 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_11 208 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

Sect. 11.2 admits a general divisor method A. Given a solution x 2 A.hI w/ the associated vote share wj is bounded in terms of the signposts that belong to the seat numbers x1;:::;x`. The Theorem in Sect. 11.3 is the main result of this chapter. For stationary divisor methods the bounds are extended as far as possible until they reach the extreme vote shares a.xj/ and b.xj/.

11.2 Vote Share Bounds: General Divisor Methods

Lemma Let A be a divisor method with signpost sequence s.n/,n 0. For every seat vector x 2 A.hI w/ and every party j Ä ` the vote weight wj is bounded according to

s.x / s.x C 1/ .11:1/ P j w j P : .11:2/ . / . 1/ Ä j Ä . 1/ . / s xj C i¤j s xi C s xj C C i¤j s xi

Equality holds in the inequalities if and only if w D .v1;:::;v`/=vC is given in (11.1) by vj WD s.xj/ and vi WD s.xi C 1/ for i ¤ j, and in (11.2) by vj WD s.xj C 1/ and vi WD s.xi/ for i ¤ j. Proof The term “for every seat vector x 2 A.hI w/” is a shortcut phrase. Spelled out in full it means “for every house size h, for every number ` of parties, for every ` vote share vector w D .w1;:::;w`/ 2 Œ0I 1 , wC D 1, and for every seat vector x 2 A.hI w/”. The divisor method A results in seat numbers xk satisfying s.xk/ Ä wk Ä s.xk C 1/,forallk Ä `.Here is a multiplier such that the house size is met: xC D h.The residuals

uk WD wk  xk 2 Œs.xk/  xkI s.xk C 1/  xk sum to uC D   h.Fromwj.  h/ D wjuC we subtract the identity wj  xj D uj to obtain X xj  wjh D wjuC  uj D.1  wj/uj C wj ui: (11.3) i¤j

Upon substituting uj by its smallest value and the other residuals ui by their largest values and vice versa, the seat excess xj  wjh is seen to be bounded from above and from below:  X xj  wjh Ä .1  wj/ xj  s.xj/ C wj .s.xi C 1/  xi/; (11.4) i¤j  X  .1  wj/ s.xj C 1/  xj  wj .xi  s.xi// Ä xj  wjh: (11.5) i¤j 11.3 Vote Share Bounds: Stationary Divisor Methods 209 P After insertion of i¤j xi D h  xj and a rearrangement of terms, (11.4) yields the lower bound (11.1). Similarly (11.5) proves equivalent to the upper bound (11.2). Equality holds in (11.4) if and only if wj  xj D uj D s.xj/  xj and wi  xi D ui D s.xi C 1/  xi for all i ¤ j. This means wj D s.xj/=,andwi D s.xi C 1/= for i ¤ j, as asserted. Equality in (11.5) follows similarly. Note that the equality characterization may involve a zero weight wi D 0 when party i gains no seat: xi D 0; or when the signpost sequence is impervious: s.1/ D 0. ut With a general signpost sequence the lemma looks rather abstract. Its message becomes concrete when specialized. Simple formulas for the extreme vote shares a.xj/ and b.xj/ emerge when the divisor method is stationary.

11.3 Vote Share Bounds: Stationary Divisor Methods

Theorem If the stationary divisor method with split r 2 Œ0I 1 apportions xj 2 f0;:::;hg seats to party j then the party’s vote share wj satisfies

x  1 C r x C r .11:6/ a.x / WD j I Ä w Ä j DW b.x /; .11:7/ j h  1 C `r j h C r  M.1  r/ j where I WD 1 when xj  1 and I WD 0 when xj D 0, and where M WD minf`  1; h  xjg. Inthecasexj  1 equality holds in (11.6) if and only if there is a vote share ` vector w D v=vC that, for some seat vector y 2 N .h/ with component yj D xj, satisfies vj D xj  1 C r and vi D yi C rfori¤ j. Equality holds in (11.7) if and only if there is a vote share vector w D v=vC that, ` for some seat vector y 2 N .h/ with yj D xj and M positive components yi for i ¤ j, satisfies vj D xj C r and, for i ¤ j, vi D yi  1 C rifyi >0and vi D 0 if yi D 0.

Proof We need to show that every vote share vector w D .w1;:::;w`/ for which there is a solution y 2 DivStar.hI w/ with component yj D xj has its jth weight bounded according to a.xj/ Ä wj Ä b.xj/. The signposts are s.xj/ D .xj  1 C r/I for all xj  0. . / . 1 / In the Lemma of Sect. 11.2 the lower boundP has numerator s yj D xj  C r I. 1 . / . . 1/ / . 1 If xj  then the denominator is s yj C i¤j s yi C  yi C yi D xj  C r/ C .`  1/r C .h  xj/ D h  1 C `r. This establishes (11.6). When xj  1 equality ` holds if and only if w D v=vC where, for some seat vector y 2 N .h/ with yj D xj, we have vj D s.yj/ D xj  1 C r and vi D s.yi C 1/ D yi C r for i ¤ j. . 1/ In the Lemma of Sect. 11.2 the upper bound has numeratorP s yj C D xj Cr.The . / . / . . // denominator is xj C r  S C h  xj ,whereS WD i¤j yi  s yi . The maximum vote share b.xj/ requires a maximum sum S.Atermyi  s.yi/ is equal to 1  r or zero according as yi  1 or yi D 0. In the case M D `  1 Ä h  xj all parties i ¤ j can gain at least one seat: S Ä .`  1/.1  r/. Otherwise at most h  xj parties i ¤ j can be successful: S Ä .h  xj/.1  r/. The two cases combine into S Ä M.1  r/. 210 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

This establishes (11.7). Equality holds if and only if w D v=vC where, for some ` seat vector y 2 N .h/ with yj D xj and M positive components i ¤ j,wehave vj D s.yj C 1/ D xj C r and vi D s.yi/ D yi  1 C r if yi  1 and vi D 0 if yi D 0 for i ¤ j. ut A given seat vector x is always reproduced by the specific vote share vector w D x=h, by exactness. Hence the seat share xj=h is sandwiched between a.xj/ and b.xj/. Minimum and maximum vote shares given xj seats define the vicinity around xj=h wherein the weight wj may realize xj seats. The obvious boundary values are reproduced of course: a.0/ D 0 and b.h/ D 1. In the case M D h  xj <` 1 most seats are apportioned to party j.There remain fewer seats than there are parties competing with party j. The case is of little interest. Practical settings satisfy M D `  1 Ä h  xj. In this case the maximum vote share given xj seats simplifies to b.xj/ D .xj C r/= .h  .`  1/ C `r/. The bounds a.xj/ and b.xj  1/ are of particular interest. The minimum vote share a.xj/ is the threshold below which party j is definitely apportioned xj  1 or fewer seats. The maximum vote share b.xj  1/ is the barrier above which party j is certain to gain xj or more seats. The region guaranteeing at most xj  1 seats must have been exited before entering the region securing at least xj seats. Hence the thresholds satisfy a.xj/ Ä b.xj  1/. Strict inequality delimits an indifference zone where party j is apportioned xj  1 or fewer seats when unlucky, and xj or more seats when lucky. Naturally the inequality a.xj/ Ä b.xj  1/ is confirmed by the formulas. Specifically, for seat numbers xj D 1;:::;h C 2  `,wehavexj  1 in a.xj/ and `  1 Ä h  .xj  1/ in b.xj  1/. With I D 1 and M D `  1 the bounds take the form x  1 C r x  1 C r a.x / D j ; b.x  1/ D j : j h  1 C `r j h  .`  1/ C `r

Equality holds, a.xj/ D b.xj  1/, if and only if there are just two parties: ` D 2. In two-party systems, the regions with xj  1 seats are adjacent to the regions with xj seats, with no indifference zone between them. The distribution of the complementary weight 1  wj among the other parties, of which there is but one, is deterministic. Neither party needs to speculate about which results are least favorable or most favorable. In general systems with `  2 parties, the minimum vote share given one seat and the maximum vote share given no seat in the case M D `  1 Ä h are r r a.1/ D ; b.0/ D : h  1 C `r h  .`  1/ C `r

For instance the divisor method with downward rounding (r D 1) has a threshold of representation equal to 1=.h1C`/ and a threshold of exclusion equal to 1=.hC1/. 11.4 Vote Share Bounds: Modified Divisor Methods 211

11.4 Vote Share Bounds: Modified Divisor Methods

Legislators occasionally modify their electoral law to make it more difficult for weaker parties to gain a first seat. The divisor method with standard rounding is sometimes modified by raising the first signpost above one half, s.1/  1=2, while maintaining the other signposts, s.n/ D n  1=2 for all n  2. As of 2014, Sweden uses the signpost s.1/ D 0:6. We term this variant the “Swedish 2014 modification” of the divisor method with standard rounding. Up to 2014 the first signpost 0.7 was employed. The modification s.1/ D 1 has also been used in practice. Here the first seat must be fully earned, in that it cannot arise from upward rounding of interim quotients below unity. The motto “Below one is none” sounds persuasive to those not affected by it. We call this variant the “full-seat modification” of the divisor method with standard rounding. The two modifications are special cases of a more general setting. Theorem Consider a stationary divisor method with split r 2 Œ0I 1 that is modified by raising the first signpost above its normal level: t WD s.1/ 2 ŒrI 1. Assuming h  2.`  1/ the minimum vote shares given one seat and the maximum vote share given no seat are

t t a.1/ D ; b.0/ D : h  .1  r/ C .`  1/t h C t  .`  1/.1  r/

PProof The formulas follow from the Lemma in Sect. 11.2.Asfora.1/ the sum S D . . 1/ / i¤j s xi C  xi is decisive. Its maximum occurs when some party i ¤ j gains all remaining seats: xi D h  xj D h  1  1;andthe`  2 parties k ¤ i; j are apportioned no seats: S D r C .`  2/tP. .0/ . . // As for b the decisive sum is S D i¤j xi  s xi . It involves vanishing terms xi  s.xi/ D 0 for xi D 0; small terms xi  s.xi/ D 1  t for xi D 1;andlargeterms xi  s.xi/ D 1  r for xi  2. In the case h  2.`  1/, the sum is a maximum if all parties i ¤ j gain two or more seats: S D .`  1/.1  r/. The omitted case h <2.` 1/ is of no practical interest. ut The thresholds for the Swedish 2014 modification and the full-seat modification of the divisor method with standard rounding are 0:6 0:6 t D 0:6 H) a.1/ D ; b.0/ D I h  1:1 C 0:6` h C 1:1  0:5` 1 1 t D 1 H) a.1/ D ; b.0/ D : h  1:5 C ` h C 1:5  0:5`

Before discussing practical implications in Sect. 11.6 we derive the correspond- ing results for the shift-quota methods. 212 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

11.5 Vote Share Bounds: Shift-Quota Methods

Shift-quota methods round the interim quotients vj=Q.s/ D .h C s/wj downwards and then conclude with a residual fit by greatest remainders (Sect. 5.4). The minimum vote share a.xj/ and the maximum vote share b.xj/ given xj seats are as follows.

Theorem If the shift-quota method with shift s 2 Œ1I 1/ apportions xj 2f0;:::;hg seats to party j then the party’s vote share wj satisfies

x  1 C 1=` C s=` x C 1  1=` C s=` .11:8/ a.x / D j I Ä w Ä j ; .11:9/ j h C s j h C s where I WD 1 when xj  1 and I WD 0 when xj D 0.If`  1 Ä h  xj then the maximum vote share given xj seats fulfills b.xj/ D xj C 1 1=` C s=` =.h C s/.If `  1>h  xj then it fulfills b.xj/< xj C 1  1=` C s=` =.h C s/. Inthecasexj  1, equality holds in (11.8) if and only if there is a vote share ` vector w D v=vC that, for some seat vector y 2 N .h/ with component yj D xj, satisfies vj D xj  1 C 1=` C s=` and vi D yi C 1=` C s=` for i ¤ j. In the case `  1 Ä h  xj, equality holds in (11.9) if and only if there is a vote ` share vector w D v=vC that, for some seat vector y 2 N .h/ with yj D xj and M positive components yi for i ¤ j, satisfies vj D xj C 1  1=` C s=` and, for i ¤ j, vi D yi  1=` C s=` if yi >0and vi D 0 if yi D 0.

Proof Let w D .w1;:::;w`/ be a vote share vector for which a solution y 2 . / shQgrRs hI w has component yj D xj. We wish to show that the jth weight has bounds (11.8) and (11.9). For all k Ä ` the seat number yk is obtained by rounding .h C s/wk downwards or upwards according to whether its remainder is small or large. We may select a split r 2 Œ0I 1 separating small from large such that the stationary divisor method with split r reproduces the current solution: y 2 DivStar .hI w/ (Sect. 5.6). The associated residuals

uk WD .h C s/wk  yk 2 Œsr .yk/  ykI sr .yk C 1/  yk sum to uC D s.Froms=` D uC=` we subtract the identity .h C s/wj  xj D uj to obtain  à s u 1 1 X x .h s/w C u 1 u u : j  C j C ` D `  j D  ` j C ` i (11.10) i¤j 11.6 Overview of Vote Thresholds 213

The latter is the right-hand side of (11.3) in Sect. 11.2 with particular weight wj D 1=`. Setting M WD minf`  1; h  xjg the arguments of Sects. 11.2 and 11.3 yield the bounds  à s 1  1 1 x .h s/w 1 1 r I .` 1/ r 1 ; j  C j C ` Ä  `  C `  Ä  ` (11.11)  à  à s 1 M  1 x .h s/w 1 r 1 r 1 : j  C j C `   `  `    ` (11.12)

In (11.11) equality holds in the second inequality if I D 1;thatis,xj  1.In(11.12) the second inequality is an equality if M D `1;thatis,`1 Ä hxj. Thus (11.11) proves (11.8), and (11.12) proves (11.9).  In the case xj  1, equality holds in (11.11) if and only if uj D sr .xj/xj D r 1    and ui D sr .yi C1/yi D r for i ¤ j. Summation yields uC D .r 1/C.`1/r .  Together with uC D s we obtain r D .1 C s/=`. Therefore the current equality characterization follows from that of (11.6) in Sect. 11.3. In the case `  1 Ä h  xj, equality holds in (11.12) if and only if uj D sr .xj C     1/  xj D r and ui D sr .yi/  yi D r  1 for i ¤ j.Nows D r C .`  1/.r  1/ yields r D 1.1s/=`. Here the equality characterization reduces to that of (11.7) in Sect. 11.3. ut The minimum vote share given one seat and the maximum vote share given no seat, in the case `  1 Ä h,are .1 C s/=` 1  .1  s/=` a.1/ D ; b.0/ D : h C s h C s For instance the Hare-quota method with residual fit by greatest remainders (s D 0) has a threshold of representation equal to 1=.h`/ and a threshold of exclusion equal to .1  1=`/=h. For the Droop-quota variant-4 method with residual fit by greatest remainders (s ! 1) the thresholds are 2= ..h C 1/`/ and 1=.h C 1/, respectively. For the Hare-quota method with residual fit by greatest remainders (s D 0)the proof implies that seat excesses are bounded: jxj  wjhjÄ1  1=`. Hence an ideal share wjh is rounded downwards when close to its floor: wjh < bwjhcC1=` ) xj D bwjhc. It is rounded upwards when close to the ceiling: wjh > bwjhcC1  1=` ) xj Ddwjhe. Therefore the identity HaQgrR.hI v/ D DivStar .hI v/ in Sect. 5.6 holds true with a split r 2 Œ1=`I 11=`. Within an `-party system the majorization relation in Sect.8.8 can be strengthened to DivSta1=`  HaQgrR  DivSta11=`.

11.6 Overview of Vote Thresholds

Authors often use the term “threshold” in a broad sense. Generally, thresholds serve to exclude groups of voters from representation in parliament. One such provision is the percentage threshold discussed in Sect. 7.3. Percentage thresholds form a first 214 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

Table 11.1 Threshold formulas Apportionment method a.1/ Natural threshold b.0/ 0:5 0:5 DivStd Divisor method with standard rounding h  1 C 0:5` h C 1  0:5` 0:6 0:6 Div0.6 Swedish 2014 modification of DivStd, 1:1 0:6` 1:1 0:5` with s.1/ D 0:6 h  C h C  1 1 Div1.0 Full-seat modification of DivStd, with 1:5 ` 1:5 0:5` s.1/ D 1 h  C h C  1 1 DivDwn Divisor method with downward 1 ` 1 rounding h  C h C 1=` 1  1=` HaQgrR Hare-quota method with grR h h 2=` 1 DQ4grR Droop-quota variant-4 method with grR h C 1 h C 1 The minimum vote share given one seat is the threshold enabling representation: a.1/.The maximum vote share given no seat is the threshold guaranteeing representation: b.0/. The formulas depend on the method, the house size h, and the size ` of the party system category of thresholds; we call them “explicit thresholds”. They are most visible to the public, and they are warmly welcomed by strong parties. A second category consists of “implicit thresholds”. They are stored somewhere in the electoral law, but are neither easily recognized nor widely publicized. An example is the Romanian threshold for indeps (Sect. 1.9). A third category are “natural thresholds”. They impose hurdles that are peculiar to the apportionment method used. Two indices suggest themselves as a quantitative measure: Should the natural thresholds be measured by the minimum vote share given one seat: a.1/? Or by the maximum vote share given no seat: b.0/? The Swiss Federal Court deliberated on the issue in a 2003 case. The Court decided that the natural threshold that is constitutionally decisive is b.0/, the critical vote share above which representation is guaranteed. The minimum vote share a.1/ promises representation only when circumstances are favorable. This vagueness is insufficient to substantiate a constitutional right. Thus it is the rightmost column in Table 11.1 that shows the constitutionally binding “natural threshold”; that is, b.0/. The dominant variable in the natural threshold b.0/ is the house size. The more seats become available, the smaller is the natural threshold. Technically some of the formulas require the house size not to be too small, h  `  1 or h  2.`  1/, as explained in the previous sections. These requirements are met by the general recommendation that the house size should meet or exceed twice the number of parties, h  2` (Sect. 7.6). Another variable in the denominator is the size of the party system: `. The dependence on ` is only weak, and sometimes absent (DivDwn, DQ4grR). If present, a growing party system causes the natural threshold to increase. 11.7 Preservation of a Straight Majority and Majority Clauses 215

The natural thresholds for the full-seat modification of the divisor method with standard rounding are illustrated by example. The full-seat modification was written into the electoral law of the German State of North Rhine-Westphalia. In the 2004 local elections four communes had councils of twenty seats, with four parties standing at the election. The natural threshold of the full-seat modification is 1=19:5 D 5:1%. Three communes had a twenty-seat council and five parties campaigning. For them the natural threshold increases to 1=19 D 5:3%. Once recognized it is easy to illustrate the effect with numbers close to actual election results. The vote vector v D .2501; 701; 501; 199/ results in the seat vector x D .13; 4; 3; 0/. The divisor interval is D.v; x/ D Œ199I 200:8;the select divisor is 200. The fourth party does not obtain representation. Their 199 votes of a total of 3902 votes constitute a share of 5.1%. The vote vector v D .1381; 301; 301; 181; 119/ leads to the seat vector x D .12; 3; 3; 2; 0/. The divisor interval is D.v; x/ D Œ119I 120:086; the select divisor is 120. The fifth party gains no seat in the council. Their 119 votes out of a total of 2283 votes amount to 5.2%. The Constitutional Court for the State of North Rhine-Westphalia, having previously barred the 5% threshold, decided that the full-seat modification is unconstitutional as well, for use in local elections in North Rhine-Westphalia.

11.7 Preservation of a Straight Majority and Majority Clauses

Thresholds aim at weak parties whether they achieve parliamentary representation or not. At the other end of the scale one may worry whether strong parties are apportioned their due share of seats. An issue of political interest is majority preservation. Does an apportionment method always ensure that a party with a straight majority of votes is apportioned a straight majority of seats? That is, if a party has more votes than its competitors accumulate, does it gain more seats than the others combined? People might want to insist that every sensible method is “majority preserving”. Alas, the opposite is true. No common apportionment method is majority preserving. Table 11.2 reproduces an example from the 1982 Bundestag records. Party A wins a straight majority of votes: 18;594;670. Its opponents total five votes less: 18;594;665. Yet the Hare-quota method with residual fit by greatest remainders awards party A only half of the seats: 248 of 496. It does not realize a straight majority. The example was presented as if the deficiency were peculiar to this particular method. However, other commonly used methods apportion the 496 seats just as well. Hence if the example were to invalidate one method, it would invalidate all of them. The sobering message is that no reasonable apportionment method is always majority preserving. The divisor method with downward rounding is a prime candidate for a majority preservation procedure. It is biased in favor of stronger parties at the expense of weaker parties (Sect. 7.4), and it prefers groups of stronger parties to groups of 216 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

Table 11.2 Majority preservation failure Party Vot es HaQgrR=DivHar=DivGeo=DivStd=DivDwn A 18;594;670 248 B 12;950;200 173 C 3;664;459 49 D 1;980;006 26 Sum 37;189;335 496 Party A wins a straight majority of 18,594,670 votes, five votes more than the cumulative 18,594,665 votes for the opponents B–D. All common methods apportion to party A only 248 of 496 seats and fail to realize a straight seat majority in parliament weaker parties in the sense of majorization (Sect. 8.6). Nevertheless it may fail to preserve a straight majority (Table 11.2). The reason is that the house size in the table is even: 496. If the house size h is odd, then the method is indeed majority preserving. More precisely, the divisor method with downward rounding is the only station- ary divisor method that is majority preserving for odd house sizes, h D 2n C 1.To seethisweevaluateb.n/, the maximum vote share given n seats. With house size 2n C 1 a party with n seats misses a straight majority by just one seat. A method is majority preserving if and only if all vote shares that earn only n seats stay below one half: b.n/ Ä 1=2. By Sect. 11.3 a stationary divisor method with split r satisfies

n C r n C r b.n/ D D : 2n C 1 C r  .`  1/.1  r/ 2.n C r/  .`  2/.1  r/

For two-party systems we obtain b.n/ D 1=2; in this case all stationary divisor methods are majority preserving. For systems with three or more parties the inequality b.n/ Ä 1=2 holds true if and only if r D 1. For odd house sizes the divisor method with downward rounding is the unique stationary divisor method that is majority preserving. In Schleswig-Holstein the former divisor method with downward rounding was replaced by the current divisor method with standard rounding. Trusting that house sizes would be mostly odd, the old law had no need for a majority clause. No majority clause was included in the new law either. The omission proved fatal. During the 2013 local elections, the community of Boostedt experienced a majority preservation failure. A straight majority of the electorate voted CDU, yet the party was apportioned only eight of seventeen seats; see Table 11.3. The case provoked a sardonic press coverage. If a straight majority of votes is to guarantee a straight majority of seats then the electoral law must include a specific majority clause. The majority clause expresses the legislator’s positive intention that a party with a majority of votes is apportioned a majority of seats in parliament, no matter how small the vote margin. Sections 11.8–11.11 present three majority clauses: the house-size augmentation clause, the majority-minority partition clause, and the residual-seat redirection clause. 11.9 Majority-Minority Partition Clause 217

Table 11.3 Council election, SH2013Boostedt Second votes Quotient DivStd Boostedt community, 2013 CDU 2815 8.49 8 SPD 2155 6.503 7 FWG 549 1.7 2 Sum (Divisor) 5519 (331.4) 17 A straight majority of the electorate voted CDU, yet the party fell short of a straight majority of seats. The law fails to provide for a majority clause. The case met with sardonic press comments

11.8 House-Size Augmentation Clause

A viable majority clause is to increase the house size in those rare cases when the majority is not preserved automatically. The “house-size augmentation clause” creates extra seats for the benefit of the majority party until a straight majority of seats is reached. The clause presupposes that the house size can be increased, if required by extraordinary circumstances. It has the advantage of not retracting a seat from a party that was expecting it. In Boostedt (Table 11.3) CDU would receive two extra seats to establish a straight majority, with ten out of nineteen seats. An exaggerated example may be drawn from Table 10.1. The last column depicts a majority preservation failure. The majority party’s 50 seats lag behind the cumulative seats of its nine opponents (54 seats) by a margin of four seats. The house-size augmentation clause would create five additional seats for the benefit of the majority party to raise it to a straight majority of 55 seats out of 109. If the house size had been 109 from the very beginning, the resulting seat allocation would have been just the same: 55 seats for party A and six seats for each party B–K. Even though the clause looks biased, the final result often adheres to proportionality. It does so in the present example and in many others cases, but not always (check h D 100 in Table 10.1).

11.9 Majority-Minority Partition Clause

Another viable majority clause maintains partial proportionality. If the applicable apportionment method equips the majority party with a straight majority of seats, then no action is needed. Otherwise the clause starts afresh by separating the seat apportionment for the majority party from the seat apportionment for the other parties. The majority party is awarded the smallest possible straight majority of seats. The other seats are apportioned among the other parties using the applicable apportionment method. We call this procedure the “majority-minority partition clause”. In Boostedt (Table 11.3) CDU would be awarded nine seats. The other eight seats would be shared proportionally among the two other parties (SPD: 6, FWG: 2). 218 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

The majority-minority partition clause extends to a coalition of parties whose partners win an aggregate straight majority of votes. The clause becomes active if the nominal apportionment denies the coalition a straight majority of seats. Then the majority-minority partition clause starts afresh and proceeds in two steps. The first step is to apportion the smallest possible straight majority of seats among the coalition partners. The second step apportions the remaining seats among the remaining parties. Both steps are to use the pertinent apportionment method. If the underlying apportionment method is a divisor method then the clause results in seat vectors that are house size monotone (Sect. 9.5). To see this we note that the clause merges two apportionment sequences. The first sequence is the sequential apportionment of seats among the majority parties, the second is the sequential apportionment of seats among the minority parties. Neither of the two sequences violates house size monotonicity. Since the two strings are joined together in a coherent fashion (Sect. 9.2), monotonicity is maintained. Table 11.4 provides an example. The majority-minority partition clause has a historical precursor known under its Latin name itio in partes (separation into parts). The clause was an inventive novelty of parliamentary decision-making codified in the Peace of Westphalia 1648. A resolution would be carried by the plenum only if carried separately by either part of the plenum. Thus it secured procedural parity between two unequal parts that were anxious to safeguard their political identities. The two parts, then, were the opposing blocs of the contracting states: the Catholic bloc (Corpus catholicorum) and the Protestant bloc (Corpus evangelicorum). In the context of contemporary proportional representation systems, the two parts are the government majority and the opposition minority.

Table 11.4 Majority-minority partition clause for a small committee, 15th German Bundestag 2002 Political group Size DivStd MMP DivStd MMP DivStd MMP DivStd Government majority parties SPD 249 5 6 6 7 7 7 7 B90/GRÜNE 55 1 1 1 1 1 2 2 Opposition minority parties CDU/CSU 247 4 4 5 5 6 6 7 FDP 47 1 1 1 1 1 1 1 Sum 598 11 12 13 14 15 16 17 Divisors 55 55j45 45 45j38.2 38:2 38.2j35 35 For committee sizes 12, 14, 16, the majority-minority partition (MMP) clause carries out two apportionments. The smallest possible seat majority is apportioned among the government majority parties. The other committee seats are apportioned among the other parties 11.10 The 2002 German Conference Committee Dilemma 219

11.10 The 2002 German Conference Committee Dilemma

The majority-minority partition clause is illustrated with a majority preservation problem in the 2002 German Bundestag. When the Bundestag and the Bundesrat (Federal Council, Second Chamber) disagree, the issue is submitted to the Confer- ence Committee. Existence and functions of the Conference Committee are detailed in the Basic Law. For the Bundesrat, the sixteen German states nominate one com- mittee member each. They are referred to as the “Bundesrat bench”. For the Bundes- tag, the Basic Law stipulates that its bench has the same size: sixteen. The issue was how to apportion the 16 seats of the Bundestag bench among four political groups. In the 2002 Bundestag the government majority was formed by the political groups of SPD (249 seats, as of 1 February 2005) and BÜNDNIS 90/DIE GRÜNEN (55). The opposition minority consisted of CDU/CSU (247) and FDP (47). The divisor method with standard rounding allocates the 16 committee seats in the division 7W1 to SPD and BÜNDNIS 90/DIE GRÜNEN, and 7W1 to CDU/CSU and FDP. Government majority and opposition minority are tied with eight seats each. When confronted with a troubled apportionment proposal, the Bundestag habitually contemplates the apportionments that are obtained by the divisor method with downward rounding and the Hare-quota method with residual fit by greatest remainders. For the 2002 data all three methods produce the troublesome tie. The government parties were displeased with the threatening tie. They decided to proportionally apportion just fifteen seats, and to award the sixteenth seat to the strongest political group: SPD. Hence the actual seat division was 8W1 for the government parties, versus 6W1 for the opposition parties. The opposition considered the decision unjust and complained to the Federal Constitutional Court. In an opaque decision the Court ruled that the minutes of the Bundestag’s Ways and Means Committee were insufficient to disclose the reasoning that led to the disputed decision. The Bundestag was ordered to reconsider the issue and to document its decision-making process more fully, so as to enable the Court to assess the constitutional merits. However, before any action was taken the legislative period dissolved early. The file was closed, as was all pending business, and stored away in the archives. Table 11.4 shows how the majority-minority partition clause would have worked out. For committee size h D 11 the divisor method with standard rounding apportions six seats to the government majority, and five seats to the remaining parties. The majority is preserved and the majority clause remains dormant. It gets activated for committee size h D 12, since otherwise the two parts are tied with six seats each. Instead seven seats are apportioned among the majority parties and, separately, five seats among the remaining parties. The differentials relative to the regular method are marked with a dot (). For a committee of size h D 16 the majority-minority partition clause yields a division of 7W2 seats for the government parties, versus 6W1 seats for the opposition parties. The opposition’s litigation would have resulted in the junior government party doubling its representation: from one committee seat to two. 220 11 Tracing Peculiarities: Vote Thresholds and Majority Clauses

11.11 Residual-Seat Redirection Clause

An actual failure of majority preservation occurred at the 2009 local elections in the German State of North Rhine-Westphalia. In the County of Coesfeld 54,233 citizens voted CDU while 54,061 citizens voted for other lists. The divisor method with standard rounding apportions CDU only half of the seats: 27 of 54. Thus CDU is denied a straight majority of seats. The North Rhine-Westphalian law for local elections rectifies the failure by granting CDU a seat that is redirected from a competing list whose interim quotient is rounded upwards (SPD, GRÜNE, VWG). The seat is retracted from the list whose quotient has a fractional part that is smallest (VWG: .7). The actual seat apportionment was .28;12;6;5;2;1/, not the unmodified allocation .27;12;6;5;3;1/.SeeTable11.5. The “residual-seat redirection clause” is unproblematic only for the Hare-quota method with residual fit by greatest remainders. Indeed, a straight majority of votes for party j forces the ideal share of seats to exceed half the house size: vj=vC > 1=2 ) .vj=vC/h > h=2, possibly by ever so little. With a tiny remainder the main apportionment falls short of a majority: yj Db.vj=vC/hc < h=2. Now the clause augments it by a residual seat, xj D yj C 1>h=2.InTable11.5 CDU has quotient 27.043. Since the fractional part :043 is too small, the clause redirects a residual seat from VWG to CDU in order to provide CDU with a straight majority of 28 seats. When an electoral law is amended to introduce a new apportionment method an existing majority clause must be reassessed. The reassessment occasionally evades the attention of the legislator. In North Rhine-Westphalia the former Hare- quota method with residual fit by greatest remainders was replaced by the current divisor method with standard rounding. The residual-seat redirection clause was left untouched. It happened to work out fine in the case of Coesfeld. We illustrate its potential deficiency with the contrived data from Table 10.1, for house size 104. The divisor method with standard rounding awards the majority party A only 50 seats. The residual-seat redirection clause transfers just one residual seat from party K to

Table 11.5 Council election, Coesfeld county, 2009 NW2009Coesfeld Vote counts Quotient DivStd Quotient HaQgrR CDU 54;233 27.1 27 27.043 27 SPD 23;648 11.8 12 11.792 12 GRÜNE 11;798 5.9 6 5.883 6 FDP 10;329 5.2 5 5.150 5 VWG 5303 2.7 3 2.644 3 LINKE 2983 1.49 1 1.487 1 Sum (divisorjsplit) 108;294 (2000) 54 (0.5) 54 CDU has more than half of the votes, but received only half of the seats: 27 of 54. Of the interim quotients that are rounded upwards (SPD, GRÜNE, VWG), the smallest (VWG, remainder 0.7) loses a seat that is redirected to CDU 11.12 Divisor Methods and Ideal Regions of Seats 221 party A. But 51 seats in a house of 104 seats still fail a straight majority. The seat redirection clause may serve its purpose, or not. It cannot be recommended. The residual-seat redirection clause was already put forward by Gfeller (1890). It was reinvented by Horst F. Niemeyer, as reported by Niemeyer/Niemeyer (Sect. 16.15 [3]). The term “Hare/Niemeyer procedure” that is popular in Germany embraces not only the Hare-quota method with residual fit by greatest remainders, but also its modification by the residual-seat redirection clause. The residual-seat redirection clause was formerly part of many German electoral laws. It more and more gives way to the house-size augmentation clause, such as in the German Federal Election Law and the Bavarian State Election Law. The two amendments took place for different reasons though. The Federal Election Law, using the divisor method with standard rounding, may miss a majority in parliament by more than one seat; see Table 10.1 with h D 100. The Bavarian Election Law still uses the Hare-quota method with residual fit by greatest remainders, but it does so in seven separate districts. Within a district, the transfer of one seat would suffice to rescue a missed majority. In the whole electoral area, more than one seat may be needed. In both cases the residual-seat redirection clause may fail to do its job, while the house-size augmentation clause will work fine.

11.12 Divisor Methods and Ideal Regions of Seats

The chapter concludes with some statements regarding whether a divisor method stays within the ideal region of seats (Sect. 10.12). In Sect. 11.2 the bounds for the weights wj given xj seats are derived via bounds for the seat excess xj  wjh.The seat excess bounds aid in determining whether the seat numbers xj are equal to the floor or the ceiling of the ideal share of seats: bwjhc or dwjhe (Sects. 3.2 and 3.5). To begin with consider a two-party system with parties i and j. The left-right disjunction prohibits that the inequalities xj  s.xj/ Ä 1 and s.xi C 1/  xi Ä 1 both hold with equality (Sect. 3.8.c). At least one of the inequalities must be strict. Then the bounds (11.4) and (11.5) in Sect. 11.2 imply 1

Theorem a. For every divisor method A every seat vector x 2 A.hI w/ is such that either all its components remain above the lower value of the ideal region: xj bwjhc for all j Ä `; or all its components remain below the upper value of the ideal region: xj Ädwjhe for all j Ä `. b. The divisor method with downward rounding is the only divisor method that always stays above the lower value of the ideal region; that is, every seat vector x 2 DivDwn.hI w/ and all parties j Ä ` satisfy xj bwjhc. c. The divisor method with upward rounding is the only divisor method that always stays below the upper value of the ideal region; that is, every seat vector x 2 DivUpw.hI w/ and all parties j Ä ` satisfy xj Ädwjhe. d. No divisor method A stays within the ideal region at all times; that is, there exist ` a house size h, a system size `, a vote share vector w 2 .0I 1/ ,wC D 1,aseat vector x 2 A.hI w/, and a party j Ä ` that satisfy xj < bwjhc or xj > dwjhe. Proof a. Let the seat vector x 2 A.hI w/ belong to a divisor method A with signpost sequence s.n/. We assume that there are two parties i and k satisfying xi < bwihc and xk > dwkhe.Let be a multiplier for x.Fromxi 2 ŒŒ  wi we obtain wi Ä s.xi C1/ Ä xi C1 ÄbwihcÄwih,and Ä h. Similarly xk 2 ŒŒ  wk gives wk  s.xk/  xk  1 dwkhewkh,and  h. Since the multiplier  D h is unique, s.xi C 1/ D xi C 1 and s.xk/ D xk  1 are tied. The ties contradict the left-right disjunction (Sect. 3.10), whence the assumption must be discarded. b. The divisor method with downward rounding has r D 1. Inequality (11.5) in Sect. 11.2 yields 1<.1  wj/ Ä xj  wjh Ä xj bwjhc. This tightens to 0 Ä xj bwjhc. Hence the divisor method with downward rounding stays above the lower value of the ideal region. It remains to establish uniqueness. Let A be a divisor method with signpost sequence s.n/. Assuming A ¤ DivDwn, there exist a house size h and a vote share vector w such that some seat vector x 2 A.hI w/ does not belong to the divisor method with downward rounding: x 62 DivDwn.hI w/. Violation of the max-min inequality means that there are two parties i and k satisfying .xi C 1/=wi < xk=wk;thatis,wixk  wk.xi C1/ > 0. Now a new problem is constructed in a system with L  2 parties, where   w L WD 1 C i : wixk  wk.xi C 1/

The house size is chosen to be H WD xi C .L  1/xk. The weights are taken to be v1 WD wi and vj WD wk for all j D 2;:::;L. Method A yields the seat vector y 2 A.hI v/ with components y1 WD xi and y2 D  D yL WD xk since the max- min inequality for x 2 A.hI w/ implies the max-min inequality for y 2 A.hI v/:

s.y / s.x / s.x C 1/ s.y C 1/ max J Ä max j Ä min j Ä min J : JÄL vJ jÄ` wj jÄ` wj JÄL vJ 11.12 Divisor Methods and Ideal Regions of Seats 223

The first party’s ideal share of seats fulfills .v1=vC/H D wi.xi C.L1/xk/=.wi C .L  1/wk/  xi C 1;thatis,L  1  wi=.wixk  wk.xi C 1//. This tightens to L  1 dwi=.wixk  wk.xi C 1//e, by choice of L. The first party stays strictly below the lower value of the ideal region: y1 D xi < xi C 1 Äb.v1=vC/Hc. Hence the apportionment method A does not stay above the lower value of the ideal region at all times. c. Part c is established by an analogous construction as in part b. d. Part d follows since parts b and c determine two distinct methods. ut Another type of problem specification is to impose minimum and maximum restrictions on the feasible seat numbers. This is the topic of the next chapter. Chapter 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

Abstract Electoral systems occasionally restrict the seat numbers to respect a guaranteed minimum or to obey a preordained maximum. Minimum requirements or maximum cappings form an obstacle for quota methods, but are accommodated easily by divisor methods. Such restrictions commonly apply when representing geographical districts. They also arise when combining proportional representation of political parties in the electoral region with the election of persons in single- seat constituencies. Various instances of restrictions are exemplified, with election data from the United Kingdom and Germany and with the issue of determining the composition of the EP.

12.1 Minimum Representation for Electoral Districts

The Constitution of the United States of America guarantees every state of the Union at least one representative when apportioning the seats of the House of Representatives between the various states (Sect. 7.5). Other countries follow suit. France grants a minimum parliamentary representation to the Départements, the Swiss Confederation to the Cantons, the European Union to the Member States. Clearly it is reasonable to ensure that territorial units which are subject to the legislation of a parliament are represented in this parliament. Unmodified apportionment methods may fail to obey a given minimum require- ment. There is a sole exception: the demand of allocating at least one seat. Any divisor method which is impervious allocates to every participant one seat or more (Sect. 4.5). With an impervious divisor method, even tiny interim quotients are rounded upwards to unity and thereby secure at least one seat. The exception emerges with the apportionment of the seats of the House of Representatives of the United States of America. The legally decreed apportionment method is the divisor method with geometric rounding, advertised by Huntington under the winning label: method of equal proportions (Sect. 10.10). This method is impervious. Therefore it complies with the constitutional guarantee that every state is allocated at least one seat. In other settings the minimum requirement asks for two seats or more. For instance the composition of the EP guarantees every Member State at least six seats. The EP even utilizes a maximum cap of at most 96 seats for every Member State. Another set of examples arises when combining proportional representation

© Springer International Publishing AG 2017 225 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_12 226 12 Truncating Seat Ranges: Minimum-Maximum Restrictions with the election of persons. The seats that the candidates of a party win in their constituencies are referred to as “direct seats”. In the proportionality calculations, the direct seats of a party induce a minimum requirement for this party’s seat number. The direct-seat requirements come with a dynamic flavor, since they become known only after the vote count has been completed. Whether dynamic or static, the aim of satisfying restrictions generally necessitates a given apportionment method to be appropriately modified. The modifications called for by minimum-maximum restrictions are difficult to achieve with quota methods (Sect. 12.2), but easy to implement with divisor methods (Sect. 12.3). This proves to be yet another reason why divisor methods are superior procedures. The direct-seat restricted variant of the divisor method with downward rounding is exemplified with the 2016 election of the London Assembly (Sect. 12.4). The direct-seat restricted divisor method with standard rounding is illustrated with a re-evaluation of the 2009 Bundestag election (Sect. 12.5). The simultaneous observance of minimum requirements and maximum cappings is demonstrated with the allocation of the EP seats between the Member States of the European Union (Sects. 12.6–12.8).

12.2 Quota Method Ambiguities

For the Hare-quota method with residual fit by greatest remainders to handle minimum requirements, a “pragmatic modification” has been met in Table 9.3. With the 1880 census data, the main apportionment of 300 seats allocates no seat to Delaware nor to Nevada. It is uncertain whether the two states would profit from the 18 seats that remain for the residual fit. In this instance Delaware would, Nevada would not. Hence the modification interrupts the transition from the main apportionment to the residual fit. First each state with no seat in the main apportionment is assigned a seat from the contingent of the eighteen remaining seats. In this way the one-seat minimum requirement is satisfied for all 38 states. Thereafter the residual fit continues with the reduced number of remaining seats, 16 instead of 18, and with only 36 states after setting aside Delaware and Nevada. In general the pragmatic modification utilizes the seats remaining after the main apportionment, as a reservoir from which to tap for the correction of any deficiencies still persisting. In the presence of maximum cappings, the pragmatic modification would fill the reservoir with the seat overflow emerging after the main apportionment. Clearly the strategy lacks precision. The reservoir may contain too few seats to even out all minimum requirements, or it may be flooded with too many overflow seats so that a residual fit by greatest remainders is no longer practicable. Such difficulties are unlikely to materialize in settings such as in Table 9.3, with many participants and with the lowest non-trivial restriction of one seat each. But the pragmatic modification would usually be unable to handle a six-seat minimum requirement as in the EP, or with dynamic direct-seat restrictions as in Sect. 12.4. 12.3 Minimum-Maximum Restricted Variants of Divisor Methods 227

A “principled modification” is proposed by Balinski and Young (1982, p. 133). It modifies the Hare-quota method with residual fit by greatest remainders. The quota is adjusted so as to accommodate minimum requirements or maximum cappings. Let party j Ä ` be restricted to at least aj seats and at most bj seats. The median of the three numbers aj;vj=Q; bj is their middle value: 8  à < bj in case vj=Q > bj; vj med a ; ; b WD v =Q in case v =Q 2 .a I b ; j Q j : j j j j aj in case vj=Q Ä aj:

The “modified Hare-quota” is defined to be a value Q which solves the equation  à X v med a ; j ; b D h: j Q j jÄ`  The term med aj;vj=Q; bj is taken to be the “ideal share of seats of party j subject to all restrictions being met”. Its floor is the main apportionment for party j.The remaining seats are meted out via a residual fit by greatest remainders. The residual fit applies only to those parties i with ai Ä vi=Q < bi. The crux is the determination of the modified Hare-quota Q which is laborious. Unfortunately the pragmatic modification and the principled modification may produce two solutions that are distinct. Thus the question of how to incorporate restrictions into a quota method admits of no clear-cut answer. Maybe no answer is needed. Neither modification is known to be part of a contemporary electoral system. A resilient approach emerges by noting that the principled modification treats the quota as a flexible quantity that solves an appropriate equation. However, if a sensible invocation of flexibility is deemed pragmatic, then the answer is divisor methods, not quota methods (Sect. 5.10).

12.3 Minimum-Maximum Restricted Variants of Divisor Methods

Divisor methods adapt to minimum-maximum restrictions quite easily. They simply truncate the underlying rounding rule as need be. That is, the truncated rounding rule never falls below the lower level aj that is specified for party j, nor does it reach above the upper level bj. It is easy enough to say this. Yet a precise description requires a few notational prerequisites. It goes without saying that the restrictions which are imposed on an appor- tionment problem must be “compliant”. That is, minimum requirements must not exceed maximum cappings: aj Ä bj. Moreover, the sum of all minimum requirements cannot exceed the house size: aC Ä h. Nor can the house size exceed 228 12 Truncating Seat Ranges: Minimum-Maximum Restrictions the sum of all maximum cappings: h Ä bC. In the sequel all restrictions are tacitly assumed to be compliant. Otherwise the solution set of the problem would be void. Suppose the divisor method A relies on the rounding rule ŒŒ  with signpost . / ŒŒ  bj sequence s n , n 2 N. The “truncated rounding rule”  aj maps a positive number >0 ŒŒ  bj t into the set t aj defined as 8 < fbjg in case t > s.bj/; b ŒŒt j WD ŒŒ  . / . / ; aj : t in case t 2 s aj I s bj fajg in case t Ä s.aj/:

0 ŒŒ 0  bj 0 For t D the no input–no output law continues to apply: aj WD f g. Now the divisor method A is modified into the “minimum-maximum restricted b .v ;:::v / Œ0 /` variant” Aa which maps a vote vector 1 ` 2 I1 into the set of seat vectors  hh ii hh ii  v b1 v b` b. v/ . ;:::; / `. / 1 ;:::; ` >0 ; Aa hI WD x1 x` 2 N h j x1 2 x` 2 for some D D a1 D a`

` where a D .a1;:::;a`/ 2 N denotes the vector of minimum requirements, and ` b D .b1;:::;b`/ 2 N denotes the vector of maximum cappings. This means that an interim quotient vj=D is rounded to the seat number xj 2 ŒŒ v j=D , except when the minimum requirement warrants more seats: xj D aj in the case vj=D Ä s.aj/;or when the maximum capping imposes fewer seats: xj D bj in the case vj=D > s.bj/. The divisor D ensures that all available seats are allocated: xC D h. The sentence that paraphrases the solution indicates that the rounding of an interim quotient may be overruled by a restriction: Every D votes justify roughly one seat, except when a minimum requirement warrants more seats or a maximum capping imposes fewer seats. The subsequent examples testify to the ease with which divisor methods accommodate minimum-maximum restrictions.

12.4 Direct-Seat Restricted Variant of DivDwn

The direct-seat restricted variant of the divisor method with downward rounding is used in the 2016 election of the London Assembly. The assembly size is 25 seats. Fourteen seats are filled from single-seat constituencies by means of a plurality vote system. In a plurality vote system, candidates of stronger parties are likely to come first. Indeed, nine direct seats are won by candidates of the Labour Party, five by candidates of the Conservative Party. These direct seats persist and restrict the final proportional apportionment of the overall 25 seats. 12.4 Direct-Seat Restricted Variant of DivDwn 229

Table 12.1 London Assembly, election 2016 London assembly 2016 Direct seats List votes Quotient DivDwn Labour Party 9 1;054;801 12.4 12 Conservative Party 5 764;230 8.99 8 Green Party 0 207;959 2.4 2 UK Independence Party 0 171;069 2.01 2 Liberal Democrats 0 165;580 1.9 1 Sum (Divisor) 14 2;363;639 (85,000) 25 In order to safeguard the 14 direct seats that are elected by plurality vote in single-seat constituencies, the direct-seat restricted variant of the divisor method with downward rounding is employed. With the 2016 data, the restrictions remain inactive

Proportionality calculations are based on London-wide list votes. A 5% threshold applies to the 2,615,676 valid votes, calling for at least 130,784 votes. As a consequence 252,037 valid votes are rendered ineffective; six parties and one indep are eliminated. This leaves 2,363,639 votes and five parties to participate in the apportionment process. The divisor method with downward rounding is used, restricted by safeguarding the direct seats. In 2016 the restrictions remained dormant. Every 85,000 votes justify roughly one seat. The Labour Party gains 12 seats, thus carrying their 9 direct seats plus filling 3 seats from their list of nominees. Similarly the eight seats for the Conservative Party support their five direct seats. See Table 12.1. Since all direct seats are considered seeded, the seat vector y D .9;5;0;0;0/ provides an initialization of the jump-and-step algorithm shortcutting the laborious tabulation which is usually employed (Sect. 4.13). Thereafter the comparative scores vj=s.yj C 1/ are evaluated as usual; see the tabulation below. Eleven (D 25  14) lines are needed; that is, one line for each seat beyond the minimum requirement. In every line the party with the highest comparative score is flagged () to receive the next seat. The last few lines calculate the divisor interval: Œ84;914:4I 85;534:5. Table 12.1 quotes the select divisor 85;000. An example with an active minimum requirement occurred at the 2016 general election for the Scottish Parliament. For the purpose of the election, Scotland is divided into eight electoral districts; in Scotland they are referred to as regions. The districts encompass between eight and ten single-seat constituencies. Voters cast two votes, a “constituency vote” and a “regional vote”. Every constituency returns a Member of the Scottish Parliament by plurality of constituency votes; in our parlance we speak of direct seats. The district magnitude is obtained by adding seven seats to the number of constituencies in the district. Every district evaluates its regional votes separately, using the direct-seat restricted variant of the divisor method with downward rounding. In seven districts the direct-seat restrictions remain inactive. In these districts the numbers of seats are identical to those obtained from the unrestricted divisor method with downward rounding. Here the seats from the unrestricted method happen to support all pertinent direct-seat restrictions. 230 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

London 2016 Labour Cons Green UKIP LibDem Sum Direct seats 9 5 0 0 0 14

List votes vj 1,054,801 764,230 207,959 171,069 165,580 2,363,639 Seat 15 v1=.9C1/ D v2=.5C1/ D v3=.0C1/ D v4=.0C1/ D v5=.0C1/ D Increment: 105,480.1 127,371.7 207,959 171,069 165,580 Green Seat 16 105,480.1 127,371.7 v3=.1C1/ D 171,069 165,580 UKIP 103,979.5 Seat 17 105,480.1 127,371.7 103,979.5 v4=.1C1/ D 165,580 LibDem 85,534.5 Seat 18 105,480.1 127,371.7 103,979.5 85,534.5 v5=.1C1/ D Cons 82,790 Seat 19 105,480.1 v2=.6C1/ D 103,979.5 85,534.5 82,790 Cons 109,175.7 Seat 20 105,480.1 v2=.7C1/ D 103,979.5 85,534.5 82,790 Labour 95,528.8 Seat 21 v1=.10C1/ D 95,528.8 103,979.5 85,534.5 82,790 Green 95,891 Seat 22 95,891 95,528.8 v3=.2C1/ D 85,534.5 82,790 Labour 69,319.7 Seat 23 v1=.11C1/ D 95,528.8 69,319.7 85,534.5 82,790 Cons 87,900.1 Seat 24 87,900.1 v2=.8C1/ D 69,319.7 85,534.5 82,790 Labour 84,914.4 Seat 25 v1=.12C1/ D 84,914.4 69,319.7 85,534.5 82,790 UKIP 81,138.6

Final seats xj 12 8 2 2 1 25 Calculation of the select divisor

vj=s.xj/ 87,900.1 95,528.8 103,979.5 85,534.5Ž 165,580 (Ž D min)

vj=s.xjC1/ 81,138.6 84,914.4 69,319.7 57,023 82,790 ( D max) Divisor interval D Œ84;914:4I 85;534:5, Midpoint D 85;224:5, Select divisor D 85;000

The direct-seat restriction is activated in just one district: Mid Scotland and Fife. The district magnitude is 16 seats. Every 17,500 votes justify roughly one seat, except when the direct-seat counts warrant more seats. The Scottish National Party wins eight direct seats, but its interim quotient 6.9 vindicates only six seats. Therefore the quotient is discarded in favor of the number of direct seats: eight. To highlight the intervention of the direct-seat restriction, the quotient is marked with a dot: 6:9. The seat column is labeled “rDivDwn” which is meant to be short for “restricted variant of the divisor method with downward rounding”. See Table 12.2. A speedy determination of the final result starts with the adjusted divisor D D vC=.h C `=2/ D 283;055=18:5 D 15;300:3 (Sect. 4.11). Since the Scottish National Party’s eight direct seats dominate their proportionate due, 8>b7:9cD7, 12.4 Direct-Seat Restricted Variant of DivDwn 231

Table 12.2 Mid Scotland and Fife region, Scottish Parliament election 2016 SP2016Mid Scotland and five Direct seats Regional votes Quotient rDivDwn Scottish national party 8 120;128 6.9 8 Conservatives 0 73;293 4.2 4 Labour 0 51;373 2.9 2 Liberal democrat 1 20;401 1.2 1 Green 0 17;860 1.02 1 Sum (Divisor) 9 283;055 (17,500) 16 The direct-seat restricted variant of the divisor method with downward rounding (rDivDwn) allocates to parties as many seats as indicated by the quotient’s integral part or, if larger, by the number of direct seats. The direct-seat proviso determines the eight seats for the Scottish National Party the initial seat distribution is found to be y D .8;4;3;1;1/. Its sum is 17, one seat too much. The comparative scores for decrementation are vj=s.yj/. Their lowest value indicates the party whose seat number is to be decremented: Labour. Now the seat vector x D .8;4;2;1;1/meets the house size h D 16, whence it is final. The last lines calculate the divisor interval: Œ17;124:3I 17;860.Table12.2 quotes the select divisor 17;500.

West Scotland and Fife 2016 SNP Cons Labour LibDem Green Sum (Divisor) Direct seats 8 0 0 1 0 9

Regional votes vj 120,128 73,293 51,373 20,401 17,860 283,055 Initial quotient 7.9 4.8 3.4 1.3 1.2 (15,300.3) Initial seats 8 4 3 1 1 17 Seat 16 v2=4 D v3=3 D v4=1 D v5=1 D Decrement: – 18,323.3 17,124.3 20,401 17,860 Labour

Final seats xj 8 4 2 1 1 16 Calculation of the select divisor vj=s.xj/ – 18,323.3 25,686.5 20,401 17,860Ž (Ž D min) vj=s.xj C 1/ – 14,658.6 17,124.3 10,200.5 8930 ( D max) Divisor interval D Œ17; 124:3I 17; 860, Midpoint D 17;492:3, Select divisor D 17;500

A slower determination of the final result follows the previous example of the 2016 London Assembly election. That is, the direct-seat results provide the apportionment of nine initial seats: .8;0;0;1;0/. Thereafter seats 10, 11, :::,16 are incremented, one after the other, as determined by the highest value of the comparative scores vj=s.yj C 1/. In the United Kingdom the direct-seat restricted variant is known as the “additional member proportional (AMP) system”. In view of the British history of plurality elections in single-seat constituencies, the direct seats are well accepted. The other “additional” seats are installed to move towards proportionality. The attribute “additional member” may be interpreted mischievously as if there were 232 12 Truncating Seat Ranges: Minimum-Maximum Restrictions two classes of deputies. The direct seat deputies are the true representatives of the people, while the additional members are deemed as decorative prima donnas. However, there is no such thing as two classes of deputies. All members of parliament are guaranteed statutory equality. The alternative label “mixed member proportional (MMP) system”, which is used in New Zealand, does not provoke such misgivings.

12.5 Proportionality Loss

In Table 12.2 the seat vector of the direct-seat restricted variant is x D .8;4;2;1;1/. In contrast, the seat vector of the unrestricted parent method, the divisor method with downward rounding, is z D .7;4;3;1;1/. The difference between the two vectors reveals the impact of the restrictions: x  z D .1; 0; 1; 0; 0/. That is, the strongest party exceeds its proportionate due by one seat, the third-strongest party falls short by one seat. The seat vectors x and z can be converted into each other by transferring one seat between the strongest party and the third-strongest party. P Let jxzj signify the modulus-norm of the difference vector: jxzjWD jÄ` jxj zjj. The count of transfers for passing back and forth between x and z equals jxzj=2. We define the “proportionality loss” of x to be 1 PL.x/ x z : WD 2 j  j

In the preceding example we obtain PL.x/ D 1. The proportionality loss PL.x/ measures the deviation of the restricted solution x from the unrestricted solution z. It does so in terms of concrete parliamentary seats, not by means of some abstract goodness-of-fit criterion as in Chap. 10.

12.6 Direct-Seat Restricted Variant of DivStd

The proportionality loss proves particularly enlightening in the electoral system for the German Bundestag. Direct seat wins based on first votes have to be matched with the proportional success based on second votes. Table 2.2 shows the 2009 evaluation, according to the then valid Federal Election Law. The sub- apportionments for SPD, FDP, LINKE, and GRÜNE support their direct seats (if any) without further ado. Only the seat apportionment for CDU is an issue. The CDU is apportioned 173 seats by the unrestricted divisor method with standard rounding. In seven states CDU wins more direct seats than its state list is entitled to, thus creating 21 overhang seats. The final number of CDU seats is 173 C 21 D 194. Table 12.3 shows that the same result as in Table 2.2 is obtained when apportioning the 194 CDU seats with the direct-seat restricted variant of the divisor 12.6 Direct-Seat Restricted Variant of DivStd 233

Table 12.3 Direct-seat restricted sub-apportionment of CDU seats to districts (17BT2009) Direct seats Second votes Quotient rDivStd DivStd Difference Sub-apportionment to districts: CDU x z x  z Schleswig-Holstein 9 518;457 7.51 9 9 0 Mecklenburg-Vorpommern 6 287;481 4.2 6 5 1 Hamburg 3 246;667 3.6 4 4 0 Lower Saxony 16 1;471;530 21.3 21 24 3 Bremen 0 80;964 1.2 1 1 0 Brandenburg 1 327;454 4.7 5 5 0 Saxony-Anhalt 4 362;311 5.3 5 6 1 Berlin 5 393;180 5.7 6 6 0 North Rhine-Westphalia 37 3;111;478 45.1 45 51 6 Saxony 16 800;898 11.6 16 13 3 Hesse 15 1;022;822 14.8 15 17 2 Thuringia 7 383;778 5.6 7 6 1 Rhineland-Palatinate 13 767;487 11.1 13 13 0 Baden-Württemberg 37 1;874;481 27.2 37 31 6 Saarland 4 179;289 2.6 4 3 1 Sum (Divisor) 173 11;828;277 (69,000) 194 194 12  12 In the direct-seat restricted variant (rDivStd) the better performance is decisive: direct seats or proportional seats. Marked quotients () are overruled by direct seats. The proportionality loss amounts to 12 seats method with standard rounding; see column “rDivStd”. As a standard of comparison column “DivStd” exhibits the unrestricted apportionment. The last column contains the difference between the restricted solution (x) and the unrestricted solution (z). The proportionality loss is found to be jx  zj=2 D 12 seats. That is, the realized apportionment x is twelve seats apart from the proportional optimum z. The political impact of active restrictions differs from country to country. In Germany an active restriction transfers seats between district lists within the same party. The super-apportionment of seats among parties remains untouched. Active restrictions do not change the political division of the Bundestag. Only the assignment of seats to candidates is affected. More direct seats of a party in a district entail more representatives from this district; other districts are represented by fewer deputies than promised by full and literal proportionality. Since all Members of the Bundestag are representatives of the whole people (Sect. 2.5), regional provenance of deputies is considered to be a subordinate issue. In the United Kingdom active restrictions trigger seat transfers between parties (Sect. 12.4). If one party sends to the London Assembly or Scottish Parliament more deputies than proportionality indicates, another party sends fewer deputies. The political composition of the Scottish Parliament differs from what is warranted by pure proportionality. On the other hand active restrictions maintain the preor- dained balance between districts. The prespecified district magnitudes are observed meticulously. Every district sends seven representatives in addition to its direct 234 12 Truncating Seat Ranges: Minimum-Maximum Restrictions seats. Here regional representation is considered a more sensitive issue than party representation. A similar sensitivity emerges when considering the composition of the EP.

12.7 Composition of the EP: Legal Requirements

The term “composition of the EP” refers to the allocation of the EP seats between the Member States of the European Union. It is governed by requirements stipulated by primary and secondary Union law. Primary Union law is set forth in “the treaties”. The treaty that is relevant for the composition of the EP is the Treaty on European Union (TEU), also known as the Lisbon Treaty; see European Union (2012). It lays conditions upon possible allocation methods. Of particular significance are the following requirements which we paraphrase to ease referencing in our exposition: 1. Citizens are directly represented in the EP (Art. 10(2) TEU). 2. The EP shall be composed of representatives of the Union’s citizens (Art. 14(2) TEU). 3. Representation of citizens shall be degressively proportional (Art. 14(2) TEU). 4. The EP size shall not exceed 751 seats (Art. 14(2) TEU). 5. Every Member State shall be allocated at least six seats (Art. 14(2) TEU). 6. Every Member State shall be allocated at most 96 seats (Art. 14(2) TEU). A key term is the oxymoron of “degressive proportionality” in requirement 3. It has a long tradition in the debates of the EP. One may have degressive representation, proportional representation, or progressive representation just as one may have degressive taxation, proportional taxation, or progressive taxation. “Degressive proportionality”, however, is a paradoxical concept. The notion is enunciated as a manifestation of solidarity in a 2007 text adopted by the EP, see European Parliament (2007): The more populous States agree to be underrepresented in order to allow the less populous States to be represented better. There is a tension between the principles of direct representation (requirement 1) and of degressive representation (requirement 3). Direct representation supports an allocation referring directly to population figures, whereas degressive representation favors an allocation weighing population figures contingent on whether they origi- nate from more or less populous Member States. As a consequence the meanings of “citizens” in requirements 2 and 3 differ significantly even though both requirements appear in the same section of Art. 14 TEU. Reference to “Union citizens” appears to place all Union citizens on an equal footing. In contrast the principle of degressive proportionality discriminates “citizens” by Member States. The citizens of more populous States agree to be underrepresented in order to allow the citizens of less populous States to be represented better. 12.7 Composition of the EP: Legal Requirements 235

In requirements 5 and 6 there is a potential ambiguity in the term “Member State” over whether it refers to government or to people. When “Member State” is interpreted to mean “government”, Art. 10(2) TEU decrees that the appropriate representative body is the Council of Ministers, not the EP. For the composition of the EP the term “Member State” means “people”; that is, a Member State’s citizenry. Secondary Union law includes the legal acts adopted by the organs of the European Union. The extended deliberations of the EP on its composition have led to detailed specifications that have found their way into Art. 1 of a 2013 European Council decision, see European Council (2013): 7. The least populous Member State shall be allocated six seats. 8. The most populous Member State shall be allocated 96 seats. 9. Any more populous Member State shall be allocated at least as many seats as any less populous Member State. 10. The principle of degressive proportionality shall require decreasing representa- tion ratios when passing from a more populous Member State to a less populous Member State, where the representation ratio of a Member State is defined to be the ratio of her population figure relative to her number of seats before rounding. Requirements 7 and 8 specify that the bounds mentioned in requirements 5 and 6 must actually be attained. Requirement 9 demands that the allocated seat distribution must be concordant. This is a natural postulate. Concordance is an organizing principle shared by all reasonable apportionment methods (Sect. 4.3). Requirement 10 enunciates the concrete specification of the abstract principle of degressive proportionality. The specification uses the notion of representation ratios. Supposing that Member State i has population figure pi and interim quotient qi, its “representation ratio” is defined to be pi=qi. Degressive proportionality demands that the representation ratios are decreasing when passing from a more a populous State i to a less populous State j:

pi pj pi > pj H) > : qi qj

The essential input for the EP composition are the population figures of the Member States. They are needed also for the qualified majority voting (QMV) rule in the Council of Ministers. Since Council and EP are constitutional organs with joint governance responsibility, the functioning of the two institutions ought to be based on the same population data. Therefore the tables’ columns “QMV2017” use the population figures valid for QMV purposes in the calendar year 2017; see European Council (2016). The tables also include a column “Repr.Ratio” in witness of degressive proportionality. For ease of interpretation the representation ratios are commercially rounded: hpi=qii. 236 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

12.8 Cambridge Compromise

The Cambridge compromise is a proposal of an allocation method for the EP composition that is transparent and conforms well with the requirements of primary and secondary Union law. The method is responsive to population changes and impartial to politics, and it is objective, fair and durable. The method results from a workshop at the University of Cambridge whence its name; see Grimmett et al. (2011). The Cambridge compromise consists of two stages: 1. Every Member State is assigned b base seats. The number of base seats is determined so that in the end the least populous Member State is allocated six seats. 2. The remaining seats are allocated proportionally to population figures using the divisor method with upward rounding with a maximum cap. The capping is deter- mined so that the most populous Member State is allocated at most 96 seats. The label for the Cambridge compromise is “b+Upw”. It is indicative of first assigning base seats, plus then adding proportionality seats. The two stages jointly establish the method. For instance, with the QMV2017 populations the Cambridge compromise pro- ceeds as follows: “Every Member State is assigned five base seats, plus one seat per 846,000 citizens or part thereof, with a maximum cap of 96 seats.” See Table 12.4. The smallest state, Malta, finishes with six seats (requirement 7). With only four base seats, Malta would have five seats. With six base seats, Malta would end up with seven seats. The initial assignment of 5 base seats to each of the 28 Member States utilizes a total of 140 seats, leaving 611 seats for the second stage of proportional allocation. Technically, a Member State i with population pi is allocated xi seats where

 91 pi xi D 5 C : 846;000 1

The formula neglects ties which in the present context are highly unlikely. The divisor D D 846;000 ensures that the maximum EP size is exhausted: xC D 751 (requirement 4). Since upward rounding is impervious, the minimum requirement of one seat (de1) would be dispensable, but is displayed as a reminder that five base seats plus at least one proportionality seat satisfy the six seat minimum (requirement 5). Given that there are 5 base seats, the maximum cap at 91 seats (de91) ensures that 5 base seats plus at most 91 proportionality seats obey the 96 seat maximum (requirement 6). With numbers of seats before rounding given by qi D bCpi=D, column “Repr.Ratio” in Table 12.4 verifies degressive proportionality (requirement 10). The Cambridge compromise complies very well with primary Union law. The initial assignment of b base seats to every Member State secures the representation 12.8 Cambridge Compromise 237

Table 12.4 Cambridge compromise CamCom-2017 QMV2017 Quotient 5+Upw Repr.Ratio 2014 Diff. Germany 82;064;489 5C97.003 96 804;531 96 0 France 66;661;621 5C78.8 84 795;520 74 10 United Kingdom 65;341;183 5C77.2 83 794;562 73 10 Italy 61;302;519 5C72.5 78 791;392 73 5 Spain 46;438;422 5C54.9 60 775;373 54 6 Poland 37;967;209 5C44.9 50 761;194 51 1 Romania 19;759;968 5C23.4 29 696;830 32 3 The Netherlands 17;235;349 5C20.4 26 679;286 26 0 Belgium 11;289;853 5C13.3 19 615;419 21 2 Greece 10;793;526 5C12.8 18 607;802 21 3 Czech Republic 10;445;783 5C12.3 18 602;157 21 3 Portugal 10;341;330 5C12.2 18 600;410 21 3 Sweden 9;998;000 5C11.8 17 594;483 21 4 Hungary 9;830;485 5C11.6 17 591;487 20 3 Austria 8;711;500 5C10.3 16 569;480 18 2 Bulgaria 7;153;784 5C8.5 14 531;642 17 3 Denmark 5;700;917 5C6.7 12 485;653 13 1 Finland 5;465;408 5C6.5 12 476;899 13 1 Slovakia 5;407;910 5C6.4 12 474;698 13 1 Ireland 4;664;156 5C5.5 11 443;648 11 0 Croatia 4;190;669 5C4.95 10 421;024 11 1 Lithuania 2;888;558 5C3.4 9 343;289 11 2 Slovenia 2;064;188 5C2.4 8 277;447 8 0 Latvia 1;968;957 5C2.3 8 268;713 8 0 Estonia 1;315;944 5C1.6 7 200;739 6 1 Cyprus 848;319 5C1.003 7 141;322 6 1 Luxembourg 576;249 5C0.7 6 101;432 6 0 Malta 434;403 5C0.5 6 78;789 6 0 Sum (Divisor) 510;860;699 (846,000) 751 – 751 33  33 Every Member State is assigned five base seats, plus one seat per 846,000 citizens or part thereof. There is a 96 seat maximum capping; it overrules the quotient for Germany (). The population figures “QMV2017” are valid for the purposes of qualified majority voting in the Council during the calendar year 2017 of her the Member State’s citizenry as a whole. The subsequent proportional allocation of the remaining seats represents the citizens as individuals. The Cambridge compromise achieves degressive proportionality without distort- ing the meaning of “citizens”. It does so in each of its two stages. The first stage, the assignment of base seats, treats all Member States alike; this is extremely degressive since it neglects population sizes entirely. The second stage, the proportional allocation of the remaining seats, embodies a mild form of degressivity through the use of the divisor method with upward rounding. This method is known to introduce 238 12 Truncating Seat Ranges: Minimum-Maximum Restrictions a bias in favor of less populous Member States at the expense of more populous Member States (Sect. 7.4). This type of bias reinforces the concept of degressive proportionality. Table 12.4 points towards some limitation of the legal demands. Requirement 8 demands that the most populous Member State shall be allocated 96 seats. The demand can be met with the current data, but breeds conflict generally. If in Table 12.4 the German population were to be seven million fewer while the other populations are unchanged, the Cambridge compromise would allocate 94 seats to Germany. An allocation of 96 seats would violate degressive proportionality (requirement 3). The table also illustrates the operational intricacy of degressive proportionality (requirement 10). The subtlety is in the denominator: the “number of seats before rounding” qi D b C pi=D. If the qualification “before rounding” were absent, the denominator would be the seat number xi and the ratio pi=xi would recover the “representative weight” of Sect. 2.8. However, representative weights are unable to underpin the principle of degressive proportionality. For instance in Table 12.4,the representative weight of a Portuguese deputy is 574,518 citizens per seat. A deputy from less populous Sweden has the larger representative weight 588,118. Generally, the reversal of representative weights may be unavoidable when comparing two states with nearly equal population figures but with unequal seat numbers. The reversal cannot materialize when employing representation ratios. Indeed, the representation ratio of Portugal is larger than the representation ratio of less populous Sweden: 600;410 > 594;483.

12.9 Power Compromise

The Cambridge compromise with power-adjusted populations, called “power com- promise” for short, is a variant that attempts to gently balance the conflicting principles of direct representation and of degressive representation (requirements 1 and 3). The power compromise observes the maximum capping in a more sensitive way. In cases when the Cambridge compromise activates the capping, the original population figures are transformed into power-adjusted population units. The power is determined so that the most populous Member State just reaches the maximum, when the divisor method with upward rounding is applied to the power-adjusted population units. When the maximum capping is inactive there is no need for a transformation; in these cases the power compromise coincides with the Cambridge compromise. The quotient 5C97:003 D 112:003 in Table 12.4 exceeds the maximum capping of 96 seats. This triggers a transition to the power compromise as shown Table 12.5. It may be paraphrased as follows: “Every Member State is assigned five base seats, plus one seat per 254,500 adjusted population units or part thereof, with adjustment power 0.93.” That is, a Member State i with population pi is allocated xi.0:93/ seats, 12.9 Power Compromise 239

Table 12.5 Power compromise PowCom-2017 QMV2017 Adjusted Quotient 5CUpwt Repr.Ratio 2014 Diff. Germany 82;064;489 22,917,350 5C90.05 96 863;396 96 0 France 66;661;621 18,888,808 5C74.2 80 841;482 74 6 United Kingdom 65;341;183 18,540,605 5C72.9 78 839;310 73 5 Italy 61;302;519 17,472,492 5C68.7 74 832;302 73 1 Spain 46;438;422 13,495,719 5C53.03 59 800;271 54 5 Poland 37;967;209 11,190,515 5C43.97 49 775;306 51 2 Romania 19;759;968 6,096,509 5C23.95 29 682;441 32 3 The Netherlands 17;235;349 5,368,719 5C21.1 27 660;481 26 1 Belgium 11;289;853 3,622,431 5C14.2 20 586;988 21 1 Greece 10;793;526 3,474,097 5C13.7 19 578;720 21 2 Czech Republic 10;445;783 3,369,885 5C13.2 19 572;648 21 2 Portugal 10;341;330 3,338,536 5C13.1 19 570;776 21 2 Sweden 9;998;000 3,235,335 5C12.7 18 564;460 21 3 Hungary 9;830;485 3,184,892 5C12.5 18 561;283 20 2 Austria 8;711;500 2,846,338 5C11.2 17 538;277 18 1 Bulgaria 7;153;784 2,369,836 5C9.3 15 499;854 17 2 Denmark 5;700;917 1,918,795 5C7.5 13 454;638 13 0 Finland 5;465;408 1,844,969 5C7.2 13 446;178 13 0 Slovakia 5;407;910 1,826,911 5C7.2 13 444;056 13 0 Ireland 4;664;156 1,592,058 5C6.3 12 414;384 11 1 Croatia 4;190;669 1,441,198 5C5.7 11 393;016 11 0 Lithuania 2;888;558 1,019,608 5C4.01 10 320;726 11 1 Slovenia 2;064;188 745,962 5C2.9 8 260;265 8 0 Latvia 1;968;957 713,904 5C2.8 8 252;265 8 0 Estonia 1;315;944 490,784 5C1.9 7 189;934 6 1 Cyprus 848;319 326,257 5C1.3 7 135;041 6 1 Luxembourg 576;249 227,702 5C0.9 6 97;757 6 0 Malta 434;403 175,082 5C0.7 6 76;373 6 0 Sum (keys) 510;860;699 (0.93) (254,500) 751 – 751 21  21 Every Member State is assigned 5 base seats plus one seat per 254,500 adjusted population units or part thereof. The adjusted units are obtained by raising the population figures to the power 0.93. The population figures “QMV2017” are valid for the purpose of qualified majority voting in the Council during the calendar year 2017

where xi.0:93/ is given by   p 0:93 x .0:93/ 5 i : i D C 254;500

The two allocation keys, the power t D 0:93 and the divisor D D 254;500,are displayed in the bottom line of the table. For ease of reading the adjusted population 0:93 units are commercially rounded to integers: h pi i. 240 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

The power compromise is labeled “b+Upwt”. The label indicates the presence of b base seats and the adjustment of the population figures by raising them to the power t. The number of seats allocated to Member State i is given by   p t x .t/ D b C i : i D

The divisor D is determined so that the house size is met: xC.t/ D 751. Two cases need to be distinguished. The first case comprises the instances when the Cambridge compromise allocates to the most populous Member State i D 1 no more seats than the maximum capping M D 96 allows: x1.1/ Ä M. Then the power parameter is taken to be unity: t D 1. In these instances the Cambridge compromise allocation persists as is. The second case has x1.1/ > M. In these instances a power t 2 .0I 1/ is calculated so that, when the divisor method with upward rounding is applied to the t power-adjusted population units pi , the most populous Member State is allocated just enough seats to reach the maximum capping: x1.t/ D 96. The theorem below ascertains the existence of such a power; the proof of the theorem indicates how to carry out the pertinent calculations. Before turning to these technicalities the power compromise is assessed with regard to the requirements demanded by primary and secondary Union law (Sect. 12.7). Since the power compromise employs a power less than or equal to unity it is straightforward to verify degressive proportionality as demanded by requirement 10. The other requirements are also satisfied (except possibly that the most populous Member State is not populous enough to be allocated the maximum of 96 seats). The more principled requirements 1–3 are more of a challenge. The power compromise interprets the term “citizens” in a rather broad sense. During its calculations it replaces real population figures, which count concrete citizens, by imaginary population units, which measure abstract units. In Table 12.5, Malta’s population of 434,403 citizens is transformed to 175,082 population units. Does this mean that only 40% of the citizenry is accounted for? Or 40% of each citizen? Neither interpretation seems helpful; the interim power-adjustments remain obscure. Their vindication lies in the final result achieving a higher degree of degressivity. There is a tension between the principles of direct representation (requirement 1) and degressive representation (requirement 3). Direct representation supports an allocation that treats Union citizens in an equal manner. Degressive representa- tion favors an allocation giving some priority to the citizens of less populous Member States. The Cambridge compromise may be viewed as prioritizing direct representation over degressivity. In contrast, the power compromise allows greater degressivity, but at some cost to direct representation. The two methods yield seat allocations that become increasingly identical as the power parameter moves closer to unity. They coincide when the power equals unity, and this could occur in the future. For instance, if in Table 12.5,theGerman population were to decline by five million to 77,064,489 while the other populations 12.9 Power Compromise 241 are unchanged, the power compromise would have power unity and hence would coincide with the Cambridge compromise. This possibility of future coincidence of the two methods mitigates the marginal disregard by the power compromise of the principle of direct representation. Finally we turn to the theorem announced above. Neglecting base seats, let h be the number of seats remaining for the proportionality stage, and let M be a prespecified maximum satisfying dh=`eÄM Ä h  .`  1/. The power- t adjusted populations pi yield seat vectors y.t/ that are increasing with regard to the majorization ordering (Chap. 8).

Theorem With decreasing population figures p1 >  > p` and arbitrary powers t t t >0, let the seat vectors y.t/ D .y1.t/;:::;y`.t// 2 DivUpw .hI . p1 ;:::;p` // t t belong to the power-adjusted population units p1 ;:::;p` . a. The seat vectors y.t/ are increasing in the majorization ordering:

t < T H) y.t/  y.T/: b. If y1.1/ > M then there exists a power t <1such that y1.t/ D M. Proof a. With t fixed we consider the seat vector y D y.t/. Its accompanying max-min inequality (Sect. 4.6)is

pt pt max i Ä min j : (12.1) iÄ` s.yi C 1/ jÄ` s.yj/  We obtain t log.pi=pj/ Ä log s.yi C 1/=s.yj/ . Imperviousness implies yi  1 and s.yi C 1/  1. In the case s.yj/ D 0 we set the right-hand side equal to infinity. Since the populations are decreasing the sign of log. pi=pj/ determines the “power interval”: Ä Â  Ã Â  Ã s.y C 1/ p s.y C 1/ p max log i log i I min log i log i : (12.2) i>j s.yj/ pj i

Every value in this interval reproduces the seat vector y.Letb be the upper limit in (11.2), and let i < j be two states with b D log s.yi C 1/=s.yj/ = log. pi=pj/. b= . 1/ b= . / Now assume that a tie emerges in (11.1): pi s yi C D pj s yj . A seat transfer from the smaller state j to the larger state i leads to the next seat vector y.b/.The two vectors are increasing in the majorization ordering: y  y.b/, by the Lemma in Sect. 8.3. By transitivity all values T with t < T now satisfy y D y.t/  y.T/. b. In view of majorization the largest state has nondecreasing seat numbers: t < T ) y1.t/ Ä y1.T/ (Sect. 8.7). The start is limt!0 y1.t/ Ddh=`e;thefinishis limt!1 y1.t/ D h  .`  1/. Indeed, for t ! 0 the power-adjusted populations become nearly equal. Hence larger states are allocated dh=`e seats and smaller 242 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

states bh=`c seats, where the division into larger and smaller states is such that the house size h is met. For t !1a single seat is accorded to each of the `  1 smaller states, while all other seats are awarded to the largest state. It follows that the equation y1.t/ D M is solvable for t. If the Cambridge compromise exceeds the maximum capping: y1.1/ > M, then all solutions t of the equation y1.t/ D M fulfill t <1. ut The power interval (2) suggests an algorithm formulating how to step up from one seat vector to the next. The select power quoted from this interval is the midpoint rounded to as few decimal places as is possible while staying in the interval’s interior. Usually several powers t1 <  < tn fit the maximum capping for the most populous Member State: y1.t1/ D  D y1.tn/ D M. The question is: Which of these powers is to be used to determine the adjusted population units? The answer is given by the Theorem: the smallest. Indeed, the Theorem states that the induced seat vectors are ordered by majorization: y.t1/    y.tn/. This indicates that the seat vectors become more preferential to groups of larger states and less preferential to groups of smaller states. The seat vector lowest in the majorization ordering is y.t1/. It is the solution whereby groups of less populous states benefit most. Since the power compromise is meant to reinforce degressivity, the natural choice to employ is the smallest power: t1. In Table 12.5 the QMV2017 population figures admit five powers: t1 D 0:93, t2 D 0:932, t3 D 0:935, t4 D 0:937,andt5 D 0:94. The power compromise selects the smallest value: 0:93. Powers larger than 0.93 involve a seat transfer from a less populous Member State to a more populous Member State: for 0.932 from Lithuania to Poland, for 0.935 from Poland to United Kingdom, for 0.937 from the Netherlands to Poland, and for 0.94 from Portugal to Italy. The selection of the smallest power is expressed by saying that the most populous Member State realizes “just” 96 seats.

12.10 Jagiellonian Compromise

The Jagiellonian compromise is a proposal of a qualified majority voting rule for use in the Council of Ministers of the European Union. The Council is the decision-making body of the governments of the Member States. Thus the Jagiellonian compromise does not belong to the systems for representing people in a parliamentary body that are the focus of this book. Yet the procedure pays due attention to the Union’s citizens. In this sense Jagiellonian compromise is in agreement with the Cambridge compromise, its later namesake. In view of their affinity we conclude the chapter with sketching the essence of the Jagiellonian compromise. 12.10 Jagiellonian Compromise 243

Assuming that an arbitrary proposal is tabled in the Council, a Member State is assumed to vote Yes or No. The Jagiellonian compromise equips every Member State with a certain “voting weight”. The states voting Yes constitute a majority provided the sum of their voting weights meets or exceeds a preordained quota. A state i is “critical” when her Yes is decisive to establishing a majority when assuming the the votes of all other states j ¤ i are given and fix. Let ni denote the number of voting outcomes in which i is critical. The “decision power of Member State i”isdefinedtobeni=nC. That is, the decision power of i is the relative frequency of how often Member State i is tipping the scales. With 28 States there are 228 D 268 million voting outcomes. Elaborate techniques are needed to count the outcomes in which the States are critical. Formerly the Council and its precursors agreed on voting weights and quota by way of negotiations. Past procedures cast doubt on the expertise of diplomats and their advisers. Diplomatic good-will cannot substitute for factual expertise. In the European Economic Community 1958–1972, Germany, France and Italy each had voting weight four, the Netherlands and Belgium two each, and Luxem- bourg one. The quota was set to be twelve. This rule gave Luxembourg no decision power whatsoever. The other states commanded twelve votes or more, or ten votes or less. Luxembourg was never critical. The decision power of Luxembourg was nil. In 1973 Luxembourg’s voting weight was doubled to two. Ireland and Denmark were assigned voting weight three each. For the ten Member States 1981–1985 the quota was set at 45. Among the 210 D 1024 possible voting outcomes Luxembourg was critical as often as were Ireland and Denmark. The three states had exactly the same decision power, despite having ostensibly distinct voting weights. Starting 1 November 2014 the Council adopted a new qualified majority voting rule. Under this procedure, when the Council votes on a proposal by the Commission or the High Representative of the Union for Foreign Affairs and Security Policy, a qualified majority is reached if two conditions are met: 1. At least 55% of the Member States vote in favor; that is, 16 or more of 28. 2. The proposal is supported by Member States representing at least 65% of the population of the Union. This procedure is known as the “double majority voting rule”. The decision powers of the Member States can only be found with much computational effort. For the year 2017 the decision powers for the double majority voting rule are listed in the penultimate column “DM2017” of Table 12.6. The Jagiellonian compromise defines the voting weight of Member State i to be p the square root of its population figure rounded to the nearest whole number: h pi i. The rounding step is superimposed for ease of reference; the numerical differences between rounded and unrounded square roots are negligible. The Jagiellonian quota J is the average of the square root of the population total and the sum of the voting weights: ! 1 p X ˝p ˛ J p p : WD 2 C C i iÄ28 244 12 Truncating Seat Ranges: Minimum-Maximum Restrictions

Table 12.6 Jagiellonian compromise JagCom-2017 QMV2017 Voting weight Decision power DM2017 Diff. Germany 82;064;489 9059 9:13 10:25 1.12 France 66;661;621 8165 8:25 8:44 0.19 United Kingdom 65;341;183 8083 8:17 8:29 0.12 Italy 61;302;519 7830 7:91 7:86 0.05 Spain 46;438;422 6815 6:89 6:18 0.71 Poland 37;967;209 6162 6:23 5:07 1.16 Romania 19;759;968 4445 4:49 3:75 0.74 The Netherlands 17;235;349 4152 4:19 3:49 0.70 Belgium 11;289;853 3360 3:39 2:89 0.50 Greece 10;793;526 3285 3:32 2:85 0.47 Czech Republic 10;445;783 3;232 3:26 2:81 0.45 Portugal 10;341;330 3216 3:25 2:80 0.45 Sweden 9;998;000 3162 3:19 2:77 0.42 Hungary 9;830;485 3135 3:17 2:75 0.42 Austria 8;711;500 2952 2:98 2:64 0.34 Bulgaria 7;153;784 2675 2:70 2:48 0.22 Denmark 5;700;917 2388 2:41 2:33 0.08 Finland 5;465;408 2338 2:36 2:31 0.05 Slovakia 5;407;910 2325 2:35 2:30 0.05 Ireland 4;664;156 2160 2:18 2:23 0.05 Croatia 4;190;669 2047 2:07 2:18 0.11 Lithuania 2;888;558 1700 1:72 2:05 0.33 Slovenia 2;064;188 1437 1:45 1:97 0.52 Latvia 1;968;957 1403 1:42 1:96 0.54 Estonia 1;315;944 1147 1:16 1:89 0.73 Cyprus 848;319 921 0:93 1:84 0.91 Luxembourg 576;249 759 0:77 1:82 1.05 Malta 434;403 659 0:66 1:80 1.14 Sum 510;860;699 99;012 100:00 100:00 6:81  6:81 Quota 60;807 61:41 The voting weights are the square root of the population figures. A group of states constitutes a majority provided the sum of their voting weights meets or exceeds the quota 60;807. The decision powers are practically identical with the normalized voting weights. Column “DM2017” contains the decision powers for the double majority voting rule in 2017

Table 12.6 exhibits the Jagiellonian quota in the last line of the column “Voting Weight”, for the population figures valid in the calendar year 2017. The quota is commercially rounded to a whole number. The quota formula is due to Słomczynski´ and Zyczkowski˙ (2006). The two authors are faculty members of the Jagiellonian University in Kraków, thus explain- ing the attribute “Jagiellonian”. 12.10 Jagiellonian Compromise 245

The Jagiellonian compromise features two surprising properties. The first is that the decision power of a Member State is proportional to its voting weight. There is no need to employ elaborate techniques to calculate decision powers. It suffices to divide the voting weight of a Member State by the sum of all voting weights. This property makes life easy, but its proof is demanding and hence omitted. Moreover, the real concern is not the decision power of a Member State, but the “decision power of a Union citizen”. Citizens contribute to Council’s decision-making process indirectly via their governments. The seminal monograph of Felsenthal and Machover (1998) outlines how to appraise a QMV rule from the citizens’ viewpoint. Decision-making is understood to be a repetitive business. Proper indices must capture the average performance of the rule, not a single realization. The citizens’ a priori power to contribute to the adoption of an act is modeled by a thought experiment. A popular vote is taken, and the government executes the majority’s will. Citizens are assumed to cast their votes independently of each other. 1 0 Let Yn D indicate a Yea of citizenP n,andYn D a Nay. In state i the decision is determined by the Yea total: Yn. According to the Central Limit Theorem nÄpi p the sum is stabilized when scaled by 1= pi; this is where the square root makes its appearance. It can then be shown that, if the decision power of Member State i is b , p i the decision power of its citizens is proportional to bi= pi. The discussion of the Jagiellonian compromise is now quickly concluded.P As p = p mentioned above the decision power of Member State i is bi Dh pi i jÄ28h pj i. p =p Since the quotientsP h pi i pi are practically equal to unity, the quotient =p 1= p bi pi D jÄ28 pj is constant. That is, the citizens of all Member States share the same indirect decision power. This is the second surprise that comes with the Jagiellonian compromise: All Union citizens have the same indirect decision power. The final conclusion is gratifying. The Cambridge compromise and the Jagiel- lonian compromise each boost the democratic foundations of the Union. There is added strength in the two procedures when viewed as a pair. The Cambridge compromise would transfer some of the representative weight from middle-sized Member States to smaller and larger Member States, and so would its variant with power-adjusted populations; see column “Diff.” in Tables 12.4 and 12.5.The Jagiellonian compromise would transfer some of the decision-power from smaller and larger Member States to middle-sized Member States; see column “Diff.” in Table 12.6. The directions of the transfers have a balancing effect, and thus the pair is in equilibrium. Each of these transfers is soundly rooted in the constitutional directive to put citizens first. The next chapter turns to a multi-goal problem of a different type, the reconcili- ation of proportional representation and the election of persons. Chapter 13 Proportionality and Personalization: BWG 2013

Abstract The 2013 amendment of the German Federal Election Law (Bun- deswahlgesetz 2013, BWG 2013) is described. The seat apportionment obeys strict proportionality by political parties. All second votes in the country acquire practically equal success values. Thereafter a party’s countrywide seats are allocated among its various state-lists of nominees. Direct seats that are won via first votes in single-seat constituencies are granted priority by using the direct-seat restricted variant of the divisor method with standard rounding. Its feasibility is ensured by an initial house size adjustment that usually raises the Bundestag size above the nominal level of 598 seats. The amended electoral system is exemplified with the September 2013 election to the 18th Bundestag.

13.1 The 2013 Amendment of the Federal Election Law

The success of Germany’s post-war Federal Election Law rests on its multi-purpose character. The law serves three goals: to achieve proportionality between political parties across the whole country, to allocate the seats of a party among the party’s various state-lists of nominees, and to combine the system’s proportionality compo- nent with plurality elections of individual candidates in single-seat constituencies. Until 1980 the combination of proportional representation with the election of persons proved adequate. There were just two or three parties that passed the 5% threshold and participated in the apportionment calculations. The subsequent diversification of the party system exposed the weaknesses of the old law (Sect. 2.4). Eventually the Federal Election Law was amended in 2013. The amendment was carried by the broad consensus of four of the five political groups in the German Bundestag, with only the LINKE party abstaining. The present chapter explains the amendment by means of the election to the 18th German Bundestag on 22 September 2013. The amended provisions introduce two novel elements. The first novelty is an initial adjustment of the size of the Bundestag beyond its nominal level of 598 seats. The adjustment is to ensure that the two components of the law, proportional representation and the election of persons, can be combined successfully. In 2013 the Bundestag size was raised to 631 seats. Thereafter the super-apportionment distributes the h D 631 seats among the parties in proportion to their countrywide second votes (Sect. 13.2).

© Springer International Publishing AG 2017 247 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_13 248 13 Proportionality and Personalization: BWG 2013

The second novelty concerns the sub-apportionments within a party. They do not use the unabridged divisor method with standard rounding, but its direct-seat restricted variant (Sect. 13.3). This innovation classifies as a novelty only within Germany. Other countries, when adopting the German two-votes system, instantly rectified its notorious deficiencies by implementing the minimum restricted variants. Scotland and London provide examples (Sect. 12.4). The initial house size adjustment strategy is cumbersome and rather generous (Sect. 13.4). There are plenty of alternative strategies that are less complicated and more parsimonious (Sect. 13.5).

13.2 Apportionment of Seats Among Parties

The very first action of the 2013 Federal Election Law is to adjust the size of the Bundestag to actual needs. We defer discussion of the adjustment procedure to Sect. 13.4. As for the September 2013 election results, the house size is raised from 598 to 631 seats. Thereafter follows the super-apportionment: the apportionment of the 631 seats among the political parties. The apportionment is proportional to the countrywide second-vote totals of the parties. A party’s second-vote total is the sum of second-votes over all sixteen German states, except that CDU campaigns in only fifteen states (all but Bavaria) and that CSU stands in just one state (Bavaria). The seat apportionment is carried out by means of the divisor method with standard rounding. Every 58,420 second votes justify roughly one seat; see Table 13.1. The divisor method with standard rounding guarantees that all second votes are treated as fairly as possible. Firstly, all second votes enjoy a practically equal success value for all voters. The realized success values minimize the squared- error deviations from the ideal 100% success (Sect. 10.2). Secondly, the seat apportionment is success-value stable. No seat transfer from one party to another can diminish the success-value disparity between any two voters. (Sect. 10.8). Thirdly,

Table 13.1 Super-apportion- 18BT2013 Second votes Quotient DivStd ment of 631 seats by second votes, election to the 18th Super-apportionment Bundestag 2013 CDU 14;921;877 255.4 255 SPD 11;252;215 192.6 193 LINKE 3;755;699 64.3 64 GRÜNE 3;694;057 63.2 63 CSU 3;243;569 55.52 56 Sum (Divisor) 36;867;417 (58,420) 631 Every 58,420 votes justify roughly one seat. The seat apportionment is performed using the divisor method with standard rounding: DivStd. All second votes acquire practically equal success values 13.3 Assignment of Candidates to Seats 249 the apportionment is ideal-share stable. All political parties are awarded their ideal share of seats as far as is practical (Sect. 10.11). Fourthly, the seat biases of all parties are practically zero. The unavoidable rounding operations entail random profits and random deficits that on average are devoid of a systematic trend (Sect. 7.4). Lastly, the procedural instruction is as simple as can be: “Scale all vote counts by a common divisor. Then round the interim quotients to whole numbers as is standard in day-to- day life”.

13.3 Assignment of Candidates to Seats

Since CSU stands only in one state, Bavaria, its seat apportionment is instantly completed. The super-apportionment allots 56 seats to CSU (Table 13.1). Moreover the party wins 45 direct seats on the basis of first votes. Therefore, CSU sends 45 constituency winners into the Bundestag, plus the eleven nominees on its Bavarian state-list that rank top when leapfrogging constituency winners. The other four parties stand in more states than just one. Again every direct seat winner is declared elected. These direct seat wins impose minimum requirements on the final sub-apportionment of a party’s countrywide seats. Therefore the apportionment is conducted using the direct-seat restricted variant of the divisor method with standard rounding (rDivStd). The seats that surpass the direct seat wins are assigned to a state-list’s top-ranked nominees who did not win a direct seat. See Table 13.2; the two-letter tabs for the states are expanded in Table 2.2. The initial adjustment of the house size ensures that every sub-apportionment commands sufficiently many seats to carry all direct seats in the form of minimum requirements. In 2013 the direct-seat restrictions remain dormant for SPD, LINKE, and GRÜNE. Their seat allocations coincide with those from the unabridged divisor method with standard rounding. Hence these apportionments enjoy the same excellent properties that are reviewed in the previous section. Note, however, that success-value optimality has two distinct meanings. The previous super- apportionment evaluates second votes by political affiliation for which the principle of electoral equality is paramount. In contrast the current sub-apportionments evaluate second votes by geographic provenance. This within-party distribution does not interfere with the countrywide division of seats between parties. Only the CDU sub-apportionment has to cope with direct-seat restrictions that are active. Every 59,700 second votes justify roughly one seat, except when a direct- seat restriction warrants more seats. In Table 13.2 the quotients that are superseded by direct seat wins are marked with a dot (). The proportionality loss between the direct-seat restricted apportionment (x) and the unrestricted apportionment (not shown in the table: z)isjx  zj=2 D 3 (Sect. 12.5). Thus the direct-seat restrictions imply that three CDU seats are assigned to state- lists other than indicated by pure proportionality. The restrictions are instrumental in avoiding overhang seats and negative voting weights that formerly troubled the law. The fact that the restrictions imply slight deviations from full and literal 250 13 Proportionality and Personalization: BWG 2013

Table 13.2 Sub-apportionments of party-seats to state-lists, election to 18th Bundestag 2013 18BT2013 Dir. Second Quot. rDivStd Dir. Second Quot. rDivStd (continued) votes votes Sub-apportionment to state-lists: CDU Sub-apportionment to state-lists: SPD SH 9 638,756 10.7 11 2 513;725 8.8 9 MV 6 369,048 6.2 6 0 154;431 2.6 3 HH 1 285,927 4.8 5 5 288;902 4.9 5 NI 17 1,825,592 30.6 31 13 1;470;005 25.1 25 HB 0 96,459 1.6 2 2 117;204 2.0 2 BB 9 482,601 8.1 9 1 321;174 5.49 5 ST 9 485,781 8.1 9 0 214;731 3.7 4 BE 5 508,643 8.52 9 2 439;387 7.51 8 NW 37 3,776,563 63.3 63 27 3;028;282 51.8 52 SN 16 994,601 16.7 17 0 340;819 5.8 6 HE 17 1,232,994 20.7 21 5 906;906 15.503 16 TH 9 477,283 8.0 9 0 198;714 3.4 3 RP 14 958,655 16.1 16 1 608;910 10.4 10 BY – – – – 0 1;314;009 22.46 22 BW 38 2,576,606 43.2 43 0 1;160;424 19.8 20 SL 4 212,368 3.6 4 0 174;592 3.0 3 Sum (Divisor) 191 14,921,877 (59,700) 255 58 11;252;215 (58,500) 193 Sub-apportionment to state-lists: LINKE Sub-apportionment to state-lists: GRÜNE SH 0 84,177 1.4 1 0 153;137 2.53 3 MV 0 186,871 3.1 3 0 37;716 0.6 1 HH 0 78,296 1.3 1 0 112;826 1.9 2 NI 0 223,935 3.7 4 0 391;901 6.47 6 HB 0 33,284 0.6 1 0 40;014 0.7 1 BB 0 311,312 5.2 5 0 65;182 1.1 1 ST 0 282,319 4.7 5 0 46;858 0.8 1 BE 4 330,507 5.51 6 1 220;737 3.6 4 NW 0 582,925 9.7 10 0 760;642 12.6 13 SN 0 467,045 7.8 8 0 113;916 1.9 2 HE 0 188,654 3.1 3 0 313;135 5.2 5 TH 0 288,615 4.8 5 0 60;511 1.0 1 RP 0 120,338 2.0 2 0 169;372 2.8 3 BY 0 248,920 4.1 4 0 552;818 9.1 9 BW 0 272,456 4.54 5 0 623;294 10.3 10 SL 0 56,045 0.9 1 0 31;998 0.53 1 Sum (Divisor) 4 3,755,699 (60,000) 64 1 3;694;057 (60,600) 63 The CDU direct seats (Dir.) overrule the interim quotient in three states (marked ), due to the direct-seat restricted variant of the divisor method with standard rounding. For the other parties the direct seats are supported by the proportionality seats whence the restrictions remain dormant 13.4 Initial Adjustment of the Bundestag Size 251 proportionality is the price to pay for the successful combination of two distinct goals of the electoral system: proportional representation and the election of persons.

13.4 Initial Adjustment of the Bundestag Size

A central novelty is the initial adjustment of the Bundestag size to ensure that the system can accommodate the direct seat wins. There are many ways to achieve this goal. Sadly the one chosen is most cumbersome. It carries some of the burden that seems inevitable when working towards a broad political consensus. An attempted 2011 amendment introduced separate apportionments in each of the states. The separate per-state apportionments resurface in the current adjustment calculation. They persist not because of technical necessity, but more as a face-saving device for the proposers of futile reform attempts of the past. The calculations to adjust the Bundestag size advance in three steps. Step 1 allocates the 598 nominal Bundestag seats between the sixteen states on the basis of population figures. Step 2 evaluates each state separately. The seats of a state are apportioned among parties in proportion to their second votes. The maximum of direct seat wins and proportionality seats is earmarked as the target seat number for this party in this state. Step 3 raises the Bundestag size until every party gets at least as many seats as are called for by the total of the party’s target seats. Step 1: Allocation of 598 Seats Between States The first step allocates the 598 nominal seats between the sixteen states. The divisor method with standard rounding is applied to the population figures of 31 December 2012. See Table 13.3. It can happen that a direct seat winner is an indep, or affiliated with a party that fails the 5% threshold. Then a seat is assigned to the winning candidate, and simultaneously deducted from the allocation of the state to which the constituency belongs. For example, in 2002 two PDS candidates won their Berlin constituencies. However, the PDS did not pass the 5% threshold. The allocation for Berlin would be reduced from 24 to 22 seats, for use in the subsequent Step 2. Step 2: Target Seats The second step is painful. It consists of 16 apportionment calculations, one for each state. The seats from Step 1 are apportioned among the state’s parties by second votes using the divisor method with standard rounding. In Table 13.4 the state divisors are collected in the CDU box. The same divisors are used also in the other boxes. For instance in Schleswig-Holstein the 638,756 CDU votes are divided by 61,000. The interim quotient 10.47 is rounded to 10 seats. At this point the direct seat wins come into play. The greater success of proportionality seats (10) over direct seats (9) dictates the target seat number: 10. For CDU, the per-state target seats sum to a target seat total of 242 seats. For SPD, LINKE and GRÜNE the target seat totals are 183, 60, and 61 seats, respectively. The CSU is not shown in the table. Its interim quotient 3;243;569=58;300 D 55:6 leads to 56 proportionality seats. In view of 45 direct CSU seats, the target for CSU is 56 seats. 252 13 Proportionality and Personalization: BWG 2013

Table 13.3 Step 1 of the house size adjustment calculations, election to the 18th Bundestag 2013 18BT2013-Step 1 German pop. 31.12.2012 Quotient DivStd Schleswig-Holstein 2;686;085 21.7 22 Mecklenburg-Vorpommern 1;585;032 12.8 13 Hamburg 1;559;655 12.6 13 Lower Saxony 7;354;892 59.3 59 Bremen 575;805 4.6 5 Brandenburg 2;418;267 19.49 19 Saxony-Anhalt 2;247;673 18.1 18 Berlin 3;025;288 24.4 24 North Rhine-Westphalia 15;895;182 128.1 128 Saxony 4;005;278 32.3 32 Hesse 5;388;350 43.4 43 Thuringia 2;154;202 17.4 17 Rhineland-Palatinate 3;672;888 29.6 30 Bavaria 11;353;264 91.52 92 Baden-Württemberg 9;482;902 76.4 76 Saarland 919;402 7.4 7 Sum (Divisor) 74;324;165 (124,050) 598 The 598 nominal seats are allocated between the states by population figures as of 31 December 2012. Every 124,050 citizens justify roughly one seat

Step 3: Adjustment of the Bundestag Size The final step interprets the target seat total of a party to be the minimum number of seats to which the party is entitled. If, using the divisor method with standard rounding, the 598 nominal seats satisfy the restrictions then the Bundestag has 598 seats. If not, the Bundestag size is increased. The first house size satisfying all minimum requirements is the Bundestag size sought. Table 13.5 exhibits the first house size when all restrictions are satisfied: 631. The qualifier “first” is to the point because divisor methods are house size monotone (Sect. 9.5). With a Bundestag size of 631 seats the super-apportionment and the ensuing sub-apportionments are evaluated as explained in Sects. 13.2 and 13.3.

13.5 Alternative House Size Adjustment Strategies

Step 2 reacts too sensitively to the variability of voter turnouts in the sixteen states. The differences are exacerbated by the non-uniform distribution of ineffective votes. The two effects manifest themselves through the unevenness of the state divisors in Table 13.4. The largest divisor is 67,000 in Saarland, the smallest 58,300 in Bavaria. It is the smallest divisor that essentially determines the final divisor 58,420 in Table 13.5. Upon second glance the uneven divisors respond to the variation of voter turnout, and the variation of ineffective votes. But voter turnout and ineffective 13.5 Alternative House Size Adjustment Strategies 253 (continued) – 242 9 9 8 9 4 6 5 1 10 28 16 20 15 59 43 Target – 8 8 8 8 3 10 6 5 1 16 15 28 59 20 43 DivStd – 10.47 1.48 8.0 8.1 8.2 8.0 3.2 6.2 4.8 27.7 16.3 20.2 15.2 59.4 42.52 Quotient 638,756 369,048 285,927 1,825,592 96,459 482,601 485,781 508,643 994,601 1,232,994 477,283 958,655 – 212,368 3,776,563 2,576,606 Second votes Dir. 9 6 1 17 0 9 9 5 37 16 17 9 14 – 38 4 CDU State divisor 61,000 60,000 60,000 66,000 65,000 60,000 60,000 62,000 63,600 61,000 61,000 60,000 63,000 58,300 60,600 67,000 Step 2 of the house size adjustment calculations, election to the 18th Bundestag 2013 18BT2013-Step 2 SH MV HH NI HB BB ST BE NW SN HE TH RP BY BW SL Target seat total Table 13.4 254 13 Proportionality and Personalization: BWG 2013 1 1 1 3 3 1 5 5 5 9 8 3 5 2 4 4 60 Target 1 3 1 3 1 5 5 5 9 8 3 5 2 4 1 4 DivStd 1:4 1:3 0:51 0:8 3:1 3:4 5:2 4:7 5:3 9:2 7:7 3:1 4:8 1:9 4:3 4:496 Quotient 84;177 78;296 33;284 56;045 186;871 223;935 311;312 282;319 330;507 582;925 467;045 188;654 288;615 120;338 248;920 272;456 Sec. votes LINKE Dir. 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 8 5 4 7 6 3 3 3 5 2 15 10 22 19 48 23 183 Target 8 3 5 2 5 4 7 6 3 3 10 15 19 DivStd 22 48 23 8:4 5:4 3:6 7:1 5:6 3:3 9:7 2:6 2:6 4:8 1:8 14:9 Quotient 22:3 22:54 19:1 47:6 513;725 154;431 288;902 117;204 321;174 214;731 439;387 340;819 906;906 198;714 608;910 174;592 Sec. votes 1;470;005 1;314;009 1;160;424 3;028;282 2 0 5 2 1 0 2 0 5 0 1 0 0 0 SPD Dir. 13 27 (continued) SH MV HH NI HB BB ST BE NW SN HE TH RP BY BW SL Target seat total Table 13.4 13.5 Alternative House Size Adjustment Strategies 255 xes. The 1 1 1 1 1 3 2 6 4 2 5 3 9 0 61 12 10 Target 1 2 6 1 1 1 4 2 5 1 3 0 3 9 12 10 DivStd 2:51 0:6 5:9 0:6 1:1 0:8 3:6 1:9 5:1 1:0 2:7 0:48 1:9 9:48 Quotient 12:0 10:3 s the target seat numbers 37;716 40;014 65;182 46;858 60;511 31;998 153;137 112;826 391;901 220;737 760;642 113;916 313;135 169;372 552;818 623;294 Sec.votes onality seats (DivStd) yield GRÜNE Dir. 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 SH MV HH NI HB BB ST BE NW SN HE TH RP BY BW SL Target seat total The evaluations operate per state;maximum of that the is, direct rowwise. seats The (Dir.) state and the divisor proporti that is shown in the top box is applied to this state throughout the other three bo 256 13 Proportionality and Personalization: BWG 2013

Table 13.5 Step 3 of the house size adjustment calculations, election to the 18th Bundestag 2013 18BT2013-Step 3 Min. Second votes Quotient DivStd Quotient DivStd Quotient DivStd CDU 242 14;921;877 241.8 242 255.3 255 255.4 255 SPD 183 11;252;215 182.4 182 192.51 193 192.6 193 LINKE 60 3;755;699 60.9 61 64.3 64 64.3 64 GRÜNE 61 3;694;057 59.9 60 63.2 63 63.2 63 CSU 56 3;243;569 52.6 53 55.49 55 55.52 56 Sum (Divisor) 602 36;867;417 (61,700) 598 (58,450) 630 (58,420) 631 The parties’ target seat totals generate minimum requirements (Min.). Up to house size 630 some restrictions remain active (), thereafter they turn dormant. Hence the Bundestag size is 631 seats votes are side issues when the task is to reconcile proportional representation and the election of persons. A reconciliation of proportionality and personalization can be better served by other adjustment strategies. Here is an overview over six alternative strategies (a)– (f):

Bundestag CDU Proportionality loss in Size adjustment strategy size 2013 seats CDU sub-apportionment (a) Direct-seat restrictions 598 242 4 (b) Direct-seat restrictions C10% 598 242 4 (c) Quota-based estimates 598 242 4 (d) BWG 2008 CDU seats 607 246 3 (e) BWG 2013 apportionment 631 255 3 (f) Zero unproportionality 661 268 0

Strategy (a) apportions the 598 nominal seats using the divisor method with stan- dard rounding. CDU is entitled to 242 seats. In order to respect the 191 direct seat wins of CDU, the CDU sub-apportionment uses the direct-seat restricted variant. It allocates four seats differently as when compared with perfect proportionality. Hence its proportionality loss is four seats. Strategy (b) makes sure that the number of seats for CDU exceeds the number of direct seats by 10%. That is, the house size must guarantee CDU at least 191C19 D 210 seats. Since this is achieved with a nominal house size of 598 seats, no further action is needed. The end result is the same as with strategy (a). Strategy (c) replaces the separate per-state evaluations in Step 2 by applying a common divisor to all eighty state-lists of nominees. The divisor used is the votes- to-seats ratio: 36;867;417=598 D 61;651:2. Interim quotients are truncated to their integral part in order to obtain parsimonious seat estimates. With target seats taken to be the maximum of direct seats and estimated seats, the target seat total for CDU turns out to be 241 seats. Again no further action is needed. 13.5 Alternative House Size Adjustment Strategies 257

Strategy (d) argues that, with the old 2008 law, the CDU seats would have consisted of 242 proportionality seats plus 4 overhang seats. This yields a minimum requirement of 246 seats for CDU. The restriction is satisfied with a Bundestag size of 607 seats. The proportionality loss is three seats, one less than before. Strategy (e) is the current 2013 law described above. The proportionality loss is three seats (Sect. 13.3). Strategy (f) raises the Bundestag size to 661 seats. This would have enabled the defunct 2008 law to accommodate all direct seats into the proportionality calculations and thus to circumvent any overhang seats. The proportionality loss is zero. It is evident that the complicated and laborious strategy (e) of the current law performs poorly. The proportionality loss in the CDU sub-apportionment is three seats. The more parsimonious strategies (a)–(c) share a proportionality loss of four seats. The one-seat difference is negligible. It does not vindicate the loss of transparency and the extra work which is incurred by strategy (e). A refined approach to accommodate the double division of the electorate, by regional provenance and by party affiliation, is double proportionality. This is the topic of the final two chapters. Chapter 14 Representing Districts and Parties: Double Proportionality

Abstract Double-proportional divisor methods take account of two dimensions: the geographical division of the electorate by district, and the political division of the electorate by party. At the outset all available seats are allocated between districts proportionally to population figures, and among parties proportionally to countrywide vote counts. The subsequent step is the essence of the method: the sub-apportionment of seats simultaneously by district and party. The result must conform to both dimensions: every district must meet its district magnitude, and every party must exhaust its countrywide seats. To this end two sets of electoral keys are employed: district divisors, and party divisors. The sub-apportionment is easily verified provided the divisors are publicized.

14.1 Double Proportionality: Practice

There are many electoral systems that subdivide their electoral area into two or more regional districts. Some of these systems treat the area’s subdivision by geographical district in a manner similar to the electorate’s division by political party. In fact the appreciation of a member of parliament as a representative of her or his local district historically precedes the view that parliament as a whole is a mirror image of the people’s division along party lines (Sects. 1.10 and 2.1). The double challenge can be met by double-proportional divisor methods. Naturally these methods are conceptually more demanding than simple-proportional divisor methods. Because of the dual objectives, the requisite notation is more elaborate, and so are the required calculations. For this reason our presentation of double-proportional divisor methods is spread over two chapters. The present chapter highlights the practice of these methods, Chap. 15 expounds the theory. The practice of double-proportional divisor methods is explained by means of two examples. The first example is a real application of double proportionality: the 2016 election of the parliament of the Swiss Canton of Schaffhausen (Sects. 14.2– 14.5). The second example is a hypothetical application of double proportionality to unionwide lists at EP elections (Sects. 14.6–14.8).

© Springer International Publishing AG 2017 259 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_14 260 14 Representing Districts and Parties: Double Proportionality

14.2 The 2016 Parliament Election in the Canton of Schaffhausen

This section reviews the election to the Cantonal Parliament (Kantonsrat) of the Swiss Canton of Schaffhausen on 25 September 2016. For the purpose of its parliamentary elections the canton is subdivided into six districts. Formerly the districts were evaluated separately. Since 2008, when parliament size was reduced from 80 to 60 seats, districts are evaluated jointly. The double-proportional variant of the divisor method with standard rounding is used. The seats are distributed in three steps: first between districts, then among parties, and finally by district and party. In Sects. 14.3 and 14.4 we comment on some quirks that are peculiar to double proportionality. Prior Allocation of Seats Between Districts: District Magnitudes The number of seats allocated to a district is called the “district magnitude”. The Schaffhausen district magnitudes also fix the number of candidates a voter may elect on the ballot sheet. The district magnitudes were decreed in October 2015, in good time before the election. They were calculated from the population figures of 31 December 2014, using the divisor method with standard rounding. Every 1310 citizens justify roughly one seat; see Table 14.1. The Canton constitution guarantees each district at least one seat. For the 2016 election the guarantee happens to be fulfilled without further ado. Should it ever happen that the smallest district fails to be represented, the law would have to be amended to use the minimum restricted variant of the divisor method. The district magnitudes extend from 27 seats in the largest district, the City of Schaffhausen, down to a single seat in the exclave Buchberg-Rüdlingen. Formerly the people in Buchberg-Rüdlingen elected their single representative by plurality vote. Votes were wasted unless cast for the eventual district winner, thus questioning the constitutional promise of a proportional representation system. The volatility

Table 14.1 District Sh2016DistrictMagn. Population Quotient DivStd magnitudes Schaffhausen 35;977 27.46 27 Klettgau 16;428 12.54 13 Neuhausen 10;318 7.9 8 Reiat 9765 7.45 7 Stein 5504 4.2 4 Buchberg-Rüdlingen 1587 1.2 1 Sum (Divisor) 79;579 (1310) 60 The 60 seats are allocated between districts proportionally to population figures of 31 December 2014. The divisor method with standard rounding is used. Every district is allocated at least one seat, even without the existing minimum requirement 14.2 The 2016 Parliament Election in the Canton of Schaffhausen 261 of the district magnitudes loses its impact when adopting a double-proportional method. Super-Apportionment of Seats Among Parties: Overall Party-Seats The votes for all candidates of a party in a district are totaled to yield the “party votes” (Parteistimmen). However, a voter in the City of Schaffhausen may elect up to 27 candidates, while a voter in Buchberg-Rüdlingen can elect only one. Since electoral equality pertains to human beings, the different scales must be standardized. Standardization is achieved through a passage from party votes to the “voter number” (Wählerzahl) of a party in a district:

party votes voter number D : district magnitude

The voter numbers are rounded to two decimal places. In contrast, the voter numbers in Zurich are rounded to whole numbers (Sect. 4.9). The higher level of precision in Schaffhausen is superficial; either way the results are mostly the same. The cantonwide voter numbers of the parties provide the basis for the appor- tionment of the 60 seats among the 15 competing parties. The divisor method with standard rounding is used; see Table 14.2. Note that we show the party votes later, in Table 14.3. For instance in Buchberg-Rüdlingen, the 456 SVP party votes persist

Table 14.2 Sh2016Super-app. Voter number Quotient DivStd Super-apportionment SVP 7032:75 17.49 17 SP 5370:96 13.4 13 FDP 3463:38 8.6 9 AL 1760:26 4.4 4 GLP 1410:06 3.51 4 ÖBS-GRÜNE 980:60 2.4 2 EDU 961:61 2.4 2 CVP 915:32 2.3 2 EVP 599:23 1.49 1 SVP/Agro 518:90 1.3 1 JSVP 498:38 1.2 1 SVP/KMU 454:80 1.1 1 JFSH 413:43 1.0 1 JUSO 325:63 0.8 1 SVP/Sen. 246:31 0.6 1 Sum (Divisor) 24;951:62 (402) 60 The apportionment of seats among parties is based on the cantonwide voter numbers. The divisor method with standard rounding is used. Every 402 units justify roughly one seat. The resulting apportionment realizes practically equal success values for all votes that take part in the super- apportionment 262 14 Representing Districts and Parties: Double Proportionality rict 500 1140 2800 1788 5400 10,100 80-0 1 3405-1 1415-0 5389-1 1121-0 3 24,078-2 13,234-1 8490-1 8548-1 e linked by a hyphen: 61,962-5. The other seat numbers are obtained similarly. The dist 1111111divisor 17 13 9 4 4 2 2 2 573-0105-0 1250-0 277-0 734-0 328-0 1280-1 236-0 958-1 79-0 178-0 442-0 62-0 177-0 257-0 164-0 196-0 210-0 261-0 178-0 456-1 98-0 62-0 26-0 7480-22656-2 4232-1 1805-2 3346-1 469-03144-0 662-0 428-0 2465-1 1378-1 2687-1 1063-0 1169-0 889-0 358-0 392-0 772-0 155-0 955-0 955-0 927-0 4990-3 4708-2 2070-1 874-0 476-0 1122-1 535-0 1742-1 6133-1 1332-0 2277-0 2660-0 2462-0 5241-1 2001-1 25,032-4 13,900-2 12,144-2 2976- 61,962-5 69,104-6 43,557-5 32,252- es 5 seats. Input and output ar Sub-apportionment ) EVP SVP/A JSVP SVP/K JFSH JUSO SVP/S District continued Klettgau Stein Reiat Neuhausen Buchberg-Rüdlingen Sh2016Sub-app.SchaffhausenKlettgauNeuhausenReiatStein 27 Buchberg-Rüdlingen SVPParty divisor 13 8 ( 1 SPSchaffhausen 7 4 FDP 1.1157Party AL divisor 1.054 GLP 0.9 ÖBS 1.2 EDU 1 0.904 0.97 CVP 0.8 1 0.7 1 0.6 1.68 1 0.37 Table 14.3 The Schaffhausen SVP partyresulting votes quotient (61,962) 5.4987 are justifi divisors divided and the by party the divisors district jointly ensure divisor that for each Schaffhausen district meets (10,100) its and district by magnitude the and that party each divisor party for exhausts its SVP overall (1.1157). party-seats The 14.2 The 2016 Parliament Election in the Canton of Schaffhausen 263 as voter number (since 456=1 D 456). In the City of Schaffhausen, the 61,962 SVP party votes yield voter number 2294.89 (since 61;962=27 D 2294:89). The sum of the SVP voter numbers over the six districts is 7032.75, as exhibited in Table 14.2. The cantonwide seat number of a party is referred to as its “overall party-seats”. Thus the strongest list, SVP, is apportioned 17 overall party-seats, the weakest list, SVP/Sen., one. All voters in the canton are treated equally, without regard to the district in which they reside. The seat apportionment among parties inherits the excellent properties that come with the divisor method with standard rounding: The seat biases of all parties vanish identically (Sect. 7.4); the success values of the voters are as equal as is feasible (Sect. 10.2); and the seat apportionment is success- value stable (Sect. 10.8) as well as ideal-share stable (Sect. 10.11). Sub-apportionment by District and Party: Final Seat Assignment The con- cluding sub-apportionment consists of the assignment of all 60 seats to the lists of nominees presented to the electorate in the districts. With six districts and 15 parties the maximum number of potential lists is 6 15 D 90. But some parties do not stand in some districts, and so only 78 lists materialize. The sub-apportionment delivers an assignment of seats by district and party aiming at three goals: 1. Every district meets its district magnitude. 2. Every party exhausts its overall party-seats. 3. Proportionality is observed among parties within a given district, and among districts within a given party. The goals are achieved by the double-proportionalvariant of the divisor method with standard rounding. Two sets of divisors are needed. The first consists of a “district divisor” Ci >0for every district i. The second set includes a “party divisor” Dj >0 for every party j. The divisors ensure that goals 1 and 2 are satisfied meticulously. Goal 3 is realized, as far as possible, in that district divisors scale the party votes within a given district, while party divisors scale the party votes within a given party. How these divisors are determined is discussed in the next chapter. Once the divisors are obtained and published it is rather easy to determine the seat numbers. The party votes vij, which in district i are cast for party j, are divided by the two associated divisors Ci and Dj to obtain the interim quotient: vij=.CiDj/. Standard rounding of the interim quotient yields the seat number xij. In the absence of ties we thus have   vij xij D : (14.1) CiDj

Table 14.3 displays the table of solutions. The inner box shows party votes and seat numbers, linked by a hyphen “-”. Verification is aided by the information arranged around the inner box. District magnitudes are on the left, overall party- seats at the top, district divisors on the right, and party divisors at the bottom. 264 14 Representing Districts and Parties: Double Proportionality

For instance, in the first district the 61,962 SVP party votes are divided by the district divisor (10,100) and by the SVP divisor (1.1157). The resulting interim quotient (5.4987, not shown in the table) justifies five seats. Similarly the 2001 SVP/Sen. party votes in Schaffhausen are divided by the divisors 10,100 and 0.37. The interim quotient (0.54) justifies one seat. In every row of the table the seat numbers sum to the district magnitudes, while in every column they sum to the overall party-seats. The seat apportionment of the 2016 Cantonal Parliament election in the Canton of Schaffhausen is complete. The merits of double proportionality vividly surface in the single-seat district Buchberg-Rüdlingen. Until 2004 the election in this district was by plurality vote. In 2004 the share of valid votes in the electorate amounted to 580=1068 D 54:3%. With double proportionality, it rose to 635=1081 D 58:7% in 2008, to 730=1136 D 64:2% in 2012, and to 722=1153 D 62:6% in 2016. People in Buchberg-Rüdlingen seem to have realized that their votes contribute to the cantonwide success of the parties of their choice as much as the votes in the other districts do, even though the strongest party in Buchberg-Rüdlingen maintains its traditional pole position.

14.3 Discordant Seat Assignments

The sub-apportionment is a table reconciling rows (districts) and columns (parties) to attain the prespecified marginal values (district magnitudes and overall party- seats). The interactions of rows and columns are complex, and occasionally entail discordant seat assignments. Any instance of discordance is irritating of course. The irritation often dissipates when also looking at the other dimension. For instance in Table 14.3, three lists in the Klettgau district are awarded a seat each with less than 3000 votes (AL: 2976; SVP/Agro: 2465; JSVP: 2687). But EVP with more votes (3144) is assigned no seat. At first glance the result is irritating. But the irritation is tempered when the within-district view is supplemented with a within-party view. An examination of the EVP performance with regard to the other districts explains what is happening. The EVP has interim quotient 6133=10;100 D 0:61 in the Schaffhausen district, but just 3144=5400 D 0:58 in the Klettgau district. Hence the performance in Schaffhausen is marginally superior. This is why the sub- apportionment assigns the one and only EVP party seat to Schaffhausen, not to Klettgau. Fortunately discordant seat assignments are rare due to the excellent votes-seats image that is accomplished by double proportionality. The few discordant seat assignments that may emerge in a practical instance, if at all, are unstructured and unpredictable. That they arise is due to the prespecified marginals: district magnitudes and overall party-seats. The marginals restrict the structure of double- proportional tables in ways that render discordances unavoidable in general. 14.5 Double Proportionality in Swiss Cantons: The Reality 265

14.4 Winner-Take-One Modification

In a single-seat district discordant seat numbers are particularly perplexing. It deprives the strongest party of the sole seat available. In these instances concor- dance can be restored, however, and we recommend doing so. Concordance may be enforced by the “winner-take-one” (WTO) imperative: “In every district the strongest party is assigned at least one seat.” The wording addresses every district, whether small or large, in order to observe the principle of electoral equality. But the relevance of the WTO modification is essentially limited to single-seat districts. In single-seat districts it preserves the merits of the plurality vote and overcomes the demerits of wasting any minority votes. A discordance accident in a single-seat district is easily manufactured. In Table 14.3 we hypothetically raise the SP votes in Buchberg-Rüdlingen from 98 to 448 votes, while keeping all other votes as they were. Now the sub-apportionment assigns the sole Buchberg-Rüdlingen seat to the minority of 448 SP voters, not to the majority of 456 SVP voters. The assignment would contradict the former plurality vote and annoy the press, the parties, and everyone’s common sense. In the Schaffhausen elections, the Buchberg-Rüdlingen district was never threat- ened by a discordant seat assignment, neither in 2008 nor in 2012 nor in 2016. Yet the WTO modification was added to the Schaffhausen electoral law in time for the 2016 election. The law includes the proviso that the WTO modification is enacted only if the party that finishes strongest in a district is apportioned at least one seat in the super-apportionment. In practice the proviso is always satisfied. The Schaffhausen proviso may be replaced by another proviso. The alternative proviso induces restrictions on the super-apportionment: Any party that is strongest in n districts must be apportioned at least n overall party-seats. The restrictions ensure that the super-apportionment and the WTO-modified sub-apportionment are compatible. In the 2016 Schaffhausen election, SVP is strongest in five districts and SP in one. Hence the super-apportionment is restricted to allocating at least five seats to SVP, and at least one seat to SP. These restrictions are visibly fulfilled: SVP is apportioned 17 seats, and SP 13 (Table 14.2). Evidently the induced minimum requirements are nothing to worry about. Practically they are always satisfied. In a single-seat district such as Buchberg-Rüdlingen the WTO modification may make a difference. For instance in the discordance accident manufactured above, the WTO modification would ascertain that the one and only seat is assigned to SVP with 456 voters, not to SP with 448 voters.

14.5 Double Proportionality in Swiss Cantons: The Reality

The very first Swiss canton that introduced a double-proportional electoral system was the Canton of Zurich. The world premiere of double proportionality took place with the election of the council of the City of Zurich on 12 February 2006. Since 266 14 Representing Districts and Parties: Double Proportionality then double proportionality has spread throughout Switzerland. Meanwhile another six cantons have followed suit: Schaffhausen in 2008, Aargau in 2009, Nidwalden in 2014, Zug in 2014, Schwyz in 2016 and, as of late, Valais in 2017 (with six super-apportionments, each comprising one up to three districts). What is so special about Switzerland that electoral reform progresses so quickly? Swiss people stand up for their political rights. National broadsheet newspapers report on the topic, as do provincial tabloids. Electoral reform is not a prerogative of the political elite. Incumbent parliaments and stronger parties occasionally despise a reform by defending the status quo as their vested right, as is familiar from everywhere else in the world. But the call of the Swiss people for electoral equality is forceful. If need be it can be brought to bear by way of a referendum.

14.6 Double Proportionality in the European Union: A Vision

The European Union functions differently. Ever since the inception of the EP and its precursors, electoral reform has been a prominent item on the agenda. The task of designing an appropriate electoral system for the EP is enormous. The system is expected to play the servant of two masters. There are two dimensions of representativity: representation of the Union citizens by Member State provenance (Sects. 12.7–12.9), and representation of the Union citizens by political affiliation (Chap. 1). Of course the five principles that constitute Europe’s electoral heritage need to be remembered, too: A democratic election must be direct and universal as well as free, equal and secret (Sect. 2.5). The two dimensions of representation are inherently distinct. It must be carefully examined whether and how the electoral principles apply. Representation by Member State provenance is neither free nor equal nor secret. The author of this book is German; he is not free to be Estonian or Greek. He is not represented equally, nor are his fellow Germans; they are underrepresented in order to allow all other Union citizens to be represented better. Nor is it a secret that the author is German; there is no need for this type of secrecy. The principles of a free, equal and secret election pertain to the representation by political affiliation, only. However, the two dimensions are also mutually interacting. Does the unequal representation of Member State provenance impair the level of equality that is achieved when representing political affiliation? The answer is a reassuring No for the prospective double-proportional system which we envision in Sect. 14.8.The answer is a disconcerting Yes for the electoral system currently practiced. The current electoral system evaluates the votes that are cast for political parties within each Member State separately. Chapter 1 undergoes the chore of analyzing the details for the 2014 EP elections. The unionwide aggregation reveals a violation of the equality principle. The strongest political group is S&D, with 40,189,841 votes. The second-strongest political group is EPP, with only 40,170,109 votes. Yet the second-strongest group dwarfs the strongest group by a sensational margin of 14.7 Degressive Compositions and Separate District Evaluations 267

30 seats; see Table 1.33. With fewer votes being awarded more seats the motto “one person, one vote” cannot possibly hold true. The votes cast for political parties do not have the same equal success value for all Union citizens. The discordant aggregate result of the 2014 elections is the consequence of a superposition of many factors, such as different turnouts, different thresholds, and different apportionment methods in the Member States. Another contributing factor is the degressive composition of the EP (Sect. 12.7). We find it instructive to elucidate the discordance potential of degressivity in a contrived toy union.

14.7 Degressive Compositions and Separate District Evaluations

Let us consider a union with three member states MS1, MS2, and MS3 where two parties A and B compete for the 15 seats of the union’s parliament. The political division for the parties AWB is 770W130 in MS1, 20W380 in MS2, and 10W190 in MS3. We adopt an automatic system wherein every 100 voters justify roughly one seat. The resulting seat apportionment is 8W1 in MS1, 0W4 in MS2, and 0W2 in MS3. We make the utopian assumption that the ensembles of citizens and voters coincide; there are no individuals who do not vote. Then the fits of district mag- nitudes and overall party-seats are perfect. The 900 MS1 citizens are represented by 9 deputies, the 400 MS2 citizens by 4 deputies, and the 200 MS3 citizens by 2 deputies. Furthermore the 800 A voters are represented by 8 deputies and the 700 B voters by 7 deputies. The union finds itself in a peaceful state of equilibrium. Yet some citizens take offense at the numbers and worry. MS1 with 900 inhabitants is one and a half times as populous as MS2 and MS3 together, which account for only 600 people. Is MS1 going to be the dominant power in the union? The worried citizens persuade the others that, bowing to the principle of solidarity and degressivity, MS1 should give up two seats in favor of MS2 and MS3. And so it is agreed. The new district magnitudes are 7 seats for MS1, 5 for MS2, and 3 for MS3. It goes without saying that the seats are apportioned separately in each member state. With just two parties present, the natural two-party apportionment method is used (Sect. 9.10). The resulting apportionment of the 15 seats thus is:

Party A Party B Vot es Quotient Seats Vot es Quotient Seats MS1(7seats) 770 5:99 6 130 1:01 1 MS2(5seats) 20 0:25 0 380 4:75 5 MS3(3seats) 10 0:15 0 190 2:85 3 Sum 800 – 6 700 – 9 268 14 Representing Districts and Parties: Double Proportionality

The new seat apportionment turns the political world in the union upside down. The popular majority of 800 A votes is worth only 6 seats, while the 700 B votes are conceded a parliamentary majority of 9 seats. How come? Partisan strongholds are highly correlated with the union’s subdivision by member state. At this juncture some A voters take offense at the numbers and complain. They propose a system that is immune against the correlation between geographical provenance and political affiliation. In the first place the parties ought to be apportioned their due number of seats: 8 seats for A and7forB. Thereafter the seats should be assigned by state and party using a double-proportional method. Now the result is this (in terms of “votes-quotient-seats”):

Party A Party B District 8 7 divisor MS1 7 770-7.0-7 130-0.45-0 220 MS2 5 20-0.6-1 380-4.1-4 72 MS3 3 10-0.4-0 190-2.9-3 50 Party divisor 0.5 1.3

Success value equality of the party votes is restored: 800 A votes lead to 8 seats and 700 B votes yield 7 seats. One person, one vote. The agreed district magnitudes are met too: 7 seats for MS1, 5 for MS2 and 3 for MS3. As far as the EP seat allocation between the Member States is concerned the label “double proportionality” sounds misleading. It is true that overall party- seats could be apportioned proportionally to unionwide vote counts. However, the current allocation of seats between the Member States is not proportional to population figures. They are simply a political fix. In order to indicate the shift of views we replace the label “double proportionality” by the term “compositional proportionality”. The envisioned application to the EP election falls into this category.

14.8 Compositional Proportionality for Unionwide Lists

In this section we illustrate by example how compositional proportionality may be put to good use at EP elections. The example proposes a way to vote for unionwide party lists while maintaining the Member States’ seat contingents. However, unionwide candidate lists did not exist at the 2014 elections, and in most Member States the political parties at European level were not visible on the ballot sheets. Instead we take recourse to the Political Groups that formed when the 2014 EP first convened. We include the non-attached members as a pseudo group NI in order not to lose their votes. We pretend that these eight surrogate “parties” submitted unionwide lists of candidates. Moreover we assume that in a given Member State the number of votes cast for a list is equal to the sum of the votes cast for those 14.8 Compositional Proportionality for Unionwide Lists 269 domestic parties that later joined the political group in question. The unionwide results are displayed in Table 14.5; the “votes” by Member State and “party” are included in Table 14.5. We continue to enclose the terms “parties” and “votes” in quotation marks in order to indicate their hypothetical and speculative complexion. The seats for the unionwide lists are garnered from the Member States seat contingents. Double proportionality eventually returns exactly as many seats to each Member State as it contributed in the beginning. Thus the procedure maintains the preordained composition of the EP. This is why we refer to it as “compositional proportionality”. Our illustrative apportionment has to deal with 36 subjects: twenty-eight Member States plus eight “parties”. For a proportional representation system to perform reasonably well we need at least about twice as many seats as subjects; see the house size recommendation in Sect. 7.6. We generate 72 seats in a fashion reminiscent of degressivity: More populous states contribute at least as many seats as less populous states. Member States with a population above sixty million contribute eight seats each (DE, FR, UK, IT); those with a population between sixty and fifteen million four seats (ES, PL, RO, NL); those with a population between fifteen and ten million two seats (BE, EL, CZ, PT); and the sixteen remaining states one seat each. Any other recipe for garnering seats would work just as well. Super-Apportionment of Seats Among Unionwide Lists The super-apportion- ment allots the 72 available seats among the “parties” proportionally to their union- wide “votes”. The divisor method with standard rounding is used. Every 2,080,000 “votes” justify roughly one seat. For example the GREENS/EFA “votes”, when divided by 2,080,000, have interim quotient 5.55 which is rounded upwards to 6 seats. The EFDD “votes” yield quotient 5.2 which is rounded downwards to 5 seats. The super-apportionment anticipates the invocation of the winner-take-one modification in the sub-apportionment. Since S&D is strongest in four Member States it needs to be apportioned a minimum of four seats. EPP is strongest in 14

Table 14.4 Super-apportionment of 72 seats among unionwide “parties” EP2014Super-app. “Votes” Min. Quotient DivStd S&D 40;189;841 4 19.3 19 EPP 40;170;109 14 19.3 19 ALDE 13;169;384 5 6.3 6 GUE/NGL 11;985;225 2 5.8 6 ECR 11;933;741 1 5.7 6 GREENS/EFA 11;534;372 0 5.55 6 EFDD 10;822;471 1 5.2 5 NI 10;224;602 1 4.9 5 Sum (Divisor) 150;029;745 28 (2,080,000) 72 The Political Groups in the EP substitute for the inactive European “parties”. The cumulative “votes” are copied from Table 1.33. The 72 seats for the “parties” are hypothetically diverted from the Member States 270 14 Representing Districts and Parties: Double Proportionality

Member States whence its needs at least 14 seats. Table 14.4 shows that all minimum requirements remain dormant; they are automatically satisfied. Sub-apportionment by State and List In the sub-apportionment the overall “party”-seats are assigned to the Member States in such a way that each Member State is returned exactly as many seats as it contributed in the beginning. The Member States’ seat contributions are recalled to the left of the inner box in Table 14.5. The sub-apportionment is conducted using the double-proportional divisor method with standard rounding. The key quantities are the state divisors and the “party” divisors. They are exhibited in the rightmost column and in the last row of Table 14.5. Given the electoral keys it is straightforward to verify the seat numbers. For instance the German S&D votes (8,003,628) are divided by the divisors for Germany (3,600,000) and for S&D (0.94). The interim quotient 2.4 justifies two seats. Input “votes” and output seats are linked in the table by a hyphen: 8,003,628-2. There are sixteen Member States that contribute just a single seat. Therefore double proportionality is modified by the winner-take-one rule. The rule makes sure that in every Member State the strongest list is assigned at least one seat. That is, the strongest list in a Member State is assigned a seat even when its interim quotient is less than one half. In Table 14.5 the winner-take-one rule applies to Austria. The strongest Austrian list is EPP, with interim quotient 761;896=.2;000;000 1:5/ D 0:2. Standard rounding would round 0.2 to zero. Due to the winner-take-one modification the Austrian EPP quotient is rounded upwards to one: 761,896-1. Assignment of Seats to Candidates The final step is to decide which candidates are going to be assigned a seat. The details will depend on the list type and vote pattern chosen, once the European parties start to operate properly. There could be a single unionwide list for each party. Or a party may prefer to propose twenty-eight lists, one for each Member State. In Germany the two list types are implemented side by side: CDU submits fifteen state-lists and CSU just one, while all other parties present their candidates on a single federal list (Sect. 1.3). Moreover lists can be closed, flexible, or open. The type of the lists will interact with the voting pattern offered to the electorate (Sect. 1.5). During the 2014 EP elections, European parties started contributing to the campaign by nominating unionwide lead candidates (Spitzenkandidaten). Assuming that every party presents a single unionwide list, the Spitzenkandidat would be the natural top nominee. If voters were allowed to mark two or more preferences, they could express their support for the unionwide Spitzenkandidat along with their support of further—possibly domestic—candidates. Thus the task of amending the EP electoral system to incorporate unionwide lists while safeguarding parliament’s composition may be solved by a method whose lengthy label reflects the complexity of the issue: “winner-take-one modification of the double-proportional variant of the divisor method with standard rounding”. Whether this or any other method is ever adopted remains to be seen once the political parties at European level start to shape EP elections. 14.8 Compositional Proportionality for Unionwide Lists 271 900;000 300;000 400;000 430;000 600;000 800;000 (continued) divisor State 1;100;000 1;000;000 1;300;000 1;300;000 2;000;000 1;776;000 3;000;000 3;600;000 2;000;000 1;000;000 2;400;000 1;880;000 1;000;000 3;600;000 2;000;000 5 3,402,87-0 505,586-0 556,835-0 NI 886,255-0 1,688,197-1 284,856-0 4,712,461-3 633,114-1 485,848-0 131,163-0 5 5,807,362-2 EFDD 79,540-0 359,248-0 4,376,635-3 6 284,980-0 GREENS 410,089-0 572,591-0 86,806-0 1,696,442-1 1,756,940-1 1,194,889-1 739,016-0 161,263-0 81,458-0 331,594-1 249,305-0 3,749,562-2 6 2,246,870-2 ECR 80,145-0 238,629-0 63,641-0 3,875,987-2 1,123,355-0 197,837-0 364,843-0 116,389-0 605,889-1 222,457-0 369,545-0 2,272,817-1 6 GUE/NGL 1,518,376-1 159,813-0 1,108,457-0 166,478-0 183,724-0 161,074-0 477,316-1 1,252,730-1 2,893,058-1 658,333-1 566,689-0 234,272-0 2,535,053-1 6 ALDE 1,087,632-0 1,307,001-1 244,501-0 229,781-0 386,725-0 528,789-0 456,642-1 37,376-0 68,986-0 137,952-0 1,415,641-0 1,884,565-1 2,087,359-0 379,582-0 1,520,431-1 234,788-0 609,615-0 EPP 19 10,380,101-2 3,943,819-1 4,382,329-1 2,752,061-1 721,766-0 1,129,739-0 1,298,948-0 392,539-1 910,647-1 728,062-0 1,193,991-1 761,896-1 825,370-1 208,262-0 390,376-0 186,912-1 369,120-0 318,203-1 6,171,129-1 2,213,046-2 S&D 19 8,003,628-2 2,650,357-1 4,102,240-2 10,986,853-4 3,614,232-1 667,319-1 2,093,234-2 446,763-0 1,269,993-1 836,136-1 214,800-1 1,034,249-1 1,103,079-1 478,837-0 680,180-0 424,037-0 435,245-0 212,781-0 135,089-0 1,241,68-0 137,952-0 Hypothetical sub-apportionment for unionwide “parties” AT 1 DEFR 8 UK 8 IT 8 ES 8 PL 4 RO 4 NL 4 BE 4 EL 2 CZ 2 PT 2 SE 2 HU 1 1 BGDK 1 FI 1 SK 1 IE 1 HR 1 1 Table 14.5 272 14 Representing Districts and Parties: Double Proportionality , y 200;000 130;000 200;000 100;000 200;000 200;000 500;000 divisor State 5 1 NI 5 0.8 EFDD 36,637-0 163,049-0 6 0.65 GREENS 41,525-0 28,303-0 43,369-0 30,597-0 75,643-0 6 63,229-0 1 ECR 92,108-0 6 GUE/NGL 69,852-0 1.1 6 ALDE 1.2 32,662-0 153,268-1 30,108-0 335,980-1 sulting quotient is 2.4 (not shown) and justifies two seats EPP 19 166,403-1 199,393-0 20,4979-1 45,765-0 97,732-1 76,736-1 100,785-0 1.5 visor (0.94). The re S&D 19 197,477-0 32,484-0 57,863-0 44,550-0 47,938-0 23,895-0 134,462-1 0.94 (continued) LTSI 1 LV 1 EE 1 CY 1 LU 1 MT 1 “Party” div. 1 Table 14.5 The double-proportional variantand of it the guarantees divisor each method(3,600,000), “party” and with its by standard the unionwide S&D rounding party-seats. di is Sample used. calculation: It The returns German S&D to votes each (8,003,628) Member are State divided the by number the of divisor seats for German contributed 14.8 Compositional Proportionality for Unionwide Lists 273

The next chapter establishes the properties of double-proportional divisor meth- ods which so far we tacitly took for granted. There are two questions that are of paramount interest for practical applications. The first is uniqueness: Is the array of seat numbers by district and party, such as in Tables 14.3 and 14.5, unique? The answer is: Yes, for all practical purposes (Sect. 15.4). There does not exist a second, distinct array whose rows sum to the same district magnitudes and whose columns sum to the same overall party-seats. The second question concerns the district divisors and party divisors: How to capture them? The answer is demanding and laborious. The divisors are determined by means of algorithms that are more involved than the jump-and-step procedure of simple-proportionality (Sects. 15.8 and 15.9). Chapter 15 Double-Proportional Divisor Methods: Technicalities

Abstract Every divisor method induces a double-proportionalvariant for situations when the electoral area is subdivided into two or more districts with prespecified district magnitudes. The resulting seat matrices are shown to be unique up to ties. They are characterized by an optimization property, and by a system of inequalities. The key quantities in determining a seat matrix are district divisors and party divisors. Two algorithms are presented to determine the divisors: the alternating scaling algorithm, and the tie-and-transfer algorithm.

15.1 Row and Column Marginals, Weight and Seat Matrices

Double-proportional apportionment methods are of interest in situations when the electoral area is subdivided into k  2 districts i D 1;:::;k with preordained district magnitudes ri. Furthermore it is assumed that there are `  2 parties j D 1;:::;` for which the overall party-seats sj have already been determined. The present chapter is devoted to the analysis of seat matrices x. A “seat matrix” x is a k ` matrix with integer entries that rowwise sum to the district magnitudes and that columnwise sum to the overall party-seats. It is conducive to adapt the language appropriately. The district magnitudes are referred to as “row marginals”: r WD k .r1;:::;rk/ 2 N .h/. The overall party-seats constitute the “column marginals”: ` s WD .s1;:::;s`/ 2 N .h/. The set of all seat matrices with given marginals r and s is denoted by ˚ ˇ « k` k` ˇ N .r; s/ WD x D ..xij// 2 N xiC D ri; i Ä k; and xCj D sj; j Ä ` :

The entry xij signifies the number of seats which in district i are assigned to party j. As usual in this book, a subscript “C” indicates summation: xiC D xi1CCxi`,etc. Hence xiC are the seats allocated to district i; they must meet the preordained district magnitude ri. Likewise xCj are the seats apportioned to party j; they must exhaust the overall party-seats sj. It follows that row marginals and column marginals share thesametotal:xCC D rC D sC D h. The current chapter assumes that the first part of the apportionment task has been completed, by allocating the seats between districts to obtain the district magnitudes ri and by apportioning the seats among parties to obtain the overall party-seats sj. It does not matter whether this has been accomplished by using a

© Springer International Publishing AG 2017 275 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_15 276 15 Double-Proportional Divisor Methods: Technicalities formal apportionment procedure, such as a divisor method or a quota method, or simply by relying on negotiation and agreement. Districts i with ri D 0 and parties j with sj D 0 are allocated no seats and do not need to be considered any further. Without loss of generality all marginals are assumed to be positive: ri  1 and sj  1. The remaining apportionment task is to assign the seats by district and party. The seat matrices of interest are those that faithfully reflect the proportions of the vote weights vij emerging from the presumed election. The ensemble of all weights is arranged into a matrix. A “weight matrix” v is taken to be a k ` matrix with nonnegative entries and with no row nor column vanishing:

k` v D ..vij// 2 Œ0I1/ ;viC >0for all i Ä k;vCj >0for all j Ä `:

An individual weight vij documents the vote count (or voter number, or any other reasonable weight) that in district i is recorded for party j. Not all parties stand in all districts, hence some weights may be zero. On the other hand every district i features at least one party that is met with support: viC >0. Likewise every party j attracts votes somewhere: vCj >0. Hence no row nor column vanishes. With these prerequisites the double-proportional apportionment problem may be specified as follows: Assuming that row marginals r, column marginals s and a weight matrix v are given, which seat matrices x 2 Nk`.r; s/ conform to the proportions in the weight matrix v in a manner that is deemed adequate? The solutions to this problem must meet some plain properties similar to those for simple-proportional apportionment rules (Sect. 4.2). Every method A claiming to solve the problem must produce solution sets consisting of integer matrices, that is A.r; sI v/ Â Nk`, fulfilling three elementary properties: (a) (Non-emptiness) The solution set A.r; sI v/ is non-empty. (b) (Inheritance of zeros) Every matrix x 2 A.hI v/ inherits all zeros of v:

vij D 0 H) xij D 0 for all i Ä k and j Ä `:

(c) (Fitted marginals) Every matrix in A.r; sI v/ has row sums that are equal to r, and column sums that are equal to s: A.r; sI v/ Â Nk`.r; s/. The next section introduces double-proportional divisor methods. Property c is part of their definition. Property b is a consequence of the no input–no output law of rounding rules (Sect. 3.9). Property a is more challenging; it is dealt with in Sect. 15.6.

15.2 Double-Proportional Divisor Methods

The double-proportional apportionment methods that are to be considered in the sequel are variants of divisor methods. As in simple-proportional settings the underlying rounding rule ŒŒ  is assumed to rely on a signpost sequence s.0/, s.1/, 15.2 Double-Proportional Divisor Methods 277 s.2/,etc.(Sect.4.5). The double use of the letter s, for signposts s.n/ and for column marginals sj, is unfortunate. We trust that the particular contexts save the readers from confusion. Definition The “double-proportional divisor method with rounding rule ŒŒ  ” A is defined to have solution sets  ˇ ÄÄ  ˇ v k` ˇ ij A.r; sI v/ WD x 2 N .r; s/ ˇ xij 2 ; i Ä k; j Ä `; CiDj 

for some C1;:::;Ck; D1;:::;D` >0 :

Thus a solution set A.r; sI v/ includes the seat matrices x whose entry xij results from scaling the weight vij by means of a “district divisor” Ci and a “party divisor” Dj, and then rounding the interim quotient vij=.CiDj/ to the seat number xij. We refer to the matrices x in A.r; sI v/ as “double-proportional seat matrices” or, for short, as “solution matrices”. ŒŒ  bij The definition also tolerates a truncated rounding rule  aij with minimum requirements aij and maximum cappings bij (Sect. 12.3). For example it is straight- forward to accommodate the winner-take-one modification (Sect. 14.4).Allthatis needed is to set the minimum requirements in district i to be aij D 1 when party j is strongest and aik D 0 when k ¤ j. The seat matrix x remains the same whether the original weights vij are used, or the row-normalized weights vij=viC, or the column-normalized weights vij=vCj, or the overall-normalized weights vij=vCC, or the unrounded voter numbers vij=ri. More generally, a rowwise rescaling of the weights according to wij WD vij=ci with ci >0, or a columnwise rescaling according to uij WD vij=dj with dj >0,giverise to weight matrices w and u with the same solution set: A.r; sI v/ D A.r; sI w/ D A.r; sI u/. This is easily verified by an appropriate adjustment of row divisors or column divisors. The fact that the solution set is not affected by a rescaling of the weight matrix v, rowwise or columnwise, opens the possibility of passing from any initially obtained divisors Ci and Dj to “select divisors” Ei and Fj that appear to be more user-friendly. The passage involves two steps. First, all columns are rescaled by the median party divisor m WD med.D1;:::;D`/. If there are several median values then m is chosen to be the lowest median. With weights wij WD vij=.Cim/,thesj overall seats of party j are allocated between districts i D 1;:::;k proportionally to w1j;:::;wkj. We choose Fj to be the select divisor for party j (Sect. 4.7). Second, with weights uij WD vij=Fj the ri seats in district i are apportioned among parties j D 1;:::;` proportionally to ui1;:::;ui`. We choose Ei to be the select divisor for district i (Sect. 4.7). By construction the select party divisors are such that some are equal to unity, while the others are greater or smaller. In Table 14.3 four party divisors are unity, four are greater, and seven are smaller. In Table 14.5 the tally is two divisors equal to unity, three greater, and three smaller. The select district divisors are of a size close to what they would be if every district were apportioned separately on its own. 278 15 Double-Proportional Divisor Methods: Technicalities

15.3 Cycles of Seat Transfers

The analysis of a given seat apportionment starts out from a plain question: What happens when transferring a single seat? The plainness of the question is contingent on the setting though. In simple-proportional settings no more than two actors are involved: a donor and a receiver. In contrast a double-proportional setting is more demanding. For instance, if party j1 transfers a seat from district i1 to district i2 then, while the overall seats of party j1 remain what they were, the magnitude of district i1 falls short by one seat and the magnitude of district i2 is one seat in excess. The excess can be balanced only by further seat transfers. These continue until the initial deficit in the cell .i1; j1/ is evened out. Altogether the succession of seat transfers forms a “cycle via rows i1;:::;iq and columns j1;:::;jq”, that is, a sequence of cells of the form

.i1; j1/; .i2; j1/; .i2; j2/; .i3; j2/; :::; .iq1; jq1/; .iq; jq1/; .iq; jq/; .i1; jq/:

The rows are taken to be distinct, and so are the columns. If q D 2 then four cells are visited, if q D 3 then six cells are visited, etc. However, cells .i; j/ with a vanishing weight vij D 0 must be excluded. Neither can they donate a seat because there is none. Nor can they receive a seat because of the no input–no output law. The cycles that visit only cells where weights are positive deserve a distinctive name. Definition Given a weight matrix v 2 Œ0I1/k`,a“v-cycle” is defined to be a ;:::; ;:::; v v cycle via rows i1 iq and columns j1 jq such that the weights ipjp and ipC1jp are positive for all p Ä q,whereiqC1 WD i1. As an illustration we consider a 3 3 weight matrix in which the diagonal weights are zero and the off-diagonal weights are positive: 0 1 0v12 v13 @ A v D v21 0v23 : v31 v32 0

There is no v-cycle via just two rows and columns since it would have to visit a zero weight on the diagonal. Yet the cycle via rows 2;3;1 and columns 1; 2; 3 is a v-cycle since it calls on the weights v21;v31;v32;v12;v13;v23 all of which are positive. The matrix z that captures the induced seat transfers is 0 1 011 z D @ 101A : 1 10 15.4 Uniqueness of Double-Proportional Seat Matrices 279

Cells with entry 1 must give up a seat, while those with entry 1 receive a seat. Cells with entry 0 remain unvisited. Since row sums and column sums of z vanish k`. ; / k`. ; / 1 we have xCz 2 N r s for all x 2 N r s which satisfy xipjp  for all p Ä q.

15.4 Uniqueness of Double-Proportional Seat Matrices

From a practical viewpoint the most important theoretical question is whether a double-proportional divisor method yields seat matrices that are unique. The question suggests itself because we know that the divisors are not unique. Maybe the seat matrices are not unique either? Do we need to worry that someone might come along with other divisors leading to a second seat matrix distinct from ours? The answer is reassuring. There is just one seat matrix that satisfies the definition in Sect. 15.2, independently of the divisors employed, for all practical purposes. The caveat “for all practical purposes” is needed because the scrupulous answer of the Uniqueness Theorem is “Yes, the seat matrix is unique up to ties”. The exception of ties is innocuous. In double-proportionalproblems ties are theoretically intriguing but practically pointless. The Schaffhausen election is free of ties (Table 14.3) as is the hypothetical application to the EP (Table 14.5). There does not exist a single empirical data set we know of that would feature ties. Hence the seat matrices that are obtained from double-proportional divisor methods are unique and final, and not up for discussion. All that is needed is that they verify the definition in Sect. 15.2. Uniqueness Theorem Consider a seat matrix x 2 A.r; sI v/ of a double- proportional divisor method A. If no more than three seat numbers xij have interim quotients vij=.CiDj/ that are equal to positive signposts then the seat matrix is unique:

fxgDA.r; sI v/:

Proof The proof is indirect. Let us assume that the set A.r; sI v/ contains two distinct matrices x ¤ y. The difference z WD y  x is non-vanishing, but has row sums equal to zero and column sums equal to zero. We select q  2 distinct rows i1;:::;iq and columns j1;:::;jq so that along the cycle

.i1; j1/; .i2; j1/; .i2; j2/; .i3; j2/; :::; .iq1; jq1/; .iq; jq1/; .iq; jq/; .i1; jq/ the entries of z alternate in sign, ; C;:::, as follows. We start with a cell .i1; j1/ where zi1j1 <0. In column j1 we pick a cell .i2; j1/ with zi2j1 >0. Next we search in row i2 a cell .i2; j2/ where zi2j2 <0. Then we look for a row i3 such that zi3j2 >0,etc. We finish when meeting a row or column already visited. The succession of cells then consists of a cycle that is possibly preceded by an initial section. Discarding the initial section the remaining cycle is preserved and relabeled. We select this cycle 280 15 Double-Proportional Divisor Methods: Technicalities

<0 <0 provided its first cell has zi1j1 . Otherwise its last cell satisfies zi1jq ;thenwe choose the cycle obtained from reversing the sequences of rows and columns. Along the cycle selected, z has sign pattern ; C;:::as desired. Moreover all weights along the cycle are positive since otherwise vij D 0 implies xij D yij D 0 and zij D 0. Hence the cycle constructed is a v-cycle. We denote the divisors of x by Ci and Dj, and those of y by Ei and Fj.The fundamental relation (Sect. 3.9) translates the relations xij 2 ŒŒ v ij=.CiDj/ and yij 2 ŒŒ v ij=.EiFj/ into

s.x / 1 s.x C 1/ s.y / 1 s.y C 1/ ij Ä Ä ij and ij Ä Ä ij : (15.1) vij CiDj vij vij EiFj vij

The above cycle entails two strings of inequalities, upon setting iqC1 WD i1:

Y Y Y Y . 1/ s.xi j / 1 1 s xi C1j C p p Ä D Ä p p ; (15.2) v C D C D v pÄq ipjp pÄq ip jp pÄq ipC1 jp pÄq ipC1jp Y . / Y Y Y s yi C1j 1 1 s.yi j C 1/ p p Ä D Ä p p : (15.3) v E F E F v pÄq ipC1jp pÄq ipC1 jp pÄq ip jp pÄq ipjp

< > 1 The sign pattern of z yields yipjp xipjp and yipC1jp xipC1jp ,thatis,yipjp C Ä xipjp 1 and xipC1jp C Ä yipC1jp . Due to signpost monotonicity (15.2)and(15.3)maybe concatenated:

Y s.x / Y s.x C 1/ Y s.y / Y s.y C 1/ Y s.x / ipjp ipC1jp ipC1jp ipjp ipjp : v Ä v Ä v Ä v Ä v pÄq ipjp pÄq ipC1jp pÄq ipC1jp pÄq ipjp pÄq ipjp Q . 1/=v >0 Hence equality holds throughout. Since pÄq s xipC1jp C ipC1jp , the common value must be positive. This entails equality in the inequalities in (15.1), giving v v . / ipjp >0 . 1/ ipC1jp >0: s xipjp D and s xipC1jp C D Cip Djp CipC1 Djp

Every cycle visits at least two rows and two columns: q  2. Therefore four or more interim quotients that are associated with the seat matrix x are tied to positive signposts. ut An equivalent formulation of the theorem states that non-uniqueness forces at least four interim quotients to be tied to positive signposts. A mere abundance of ties is not enough though. The ties must form a pattern that includes a cycle alternating between decrement options and increment options:

ŒŒ . / 1; ; s xipjp Dfxipjp  xipjp gDxipjp  ŒŒ . 1/ ; 1 : s xipC1jp C DfxipC1jp xipC1jp C gDxipC1jp C 15.5 Characterization of Double-Proportional Seat Matrices 281

The trailing plus and minus signs indicate the rounding options (Sect. 4.8). They are congruent with the entries of the matrix z in the proof. It is extremely unlikely that many ties arise simultaneously and, in addition, exhibit a pattern that allows several distinct seat matrices. This confirms that, for all practical purposes, double- proportional divisor methods produce seat matrices which are unique. The proof of existence is deferred to Sect. 15.6.

15.5 Characterization of Double-Proportional Seat Matrices

The following theorem presents useful necessary and sufficient conditions for a seat matrix x to belong to the solution set A.r; sI v/. It is the double-proportional analogue of the simple-proportional max-min inequality. The seat matrices x under consideration are assumed to “preserve the zeros of v”. This is an umbrella phrase aimed at encompassing pervious and impervious methods. For pervious methods the phrase means that x inherits all zeros of v: vij D 0 ) xij D 0. For impervious methods the meaning is that x and v have the same zeros: vij D 0 ” xij D 0. In particular property b in Sect. 15.1 is satisfied. Part b of the following theorem introduces a minimization problem. Given the weight matrix v, the objective function is defined for matrices y 2 Nk` by 0 1 Y Y s.n/ f .y/ WD @ A vij iÄk;jÄ` Wvij>0 nÄyij W s.n/>0 for the case when y preserves the zeros of v. The definition is f .y/ WD 1 when y fails to preserve the zeros of v. The parenthesized product has range n D 1;:::;yij in the case of perviousness, and n D 2;:::;yij in the case of imperviousness. Empty products equal unity. Part c runs x through a system of inequalities. Simple-proportionality needs just one inequality: the max-min inequality in Sect. 4.6. Double-proportionality requires satisfaction of the “system of v-cycle inequalities” as given in part c. Characterization Theorem Let A be a double-proportional divisor method with underlying signpost sequence s.n/,n2 N, and let x 2 Nk`.r; s/ be a seat matrix that preserves the zeros of v. Then the three statements a, b, and c are equivalent: a. x belongs to the solution set A.r; sI v/:

x 2 A.r; sI v/: b. x minimizes the above function f among all seat matrices y 2 Nk`.r; s/:

f .x/ Ä f .y/ for every y 2 Nk`.r; s/: 282 15 Double-Proportional Divisor Methods: Technicalities c. x satisfies the system of v-cycle inequalities:

Y . / Y . 1/ s xipjp s xipC1jp C for every v-cycle via i1;:::;i and j1;:::;j : v Ä v q q pÄq ipjp pÄq ipC1jp

Proof Parts I–III show that a implies b (I), b implies c (II), and c implies a (III). I. Let x be a solution matrix of the method A: x 2 A.r; sI v/ and let y be an integer matrix with fitted rows and columns: y 2 Nk`.r; s/. We claim that f .x/ Ä f .y/.Sincex preserves the zeros of v we have f .x/<1. Hence the inequality is immediate in case f .y/ D1. The case to be investigated assumes that y preserves the zeros of v too. Consider the cells .i; j/ where x and y differ: xij ¤ yij. Since either xij > yij or xij < yij, the weight must be positive: vij >0. With divisors Ci and Dj for x, . /=v 1 1 . 1/=v the fundamental relation yields s xij ij Ä Ci Dj  Ä s xij C ij.This and signpost monotonicity m Ä xij ) s.m/ Ä s.xij/ allow estimation of the parenthesized product in the definition of f (omitting the condition s.n/>0for ease of reading): 0 1 0 1 Y Y Yxij Y s.n/ s.n/ s.m/ s.n/ x y x > y ) D @ A Ä @ A C1D1 ij ij ; ij ij v v v v i j Ä ij Ä ij ij Ä ij n xij n yij mDyijC1 n yij 0 1 0 1 Y Y Yyij Y s.n/ s.n/ vij s.n/  x < y @ A @ A .C D /yij xij : ij ij ) v D v . / Ä v i j Ä ij Ä ij s m Ä ij n xij n yij mDxijC1 n yij

yijxij In cells .i; j/ where xij D yij or vij D 0 the power yij xij is zero and .CiDj/ D 1. Altogether f .x/ is less than or equal to f .y/ times ! 0 1 Y Y Y Y Y s s . /yijxij . /yijxij riri @ j j A 1: CiDj D CiDj D Ci Dj D iÄk;jÄ` Wvij>0 iÄk jÄ` iÄk jÄ`

This yields f .x/ Ä f .y/. The claim is proved. II. Let x 2 Nk`.r; s/ be a matrix preserving the zeros of v and minimizing the function f .Givensomev-cycle via i1;:::;iq and j1;:::;jq we claim that x satisfies the cycle inequality in part c of the theorem. The inequality is immediate if the left-hand side in c is zero. With a pervious

rounding rule this happens if some seat number xipjp is zero. With an impervious

rounding rule this occurs if one of the seat numbers xipjp equals unity. The case to be investigated assumes the left-hand side in c to be positive. Then all seat

numbers xipjp are large enough to allow a one-seat decrement. We introduce the seat matrix y WD x C z where the “cycle matrix” z has 1 1 0 entries zipjp WD  and zipC1jp WD for all p Ä q,andzij WD otherwise. 15.5 Characterization of Double-Proportional Seat Matrices 283

. / . / The parenthesized products in f y differ from those in f x because of yipjp D 1 1 xipjp  and yipC1jp D xipC1jp C . We obtain

Y Y . 1/ s.xi j / s xi C1j C f .y/ p p f .x/ p p : v D v pÄq ipjp pÄq ipC1jp

Minimality of x secures f .x/ Ä f .y/. Cancellation of f .y/>0establishes the desired cycle inequality. The claim is proved. III. Let x satisfy the system of cycle inequalities. We claim that there exist divisors Ci and Dj fulfilling s.xij/ Ä vij=.CiDj/ Ä s.xij C 1/. Vanishing weights are non- informative; they always fulfill s.0/ D 0 D vij Ä s.1/. For positive weights we divide by vij >0and take logarithms:

s.xij/ s.xij C 1/ log Ä ai C bj Ä log ; vij vij

where log.0=vij/ WD 1. The existence of ai WD  log Ci and bj WD  log Dj is at issue. For a concise problem formulation we pass to the linear space Rk` of real k ` matrices. Let the linear subspace L Rk` consist of the matrices y k` having entries yij D ai C bj for some ai; bj 2 R. Define the rectangle R R to be the Cartesian product of the intervals 8 Ä  ˆ s.x / s.x C 1/ ˆ log ij I log ij in case v >0and s.x />0; .1/ ˆ v v ij ij < ij ij# 1 Iij WD s xij C ˆ 1I log in case vij >0and s.xij/ D 0; .2/ ˆ v :ˆ ij .1I 1/ in case vij D 0: .3/

The intervals are non-empty. This is evident in (1) and (3). In (2) non-emptiness follows from s.xij C 1/=vij >0since x preserves the zeros of v. Hence the rectangle R is non-empty, too. There are two cases. In the first case the sets L and R intersect. Then there are ai bj numbers ai and bj fulfilling ai Cbj 2 Iij. The divisors Ci WD e and Dj WD e prove the claim. In the second case the sets L and R are disjoint. We show that the second case entails a contradiction. ? To see this, we investigate the orthogonal complementP L P. Orthogonality ; 0 refers to the Euclidean inner product hy ziDtracey z D iÄk jÄ` yijzij in the k` 0 0 space R , where the prime indicates transposition. Let 1` WD .1;:::;1/ 2 ` ` 0 0 R designate the “unity vector” in R . The original space is L Dfa1` C 1kb j a 2 Rk; b 2 R`g. The inner product with a matrix z 2 Rk` decomposes into k ` 0 0; 0 0 inner products in R and R : ha1` C 1kb ziDtrace1`a z C traceb1kz D 0 k ` ha; z1`iChz 1k; bi. The latter vanish for all a 2 R and b 2 R if and only if 284 15 Double-Proportional Divisor Methods: Technicalities

0 ? z1` D 0 and z 1k D 0. Thus the subspace L consists of the matrices z whose ? k` 0 rows and columns sum to zero: L Dfz 2 R j z1` D 0; z 1k D 0g.All cycle matrices lie in L?. Note that the negative of a cycle matrix is the cycle matrix of the “reverse cycle” via i1; iq; iq1;:::;i4; i3; i2 and jq;:::;j1. The “support” of a matrix is defined to consist of the cells with non-zero entries. An “elementary matrix” of a subspace K is a non-zero matrix in K whose support does not properly contain the support of any other non-zero matrix in K.Ifu; z 2 K are elementary with the same support then they are scalar multiples of each other. Indeed, choose a common support cell .i; j/ and set c WD uij=zij ¤ 0. Then the support of w WD u  cz 2 K is properly contained in the support of z. Hence w D 0,andu D cz. There are only finitely many distinct support sets and so there are only finitely many elementary matrices in K, up to scalar multiples. It is easily seen that the elementary matrices of the subspace L? are the cycle matrices, up to scalar multiples. Now we invoke Theorem 22.6 of Rockafellar (1970, p. 206). Its part (b), applying when the sets L and R are disjoint, states that there exists an elementary matrix z in L? such that hy; zi >0for all y 2 R.Thatis,thereisa cycle matrix z of a cycle via i1;:::;iq and j1;:::;jq such that X X > : yipjp yipC1jp for all y 2 R pÄq pÄq

The cycle is a v-cycle. Indeed, since the left-hand side is bounded from below

by the right-hand sum, the terms yipjp 2 Iipjp stay finite and the intervals Iipjp

stem from definition (1). The intervals IipC1jp , being bounded from above, may stem from (1) or (2). Since all values in (1) and (2) are positive, the current v cycle is a -cycle. Furthermore a specific vector y 2 R is given by yipjp WD . /=v . /=v log s xipjp ipjp and yipC1jp WD log s xipC1jp ipC1jp . Exponentiation yields

Y s.x / Y s.x C 1/ ipjp > ipC1jp : v v pÄq ipjp pÄq ipC1jp

This contradicts the premise that x satisfies every v-cycle inequality. Hence the sets L and R cannot be disjoint. The proof is complete. ut Characterization c allows a refinement of the Uniqueness Theorem in Sect. 15.4 in parallel to what Sect. 4.8 establishes for simple-proportional uniqueness. Suppose we are given a solution matrix x 2 A.r; sI v/. If there exists another solution matrix y ¤ x, then the proof of the Uniqueness Theorem uses y to construct a v-cycle such that x satisfies the associated cycle inequality with equality. Conversely, if there is a v-cycle for which x satisfies the cycle inequality with equality, then part II of the foregoing proof shows that y WD xCz is another solution. Hence multiplicities may be summarized thus: There exists some solution matrix other than x if and only if x satisfies some v-cycle inequality with equality. 15.6 Existence of Double-Proportional Seat Matrices 285

The opposing statements mark uniqueness: The double-proportional seat matrix x is unique if and only if x satisfies every v-cycle inequality with strict inequality.

15.6 Existence of Double-Proportional Seat Matrices

We still need to deal with the elementary property a (Sect. 15.1) that the solution sets A.r; sI v/ are non-empty. In other words the question is whether there exists a double-proportional seat matrix x 2 A.r; sI v/. The answer follows from the characterization of x by means of the minimization problem in Theorem b in Sect. 15.5. The point is that the objective function f is minimized over a set with only finitely many elements: Nk`.r; s/. Clearly one of them, x say, must be a minimizer: f .x/ Ä f .y/ for all y 2 Nk`.r; s/. The awesome format of the function f may be entirely ignored. There are two cases. Either the minimum value is equal to infinity: f .x/ D1. Then x does not preserve the zeros of v, nor does any other matrix y in Nk`.r; s/. This is the unpleasant case when property a fails: The solution set A.r; sI v/ is empty. A trivial non-existence example emerges with a diagonal weight matrix v (assuming k D `). Then the seat matrix x must feature r on its diagonal, as well as s.Ifthe marginals are distinct then no such matrix can exist: r ¤ s ) A.r; sI v/ D;. In the second case the minimum value of f is finite: f .x/<1.Thenx preserves the zeros of v and hence belongs to the solution set A.r; sI v/,bytheTheoremin Sect. 15.5. This is the pleasant case where property a holds true: The solution set A.r; sI v/ is non-empty. The two cases may be identified without actually evaluating the minimum f .x/. It suffices to find a matrix y such that its value f .y/ finite. Then the minimum cannot be but finite too. By definition the function f is finite for matrices y that have fitted row and column marginals and that preserve the zeros of v. Thus we may circumvent even the objective function f itself. This makes life easier. The question is this: Does thereexistamatrixy in Nk`.r; s/ that preserves the zeros of v? If all weights are positive, the answer is: Yes. If one or more weights are zero, the answer is: It depends. The two answers are expounded in the sequel. The first answer relies on a “greedy construction” of a suitable matrix z. With no zero weights all cells are eligible. There is no loss of generality in supposing that the first row and the first column have marginals r1  s1. The process starts by filling cell .1; 1/ of y with s1. Then cell .1; 2/ is filled with the minimum of r1  y11 and s2. The process continues filling cells in the first row, up to column j2, until the first row has a sum equal to r1. Then it continues filling cells in column j2, from cell .2; j2/ downwards to row i2, until column j2 has a sum equal to sj2 . The process continues until it finishes in cell .k;`/. All cells not encountered are filled with zeros. See Table 15.1. For pervious divisor methods the matrix y works fine since entries yij D 0 are permitted. For impervious divisor methods the process must be modified to secure 286 15 Double-Proportional Divisor Methods: Technicalities

Table 15.1 Greedy ⎛ 17139442221111111⎞ construction of an integer 27 17 10 13⎜ 3 9 1 ⎟ matrix with given marginals ⎜ ⎟ 8⎜ 341 ⎟ when all cells are eligible ⎜ ⎟ 712211⎝ ⎠ 41111 11

The process starts by filling cell .1; 1/ and then discards the first column. Next it fills cell .1; 2/ and then discards the first row, etc. All cells not touched upon are filled with zero

yij  1. The greedy construction is used to obtain an integer matrix z with row marginals ri  ` and column marginals sj  k.Theny is defined to have entries yij WD 1 C zij. Having generated an appropriate matrix y 2 Nk`.r; s/ we now refer to Theorem in Sect. 15.5 to confirm the existence of a double-proportional seat matrix x 2 A.r; sI v/: If the weight matrix v has no zeros then the solution set A.r; sI v/ is non- empty. The second answer pertains to situations permitting some weights to be zero. Often the greedy construction still works fine. At times it may fail when trying to fill a cell that is barred due to weight zero. Fortunately the theory of graphs and networks provides an applicable check, under the heading of “flow inequalities”. The flow scheme pretends that the seats enter through the row marginals, pass only through cells where the weights are positive, and leave through the column marginals. Let I Âf1;:::;kg be a subset of rows, and J Âf1;:::;`g a subset of columns. TheP volume which flowsP through the subsets I and J is given by the partial sums: rI WD i2I ri and sJ WD j2J sj. The columns accessible in v from the rows in I are assembled in the set Jv.I/; that is, Jv.I/ WD f j Ä ` j vij >0for some i 2 Ig. These columns are said to be 0 “connected in v with I”. The columns in the complement Jv.I/ are inaccessible: 0 vij D 0 for all i 2 I and j 2 Jv.I/ . With this model in mind, the flow volume that enters the system through the rows in I cannot exceed the flow volume that may leave the system through the columns in Jv.I/:

rI Ä sJv.I/ for all I Âf1;:::;kg: (15.4)

These flow inequalities are necessary and sufficient for the existence of a matrix y in Nk`.r; s/ that inherits all zeros of v. For pervious methods this implies the existence of a double-proportional seat matrix x: For pervious divisor methods the solution set A.r; sI v/ is non-empty if and only if r, s, and v satisfy the flow inequalities (15.4). The flow inequalities evidently apply to the specific situation when the weight matrix v has no zeros. Then every non-empty row subset I is connected in v with all columns: Jv.I/ Df1;:::;`g. Hence the inequalities rI Ä sC D h are immediate. The flow inequalities attain a slightly different format in the presence of minimum requirements aij and maximum cappings bij. For example imperviousness 15.7 A Dual View 287 is accommodated by setting aij D 1 when vij >0and bij D 0 when vij D 0. Restrictions may surface also for other reasons, such as the winner-take-one modification. Of course we assume the restrictions to be compliant: aij Ä bij, aiC Ä ri Ä biC, and aCj Ä sj Ä bCj. The flow inequalities now attain the following form:

rI C aI0J Ä sJ C bIJ0 for all I Âf1;:::;kg and J Âf1;:::;`g: (15.5) P 0 0 Here aI J WD .i;j/2I0J aij denotes the sum of the entries of a in the block I J, etc. Again the flow inequalities turn out to be necessary and sufficient for existence: ŒŒ  bij . ; v/ For a truncated rounding rule  aij the solution set A r sI is non-empty if and only if r, s, a, and b satisfy the flow inequalities (15.5). Both versions of the flow inequalities can be efficiently implemented for machine computation, using the max-flow min-cut theorem of graph theory. In summary there are various ways to ascertain that the solution set A.r; sI v/ is non-empty. What does this mean in practice? Not much, if anything. Empirical election data are trouble-free for the reason that their vote matrices are rather homogeneous and well-balanced. In no event are zero weights frequent enough and awkwardly patterned to raise a fear that the solution set A.r; sI v/ might be empty.

15.7 A Dual View

Double-proportional seat matrices x 2 A.r; sI v/ are distinguished by two features. On the one hand they are members of the set of integer matrices with given row and column marginals: x 2 Nk`.rI s/. On the other hand they are members of the set of integer matrices having a “double-proportional structure”:  ˇ ÄÄ   ˇ v k` ˇ ij x 2 y 2 N ˇyij 2 ; i Ä k; j Ä `; for some E1;:::;Ek; F1;:::;F` >0 : EiFj

Which feature is placed first, which second? The minimization problem in Theorem binSect.15.5 suggests that we search the set of matrices with marginals r and s until we find a matrix x that minimizes the objective function f . The theorem tells us that the minimizer x has the desired double-proportional structure. We may call this the “primal view”. The “dual view” interchanges one feature for the other. We search through the set of matrices with double-proportional structure until we find a matrix x that realizes the marginals r and s. Accordingly, there are two broad types of algorithms, primal algorithms and dual algorithms. Primal algorithms maintain the row and column marginals while approaching the double-proportional structure. Dual algorithms maintain the double-proportional structure while approaching the desired row and column marginals. 288 15 Double-Proportional Divisor Methods: Technicalities

The two algorithms to be discussed below are of dual type: the alternating scaling algorithm (Sect. 15.8) and the tie-and-transfer algorithm (Sect. 15.9). Throughout the sequel A denotes a double-proportional divisor method, with rounding rule ŒŒ  . As before, r and s are prespecified row and column marginals, and v is a given weight matrix. We assume the solution set to be non-empty: A.r; sI v/ ¤; (Sect. 15.6).

15.8 Alternating Scaling Algorithm

The alternating scaling algorithm is easy to understand. It repeats a simple- proportional divisor method many times until the solution is reached. Firstly, all districts are handled separately. This routine is called a “scaling of rows”. Row marginals are met, but column marginals possibly not. The original weights are re- scaled with the district divisors. Secondly, all parties are treated separately. This routine is called a “scaling of columns”. Column marginals are met, row marginals possibly not. The current weights are re-scaled using the party divisors. Thirdly, again all districts are handled on their own. This constitutes another scaling of rows, to be followed by another scaling of columns, and so on. The alternation terminates as soon as row and column marginals are met simultaneously. Upon termination the then current seat matrix is the double-proportional seat matrix sought. The Schaffhausen example in Table 14.3 requires seven scaling steps. The odd steps 1, 3, 5, 7 scale rows; they call for 4 6 D 24 applications of the simple-proportional divisor method with standard rounding. Steps 2, 4, 6 scale columns; they need 3 15 D 45 simple-proportional apportionments. Altogether the alternating scaling algorithm executes simple proportionality 24C45 D 69 times. It terminates with the double-proportional seat matrix that is displayed in Table 14.3. The select divisors are obtained as outlined in Sect. 15.2. The hypothetical EP example in Table 14.5 conducts six scaling steps. Here simple proportionality is applied 3 28C3 8 D 108 times. The repetitive execution of simple proportionality is hard work for a human being, but nothing for a computer. Generally, the alternating scaling algorithm constructs in a step t two matrices: x.t/ and v.t/. The matrix x.t/ records transient seat numbers. They have a double- proportional structure, but alternate in the verification of the marginals. The matrix v.t/ records interim quotients. Let t D 0;2;::: be even. The starting values are x.0/ WD 0 and v.0/ WD v. Routine 1 of the algorithm applies to odd steps t C 1 D 1; 3; 5; etc., routine 2 to even steps t C 2 D 2;4;6; etc.: 1. Odd steps t C 1 compose the seat matrix x.t C 1/ rowwise. If row i of the predecessor matrix x.t/ fits then it is copied into x.t C 1/ and its row divisor is set to unity: i.t C 1/ D 1.Otherwiserowi of x.t C 1/ is an apportionment of ri seats proportional to row i of the previous weight matrix v.t/. With select 15.8 Alternating Scaling Algorithm 289

divisor i.t C 1/, the associated interim quotients are stored in the weight matrix v.t C 1/:

row i of x.t C 1/ 2 A .riI row i of v.t// ;

vij.t/ vij.t C 1/ WD ; i.t C 1/

for rows i Ä k with xiC.t/ ¤ ri, and for all columns j Ä `. 2. Even steps t C 2 compose the seat matrix x.t C 2/ columnwise. If column j of x.t C 1/ fits then it is copied into x.t C 2/ and its column divisor is set to unity: j.t C 2/ D 1. Otherwise column j of x.t C 2/ is an apportionment of sj seats proportional to column j of the weight matrix v.t C 1/. With select divisor j.t C 2/, the ensuing interim quotients are stored in the weight matrix v.t C 2/:  column j of x.t C 2/ 2 A sjI column j of v.t C 1/ ;

vij.t C 1/ vij.t C 2/ WD ; j.t C 2/

for all columns j Ä ` with xCj.t C 1/ ¤ sj, and for all rows i Ä k.

The products of the “incremental row divisors” i.t C 1/ generate “cumulative row divisors”, and the products of the “incremental column divisors” j.t C 2/ give rise to “cumulative column divisors”:

i.1/ i.3/  i.t C 1/ DW Ci.t C 1/ DW Ci.t C 2/;

j.2/ j.4/  j.t C 2/ DW Dj.t C 2/ DW Dj.t C 3/:

The cumulative divisors equip the seat matrices with a double-proportional struc- ture: ÄÄ  vij xij.t/ 2 for all i Ä k; j Ä `; and for all t  1: Ci.t/Dj.t/

The algorithm terminates when a seat matrix x.T/ fits both marginals, rows and columns. It is a solution matrix sought: x.T/ 2 A.r; sI v/. Givenanarbitrarymatrixy 2 Nk` we measure its failure to meet the marginals r and s by the “flaw count” function c.y/: X X c.y/ WD jyiC  rijC jyCj  sjj: iÄk jÄ`

The function c.y/ counts how many seats are malapportioned with regard to the target marginals. Every “overrepresented” row (yiC > ri) increases the flaw count by its surplus seats (yiC  ri). Every “underrepresented” row (yiC < ri) increases 290 15 Double-Proportional Divisor Methods: Technicalities it by its deficit seats (ri  yiC). So do the columns. Since every malapportioned seat is counted twice, once as a surplus seat and once as a deficit seat, all flaw counts are even. We endeavor to show that the matrices x.1/; x.2/;:::;x.T/ have non-increasing flaw counts: c .x.1//  c .x.2// c .x.T//. Unfortunately ties interfere and may create problems. We demonstrate the obstacle by example. Suppose there are two districts with ten seats each, and four parties with overall seats 6; 6; 4; 4. The transient result of routine 1, in an odd step when rows are fitted, may yield in both districts the tied seat vector 3C;3;2C;2. Then the 2 4 seat matrix y with both rows equal to 4;2;3;1 has flaw count c.y/ D 8. However, the seat matrix x with both rows equal to 3; 3; 2; 2 has flaw count zero: c.x/ D 0. It is nonsensical to continue with a seat matrix y with many flaws when there is an equally justified seat matrix x with fewer flaws. The proper handling of ties requires a third routine. To this end a seat matrix y 2 Nk` is called a “tie variant” of a seat matrix x.t/, denoted by y Á x.t/,when step t is odd and all rows of both matrices solve routine 1, or when step t is even and all columns of both matrices solve routine 2. Routine 3 makes sure that the algorithm continues with a tie variant which has smallest flaw count among all tie variants of x.t/: 3. Every seat matrix x.t/ of the alternating scaling algorithm has a flaw count that is smallest among its tie variants:

c .x.t// D min c.y/: yÁx.t/

If no tie variant y ¤ x.t/ exists then no action is needed. Otherwise routine 3 may be executed by inspection since the multiplicities of simple-proportional apportionment problems are well explored (Sect. 4.8). Monotonicity Lemma The sequence of seat matrices x.1/; x.2/; : : : has non- increasing flaw counts: c .x.t//  c .x.t C 1// for all t  1.

Proof For anP even step t the matrix x.t/ has fitted column sums. Its flaw count . . // . / 1 is c x t D iÄk jxiC t  rij.Stept C is odd and fits districts. Consider a district i Ä k.Wehaveri D yiC for all tie variants y of x.t C 1/, including x.t C 1/ itself. We claim that there exists a variant y such that in every district i the differences xij.t/  yij for j Ä ` have the same sign. If district i is fitted, its row in x.t C 1/ is a copy of the row in x.t/ and all the differences are zero anyway. If district i is overrepresented .xiC.t/>ri/, its row divisor i.t C 1/ cannot be strictly smaller than unity; otherwise the district becomes even more overrepresented than before. This leaves two possibilities: i.t C 1/ > 1 or i.t C 1/ D 1.If i.t C 1/ > 1 then all interim quotients decrease: vij.t/>vij.t/= i.t C 1/ D vij.t C 1/. Monotonicity of rounding rules implies xij.t/  yij, whence all desired differences are non-negative. If i.t C1/ D 1 then row i must be tied. We select a tie variant y such that in row i as many tied seat numbers (that is xij.t/ 2 xij.t/Dfxij.t/  1; xij.t/g) are decremented 15.8 Alternating Scaling Algorithm 291 to yij D xij.t/  1 as are needed to obtain yiC D ri. This choice of y has all differences xij.t/  yij non-negative. If district i is underrepresented .xiC.t/

The second inequality is a consequence of routine 3. For odd steps tC1 monotonicity is established similarly. ut The alternating scaling algorithm is sensitive to the choice of divisors. It may even fail to find a solution when too many ties exhibit a suitable pattern. The following 3 3 example is taken from Oelbermann (2013, p. 48):

0 1111 0 1 0 1 1 20 50 50 100 0:08 0:5 0:5 01 0C 1 @ 50 20 20 A 40 ; @ 0:5 0:5 0:5 A ; @ 1 0C 0C A : 1 50 20 20 40 0:5 0:5 0:5 0C 0C 1 2:5 1 1

The first matrix lists the weights. It is framed by the marginals unity (left and top) and by the row and column divisors (right and bottom). The second matrix shows final interim quotients. The third matrix exhibits the final seat numbers, with trailing plus and minus signs indicating tie options. This particular tie pattern allows the completion of another three cycles, whence the set A.r; sI v/ contains four solutions. In this example it matters which incremental divisors i.t C 1/ and j.t C 2/ are used. With select divisors as in routines 1 and 2 the algorithm terminates at step T D 33. If, for overrepresented rows and columns, the largest feasible divisors are applied and, for underrepresented rows and columns, the smallest, then the algorithm reaches a solution at step T D 12. If the largest divisor is always used then routine 2 returns to routine 1. In this case the algorithm stalls and oscillates endlessly. For all practical purposes the alternating scaling algorithm works fine. Simulation studies show that it produces a sequence of seat matrices x.1/; x.2/;:::;x.T/ whose flaw counts decrease rapidly in the beginning. Progress towards the end may slow down. The remedy against long spells of stagnation is the tie-and-transfer algorithm. 292 15 Double-Proportional Divisor Methods: Technicalities

15.9 Tie-and-Transfer Algorithm

The tie-and-transfer algorithm finds single seats that are malpositioned and transfers them from a bad location to a better place. The algorithm is initialized with a scaling of columns. Thus the initial seat matrix x.0/ fulfills xCj.0/ D sj for all j Ä `.We denote its column divisors by j.0/ and its interim quotients by vij.0/ WD vij= j.0/. The initialization promises to allocate many seats correctly. The algorithm maintains fitted columns for all subsequent seat matrices x.t/. Hence the flaw count originates from rows only: X X X c .x.t// D jxiC.t/  rijD .xiC.t/  ri/ C .ri  xiC.t// ;  iÄk i2IC.t/ i2I .t/ where the summation ranges are the rows that are overrepresented or underrepre- sented:

C  I .t/ WD fi Ä k j xiC.t/>rig; I .t/ WD fi Ä k j xiC.t/

D The remaining rows are fitted: I .t/ WD fi Ä k j xiC.t/ D rig. Seats are transferred, one at a time, from an overrepresented row to an under- represented row. A “path via rows i1;:::;iq and columns j1;:::;jq1”isdefinedto C be a list of cells starting in an overrepresented row i1 2 I .t/ andfinishinginan  underrepresented row iq 2 I .t/:

.i1; j1/ ! .i2; j1/ ! .i2; j2/ ! .i3; j2/ !!.iq1; jq1/ ! .iq; jq1/:

The rows i1;:::;iq are taken to be distinct, as are the columns j1;:::;jq1. In steps t D 0; 1; 2; etc. the algorithm advances by executing two routines. The first finds a feasible transfer path, the second updates the seat matrix x.t/:

1. The “tie-update routine” calculates row divisors i.t C 1/ and column divisors j.t C 1/ such that the new interim quotients

vij.t/ vij.t C 1/ WD ; i.t C 1/ j.t C 1/

retain the seat numbers: xij.t/ 2 ŒŒ v ij.tC1/; but also generate a path via i1;:::;iq and j1;:::;jq1 that alternates between decrement and increment options:   v . 1/ . / v . 1/ . / 1 < ; ipjp t C D s xipjp t and ipC1jp t C D s xipC1jp t C for all p q

C  with a start in a row i1 2 I .t/ and a finish in a row iq 2 I .t/. 15.9 Tie-and-Transfer Algorithm 293

2. The “seat-transfer routine” updates the seat matrix by setting

. 1/ . / 1 . 1/ . / 1 xipjp t C WD xipjp t  and xipC1jp t C WD xipC1jp t C

for all p < q,andxij.t C 1/ WD xij.t/ otherwise. The path construction makes sure that a seat is retracted from an overrepresented row i1 and transferred to an underrepresented row iq, and that the net effect on the other row sums and column sums is zero. Thus the flaw count is strictly decreasing: c .x.t C 1// D c .x.t//  2. The algorithm terminates when no flaws are left: c .x.T// D 0. There is not much to say about the seat-transfer routine. It simply transfers a seat along the path that is provided by the tie-update routine. In the sequel the tie-update routine is discussed in detail, and finally illustrated by example. The tie-update routine relies on a paradigm shift. Previously select divisors served to keep interim quotients away from their framing signposts (Sect. 4.7). Now divisors are manipulated until these quotients do hit a signpost. When a lower signpost is hit vij.t/ D s.xij/ a decrement option emerges: xij. When an upper signpost is hit vij.t/ D s.xij C 1/ , an increment option is met: xijC.Oncemany ties are created we may hope to spot a transfer path. Hence the tie-update routine consists of two subroutines: the “path-finding subroutine” and the “tie-creating subroutine”. The path-finding subroutine operates on two sets. The first is the “set of labeled rows” IL that can be reached by a path starting in some overrepresented row in C I .t/. The second set is the “set of labeled columns” JL that are visited by these C paths. Initially all overrepresented rows are labeled but no columns: IL D I .t/ and 0 JL D;. For every labeled row i 2 IL we scan the unlabeled columns j 2 JL to check whether the cell .i; j/ contains a decrement option xij.Ifso,columnj is adjoined to the set JL. Next, for every labeled column j 2 JL, we scan the unlabeled rows 0 . ; / i 2 IL to check whether the cell i j features an increment option xijC.Ifso,rowi is adjoined to the set IL. Then the process iterates, scanning columns, then rows, and so on. The path-finding subroutine pauses when the process of labeling rows or columns stalls. There are two possibilities. Either the set of labeled rows contains an  underrepresented row: IL \ I .t/ ¤;. Then the labeling procedure identifies a path from an overrepresented row to an underrepresented row. The job is done.  Or there is no such path: IL \ I .t/ D;. Then the tie-creating subroutine is invoked. Upon invocation the state of affairs is this:

0 JL J 0  L 1 I s xij.t/

The block IL JL consists of the rows and columns labeled so far; the presence of 0 0 labels confirms that the path-finding strategy works fine. The block IL JL conveys no worthwhile information. The informative blocks are the two off-diagonal blocks. 0 . / < In block IL JL lower signposts are smaller than interim quotients: s xij t v . / 0 ij t . If this were not so, some column j 2 JL would contain a decrement option xij.t/. This column would have been labeled before this juncture: j 2 JL;thisisa 0 contradiction. A similar argument applies to block IL JL. Here interim quotients are smaller than upper signposts: vij.t/0 Œ0 1/; WD max i 2 IL j 2 JL s xij t 2 I vij.t/ ˇ ( ˇ ) ˇ s x .t/ C 1 ˇ ˇ ; ij ˇ 0 ; ;v . />0 .1 : WD min 1 i 2 IL j 2 JL ij t 2 I1 vij.t/ ˇ

The scene is now set for the creation of new ties. The case ˛  1=ˇ uses update factor ˛. The row divisors of the labeled rows in IL are divided by ˛;thecolumn divisors of the labeled columns in JL are multiplied by ˛. The other divisors stay as they were. The effect on a weight is block dependent. In block IL JL, the factors 1=˛ ˛ 0 0 and cancel out; interim quotients stay as is. In block IL JL, interim quotients are 0 . / ˛v . / v . / . / 1 unaffected. In block IL JL, we obtain s xij t Ä ij t Ä ij t Ä s xij t C ; that is: xij.t/ 2 ŒŒ ˛ v ij.t/. The first inequality follows from the definition of ˛;there . / ˛v . / is at least one equality s xi0j0 t D i0j0 t . Therefore, a new decrement option 0 0 . / v . / xi0j0  surfaces in block I L JL.Inblock IL JL, we obtain s xij t Ä ij t Ä .1=˛/vij.t/ Ä ˇvij.t/ Ä s xij.t/ C 1 ;thatis:xij.t/ 2 ŒŒ v ij.t/=˛. Altogether the new weights still justify the old seats. The case 1=ˇ  ˛ uses update factor 1=ˇ.The 0 updated quotients create a new increment option xi0j0 C in block IL JL. When finished, the tie-creating subroutine returns control to the path-finding subroutine. In this way unlabeled rows and columns are processed until a path emerges that enables another flaw-reducing seat transfer. This completes the discussion of the tie-and-transfer algorithm. 15.9 Tie-and-Transfer Algorithm 295

As an illustration we apply the tie-and-transfer algorithm to the 3 3 example from Sect. 15.8. The first matrix recalls the input weights and marginals, but now with the initializing column divisors at the bottom:

0 1111 0 1 0 1 1 20 50 50 1=5 5=7 5=7 011 1 @ 50 20 20 A ; @ 1=2 2=7 2=7 A ; @ 1 00A : 1 50 20 20 1=2 2=7 2=7 0C 01 100 70 70

The second matrix shows the interim quotients v.0/, the third matrix the seat matrix x.0/. Its columns are fitted, its rows are not. The first row is overrepresented, the third row is underrepresented, the second row is fitted. The flaw count is c .x.0// D 2. The tie-update routine begins with its path-finding subroutine. It labels the first C row but no column: IL D I .0/ Df1g and JL D;. In the first row no unlabeled column j D 1; 2; 3 contains a decrement option. The sets IL and JL stall right away. The path-finding subroutine pauses and waits for new ties to be created. 0 1 1; 2; 3 The tie-creating subroutine starts its first pass. The block IL JL Df g f g consists of the first row. The decisive factors are found to be ˛ D 7=10 and ˇ D1. The divisor update in the first row yields updated quotients and seats: 0 1 0 1 7=50 1=2 1=2 01 1 @ 1=2 2=7 2=7 A ; @ 1 00A : 1=2 2=7 2=7 0C 00

The two decrement options 1 in the first row are new. The path-finding subroutine resumes work again. It labels the columns 2 and 3: JL Df2;3g.Thenitpauses again. The tie-creating subroutine takes over for a second pass. The decisive factors turn out to be ˛ D 0 and ˇ D 7=4. The updated quotients and seats are 0 1 0 1 2=25 1=2 1=2 01 1 @ 1=2 1=2 1=2 A ; @ 1 0C 0C A : 1=2 1=2 1=2 0C 0C 0C

Four increment options 0C are new. Now the path-finding subroutine labels the second and third row, and the first column. The labeling of the underrepresented third row stops the tie-update routine. The path via rows 1; 3 and column 3 is handed over to the seat-transfer routine. The final seat matrix is the one known from Sect. 15.8: 0 1 01 0C @ 1 0C 0C A : 0C 0C 1 Chapter 16 Biographical Digest

Abstract Homage is paid to individuals who contributed to the genesis of appor- tionment methods. The sequence is arranged by year of birth. The reference format “Œn” is local to a section.

16.1 Thomas Jefferson 1743–1826

©traveler1116/Getty Images/iStock

Thomas Jefferson is best remembered for serving as third president of the United States of America 1801–1809. Jefferson’s involvement with the apportionment problem dates from the first time in 1791 when the seats of the House of Representatives had to be allocated between the States of the Union. Jefferson favored the divisor method with downward

© Springer International Publishing AG 2017 297 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4_16 298 16 Biographical Digest rounding on the grounds that use of a common divisor and neglect of the interim quotients’ fractional parts were prescribed by the Constitution: I answer, then, that taxes must be divided exactly and representatives as nearly as the nearest ratio will admit; and the fractions must be neglected because the Constitution calls absolutely that there be an apportionment or common ratio, and if any fractions result from the operation, it has left them unprovided for. The quotation is taken from Chap. 3 of Balinski and Young (1982, p. 18). In the Anglo-Saxon literature the divisor method with downward rounding is also known as the “Jefferson method” or the “method of rejected fractions”.

16.2 Alexander Hamilton 1755–1804

©GeorgiosArt/Getty Images/iStock

During the 1791 debate Alexander Hamilton proposed another approach. The method would fix the house size, rely on the rule of three, and fit the residue seats by largest remainders: 1. The aggregated numbers of the United States are divided by 30,000 which gives the total number of representatives, or 120. 2. This number is apportioned among the several states by the following rule. As the aggregate numbers of the United States are to the total number of representatives found as above, so are the particular numbers of each state to the number of representatives of such state. But 3. As this second process leaves a residue of Eight out of the 120 members unapportioned, these are distributed among those states which upon that second process have the largest fractions or remainders. 16.3 Daniel Webster 1782–1852 299

The quotation is taken from Chap. 3 of Balinski and Young (1982, pp. 16–17). Ever since the process is referred to as the Hamilton method, or the method of largest remainders. In our terminology the label is more verbose: Hare-quota method with residual fit by largest remainders. The apportionment of the Hamilton method concurred with the compromise bill that was passed by House and Senate in March 1792. The bill was vetoed by President Washington for want of agreeing with the constitutional requirements, in favor of the Jefferson method. This was the very first exercise of a presidential veto in US history.

16.3 Daniel Webster 1782–1852

©mashuk/Getty Images/iStock

Daniel Webster was among the leading politicians of his time. In 1832 he chaired a Senate committee to review the House apportionment. Not persuaded by the views presented to the committee, Webster advanced a proposal of his own: Let the rule be that the population of each State be divided by a common divisor, and, in addition to the number of members resulting from such division, a member shall be allowed to each State whose fraction exceeds a moiety of the divisor. The quotation is taken from Chap. 4 of Balinski and Young (1982, p. 32). The proposal became known as the “method of major fractions”. A “major fraction” 300 16 Biographical Digest means a fractional part greater than one half; it entitles a party to another seat. A “minor fraction” is a fractional part less than one half; it is neglected. Hence Webster’s method of major fractions is synonymous with the divisor method with standard rounding. Application of the Webster method had to wait until 1842 though. In 1832 the Jefferson method succeeded once again.

16.4 Thomas Hare 1806–1891

©National Portrait Gallery, London

Thomas Hare was an English barrister. In 1853 he became an Inspector of Charities at the Charity Commission. As new schemes for charities had to be laid before parliament, this interaction aroused Hare’s interest what sort of people became members of parliament and how the electoral machinery worked. These experiences motivated Hare to propose a novel electoral system: the single transferable vote (STV) scheme [2]. The STV scheme was invented independently in 1855 by the Danish statesman and mathematician Carl Christopher Georg Andræ (1812–1893), as recounted by his son in a 1905 review of which the essential chapters were later translated into English [1]. The STV scheme was eventually practiced in Denmark, Ireland and Malta, though not in England. 16.5 Henry Richmond Droop 1832–1884 301

Hare elaborated on his proposal in subsequent decades [4]. A crucial role in Hare’s draft is played by the “quota” of votes which a candidate must win in order to be declared elected [4, p. 25]:

It will be convenient at this point to begin the draft of the proposed ELECTORAL LAW, with the clauses which specify the manner in which the quota or number of votes sufficient for the election of a member is to be ascertained and made known at every general election. ::: The registrars ::: shall compute and ascertain the total number of votes polled at such election, and shall divide such total number by 654, rejecting any fraction of the dividend which may appear after such division, and the number of the said quotient found by such division shall be the quota or number of votes entitling the candidates respectively, for whom such quota shall be given, to be returned at the said general election as members to serve in Parliament. The number 654 is the then applicable house size. In our terminology, Hare introduced the Hare-quota variant-1: HQ1 DbvC=hc (Sect. 5.8). Sources

[1] Poul ANDRAE: Andrae and his Invention “The Proportional Representation Method”. A Memorial Work on the Occasion of the Fifty-Year Anniversary of the Introduction of “The Proportional Representation Method”. Published by the author, Copenhagen, 1926. [2] Thomas HARE: The Machinery of Representation. Second Edition. Maxwell, London, 1857. [3] Thomas HARE: On the application of a new statistical method to the ascertain- ment of the votes of majorities in a more exhaustive manner. Journal of the Statistical Society of London 23 (1860) 337–356. Reprinted in: Voting matters 25 (2008) 1–12. [4] Thomas HARE: The Election of Representatives, Parliamentary and Municipal. A Treatise, Third Edition, with a Preface, Appendix, and other Additions. Longman, Green, Longman, Roberts & Green, London, 1865. [5] Jules CARLIER: Nécrologie: Thomas Hare. Représentation proportionnelle – Revue mensuelle 10 (1891) 168–169. [6] Jenifer HART: Proportional Representation. Critics of the British Electoral System 1820–1945. Clarendon Press, Oxford, 1992.

16.5 Henry Richmond Droop 1832–1884

Henry R. Droop was a contemporary of Hare and Andræ and propagated their STV scheme [3]. He noticed that the quota of Hare and Andræ admits a smaller value without apportioning more than the available h seats. The value he proposed is what nowadays is referred to as the Droop-quota: DrQ DbvC=.h C 1/cC1 [3, p. 173]. In Sect. 5.10 we essentially repeat Droop’s reasoning to show that the Hare-quota and its variants are greater than or equal to the Droop-quota. 302 16 Biographical Digest

Sources

[1] Henry Richmond DROOP: On Methods of Electing Representatives. Macmillan, London, 1868. [2] Henry Richmond DROOP: On the Political and Social Effects of Different Methods of Electing Representatives. Maxwell, London, 1869. [3] Henry Richmond DROOP: On methods of electing representatives. Journal of the Statistical Society of London 44 (1881) 141–196. Reprinted in: Voting Matters 24 (2007) 7–46.

16.6 Eduard Hagenbach-Bischoff 1833–1910

©UB Basel, Portr BS Hagenbach E 1833, 11, Fotograph: August Höflinger

Eduard Hagenbach-Bischoff was a professor of physics at the University of Basel. He initiated the scientific studies of the physical characteristics of ice crystals in the Rhone glacier [11]. Hagenbach-Bischoff was also an active politician in the Swiss Canton of Basel. He published extensively on the divisor method with downward rounding [1–9]. In 1905 Hagenbach-Bischoff succeeded to introduce proportional representation into the Great Council of the Canton of Basel. In Switzerland the procedure became known as the “Hagenbach-Bischoff method”. Hagenbach-Bischoff did not approve 16.6 Eduard Hagenbach-Bischoff 1833–1910 303 of having the system named after him [4, p. 3]: :::muss ich mich verwahren gegen den Ausdruck: Hagenbach’sches System. :::I must object to the term: Hagenbach’s system. In all his publications Hagenbach-Bischoff justified the divisor method with downward rounding by an intriguing verbalization of the max-min inequality (Sect. 4.6). He reasoned as follows. If a certain list i with vi votesisassignedxi seats, then the quotient Di WD vi=xi provides a divisor that works fine at least for list i. Hagenbach-Bischoff’s understanding of electoral equality demands that the divisor Di must not be to the detriment of any other list j [1, p. 30]: Wenn man also mit dem für eine bestimmte Liste erhaltenen Quotienten in die Stimmen- zahlen der anderen Listen dividirt, so dürfen die durch die Division erhaltenen ganzen Zahlen nicht grösser sein als die Zahlen der zugetheilten Vertreter.

When the quotient Di that belongs to a certain list i thus is divided into the vote counts vj of the other lists j, the resulting whole numbers yj must not be larger than the seat numbers xj actually apportioned.

Due to downward rounding the tentative seat numbers are yj Dbvj=Dic. Hence the inequality yj Ä xj holds true if and only if vj=Di < xj C1. Insertion of Di D vi=xi yields v v j < i for all i; j Ä `: xj C 1 xi

This is the max-min inequality for the divisor method with downward rounding, except for admitting the occurrence of ties. When applying the jump-and-step procedure (Sect. 4.7) to the divisor method with downward rounding, Hagenbach-Bischoff noted that the procedure finishes faster when initialized with divisor bvC=.h C 1/cC1 rather than with vC=h [3, p. 165]. The note misled many political science textbooks to classify Hagenbach’s system as a quota method. This classification is erroneous; see [13] or Mora (2013). Hagenbach-Bischoff was well aware that the procedure he was proposing was a divisor method and hence superior to a quota method [8, 9]. Being misread is not the only misfortune of Hagenbach-Bischoff. A dissertation at the Law School of the University of Zurich mistakes the double name and refers to Hagenbach-Bischoff in the plural as if there had been two authors [12, pp. 79–80]. Sources

[1] Eduard HAGENBACH-BISCHOFF/Heinrich STUDER/Raoul PICTET: Berech- tigung und Ausführbarkeit der proportionalen Vertretung bei unseren poli- tischen Wahlen – Ein freies Wort an das Schweizervolk und seine Räthe. Schweighauserische Verlagsbuchhandlung, Basel, 1884. 304 16 Biographical Digest

[2] Eduard HAGENBACH-BISCHOFF: Die Frage der Einführung einer Propor- tionalvertretung statt des absoluten Mehres. H. Georg’s Verlag, Basel, 1888. [3] Eduard HAGENBACH-BISCHOFF: La solution du problème de la répartition proportionnelle. Représentation proportionnelle – Revue mensuelle 9 (1890) 159–171. [4] Eduard HAGENBACH-BISCHOFF: Zur Wahlreform. Separatabdruck aus der “Züricher Post” Nr. 53 und 54 vom 4. und 5. März 1891. Schabelitz, Zürich, 1891. [5] Eduard HAGENBACH-BISCHOFF: Die Anwendung der Proportionalvertretung bei den Schweizerischen Nationalratswahlen. Gutachten aus Auftrag des leitenden Comités des schweizerischen Wahlreformvereins. Vereinsbuchdruck- erei, Basel, 1892. [6] Eduard HAGENBACH-BISCHOFF: Emploi de listes associées. Anwendung gekoppelter Listen. Bulletin de la Société suisse pour la Représentation Proportionnelle—Bulletin des Schweizerischen Wahlreform-Vereins für Pro- portionale Volksvertretung 12 Nos. 10 & 11 (1896) 78–85. [7] Eduard HAGENBACH-BISCHOFF: Die Verteilungsrechnung beim Basler Gesetz nach dem Grundsatz der Verhältniswahl. Buchdruckerei zum Basler Berichthaus, Basel, 1905. [8] Eduard HAGENBACH-BISCHOFF: Verteilung der Sitze bei der Proportional- vertretung. Zeitschrift für die gesamte Staatswissenschaft 64 (1908) 344–346. [9] N.N.: Le chiffre diviseur D’Hondt et la règle de trois – Der D’Hondt’sche Divisor und die Regel de tri. Bulletin de la Societé suisse pour la représen- tation proportionnelle – Bulletin des Schweizerischen Wahlreform-Vereins für Proportionale Volksvertretung 3 (1886) 139–142. [10] Paul HUBER: Hagenbach, Jakob Eduard (H.-Bischoff). Neue Deutsche Biogra- phie 7 (1966) 485–486. [11] Eduard BRÜCKNER: Eduard Hagenbach-Bischoff Ž. Zeitschrift für Gletscherkunde, für Eiszeitforschung und Geschichte des Klimas 5 (1911) 235. [12] Peter Felix MÜLLER: Das Wahlsystem. Neue Wege der Grundlegung und Gestaltung. Mit einer ergänzenden Abhandlung zum mathematischen Teil von Professor Dr. Fridolin Bäbler. Polygraphischer Verlag, Zürich, 1959. [13] Vladimír DANCIŠINˇ : Notes on the misnomers associated with electoral quotas. European Electoral Studies 8 (2013) 160–165. 16.7 Victor D’Hondt 1841–1901 305

16.7 Victor D’Hondt 1841–1901

©Carlier, 1901

Victor D’Hondt, a founding member in 1881 of the Association belge pour la représentation proportionnelle, was a prominent protagonist of the proportional representation movement in continental Europe. Since 1885 he served as professor of civil and fiscal law at the University of Ghent. D’Hondt spelled his name with a capital letter D. Libraries file the name under letter H. D’Hondt proposed the divisor method with downward rounding in [4, p. 5]: Une exacte représentation exige la division de tous les chiffres électoraux par un diviseur qui donne des quotients dont la somme soit égale au nombre des sièges vacants. An exact representation requires the division of all vote counts by a divisor that yields quotients whose sum becomes equal to the number of vacant seats. The citation harbors a terminological subtlety. D’Hondt’s “quotients” are not the interim quotients vj=D that result from the division of the vote counts vj by a divisor D. What D’Hondt means [2, p. 17] are the interim quotients en laissant tomber les fractions. upon neglecting fractions. 306 16 Biographical Digest

That is, D’Hondt’s “quotients” are the interim quotients rounded downwards: bvj=Dc. The practical merit of this meaning is that the labor of division terminates at the decimal point. The true merit, though, is conceptual. The fractions after the decimal point would designate parts of a deputy. D’Hondt finds that this would lead to a curious conclusion since deputies are human beings and hence indivisible [2, p. 13]. Sources

[1] Par un électeur (Victor D’HONDT): Question électorale. La représentation proportionnelle des partis. Bruylant-Christophe, Bruxelles, 1878. [2] Victor D’HONDT: Système pratique et raisonné de représentation proportion- nelle. Muquardt, Bruxelles, 1882. [3] Victor D’HONDT: Formule du minimum dans la représentation proportionnelle. Représentation proportionnelle – Revue mensuelle 2 (1883) 117–130. [4] Victor D’HONDT: Exposé du système pratique de représentation proportion- nelle adopté par le Comité de l’Association Réformiste Belge. Van der Haeghen, Gand, 1885. [5] Jules CARLIER: Victor D’Hondt – Notice et portrait. Représentation propor- tionnelle – Revue mensuelle 20 (1901) 29–41. [6] Georges BEATSE: Victor D’Hondt (1885). Université de Gand Liber Memori- alis Notices Biographiques 1 (1913) 428–429.

16.8 Joseph Adna Hill 1860–1938

©HUP Hill, Joseph A. (1), olvwork293908. Harvard University Archives 16.8 Joseph Adna Hill 1860–1938 307

In 1911 Joseph A. Hill submitted a proposal for the apportionment of representatives to the Committee on Census of the House of Representatives of the United States of America [1]. He did this as a private citizen, but with the expertise as Chief Statistician of the Division of Revision and Results with the Bureau of the Census. Hill’s orienting principle was the pairwise comparison of the representative weights of any two states. By meticulous step-by-step arguments Hill derived the procedure that satisfies his principle. He named it the “method of alternate ratios” [1, p. 51]. It turns out to be the same as the divisor method with geometric rounding. Hill’s principle of pairwise comparisons was adopted and perfected by Edward V. Huntington. In a later account Hill reported [2, p. 42] that, since in the application of that principle there was a minor defect which I did not perceive at that time, Professor Huntington took this subject up at my suggestion, indirectly, and worked out the method of equal proportions, which is a logical and consistent application of the same principle that I had tried to apply. Hill never revealed which “minor defect” had intervened, nor did anybody else. A few years later, when reviewing his professional achievements for the fiftieth graduation anniversary, Hill makes no mentioning of apportionment methods [3]. The history of the divisor method with geometric rounding is reflected in the var- ious names assigned to it: method of alternate ratios, Hill method, Hill/Huntington method, Huntington/Hill method, Huntington method, or – the label which Hunt- ington favored most – method of equal proportions. Sources

[1] Joseph Adna HILL: Method of apportioning representatives. House Reports, 62d Congress, 1st Session, Vol. 1, No. 12 (1911) 43–108. [2] Joseph Adna HILL: Apportionment methods described. Congressional Digest, February 1929, 42–44. [3] Joseph Adna HILL: Class report. Pages 121–123 in: Class of 1885, Harvard College, Fiftieth Anniversary. The University Press, Cambridge MA, 1935. [4] Emanuel Alexandrovich GOLDENWEISER: Joseph A. Hill. Journal of the American Statistical Association 34 (1939) 407–411. 308 16 Biographical Digest

16.9 Walter Francis Willcox 1861–1964

©William R. Leonard, The American Statistician, Feb 1, 1961, Taylor & Francis, reprinted by permission of the publisher (Taylor & Francis Ltd, http://www.tandfonline.com) Reprinted by permission of the American Statistical Association, www.amstat.org

During the first half of the twentieth century, the United States of America experienced a fierce debate over seat biases of apportionment methods. The main contestants were statistician Walter F. Willcox from Cornell University and mathematician Edward V. Huntington from Harvard University. The tone of the dispute, at times vicious, was caused perhaps not only by a disagreement over bias, but also by the authors’ affiliation with the competing scientific fields of statistics and mathematics, and with the competing academic centers of Cornell and Harvard. Quite apart from the bias subject, the dispute is a fascinating sample from the sociology of sciences. By evaluating data from former apportionments Willcox amassed overwhelming evidence that Webster’s method of major fractions is unbiased [1–6]. Hence he fought for a return to Webster [4], thus strongly arguing in favor of the divisor method with standard rounding. Sources

[1] Walter Francis WILLCOX: Apportionment of representatives. House Reports, 62d Congress, 1st Session, Vol. 1, No. 12 (1911) 7–42. [2] Walter Francis WILLCOX: The apportionment of representatives. American Economic Review, Supplement 6 (1916) 3–16. [3] Walter Francis WILLCOX: The recent apportionment in New York State. Cornell Law Quarterly 2 (1916) 1–19. [4] Walter Francis WILLCOX: The apportionment problem and the size of the House: A return to Webster. Cornell Law Quarterly 35 (1950) 367–389. [5] Walter Francis WILLCOX: Methods of apportioning seats among the states in the American House of Representatives. International Statistical Institute 1947 Conference Proceedings 3 (1951) 858–867. 16.10 Ladislaus von Bortkiewicz 1868–1931 309

[6] Walter Francis WILLCOX: Last words on the apportionment problem. Law and Contemporary Problems 17 (1952) 290–301. [7] Edward Dana DURAND: Tribute to Walter F. Willcox. Journal of the American Statistical Association 42 (1947) 5–10. [8] William R. LEONARD: Walter F. Willcox: Statist. American Statistician 15 (1961) 16–19.

16.10 Ladislaus von Bortkiewicz 1868–1931

©Humboldt-Universität zu Berlin

Ladislaus von Bortkiewicz was an eminent German statistician. Three of his papers analyze electoral systems and apportionment methods from a mathematical- methodological viewpoint [1–3]. In particular he pointed out that the divisor method with standard rounding may be characterized by a pairwise comparison of success values (Sect. 10.8). The requirement demands that the success values of any two parties differ as little as possible [2, p. 608]: Man präzisiere sodann diese Forderung dahin, daß sich die (positiv genommene) Differenz zwischen den Vertretungskoeffizienten zweier beliebiger Parteien stets niedriger stelle, als sie ausfallen würde, wenn man der einen der beiden Parteien (auch nur) einen Sitz entzöge, um ihn der anderen zuzuwenden, drücke schließlich den so charakterisierten Sachverhalt algebraisch aus – und das System der ungeraden Divisoren geht aus der maßgebenden Ungleichung fertig hervor. Specifying this requirement in such a way that the difference (turned positive) between the success values of any two parties always is lower than it would be if (just) a (single) seat were withheld from one of the two parties in order to grant it to the other, and expressing these facts algebraically – the odd-number method readily emerges from the resulting inequality. 310 16 Biographical Digest

Sources

[1] Ladislaus VON BORTKIEWICZ: Mathematisch-statistisches zur preussi- schen Wahlrechtsreform. Jahrbücher für Nationalökonomie und Statistik, III. Folge 39 (1910) 692–699. [2] Ladislaus VON BORTKIEWICZ: Ergebnisse verschiedener Verteilungssysteme bei der Verhältniswahl. Annalen für soziale Politik und Gesetzgebung 6 (1919) 592–613. [3] Ladislaus VON BORTKIEWICZ: Zur Arithmetik der Verhältniswahl. Sitzungs- berichte der Berliner Mathematischen Gesellschaft 18 (1920) 17–24. [4] Emil Julius GUMBEL: L. von Bortkiewicz Ž. Deutsches Statistisches Zentral- blatt 24 (1931) 232–236. [5] Marcel NICOLAS: Bortkiewicz, Ladislaus von. Neue Deutsche Biographie 2 (1955) 478.

16.11 Edward Vermilye Huntington 1874–1952

©Mathematical Association of America, 2017. All rights reserved

Edward V. Huntington became full professor of mechanics at Harvard University in 1919. His interest in the apportionment problem stemmed from contacts with Joseph A. Hill. The method that Hill had proposed in 1910 was revived by Huntington as the “method of equal proportions” in a 1921 paper entitled “A new method of 16.11 Edward Vermilye Huntington 1874–1952 311 apportionment of representatives”. Hill’s priority is acknowledged in a footnote [1, p. 48]: In fact, the method of equal proportions, though arrived at through quite independent considerations, is in effect a carrying out to its logical conclusion of the fundamental idea of the method of alternate ratios as proposed by Dr. Hill in 1910. Further research encouraged Huntington to view the divisor method with geo- metric rounding to be superior to any other apportionment method [2]. In contrast Francis F. Willcox rated the divisor method with standard rounding highest, a judgment with which Huntington strongly disagreed [3, 4]. A particular contentious point was the question which method was unbiased and neutral to the sizes of the states. The bias controversy raged in a series of exchanges in the pages of the prestigious journal “Science”; see the recount by Balinski and Young (1982, pp. 54– 55). Sources

[1] Edward Vermilye HUNTINGTON: A new method of apportionment of repre- sentatives. Journal of the American Statistical Association 17 (1921) 859–870. Abstract “The mathematical theory of the apportionment of representatives” in: Proceedings of the National Academy of Sciences of the United States of America 7 (1921) 123–127. [2] Edward Vermilye HUNTINGTON: The apportionment of representatives in Congress. Transactions of the American Mathematical Society 30 (1928) 85– 110. [3] Edward Vermilye HUNTINGTON: Methods of apportionment in Congress. American Political Science Review 25 (1931) 961–965. [4] Edward Vermilye HUNTINGTON: The rôle of mathematics in congressional apportionment. Sociometry—A Journal of Inter-Personal Relations 4 (1941) 278–282. Reply by W.F. Willcox, 283–298. Rejoinder by E.V.Huntington, 299– 301. [5] Thomas L. BARTLOW: Mathematics and Politics: Edward V. Huntington and Apportionment of the United States Congress. Proceedings of the Canadian Society for History and Philosophy of Mathematics 19 (2006) 29–54. 312 16 Biographical Digest

16.12 Siegfried Geyerhahn 1879–1960

©John Siegfried Geyerhahn

Siegfried Geyerhahn was a law student at the University of Vienna during 1898/1899–1902. In March 1903 he was conferred a doctoral degree [2]. Geyerhahn stayed in Vienna and practiced as a lawyer. Of Jewish faith and active in a Viennese Free Mason lodge, he was arrested by the Gestapo in 1938 and interned in the concentration camps at Dachau and Buchenwald. Released in 1939 he and his family managed to emigrate to the United States [3]. There, Geyerhahn resumed his profession as a lawyer. He also served as a liaison person for the American Association of Former Austrian Jurists. Geyerhahn [1] is the first paper to devise a system close to the current electoral system for the German Bundestag. Few publications mention Geyerhahn’s prece- dence, Kopfermann (1991, p. 232) does. Geyerhahn’s intention was to reconcile proportionality and personalization. Proportionality would merely serve to calculate the number of seats apportioned to a party. The filling of these seats was to emphasize personalization. First of all, the seats were to be assigned to candidates who had won single-seat constituencies. Only the remaining seats were to be filled from a ranked list of party candidates. Geyerhahn recommended to create half as many constituencies as there were total seats. Although foreseeing the possibility that a party could win more constituency seats than proportionality would justify, he felt this possibility to be 16.13 Jean-André Sainte-Laguë 1882–1950 313 no more than a rare eventuality [1, pp. 29–30]: Man wird zugestehen, dass derartiges nur sehr selten vorkommen wird, aber es kann vorkommen, und deshalb muss für diese Eventualität ein Ausweg geschaffen werden. One will concede that such an event will occur only very rarely, but it can occur and therefore a course of action for this eventuality needs to be devised. In modern terminology Geyerhahn foresaw the occurrence of overhang seats. As a counterbalance he discussed two courses of action. The first proposal was to increase the house size; indeed, this is what the current law does (Sect. 13.4). Geyerhahn warned that the increase may be tremendous. The second proposal was to deny as many seats to constituency winners as are needed to avoid any overhang seats. Sources

[1] Siegfried GEYERHAHN: Das Problem der verhältnismässigen Vertretung – Ein Versuch seiner Lösung. Wiener Staatswissenschaftliche Studien, herausgegeben von Edmund Bernatzik/Eugen von Philippovich, dritter Band, viertes Heft. Mohr, Tübingen, 1902. [2] University of Vienna Archives: Information by courtesy of Thomas Maisel, Head of Archives, 27 March 2017. [3] Claims Resolution Tribunal: Holocaust Victim Assets Litigation, Case No CV96-4849 of Siegfried Geyerhahn, 31 March 2005.

16.13 Jean-André Sainte-Laguë 1882–1950

©Friedrich Pukelsheim 314 16 Biographical Digest

André Sainte-Laguë’s parents were teachers who worked during part of his youth in Tahiti (not: Haïti [4, p. 522]). The name is pronounced `a’gy, as in similar French words like aiguë, ambiguë, ciguë. Sainte-Laguë commenced his career as a high-school teacher of mathematics 1906–1914. During World War I he served in the army and was awarded several distinctions. Resuming his job, he authored a thesis entitled “Les résaux” (engl. Graph theory) which earned him a doctorate degree in mathematics in 1924. In 1938 he was appointed professor of applied mathematics at the Conservatoire national des arts et métiers in Paris. He was an unusually successful academic teacher. One of his courses drew over 2500 audience. Since the largest lecture hall seated just 900, he presented the course in three repeat sessions. In World War II he was a distinguished member of the underground résistance against German occupation [4]. Sainte-Laguë’s proportional representation papers are remarkable [1, 2]. Natu- rally his mathematical approach aims to minimize some criterion that measures the unavoidable deviation from perfect proportionality (Chap. 10). What is exceptional about Sainte-Laguë is that he introduces these criteria with a clear understanding whether they originate from the viewpoint of the voters, or of the deputies, or of the parties (Sect. 2.7). His favorite is the voter-oriented criterion; it leads to the divisor method with standard rounding, which he calls the “method of least squares”. Working through the other criteria he also discusses, in turn, the divisor method with downward rounding, the divisor method with geometric rounding, and the Hare- quota method with residual fit by greatest remainders [1]. Besides the optimization approach Sainte-Laguë illustrates the methods by means of empirical data [2]. Sources

[1] André SAINTE-LAGUË: La représentation proportionnelle et la méthode des moindres carrés. Annales scientifiques de l’École normale supérieure, Troisième série 27 (1910) 529–542. Résumé: Comptes rendus hebdomadaires des Séances de l’Académie des Sciences 151 (1910) 377–378. [2] André SAINTE-LAGUË: La représentation proportionnelle et les mathéma- tiques. Revue générale des Sciences pures et appliquées 21 (1910) 846–852. [3] André FOUILHÉ: Nécrologie: André Sainte-Laguë. Agrégation – Bulletin Offi- ciel de la Société des Agrégés de l’Université, Nouvelle série 13 (1950) 147–148. [4] Jerôme CHASTENET DE GÉRY: Sainte-Laguë, André (1882–1950), Professeur de Mathématiques générales en vue des applications. Pages 522–525 in: Les Professeurs du Conservatoire national des arts et métiers, Dictionnaire biographique 1794–1955, Tome II. Institut national de recherche pédagogique, Conservatoire national des arts et métiers, Paris, 1994. [5] Harald GROPP: On configurations and the book of Sainte-Laguë. Discrete Mathematics 191 (1998) 91–99. 16.14 George Pólya 1887–1985 315

16.14 George Pólya 1887–1985

©ETH-Bibliothek Zürich/Bildarchiv/Fotograf: Schmelhaus, Franz/Portr_14226-S014-AL/CC BY-SA 4.0

George Pólya is a famous mathematician of the twentieth century. Hungarian by birth he took his first regular appointment at the Eidgenössische Technische Hochschule in Zurich in 1914. He became a Swiss citizen and married Stella Vera Weber from Neuchâtel in 1918; his wife survived him by four years. Pólya came to the United States in 1940 and, in 1942, joined the faculty at Stanford University. Pólya published in Hungarian, German, French, Italian, Danish, and English [7]. 316 16 Biographical Digest

While in Switzerland Pólya authored a handful of papers on proportional representation methods [1–5]. He emphasized his intention to reach out beyond the field of mathematics. These papers are gems of scientific writing, providing another evidence of Pólya’s leitmotif to teach mathematics by conveying the excitement of discovery [6]. To no avail: outside mathematics the reception of Pólya’s proportional representation papers has been practically nil. In his papers Pólya always took great pains to start from political requirements before turning to formal investigations, as did his predecessor André Sainte-Laguë (Sect. 2.7). Pólya was first to cast the issue of seat biases into a sound framework and to obtain explicit results for the divisor method with downward rounding in three-party systems (Sect. 7.5). Sources

[1] George PÓLYA: Über die Verteilungssysteme der Proportionalwahl. Zeitschrift für schweizerische Statistik und Volkswirtschaft 54 (1918) 363–387. [2] George PÓLYA: Über Sitzverteilung bei Proportionalwahlverfahren. Schwei- zerisches Zentralblatt für Staats- und Gemeinde-Verwaltung 20 (1919) 1–5. [3] George PÓLYA: Über die Systeme der Sitzverteilung bei Proportionalwahl. Wissen und Leben—Schweizerische Halbmonatsschrift 12 (1919) 259–268 and 307–312. [4] George PÓLYA: Proportionalwahl und Wahrscheinlichkeitsrechnung.Zeitschrift für die gesamte Staatswissenschaft 74 (1919) 297–322. [5] George PÓLYA: Sur la représentation proportionnelle en matière électorale. Enseignement Mathématique 20 (1919) 355–379. Reprinted in: George Pólya Collected Papers, Volume IV. MIT Press, Cambridge MA, 1984, 32–56. [6] Albert PFLUGER: George Pólya. Journal of Graph Theory 1 (1977) 291–294. [7] Gerald Lee ALEXANDERSON/Lester Henry LANGE: Obituary George Pólya. Bulletin of the London Mathematical Society 19 (1987) 558–608. [8] Harold TAYLOR/Loretta TAYLOR: George Pólya – Master of Discovery 1887– 1985. Dale Seymour Publications, Palo Alto CA, 1993. [9] Ingram OLKIN/Friedrich PUKELSHEIM: Pólya, George (György). Neue Deutsche Biographie 20 (2001) 610–611. 16.15 Horst Friedrich Niemeyer 1931–2007 317

16.15 Horst Friedrich Niemeyer 1931–2007

©Alice Niemeyer

Horst Friedrich Niemeyer was professor of mathematics at the University of Marburg 1967–1973. Thereafter he took a position at the Rheinisch-Westfälische Technische Hochschule Aachen 1973–1996. Niemeyer reacted to a Frankfurter Allgemeine Zeitung article of 14 October 1970 [4]. Discussing an undesirable effect of the divisor method with downward rounding, the article stated that it would be hard to find any other apportionment method. In a letter to the presidency of the Bundestag Niemeyer pointed out that it was possible to evade the effect in question by using the Hare-quota method with residual fit by greatest remainders. Furthermore he recommended to adjoin the residual-seat redirection clause (Sect. 11.11). Thanks to Niemeyer’s intervention the Bundestag as well as several German state parliaments amended their electoral laws to adopt the “Hare/Niemeyer procedure”: the Hare-quota method with residual fit by greatest remainders together with the residual-seat redirection clause [3, p. 246]. Sources

[1] Horst Friedrich NIEMEYER/Gottfried WOLF: Über einige mathematische Aspekte bei Wahlverfahren. Zeitschrift für Angewandte Mathematik und Mechanik 64 (1984) T340–T343. [2] Horst Friedrich NIEMEYER: Verhältniswahlverfahren. Mathematik lehren 88 (1998) 59–65. 318 16 Biographical Digest

[3] Horst Friedrich NIEMEYER/Alice Catherine NIEMEYER: Apportionment meth- ods. Mathematical Social Sciences 56 (2008) 240–253. [4] Friedrich Karl FROMME: Regierungsmehrheit heißt nicht Ausschußmehrheit. Frankfurter Allgemeine Zeitung Nr. 238 (14. Oktober 1970) 3. [5] Ilka AGRICOLA/Friedrich PUKELSHEIM: Horst F. Niemeyer und das Propor- tionalverfahren. Mathematische Semesterberichte 64 (2017). Notes and Comments

The pertinent literature is reviewed and further perspectives are mentioned. The reference format “Author (Year)” points to the reference chapter at the end of the book. The reference format “Author (Sect. 16.b [n])” designates source Œn in Sect. 16:b.

Chapter 1: Exposing Methods: The 2014 European Parliament Elections

The 2014 EP election data are taken from the paper Oelbermann and Pukelsheim (2015). Oelbermann et al. (2010) gather the data of the 2009 elections. We do not know of an official source reporting the data of all Member States to the depth that the calculations for the translation of votes into seats could be repeated. Available data in the Internet need to be treated with caution whether they are preliminary, semi-official, or indeed official and final. Nor do the Member States use a uniform terminology to describe their electoral systems. The European Union speaks with twenty-three official languages and writes with three alphabets: Latin, Greek and Cyrillic. Some Member States exhibit their electoral provisions in their mother tongues only. Others provide unofficial translations into English, occasionally with an irritating lack of proficiency concerning the conversion of votes into seats. Much of the European diversity reflects in the electoral provisions even though it is one and the same political body that is being elected: the EP. Therefore, Chap. 1 not only reports the data, but also attempts to identify a common terminological foundation. Naturally electoral systems are also a prime topic of the field of political science. Many political science books and monographs are devoted to a systematic study of proportional representation systems, such as Rose (1974), Grofman and Lijphart (1986), Taagepera and Shugart (1989), Behnke (2007), Gallagher and Mitchell (2008), Nohlen (2009), Robinson and Ullman (2011), and many others. Of course those authors aim at a unification of electoral terms as we do. But the goals are

© Springer International Publishing AG 2017 319 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4 320 Notes and Comments different, and so is the outcome. The political science viewpoint emphasizes the political consequences of electoral procedures. Central topics are political power and its distribution, and democratic government and its establishment. The theme of the present book is more restrictive: methodological analysis. The aim is to study apportionment methods, how they function, and which quantitative consequences they entail.

Chapter 2: Imposing Constitutionality: The 2009 Bundestag Election

The definitive history of electoral experiences in imperial Germany 1871–1918 is told by historian Margaret Anderson (2000). Recounting innumerable election incidents from the time, the author develops her thesis that these decades prepared the ground for German democracy to grow and later strike roots after 1918. Richter (2016) presents an intriguing comparison of electoral reforms in the Old World and the New World around 1900. However, the new proportional representation system was soon blamed for not producing proper leaders, as voiced in articles with such titles as “Verhältniswahl und Führerauslese” (Proportional Representation and the Selection of Leaders) of renowned jurist Jellinek (1926) and proportional representation activist Schmidt (1929). Another activist, Erdmannsdörffer (1932), dreamed of a leadership figure such as Thomas Mann. After the victory of the Allies over the leader regime of Nazi Germany, the discussions of electoral matters during the former Weimar Republic bore fruit when the electoral system was to be newly installed. The political body to draft a Basic Law was the Parliamentary Council. It delegated the design of the electoral system to its Committee on Electoral Procedure. The committee’s records are meticulously edited by Rosenbach (1994). Unfortunately they do not fully explain how the current two-votes system for the Bundestag came into being. The German two-votes system was used for the first time in 1957. Since then it has acquired a high reputation internationally. Shugart and Wattenberg (2001) provocatively pose the question whether the system implements the “best of both worlds”: of the world of proportional representation and of the world of the election of persons. The two-votes system right away led to some irritation because of the occurrence of overhang seats. The 1957 incident was convincingly explained by a malappor- tionment of constituencies between states. Much of the ensuing discussion conveys the impression that malapportionment, having been the cause once, was the only conceivable cause forever. Other than for the reason of malapportionment, overhang seats were declared to be a “necessary consequence” (notwendige Folge) of a proportional representation system that is combined with the election of persons. No mentioning was made that Geyerhahn (Sect. 16.12 [1]) had already proposed options to avoid overhang seats. Nor was it acknowledged that the states that Notes and Comments 321 imported the German system adjoined amendments avoiding overhang seats; see the examples of London and Scotland in Sect. 12.4. In Germany the issue of overhang seats has grown particularly virulent after the 1990 re-unification when the party system began to diversify. The fascinating story of the electoral heritage over the last 2000 years is told by Szpiro (2010)andRza¸˙zewski et al. (2014). Elections in medieval Augsburg are studied by Rogge (1994) who mentions the rejection of the election of carpenter Marx Neumüller (Sect. 2.5). The material on Ramon Llull and Nicolaus Cusanus is taken from Hägele and Pukelsheim (2001, 2008); see also McLean and Urken (1995). An Internet edition www.uni-augsburg.de/llull of Llull’s three electoral tracts was produced by Drton et al. (2004). The role of electoral equality in the decisions of German constitutional courts is studied by Pukelsheim (2000a,b,c). Emphasis is on equality of the success values of the voters, of the representative weights of the deputies, and of the ideal share of seats of the parties (Sects. 2.7–2.9).

Chapter 3: From Reals to Integers: Rounding Functions and Rounding Rules

The task of rounding real numbers to integers is centuries-old. In a witty letter to the editor, Seal (1950) conjectures that for every allegedly novel result on rounding properties one can find a prior reference that precedes it. Floor brackets for downward rounding (bc) and ceiling brackets for upward rounding (de) are attributed to Iverson (1962, p. 12). Angle brackets for standard rounding (hi) are employed for instance by Abramowitz and Stegun (1970, p. 223). Commercial rounding, also known as financial rounding, is the accepted round- ing procedure for business and commerce; the term is a translation from German “kaufmännische Rundung”. We find it convenient to use the term “commercial rounding” for the real-valued rounding function; Kopfermann (1991, p. 108) calls it “Standardrundung”. We reserve the term “standard rounding” for the set-valued rounding rule that is induced by the commercial rounding function. The even- number rounding function is recommended by Wallis and Roberts (1956, p. 175) specifically for statistical purposes, and by Bronstein and Semendjajew (1991, p. 98) quite generally. The possible occurrence of ties necessitates the handling of set-valued rounding rules. For the sake of clarity we use different notation for rounding functions and for rounding rules. Rounding functions are visualized through simple delimiting brackets Œ, rounding rules through double delimiting brackets ŒŒ  . Double brackets first appear in Balinski and Young (1978a, p. 307); see also Pukelsheim (1993). The name “signpost” stems from Balinski and Young (1982, p. 62). The attribute “stationary” for the signpost family in Sect. 3.10 is due to Balinski and Rachev (1993, 1997). Kopfermann (1991, p. 202) and Janson (2012, p. 29) speak of “linear” divisor methods. Other authors use the term “parametric” divisor methods: Oyama 322 Notes and Comments

(1991), Oyama and Ichimori (1995), Palomares and Ramírez (2003). While we restrict attention to the two families of stationary and power-mean signposts, other signpost sequences are conceivable; see, for instance, Dorfleitner and Klein (1999), Marshall et al. (2002), Agnew (2008), Ichimori (2012).

Chapter 4: Divisor Methods of Apportionment: Divide and Round

The cardinality of the set of seat vectors N`.h/ is an occupancy number as discussed by Feller (1971, p. 38). The apportionment rule from Sect. 4.2 is called “allotment method” by Hylland (1978,p.5). Chapters 4 and 5 heavily rely on the seminal monograph Balinski and Young (1982). The dividing points d.a/ of Balinski and Young (1982, p. 99) are related to the signpost sequence by way of a shift of argument: d.a/ D s.a C 1/.The five organizing principles in Sect. 4.3 also appear in Balinski and Young (1982, pp. 97, 144, 147). Those authors refer to anonymity as symmetry; balancedness has the same name; concordance is called weak population monotonicity; decency is homogeneity; and exactness is termed weak proportionality. Subsequent papers by Balinski and coauthors, and by other authors, use further terminological alternatives. The definition of a divisor method in Sect. 4.4 emphasizes the flexible nature of the divisor. For other authors the term “divisor” is synonymous with the term “signpost”. The reason is that the signposts s.n/ serve as divisors in the calculation of the comparative scores vj=s.n/; see Sect. 4.12. The formulation of the jump-and-step procedure in Sect. 4.7 is inspired by work of Happacher and Pukelsheim (1996). However, the procedure was known and discussed ever since the inception of the divisor method with downward rounding; see Gfeller (1890), Hagenbach-Bischoff (Sect. 16.6 [5, p. 20]).

Chapter 5: Quota Methods of Apportionment: Divide and Rank

The multitude of quotas that are in use is bewildering (Sect. 5.8). Over time we invented the three-letter codes for them as they came across in empirical data; the codes are somewhat at odds with the ordering in Sect. 5.10. The max-min inequality in Sect. 5.5 appears in Kopfermann (1991, p. 196). The family of shift-quotas .Q.s/ D vC=.h C s// is studied also by Hylland (1978, p. 14), Kopfermann (1991, p. 195), Janson (2012, p. 70). Only the shift value s D 0 (Hare-quota) and the limiting case s D 1 (Droop-quota variant-4) are of practical interest. Theoretically, the embedding of some of the Hare- and Droop- quotas into a single one-parameter family is pleasing. Notes and Comments 323

The original connotation of the term “quota” signifies a fixed number of voters to be represented by a Member of Parliament; see Hare (Sect. 16.4), Droop (Sect. 16.5). The then novel proportional representation system was pitted against the established plurality vote. Since a plurality winner is determined by count of votes, a “quota” originally had to be a whole number, too. Fractions were deemed unacceptable, as is testified by the following quotes: :::rejecting the fractional numbers of the dividend; in Hare (Sect. 16.4 [2, p. 17]). Les fractions ne comptent pas; in Morin (1862, p. 26). Brüche werden nicht gerechnet; in Getz (1864, p. 50). Nowadays the term “quota” signifies any quantity, possibly fractional, provided only that it is computationally convenient. Section 10.12 has more to say on the ambiguous usage of the term “quota”. Methods with integral quotas lose the property of decency (Sect. 5.10). An example is presented by Dancišinˇ (2013, p. 153). In this example, if the votes of four parties are doubled, the apportionment changes from 8W4W2W0 to 8W5W1W0.

Chapter 6: Targeting the House Size: Discrepancy Distribution

The seat-total distributions in Sect. 6.5 and the discrepancy distribution in Sect. 6.7 are due to Happacher (1996, 2001). The alternating-sum formula for the volume of higher dimensional rectangles, which is a crucial step in part II of the proof in Sect. 6.5, is pictured in Billingsley (1995, p. 177). For three-party systems and the divisor method with downward rounding, Hagenbach-Bischoff (Sect. 16.6 [6]) carried out similar calculations. Related results are obtained by Mosteller et al. (1967), Diaconis and Freedman (1979), Kopfermann (1991, p. 185). The practical examples in Sect. 6.8 are taken from Happacher (2001) and Happacher and Pukelsheim (1998, 2000). The statement that rounding residuals tend to a uniform distribution irrespective of the underlying weight distribution is part of the folklore of the analysis of rounding effects; see Seal (1950). The earliest formal proof we are aware of is Tukey (1938). The invariance principle in Sect. 6.10 is proved for Riemann-integrable den- sities by Heinrich et al. (2004, 2005). The present version for Lebesgue-integrable densities is due to Janson (2014). Proofs that the variance of the discrepancy distribution is equal to `=12 (Sect. 6.11) are provided by Heinrich et al. (2017).

Chapter 7: Favoring Some at the Expense of Others: Seat Biases

The issue of seat biases is addressed already by Henri Poincaré in the preface to Lachapelle (1911). Pólya (Sect. 16.14 [1]) is first to derive seat bias formulas in three-party systems. Pólya emphasizes that the seat biases vanish uniformly only for 324 Notes and Comments the Hare-quota method with residual fit by greatest remainders and for the divisor method with standard rounding. The seat bias formula in Sect. 7.4 with no threshold (t D 0) is due to Schuster et al. (2003). Schwingenschlögl and Pukelsheim (2006) adjoin the threshold factor 1  `t. A stringent proof for the whole formula is obtained by Heinrich et al. (2005). The invention of list alliances is due to Hagenbach-Bischoff (Sect. 16.6 [5]). His intention was to provide a means for weaker parties to dampen the seat biases of the divisor method with downward rounding. The seat bias results for list alliances are taken from Pukelsheim and Leutgäb (2009). An empirical study of the occurrence of list alliances and their effects is Bischof et al. (2016). In the present book the term “seat bias” designates the average of seat excesses, where the average is evaluated under the assumption that all vote shares are equally likely or that the vote shares follow an absolutely continuous distribution. Other concepts of the term “seat bias” are investigated by Balinski and Young (1982, pp. 118–128). Schwingenschlögl (2008) shows that those results conform with the seat bias formula in Sect. 7.4. In essence, conformance is a consequence of the invariance principle in Sect. 6.10.Janson(2014) establishes a complementary asymptotic analysis that includes the results in Sects. 7.4 and 7.12. His approach is first to average over equally likely house sizes h in a finite range f0;:::;Hg,and then to expand the range beyond limits: H !1. During the first half of the twentieth century the United States experienced a fierce and at times vicious dispute over seat biases; see Chap. 6 in Balinski and Young (1982). The bias issue was raised again during a more recent law suit; see Ernst (1994).

Chapter 8: Preferring Stronger Parties to Weaker Parties: Majorization

Majorization is often introduced to be an ordering that captures increasing disparity, from pure equality to pure inequality. This view focuses on the vectors x D .n;:::;n/ and y D .`n;0;:::;0/. The first vector equips every party with the same number of seats, the second assigns all seats to the strongest party. Evidently x is majorized by y. But these vectors, being insensitive to the underlying vote shares wj, never surface in the context of proportional representation problems. Yet majorization has its merits in apportionment analysis. It allows a comparison of seat vectors closed to the ideal share vector .hw1;:::;hw`/. The comparison confronts an arbitrary group of stronger parties with the complementary group of weaker parties. The juxtaposition of the two groups is what makes majorization so useful for proportional representation applications. Notes and Comments 325

The majorization concept boasts an impressive range of applications, as expound- ed in Marshall and Olkin (1979) and Marshall et al. (2011). Our exposition of its application to apportionment methods follows Marshall et al. (2002). Table 8.1 elaborates an example of Balinski and Rachev (1997). The majorization result for the shift-quota methods is inspired by Lauwers and Van Puyenbroeck (2006a,b); our presentation follows Janson (2012, p. 112). Balinski and Young (1982, p. 118) study a similar relation wherein a seat is transferred from a smaller state to a larger state, akin to our Sect. 8.3. Either way monotonicity of the signpost ratios is the necessary and sufficient condition to be checked (Sect. 8.5).

Chapter 9: Securing System Consistency: Coherence and Paradoxes

The coherence theorem has forerunners in Hylland (1978) and Balinski and Young (1978b). The term “coherence” is coined by Balinski (2003), Balinski and Young (1982, p. 141) speak of “uniformity”, Young (1994, p. 171) of “consistency”. Since “uniformity” and “consistency” are employed in science often with other meanings, we abide by the term “coherence”. It nicely alludes to the interaction of the whole and its parts, and to the logical compatibility of abstract premises with concrete properties. The exposition of the coherence theorem and its proof (Sects. 9.4–9.8) follows Palomares et al. (2016). We note that vote ratio monotonicity plays a pivotal role in the exposition of Balinski and Young (1982, pp. 108, 117). The characterization of stationary divisor methods in Sect. 9.9 is from Balinski and Ramírez (2015). The coherent extension of the natural two-party apportionment to the divisor method with standard rounding (Sect. 9.10) is due to Young (1994, pp. 50, 190). Incompatibility of abstract expectations with concrete outcomes is indicated by the term “paradox”. Hence the chapter also reviews the paradoxes generally attributed to quota methods. However, paradoxes that can be explained as easily as these, namely as breaches of monotonicity, hardly qualify to be paradoxes of any depth in the logical sense. The five-party example in Sect. 9.12 is due to Janson and Linusson (2012). The example in Table 9.4 is taken from Young (1994, p. 60). A similar example is given by Brams and Straffin (1982). The Hamburg example of the new states paradox in Sect. 9.11 is mentioned by Röper (1998). The population paradox prompted the German Bundestag to adopt the divisor method with standard rounding (Sect. 9.13). With the same argument Hix et al. (2010) recommend the method for districting purposes in the United Kingdom; see also McLean (2008, 2014). In Sect. 9.14, our notion of “voter monotonicity” appears under the label “order preservation” in Balinski and Rachev (1997). The no-show example in Table 9.5 is taken from Cechlárová and Dancišinˇ (2017). The authors 326 Notes and Comments prove that quota methods fall victim to the no-show paradox only for integer quota, not with the unrounded quotas HaQ and DQ4. See also Janson (2011).

Chapter 10: Appraising Electoral Equality: Goodness-of-Fit Criteria

The search for optimality properties of apportionment methods is as old as the pro- portional representation movement. The optimality results for the divisor methods with standard rounding (Sect. 10.2) and with geometric rounding (Sect. 10.5) are due to Sainte-Laguë (Sect. 16.13 [1]). Rouyen (1901) established the optimality property of the divisor method with downward rounding (Sect. 10.3). The strengthening to necessary and sufficient conditions in Sects. 10.3 and 10.4 is due to Mora (2013). Pólya (Sect. 16.14 [5]) proved the optimality theorem for the Hare-quota method with residual fit by greatest remainders (Sect. 10.6). A pairwise-comparison approach was proposed by Hill (Sect. 16.8). The specific version that substantiates success-value stability of the divisor method with standard rounding (Sect. 10.8) is treated by von Bortkiewicz (Sect. 16.10). The pairwise- comparison approach was perfected by Huntington (Sect. 16.11). Huntington’s research led him to favor the divisor method with geometric rounding (Sect. 10.10). In 1941, after one and a half century of apportionment struggle in the US House of Representatives, the divisor method with geometric rounding became the legal procedure for the apportionment of the House of Representatives seats among the States of the Union; see the narrative of Balinski and Young (1982, p. 58). The result on ideal-share stability of the divisor method with standard rounding is due to Balinski and Young (1982, p. 132). Further optimization approaches are developed by Grilli di Cortona et al. (1999) and many others. Edelman (2006) studies minimal total deviation apportionments. Edelman (2015) devises a criterion so as to equalize the voting power of the individual voters; the optimum solution is the divisor method based on signposts that are the average of geometric rounding and standard rounding: s.n/ D .es0.n/ Ces1.n// =2. In the field of optimal design of statistical experiments, efficiency optimization leads to the divisor method with upward rounding; see Pukelsheim (1993, p. 304).

Chapter 11: Tracing Peculiarities: Vote Thresholds and Majority Clauses

The results in Sects. 11.1–11.6 belong to the usual repertoire of apportionment methodology. The arrangement of the material is a mixture of Balinski and Young (1982), Kopfermann (1991), Gallagher (1992), Balinski and Rachev (1997), Notes and Comments 327

Palomares and Ramírez (2003), Janson (2012). Thresholds for general divisor methods are determined by Jones and Wilson (2010). Ruiz-Rufino (2011) studies empirical applications of thresholds. The majority preservation material in Sects. 11.7–11.11 is less widespread. The discussion of the three majority clauses follows Pukelsheim and Maier (2006, 2008). However, the residual seat redirection clause (Sect. 11.11) was recommended long ago by Gfeller (1890), as pointed out in Sect. 11.10. The example in Table 11.2 is elaborated in Bundestag (1982). The North Rhine-Westphalian examples in Sect. 11.8 are from Pukelsheim et al. (2009). For an in-depth study of the Con- ference Committee dilemma see Alex (2017). Miller (2015) and Felsenthal and Miller (2015) exemplify breaches of majority preservation and related effects with data from Denmark, Israel, and the Netherlands. Hermsdorf (2009) proposes an algorithm designed for majority preservation.

Chapter 12: Truncating Seat Ranges: Minimum-Maximum Restrictions

Balinski (2004, pp. 192–193) points out that it is ambiguous in which way quota methods incorporate side restrictions, and emphasizes that the resulting seat apportionments depend on the way chosen (Sect. 12.2). The Cambridge compromise (Sect. 12.8) emerged from a January 2011 workshop at the University of Cambridge; see Grimmett et al. (2011). The power compromise is a variant proposed by Grimmett et al. (2012), Pukelsheim and Grimmett (2017). Arndt (2008) advocates a similar approach, as do Theil (1969) and Theil and Schrage (1977). Other ways to adjust the population figures are overviewed by Martínez-Aroza and Ramírez-González (2008) and Ramírez González (2017), as well as by Słom- czynski´ and Zyczkowski˙ (2012, 2017). As for the Jagiellonian compromise: see the references in Sect. 12.10. Birkmeier (2011) extends the approach by combining the one person-one vote principle with the one state-one vote principle.

Chapter 13: Proportionality and Personalization: BWG 2013

For a review of the German electoral system including an analysis from the quantitative viewpoint see Behnke (2007). For a compendium of the full history and all details of the German Federal Election Law see Schreiber (2009). Constitutional jurist Meyer (2010) reviews the long and agonizing struggle to deal with overhang seats in a satisfactory manner. English translations of major decisions of the German Federal Constitutional Court are compiled by Kommers and Miller (2012). 328 Notes and Comments

Our description of the 2013 amendment of the Federal Election Law follows Pukelsheim and Rossi (2013). Section 13.5 contains a proposal to set the minimum number of seats for a party to be equal to the number of direct seats plus a 10% overhead. The proposal is due to Peifer et al. (2012) who call it the “direct-seat oriented proportionality adjustment” (direktmandatsbedingte Proporzanpassung). Further alternatives are proposed by Behnke (2013). Already Brams and Fishburn (1984) discuss proportional representation in variable-size legislatures. Bochsler (2012) points out that the two-votes system with a variable-size legislature bears a potential of misuse. He exemplifies his thesis with data from the election of the Albanian parliament.

Chapter 14: Representing Districts and Parties: Double Proportionality

Pukelsheim and Schuhmacher (2004, 2011) review the introduction of the double- proportional divisor method with standard rounding in the Swiss Cantons of Zurich, Aargau and Schaffhausen. Meanwhile double proportionality has been adopted also in further cantons (Sect. 14.5). For a general exposition of the critical inter- play between territorial representation versus political representation see Bochsler (2010). There are quite a few papers that illustrate the hypothetical usefulness of double proportionality: Gassner (2000) for Belgium; Balinski and Ramírez (1999)and Balinski (2002) for Mexico; Zachariassen and Zachariassen (2006)fortheFarœ Islands; Ramírez et al. (2008) and Ramírez González (2013) for Spain; Pennisi et al. (2006, 2009) for Italy, Oelbermann and Pukelsheim (2011)fortheEP. Double-proportional divisor methods are not the only way to give the idea of biproportionality a concrete form. Alternative approaches are proposed by Cox and Ernst (1982) and Serafini (2012).

Chapter 15: Double-Proportional Divisor Methods: Technicalities

Double-proportional divisor methods originate with the papers Balinski and Demange (1989a,b). They are further elaborated by Balinski and Rachev (1993, 1997), Balinski (2006), Balinski and Pukelsheim (2007). Network flow algorithms are studied by Rote and Zachariasen (2007); see also the overview Pukelsheim et al. (2012). The characterization theorem in Sect. 15.5 is due to Gaffke and Pukelsheim (2008a), including the analysis of the critical inequalities. The objective function f . y/ in Sect. 15.5 is akin to a criterion investigated by Carnal (1993). A proof Notes and Comments 329 that the flow inequalities are necessary and sufficient for the existence of a matrix y 2 N`.r; s/ that preserves the zeros of v (Sect. 15.6) is included in many textbooks on network theory, such as Ford and Fulkerson (1962, p. 51). The flow inequalities are adapted to double-proportional apportionment problems by Joas (2005), Maier (2009), Oelbermann (2013). A generic structure of appropriate algorithms is developed by Gaffke and Pukelsheim (2008b). The tie-and-transfer algorithm is succinctly described and analyzed by Zachariasen (2006), Maier (2009). Maier et al. (2010) demonstrate the effectiveness of the alternating scaling algorithm and the tie-and-transfer algorithm by means of an extensive benchmark study. Oelbermann (2013, 2016)presentsan in-depth analysis of the alternating scaling algorithm. The alternating scaling algorithm, the tie-and-transfer algorithm, hybrid combi- nations of the two, and other algorithms are implemented in the free software BAZI that is available on the Internet site www.uni-augsburg.de/bazi. References

Every reference concludes with a list ŒA:b; C:d;::: of Sects. A:b, C:d,etc.where the reference is quoted. The chapter “Notes and Comments” is indicated by “N”. The reference sources in the individual sections of Chap. 16 are not repeated here.

Abramowitz, M./Stegun, I.A. (1970): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. [N.3] Agnew, R.A. (2008): Optimal congressional apportionment. American Mathematical Monthly 115, 297–303. [N.3] Alex, K. (2017): Patt im Ausschuss. Die Abbildung politischer Mehrheitsverhältnisse in repräsentativen Gremien im Spannungsfeld von Staatsrecht und Staatspraxis. Peter Lang, Frankfurt am Main. [N.11] Anderson, M.L. (2000): Practicing Democracy. Elections and Political Culture in Imperial Germany. Princeton University Press, Princeton NJ. [N.2] Arndt, F. (2008): Ausrechnen statt aushandeln: Rationalitätsgewinne durch ein formalisiertes Mo- dell für die Bestimmung der Zusammensetzung des Europäischen Parlaments. Zeitschrift für ausländisches öffentliches Recht und Völkerrecht—Heidelberg Journal of International Law 68, 247–279. [N.12] Balinski, M.L. (2002): Une “dose” de proportionnelle: Le système électoral mexicain. Pour la Science, Avril 2002, 58–59. [N.14] Balinski, M.L. (2003): Quelle équité? Pour la Science, Septembre 2003, 82–87. Deutsch: Die Mathematik der Gerechtigkeit. Spektrum der Wissenschaft, März 2004, 90–97. English: What is just? American Mathematical Monthly 112, 2005, 502–511. [N.9] Balinski, M.L. (2004): Le suffrage universel inachevé. Belin, Paris. [7.5, N.12] Balinski, M.L. (2006): Apportionment: Uni- and bi-dimensional. Pages 43–53 in: Mathematics and Democracy. Recent Advances in Voting Systems and Collective Choice. Springer, Berlin. [N.15] Balinski, M.L./Demange, G. (1989a): An axiomatic approach to proportionality between matrices. Mathematics of Operations Research 14, 700–719. [N.15] Balinski, M.L./Demange, G. (1989b): Algorithms for proportional matrices in reals and integers. Mathematical Programming 45, 193–210. [N.15] Balinski, M.L./Pukelsheim, F. (2007): Die Mathematik der doppelten Gerechtigkeit. Spektrum der Wissenschaft, April 2007, 76–80. [N.15] Balinski, M.L./Rachev, S.T. (1993): Rounding proportions: Rules of rounding. Numerical Func- tional Analysis and Optimization 14, 475–501. [N.3, N.15] Balinski, M.L./Rachev, S.T. (1997): Rounding proportions: Methods of rounding. Mathematical Scientist 22, 1–26. [N.3, N.8, N.9, N.11, N.15]

© Springer International Publishing AG 2017 331 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4 332 References

Balinski, M.L./Ramírez, V. (1999): Parametric methods of apportionment, rounding and produc- tion. Mathematical Social Sciences 37, 107–122. [N.14] Balinski, M.L./Ramírez, V. (2015): Parametric vs. divisor methods of apportionment. Annals of Operations Research 215, 39–48. [N.9] Balinski, M.L./Young, H.P. (1978a): Stability, coalitions and schisms in proportional representation systems. American Political Science Review 72, 848–858. [N.3] Balinski, M.L./Young, H.P. (1978b): The Jefferson method of apportionment. SIAM Review 20, 278–284. [N.9] Balinski, M.L./Young, H.P. (1982): Fair Representation. Meeting the Ideal of One Man, One Vote. Yale University Press, New Haven CT. Second Edition (with identical pagination), Brookings Institution Press, Washington DC, 2001. [3.8, 7.5, 7.8, 9.1, 9.11, 9.12, 9.13, 12.2, 16.1, 16.2, 16.3, 16.11, N.3, N.4, N.7, N.8, N.9, N.10, N.11] Bauer, H. (1991): Wahrscheinlichkeitstheorie. 4. Auflage. De Gruyter, Berlin. [6.10] Behnke, J. (2007): Das Wahlsystem der Bundesrepublik Deutschland. Logik, Technik und Praxis der Verhältniswahl. Nomos, Baden-Baden. [N.1, N.13] Behnke, J. (2013): Das neue Wahlgesetz, sicherlich nicht das letzte. Recht und Politik 49, 1–10. [N.13] Billingsley, P. (1995): Probability and Measure, Second Edition. Wiley, New York. [N.6] Birkmeier, O. (2011): Machtindizes und Fairness-Kriterien in gewichteten Abstimmungssystemen mit Enthaltungen. Logos Verlag, Berlin. [N.12] Bischof, W./Hindinger, C./Pukelsheim, F. (2016): Listenverbindungen – ein Relikt im bayerischen Gemeinde- und Landkreiswahlgesetz. Bayerische Verwaltungsblätter 147, 73–76. [N.7] Bochsler, D. (2010): Territory and Electoral Rules in Post-Communist Democracies. Palgrave Macmillan, Basingstoke. [N.14] Bochsler, D. (2012): A quasi-proportional electoral system “only for honest men”? The hidden potential for manipulating mixed compensatory electoral systems. International Political Science Review 33, 401–420. [N.13] Brams, S.J./Fishburn, P.C. (1984): Proportional representation in variable-size legislatures. Social Choice and Welfare 1, 211–229. [N.13] Brams, S.J./Straffin, P.D. (1982): The apportionment problem. Science 217, 437–438. [N.9] Bronstein, I.N./Semendjajew, K.A. (1991): Taschenbuch der Mathematik. 25. Auflage. Teubner, Leipzig. [N.3] Bundestag (1982): Bundestagsdrucksache 9/1913, 12. August 1982. [N.11] Carnal, H. (1993): Mathématiques et politique. Elemente der Mathematik 48, 27–32. [N.15] Cechlárová, K./Dancišin,ˇ V. (2017): On the no-show paradox in quota methods of apportionment. Typescript. [N.9] Cox, L.H./Ernst, L.R. (1982): Controlled rounding. INFOR—Information Systems and Operational Research 20, 423–432. [N.14] Dancišin,ˇ V. (2013): Metódy prerozdel’ovania mandátov v pomernom volebnom systéme. Filo- zofická fakulta Prešovskej univerzity v Prešove, Prešov. [N.5] Diaconis, P./Freedman, D. (1979): On rounding percentages. Journal of the American Statistical Association 74, 359–364. [N.6] Dorfleitner, G./Klein, T. (1999): Rounding with multiplier methods: An efficient algorithm and applications in statistics. Statistical Papers 40, 143–157. [N.3] Drton, M./Hägele, G./Haneberg, D./Pukelsheim, F./Reif, W. (2004): Ramon Llulls Traktate zu Wahlverfahren: Ziele und Realisierung einer Internet-Edition. Pages 131–140 in: Mediaevistik und Neue Medien, Thorbecke, Stuttgart. [N.2] Edelman, P.H. (2006): Minimum total deviation apportionments. Pages 55–64 in: Mathematics and Democracy. Recent Advances in Voting Systems and Collective Choice. Springer, Berlin. [N.10] Edelman, P.H. (2015): Voting power apportionments. Social Choice and Welfare 44, 911–925. [N.10] References 333

Erdmannsdörffer, H.G. (1932): Sehr geehrter Herr Thomas Mann! Brief vom 24. August 1932. Thomas-Mann-Archiv, ETH Zürich (www.uni-augsburg.de/bazi/Erdmannsdoerffer1932.html). [N.2] Ernst, L.R. (1994): Apportionment methods for the House of Representatives and the court challenges. Management Science 40, 1207–1227. [N.7] European Council (2013): European Council decision of 28 June 2013 establishing the compo- sition of the European Parliament. Official Journal of the European Union L 181, 29.6.2013, 57–58. [12.7] European Council (2016): Council decision of 8 December 2016 amending the Council’s Rules of Procedure. Official Journal of the European Union L 348, 12.12.2016, 27–29. [12.7] European Parliament (2007): European Parliament resolution of 11 October 2007 on the compo- sition of the European Parliament. Official Journal of the European Union C 227 E, 4.9.2008, 132–138. [12.7] European Union (2012): Consolidated version of the Treaty on European Union. Official Journal of the European Union C 326, 26.10.2012, 13–45. [12.7] Feller, W. (1971): An Introduction to Probability Theory and Its Applications,Volume I, Third Edition. Wiley, New York. [6.7, N.4] Felsenthal, D.S./Machover, M. (1998): The Measurement of Voting Power. Theory and Practice, Problems and Paradoxes. Elgar, Cheltenham. [12.10] Felsenthal, D.S./Miller, N.R.: What to do about election inversions under proportional representa- tion? Representation 51, 173–186. [N.11] Ford, L.R./Fulkerson, D.R. (1962): Flows in Networks. Princeton University Press, Princeton NJ. [N.15] Gaffke, N./Pukelsheim, F. (2008a): Divisor methods for proportional representation systems: An optimization approach to vector and matrix apportionment problems. Mathematical Social Sciences 56, 166–184. [N.15] Gaffke, N./Pukelsheim, F. (2008b): Vector and matrix apportionment problems and separable convex integer optimization. Mathematical Methods of Operations Research 67, 133–159. [N.15] Gallagher, M. (1992): Comparing proportional representation electoral systems: Quotas, thresh- olds, paradoxes and majorities. British Journal of Political Science 22, 469–496. [N.11] Gallagher, M./Mitchell, T. (2008): The Politics of Electoral Systems. Oxford University Press, Oxford. [N.1] Gassner, M.B. (2000): Représentations parlementaires. Méthodes mathématiques biproportion- nelles de répartition des sièges. Édition de l’Université de Bruxelles, Bruxelles. [N.14] Gauss, C.F. (1808): Theorematis Arithmetici Demonstratio Nova. Carl Friedrich Gauss Werke 2, 3–8. [3.2] Gauss, C.F. (1821): Anzeige, Theoria Combinationis observationum erroribus minimis obnoxiae, pars prior. Carl Friedrich Gauss Werke 4, 95–100. [10.2] Getz, G. (1864): Zum Wahlgesetz. Frankfurter Reform Zeitung, 29. January 1864, 50–51. [N.5] Gfeller, J. (1890): Du transfert des suffrages et de la répartition des sièges complémentaires. Représentation proportionnelle—Revue mensuelle 9, 120–131. [6.3, 6.6, 11.11, N.4, N.11] Grilli di Cortona, P./Manzi, C./Pennisi, A./Ricca, F./Simeone, B. (1999): Evaluation and Optimiza- tion of Electoral Systems. SIAM, Philadelphia PA. [N.10] Grimmett, G.R./Oelbermann, K.-F./Pukelsheim, F. (2012): A power-weighted variant of the EU27 Cambridge Compromise. Mathematical Social Sciences 63, 136–140. [N.12] Grimmett, G.R./Laslier, J.-F./Pukelsheim, F./Ramírez González, V./Rose, R./Słomczynski,´ W./ Zachariasen, M./Zyczkowski,˙ K. (2011): The Allocation between the EU Member States of the Seats in the European Parliament. European Parliament, Directorate-General for Internal Policies, Policy Department C: Citizen’s Rights and Constitutional Affairs, Note 23.03.2011, PE 432.760. [12.8, N.12] Grofman, B./Lijphart, A. (1986): Electoral Laws and Their Political Consequences. Agathon Press, New York. [N.1] 334 References

Hägele, G./Pukelsheim, F. (2001): Llull’s writings on electoral systems. Studia Lulliana 41, 30–38. [N.2] Hägele, G./Pukelsheim, F. (2008): The electoral systems of Nicholas of Cusa in the Catholic Concordance and beyond. Pages 229–249 in: The Church, the Councils, & Reform. The Legacy of the Fifteenth Century. Catholic University of America Press, Washington. [N.2] Happacher, M. (1996): Die Verteilung der Diskrepanz bei stationären Multiplikatorverfahren zur Rundung von Wahrscheinlichkeiten. Logos Verlag, Berlin. [6.3, 6.7, 6.8, N.6] Happacher, M. (2001): The discrepancy distribution of stationary multiplier rules for rounding probabilities. Metrika 53, 171–181. [N.6] Happacher, M./Pukelsheim, F. (1996): Rounding probabilities: Unbiased multipliers. Statistics & Decisions 14, 373–382. [N.4] Happacher, M./Pukelsheim, F. (1998): And round the world away. Pages 93–108 in: Proceedings of the Conference in Honor of Shayle R. Searle. Cornell University, Ithaca NY. [N.6] Happacher, M./Pukelsheim, F. (2000): Rounding probabilities: Maximum probability and mini- mum complexity multipliers. Journal of Statistical Planning and Inference 85, 145–158. [N.6] Heinrich, L./Pukelsheim, F./Schwingenschlögl, U. (2004): Sainte-Laguë’s chi-square divergence for the rounding of probabilities and its convergence to a stable law. Statistics & Decisions 22, 43–60. [N.6] Heinrich, L./Pukelsheim, F./Schwingenschlögl, U. (2005): On stationary multiplier methods for the rounding of probabilities and the limiting law of the Sainte-Laguë divergence. Statistics & Decisions 23, 117–129. [N.6, N.7] Heinrich, L./Pukelsheim, F./Wachtel, V. (2017): The variance of the discrepancy distribution of rounding procedures, and sums of uniform random variables. Metrika 80, 363–375. [N.6] Hermsdorf, F. (2009): Demokratieprinzip versus Erfolgswertgleichheit. Verfahren der Mehrheits- treue bei Parlamentswahlen. Zeitschrift für Parlamentsfragen 40, 86–95. [N.11] Hix, S./Johnston, R./McLean, I./Cummine, A. (2010): Choosing an Electoral System. British Acad- emy, London. [N.9] Hylland, A. (1978): Allotment Methods: Procedures for Proportional Distribution of Indivisible Entities. Mimeographed Report, John F. Kennedy School of Government, Harvard University. Published as: Working Paper 1990/11. Research Centre of the Norwegian School of Manage- ment, Sandvika. [N.4, N.5, N.9] Ichimori, T. (2012): A note on relaxed divisor methods. Operations Research Society of Japan 55, 225–234. [N.3] Iverson, K.E. (1962): A Programming Language. Wiley, New York. [N.3] Janson, S. (2011): Another note on the Droop quota and rounding. Voting matters 29, 32–34. [N.9] Janson, S. (2012): Proportionella valmetoder. Typescript, Uppsala, 20.08.2012. [6.10, N.3, N.5, N.8, N.11] Janson, S. (2014): Asymptotic bias of some election methods. Annals of Operations Research 215, 89–136. [N.6, N.7] Janson, S./Linusson, S. (2012): The probability of the Alabama paradox. Journal of Applied Probability 49, 773–794. [N.9] Jellinek, W. (1926): Verhältniswahl und Führerauslese. Archiv des öffentlichen Rechts 50, 71–99. [N.2] Joas, B. (2005): A graph theoretic solvability check for biproportional multiplier methods. Thesis. Institut für Mathematik, University of Augsburg. [N.15] Jones, M.A./Wilson, J.M. (2010): Evaluation of thresholds for power mean-based and other divisor methods of apportionment. Mathematical Social Sciences 59, 343–348. [N.11] Kommers, D.P./Miller, R.A. (2012): The Constitutional Jurisprudence of the Federal Republic of Germany. Third Edition, Revised and Expanded. Duke University Press, Durham. [N.13] Kopfermann, K. (1991): Mathematische Aspekte der Wahlverfahren. Mandatsverteilung bei Abstimmungen. Bibliographisches Institut, Mannheim. [3.12, 16.12, N.3, N.5, N.6, N.11] Lachapelle, G. (1911): La représentation proportionnelle en France et en Belgique. Préface de Henri Poincaré. Maisons Félix Alcan et Guillaumin Réunies, Paris. [N.7] References 335

Lauwers, L.L./Van Puyenbroeck, T. (2006a): The Hamilton apportionment method is between the Adams method and the Jefferson method. Mathematics of Operations Research 31, 390–397. [N.8] Lauwers, L.L./Van Puyenbroeck, T. (2006b): The Balinski-Young comparison of divisor methods is transitive. Social Choice and Welfare 26, 603–606. [N.8] Maier, S. (2009): Biproportional Apportionment Methods: Constraints, Algorithms, and Simula- tion. Dr. Hut, München. [N.15] Maier, S./Zachariassen, P./Zachariasen, M. (2010): Divisor-based biproportional apportionment in electoral systems: A real-life benchmark study. Management Science 56, 373–387. [N.15] Marshall, A.W./Olkin, I. (1979): Inequalities: Theory of Majorization and Its Applications. Academic Press, New York. [8.6, N.8] Marshall, A.W./Olkin, I./Arnold, B.C. (2011): Inequalities: Theory of Majorization and Its Applications. Springer, New York. [N.8] Marshall, A.W./Olkin, I./Pukelsheim, F. (2002): A majorization comparison of apportionment methods in proportional representation. Social Choice and Welfare 19, 885–900. [N.3, N.8] Martínez-Aroza, J./Ramírez-González, V. (2008): Several methods for degressively proportional allotments. A case study. Mathematical and Computer Modelling 48, 1439–1445. [N.12] McLean, I. (2008): Don’t let the lawyers do the math: Some problems of legislative districting in the UK and the USA. Mathematical and Computer Modelling 48, 1446–1454. [N.9] McLean, I. (2014): Three apportionment problems, with applications to the United Kingdom. Pages 363–380 in: Voting Power and Procedures – Essays in Honour of Dan Felsenthal and Moshé Machover. Springer, Cham. [N.9] McLean, I./Urken, A.B. (1995): Classics of Social Choice. University of Michigan Press, Ann Arbor. [N.2] Meyer, H. (2010): Die Zukunft des Bundestagswahlrechts. Zwischen Unverstand, obiter dicta, Interessenkalkül und Verfassungsverstoß. Nomos, Baden-Baden. [N.13] Miller, N.R. (2015): Election inversions under proportional representation. Scandinavian Political Studies 38, 4–25. [N.11] Mora, X. (2013): La regla de Jefferson-D’Hondt i les seves alternatives. Materials Matemàtics 2013/4. [16.6, N.10] Morin, A. (1862): De la représentation des minorités. Fick, Genève. [N.5] Mosteller, F./Youtz, C./Zahn, D. (1967): The distribution of sums of rounded percentages. Demo- graphy 4, 850–858. [N.6] Nohlen, D. (2009): Wahlrecht und Parteiensystem. Zur Theorie und Empirie der Wahlsysteme. 6. Auflage. Budrich, Opladen. [N.1] Oelbermann, K.-F. (2013): Biproportionale Divisormethoden und der Algorithmus der alternieren- den Skalierung. Logos Verlag, Berlin. [15.8, N.15] Oelbermann, K.-F. (2016): Alternate scaling algorithm for biproportional divisor methods. Mathe- matical Social Sciences 80, 25–32. [N.15] Oelbermann, K.-F./Pukelsheim, F. (2011): Future European Parliament elections: Ten steps towards uniform procedures. Zeitschrift für Staats- und Europawissenschaften—Journal for Comparative Government and European Policy 9, 9–28. [N.14] Oelbermann, K.-F./Pukelsheim, F. (2015): European elections 2014: From voters to representa- tives, in twenty-eight ways. Evropská volební studia—European Electoral Studies 10, 91–124. [N.1] Oelbermann, K.-F./Palomares, A./Pukelsheim, F. (2010): The 2009 European Parliament elections: From votes to seats in 27 ways. Evropská volební studia—European Electoral Studies 5, 148– 182 (Erratum, ibidem 6, 2011, 85). [N.1] Oyama, T. (1991): On a parametric divisor method for the apportionment problem. Journal of the Operations Research Society of Japan 34, 187–221. [N.3] Oyama, T./Ichimori T. (1995): On the unbiasedness of the parametric divisor method for the apportionment problem. Journal of the Operations Research Society of Japan 38, 301–321. [N.3] 336 References

Palomares, A./Ramírez, V. (2003): Thresholds of the divisor methods. Numerical Algorithms 34, 405–415. [N.3, N.11] Palomares, A./Pukelsheim, F./Ramírez, V. (2016): The whole and its parts: On the Coherence Theorem of Balinski and Young. Mathematical Social Sciences 83, 11–19. [N.9] Peifer, R./Lübbert, D./Oelbermann, K.-F./Pukelsheim, F. (2012): Direktmandatsorientierte Pro- porzanpassung: Eine mit der Personenwahl verbundene Verhältniswahl ohne negative Stimm- gewichte. Deutsches Verwaltungsblatt 127, 725–730. [N.13] Pennisi, A./Ricca, F./Simeone B. (2006): Bachi e buchi della legge elettorale italiana nell’alloca- zione biproporzionale di seggi. Sociologia e ricerca sociale 79, 55–76. [N.14] Pennisi, A./Ricca, F./Simeone B. (2009): Una legge elettorale sistematicamente erronea. Polena: Political and Electoral Navigations—Rivista Italiana di Analisi Elettorale 6, 65–72. [N.14] Pukelsheim, F. (1993): Optimal Design of Experiments. Wiley, New York. Reprinted as: Classics In Applied Mathematics, 50, SIAM, Philadelphia PA, 2006. [N.3, N.10] Pukelsheim, F. (2000a): Mandatszuteilungen bei Verhältniswahlen: Erfolgswertgleichheit der Wählerstimmen. Allgemeines Statistisches Archiv—Journal of the German Statistical Society 84, 447–459. [N.2] Pukelsheim, F. (2000b): Mandatszuteilungen bei Verhältniswahlen: Vertretungsgewichte der Man- date. Kritische Vierteljahresschrift für Gesetzgebung und Rechtswissenschaft 83, 76–103. [N.2] Pukelsheim, F. (2000c): Mandatszuteilungen bei Verhältniswahlen: Idealansprüche der Parteien. Zeitschrift für Politik—Organ der Hochschule für Politik München 47, 239–273. [N.2] Pukelsheim, F./Grimmett, G. (2017): Linking the permanent system of the distribution of seats in the European Parliament with the double-majority voting in the Council of Ministers. Pages 5–22 in: The Composition of the European Parliament. Workshop 30 January 2017. European Parliament, Directorate-General for Internal Policies, Policy Department C: Citizens’ Rights and Constitutional Affairs, PE 583.117. [N.12] Pukelsheim, F./Leutgäb, P. (2009): List apparentements in local elections: A lottery. Homo Oeconomicus 26, 489–500). [N.7] Pukelsheim, F./Maier, S. (2006): A gentle majority clause for the apportionment of committee seats. Pages 177–188 in: Mathematics and Democracy. Recent Advances in Voting Systems and Collective Choice. Springer, Berlin. [N.11] Pukelsheim, F./Maier, S. (2008): Parlamentsvergrößerung als Problemlösung für Überhangman- date, Pattsituationen und Mehrheitsklauseln. Zeitschrift für Parlamentsfragen 39, 312–322. [N.11] Pukelsheim, F./Maier, S./Leutgäb, P. (2009): Zur Vollmandat-Sperrklausel im Kommunalwahlge- setz. Nordrhein-Westfälische Verwaltungsblätter 22, 85–90. [N.11] Pukelsheim, F./Ricca, F./Scozarri, A./Serafini, P./Simeone, B. (2012): Network flow methods for electoral systems. Networks 59, 73–88. [N.15] Pukelsheim, F./Rossi, M. (2013): Imperfektes Wahlrecht. Zeitschrift für Gesetzgebung 28, 209– 226. [N.13] Pukelsheim, F./Schuhmacher, C. (2004): Das neue Zürcher Zuteilungsverfahren für Parlamentswahlen. Aktuelle Juristische Praxis—Pratique Juridique Actuelle 13, 505–522. [N.14] Pukelsheim, F./Schuhmacher, C. (2011): Doppelproporz bei Parlamentswahlen: Ein Rück- und Ausblick. Aktuelle Juristische Praxis—Pratique Juridique Actuelle 20, 1581–1599. [N.14] Ramírez González, V. (2013): Sistema electoral para el Congreso de los Diputados. Editorial Universidad de Granada, Granada. [N.14] Ramírez González, V. (2017): The r-DP methods to allocate the EP seats to Member States. Pages 23–36 in: The Composition of the European Parliament. Workshop 30 January 2017. European Parliament, Directorate-General for Internal Policies, Policy Department C: Citizens’ Rights and Constitutional Affairs, PE 583.117. [N.12] Ramírez, V./Pukelsheim, F./Palomares, A./Martínez, M.L. (2008): A bi-proportional method applied to the Spanish Congress. Mathematical and Computer Modelling 48, 1461–1467. [N.14] References 337

Richter, H. (2016): Transnational reform and democracy: Election reforms in New York City and Berlin around 1900. Journal of the Gilded Age and Progressive Era 15, 149–175. [N.2] Riker, W.H. (1982): Liberalism Against Populism. A Confrontation Between the Theory of Democracy and the Theory of Social Choice. Freemann, San Francisco. [7.6] Robinson, A./Ullman, D.H. (2011): A Mathematical Look at Politics. CRC Press, Boca Raton. [N.1] Rockafellar, R.T. (1970): Convex Analysis. Princeton University Press, Princeton NJ. [15.5] Rogge, J. (1994): Ir freye wale zu haben. Möglichkeiten, Probleme und Grenzen der politischen Partizipation in Augsburg zur Zeit der Zunftverfassung (1368–1548). Bürgertum—Beiträge zur europäischen Gesellschaftsgeschichte 7, 243–277. [N.2] Röper, E. (1998): Ausschußsitzverteilung nur unter den stimmberechtigten Mitgliedern. Zeitschrift für Parlamentsfragen 29, 313–316. [N.9] Rose, R. (1974): Electoral Behavior: A Comparative Handbook. Free Press, New York. [N.1] Rosenbach, H. (1994): Der Parlamentarische Rat 1948–1949. Akten und Protokolle. Band 6: Aus- schuß für Wahlrechtsfragen. Boldt, Boppard am Rhein. [N.2] Rote, G./Zachariasen, M. (2007): Matrix scaling by network flow. Pages 848–854 in: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms. Proceedings in Applied Mathematics, Volume 125. SIAM, Philadelphia PA. [N.15] Ruiz-Rufino, R. (2011): Characterizing electoral systems: An empirical application of aggregated threshold functions. West European Politics 34, 256–281. [N.11] Rouyer, L. (1901): Théorie mathématique de la représentation proportionnelle. Typescript, 12 pages. Ligue pour la Représentation Proportionnelle, Paris. [N.10] Rza¸˙zewski, K./Słomczynski,´ W./Zyczkowski,˙ K. (2014): Kazdy˙ głos sie¸ liczy! Wydawnictwo Sejmowe, Warszawa. [N.2] Schmidt, R. (1929): Verhältniswahl und Führerauslese. Schulze, Gleiwitz. [N.2] Schreiber, W. (2009): Kommentar zum Bundeswahlgesetz. 8. Auflage. Heymann, Köln. [N.13] Schuster, K./Pukelsheim, F./Drton, M./Draper, N.R. (2003): Seat biases of apportionment methods for proportional representation. Electoral Studies 22, 651–676. [7.6, N.7] Schwingenschlögl, U. (2008): Asymptotic equivalence of seat bias models. Statistical Papers 49, 191–200. [N.7] Schwingenschlögl, U./Pukelsheim, F. (2006): Seat biases in proportional representation systems with thresholds. Social Choice and Welfare 27, 189–193. [N.7] Seal, H.L. (1950): Spot the prior reference. Journal of the Institute of Actuaries Students’ Society 10, 255–258. [N.3, N.6] Serafini, P. (2012): Allocation of the EU Parliament seats via integer linear programming and revised quotas. Mathematical Social Sciences 63, 107–113. [N.14] Shugart, M.S./Wattenberg, M.P. (2001): Mixed-Member Electoral Systems: The Best of Both Worlds? Oxford University Press, Oxford. [N.2] Słomczynski,´ W./Zyczkowski,˙ K. (2006): Penrose voting system and optimal quota. Acta Physica Polonica B 37, 3133–3143. [12.10] Słomczynski,´ W./Zyczkowski,˙ K. (2012): Mathematical aspects of degressive proportionality. Mathematical Social Sciences 63, 94–101. [N.12] Słomczynski,´ W./Zyczkowski,˙ K. (2017): Degressive proportionality in the European Union. Pages 37–48 in: The Composition of the European Parliament. Workshop 30 January 2017. European Parliament, Directorate-General for Internal Policies, Policy Department C: Citizens’ Rights and Constitutional Affairs, PE 583.117. [N.12] Szpiro, G.G. (2010): Numbers Rule. The Vexing Mathematics of Democracy, from Plato to the Present. Princeton University Press, Princeton NJ. [N.2] Taagepera, R./Shugart, M.S. (1989): Seats and Votes. The Effects and Determinants of Electoral Systems. Yale University Press, New Haven CT. [N.1] Theil, H. (1969): The desired political entropy. American Political Science Review 63, 521–525. [N.12] Theil, H./ Schrage, L. (1977): The apportionment problem and the European Parliament. European Economic Review 9, 247–263. [N.12] 338 References

Tukey, J.W. (1938): On the distribution of the fractional part of a statistical variable. Recueil Mathé- matique Nouvelle Série (Matematicheskii Sbornik Novaya Seriya) 4, 461–462. [N.6] Wallis, W.A./Roberts, H.V. (1956): Statistics. A new approach. Methuen, London. [N.3] Young, H.P. (1994): Equity. In Theory and Practice. Princeton University Press, Princeton NJ. [N.9] Zachariasen, M. (2006): Algorithmic Aspects of Divisor-Based Biproportional Rounding. Techni- cal Report no. 06/05, Department of Computer Science, University of Copenhagen. [N.15] Zachariassen, P./Zachariassen, M. (2006): A comparison of electoral formulae for the Farœse Parliament. Pages 235–251 in: Mathematics and Democracy. Recent Advances in Voting Systems and Collective Choice. Springer, Berlin. [N.14] Index

Items are indexed by section numbers. absolutely continuous distribution, 6.4, 6.10, compatibility, 3.9, 6.4, 9.3, 14.4 6.11, 7.10, 10.2 completeness, 9.3 additional member proportional (AMP) compliant restrictions, 12.3, 15.6 system, 12.4 componentwise ordering, 9.5 adjusted divisor, 4.11, 6.1, 6.4, 12.4 compositional proportionality, 14.7, 14.8 adjusted house size, 6.1, 6.4 concordance, 4.3, 8.1, 9.1, 12.7, 14.4 alliance of lists, 4.3, 7.9, 7.11 Croatia, 1.5, 1.9, 1.11 anonymity, 4.3, 8.1, 9.1, 9.2 cross voting, 4.9 apportionment method, 4.4, 5.10 cycle of cells, 15.3, 15.5 apportionment rule, 4.2, 4.4 Cyprus, 1.3, 1.5, 1.9 Augsburg, 2.5, 6.8 Czech Republic, 1.3, 1.5, 1.9 Austria, 1.2, 1.5, 1.9, 3.3, 4.11, 5.8, 6.1, 14.8

decency, 4.3, 5.10, 9.1, 9.5 balancedness, 4.3, 9.1 decision power, 12.10 base seats, 12.8 decrement option, 4.6, 4.7, 4.8, 15.4 Basel, 16.6 degressive proportionality, 12.7, 12.9, 14.7 Bavaria, 7.6, 7.7, 7.9, 10.10, 13.2, 13.3 Denmark, 1.4, 1.5, 1.9, 12.10, 16.4 Belgium, 1.2, 1.5, 1.9, 1.10, 1.11, 2.3, 12.10 direct seats, 12.1 Boostedt, 11.7, 11.8, 11.9 direct-seat restricted variant, 12.4, 12.6, Buchberg-Rüdlingen, 14.2, 14.4 13.1, 13.3 Bulgaria, 1.2, 1.5, 1.9, 5.2, 5.3, 5.8, 5.9 discordance, 4.3, 7.9, 14.3, 14.4, 14.6 discrepancy, 3.12, 6.1, 6.2, 6.6, 6.9, 6.10 disparity function, 10.7, 10.9 Cambridge compromise (CamCom), 12.8, district divisor, 14.2, 15.2 12.10 district magnitude, 4.9, 14.2 ceiling function, 3.5 districts, 7.7, 9.13, 12.4, 14.1 chi-square statistic, 10.2, 10.5 divisor interval, 4.6 Code of Good Practice in Electoral Matters, divisor method, 4.5, 4.15, 5.5, 5.10, 6.2, 8.5, 2.5 9.3, 9.4, 9.8, 11.2, 12.3 Coesfeld, 11.11 with downward rounding (DivDwn), 4.2, coherence, 9.1, 9.2, 9.3, 9.5, 9.8, 11.9 4.11, 4.14, 6.3, 6.6, 7.5, 7.7, 7.9, 8.7, commercial rounding function, 3.6 10.3, 11.3, 11.7, 11.10, 11.12, 12.4, comparative scores, 4.12, 4.14 16.1, 16.6, 16.7, 16.13, 16.14

© Springer International Publishing AG 2017 339 F. Pukelsheim, Proportional Representation, DOI 10.1007/978-3-319-64707-4 340 Index

with geometric rounding (DivGeo), 7.5, Hagenbach-Bischoff method, 16.6 10.5, 10.9, 10.10, 12.1, 16.8, 16.11, Hamburg, 9.11 16.13 Hamilton method, 5.7 with harmonic rounding (DivHar), 10.9, Hare-quota, 5.2, 5.7, 5.8, 5.10, 12.2 10.10 Hare-quota method with residual fit by greatest with standard rounding (DivStd), 4.14, 5.3, remainders (HaQgrR), 4.10, 5.2, 7.4, 8.7, 9.10, 9.13, 10.2, 10.8, 10.9, 5.3, 5.4, 5.7, 7.5, 7.7, 7.12, 9.10, 10.10, 11.4, 11.10, 11.11, 12.6, 13.1, 9.13, 10.6, 11.5, 11.10, 11.11, 12.2, 13.2, 14.2, 14.8, 16.3, 16.9, 16.10, 16.13 16.13 Hare-quota variants, 5.8, 16.4 with upward rounding (DivUpw), 7.5, 8.7, harmonic rounding, 3.11 10.4, 11.12, 12.8 highest averages, 4.14 doppelter Pukelsheim, see double-proportional homogeneity, 4.3 divisor method house size, 4.1 double majority (DM) voting rule, 12.10 house size monotonicity, 9.5, 13.4 double-proportional divisor method, 14.1, house size recommendation, 7.6, 14.8 14.2, 14.8, 15.2 Hungary, 1.5, 1.6, 1.9 downward rounding, 3.4, 3.10, 3.11, 6.3 Droop-quota, 5.8, 5.10, 6.6 Droop-quota method with residual fit by ideal share of seats, 4.5, 5.2, 5.3, 6.1, 6.2, 7.1, greatest remainders (DrQgrR), 7.12 9.10, 12.2 Droop-quota variants, 5.8, 5.9, 9.14 imperative dual algorithms, 4.10, 15.7 loser-get-nil, 5.9 loser-take-all, 6.8 winner-take-all (WTA), 5.9, 6.8 effective votes, 1.5, 7.3 winner-take-one (WTO), 14.4, 14.8, 15.2 Estonia, 1.4, 1.5, 1.9 imperviousness, 3.8, 3.9, 4.5, 12.1, 15.6, 15.9 even-number method, 4.14 increment option, 4.6, 4.7, 4.8, 15.4 even-number rounding function, 3.6 indep (independent candidate), 1.4, 1.9, 12.4 exactness, 4.3, 9.1 integral part, 3.2 exchangeability, 6.9 invariance principle for rounding residuals, 6.9, 6.10, 7.10, 7.12 Ireland, 1.5, 1.6, 1.9, 1.10, 1.11, 2.3, 5.8, 12.10, fairness, see coherence 16.4 finitary vectors, 4.2 Italy, 1.5, 1.6, 1.9, 1.11, 2.3, 5.8, 12.9, 12.10 Finland, 1.5, 1.9 itio in partes, 11.9 floor function, 3.2 flow inequality, 15.6 fractional part, 3.2 Jagiellonian compromise (JagCom), 12.10 France, 1.5, 1.9, 1.10, 1.11, 2.1, 2.3, 7.5, 12.1, Jefferson method, 16.1 12.10 journalists’ apportionment method, 6.8 Friedberg, 7.9 jump-and-step procedure, 4.7, 4.10, 4.14, 7.10, full-seat modification, 5.9, 11.4, 11.6 14.8, 16.6 fundamental relation, 3.9

largest remainder (LR) method, 5.7 Gauss brackets, 3.2 Latvia, 1.5, 1.7, 1.9 geometric rounding, 3.11 left-right disjunction, 3.8, 4.5, 9.8, 11.12 Germany, 1.3, 1.5, 1.9, 1.10, 1.11, 2.1, 2.3, 2.4, Lithuania, 1.5, 1.7, 1.9, 5.8, 5.9, 12.9 2.5, 2.7, 2.11, 3.12, 5.7, 9.11, 9.13, localization property, 3.8, 4.5, 8.5, 9.7, 9.8 11.7, 11.10, 11.11, 12.6, 12.6, 12.8, London, 12.4, 12.6, 13.1 12.10, 13.1, 14.8, 16.12 Lower Saxony, 10.10 Greece, 1.4, 1.5, 1.7, 5.9 Luxembourg, 1.5, 1.7, 1.9, 12.10 Index 341 main apportionment, 5.1, 5.2, 5.3, 5.7, 5.8, party divisor, 14.2, 15.2 12.2 party system major fraction, 3.6, 16.3 bipartition by vote strengths, 8.1 majority clauses 11.7 partition by government majority and house-size augmentation clause, 11.8, opposition minority, 11.9 11.11 partitioning into list alliances, 7.9 majority-minority partition clause, 11.9 size, 4.1 residual-seat redirection clause, 11.11 tripartition by vote strengths, 7.8 majorization, 8.1, 10.3, 11.5, 12.9 party votes, 14.2 Malta, 1.5, 1.8, 1.9, 1.11, 12.8, 12.9, 16.4 Peace of Westphalia, 11.9 max-flow min-cut theorem, 15.6 perviousness, 3.8, 3.9, 4.5, 15.6, 15.9 max-min inequality, 4.6, 4.8, 5.5, 15.5, 16.6 Poland, 1.5, 1.8, 1.9, 1.10, 1.11, 2.3, 7.5, 12.9 method of alternate ratios, 16.8, 16.11 porportionality loss, 12.5 method of equal proportions, 12.1, 16.11 Portugal, 1.5, 1.9, 12.8, 12.9 method of least squares, 16.13 power compromise (PowCom), 12.9 method of major fractions, 16.3, 16.9 power interval, 12.9 method of rejected fractions, 16.1 power-adjusted population units, 12.9 minimum requirements, 14.2, 14.4 power-mean divisor methods (DivPwrp), 8.6, minimum-maximum restrictions, 12.1, 12.3 8.7 minor fraction, 3.6, 16.3 power-mean signposts, 3.11 mixed member proportional (MMP) system, preservation of zeros, 15.5 12.4 primal algorithms, 4.10, 15.7 motto probability simplex, 6.4, 6.5, 6.8, 6.10, 6.11, Below one is none, 11.4 7.4, 7.10, 7.11, 7.12 Divide and rank, 1.9, 5.1, 5.6 proportionality loss, 12.5 Divide and round, 1.9, 4.4 Divide and split, 5.6 One person, one vote, 14.6, 14.7 qualified majority voting (QMV) rule, 12.7 Stronger parties first, 4.9, 9.3 quota, 5.1, 5.8, 5.10, 12.10, 16.4 multiplier method, 4.5 quota methods, 4.15, 5.1, 5.5, 5.10, 6.6 natural two-party apportionment, 9.10, 14.7 rank score, 7.2, 7.8 negative voting weight, 2.4, 9.14, 13.3 representation ratio, 12.7 The Netherlands, 1.5, 1.8, 1.9, 1.11, 7.5, 12.9, representative weight, 10.5, 12.8 12.10 residual fit, 5.1, 5.2, 5.3, 5.4, 5.7, 5.8, 5.9, 5.10, no input–no output law, 3.9, 12.3, 15.1, 15.3 12.2 North Rhine-Westphalia, 11.6, 11.11 Romania, 1.5, 1.9, 11.6 rounding function, 3.1, 3.9 rounding rule, 3.9, 4.4, 12.3, 15.2 odd-number method, 4.14, 16.10 run-off election, 4.9 overall party-seats, 14.2 Russian Federation, 6.8 overhang seat, 2.1, 2.2, 2.3, 12.6, 13.3, 13.5, 16.12 overrepresentation, 7.5, 10.3, 10.7, 15.9 Schaffhausen, 14.2, 15.8 Schleswig-Holstein, 11.7 Scotland, 12.4, 12.6, 13.1 pairwise comparisons, 10.7, 16.10 seat bias, 7.4, 7.8, 7.9, 7.10, 7.10, 10.3, 11.8, panachage, 4.9 12.8, 14.2, 16.9, 16.11, 16.14 paradoxes 9.4 seat excess, 7.1, 7.10, 11.5 Alabama paradox, 9.12 seat matrix, 15.1 new states paradox, 9.11 seat number, 4.1 no-show paradox, 9.14 seat total, 6.1 population paradox, 9.13 seat total distributions, 6.5 342 Index seat vector, 4.1 traditional divisor methods, 4.5, 8.6, 10.10 select divisor, 4.7, 4.11, 5.3, 15.2, 15.8, 15.9 traditional rounding rules, 3.11 select powers, 8.7 transfer-proof method, 9.9 select split, 5.3, 8.7 truncated rounding rule, 12.3, 15.2 shift-quota methods (shQgrRs ), 5.4, 6.1, 8.8, 11.5 signpost sequence, 3.8, 4.4, 8.5, 9.7, 11.2, 15.2 unbiased seat numbers, 7.4, 7.5, 7.10, 7.12 simple rounding, 3.12 underrepresentation, 7.5, 10.4, 10.7, 12.7, 15.9 single transferable vote (STV) scheme, 5.7 uniform distribution, 6.4, 6.5, 6.7, 7.4, 7.12 Slovakia, 1.5, 1.9, 1.10, 5.8 United Kingdom, 1.5, 1.9, 1.10, 1.11, 2.3, 12.4, Slovenia, 1.5, 1.9, 1.10 12.6, 12.9 Solothurn, 1.7, 5.8, 7.6 United States of America, 2.5, 6.8, 7.5, 12.1 solution set, 4.2, 4.7, 5.6 universal initialization, 4.12, 6.1 Spain, 1.5, 1.9, 1.11, 4.9, 7.7, 9.3 upward rounding, 3.5, 3.10, 3.11, 6.3 split interval, 5.5, 5.6 Uster, 4.9 standard rounding, 3.7, 3.10, 3.11, 6.3, 9.10 stationary divisor methods (DivStar), 4.5, 5.6, 6.4, 6.5, 6.7, 6.9, 6.10, 6.11, 7.4, 8.6, valid votes, 7.3 8.7, 9.9, 11.3 vote count, 4.3, 4.5, 6.4, 8.1 stationary signposts, 3.10 vote pattern, 14.8 success value of a voter’s vote, 2.7, 2.10, 10.2, vote ratio monotonicity, 9.9, 9.14 10.3, 10.4, 10.7, 10.8, 13.2, 14.2, vote sequence, 4.2 14.6, 14.7 vote share, 4.1, 4.3, 4.5, 6.4, 8.1 Sweden, 1.5, 1.9, 11.4, 12.8 vote vector, 4.1 Swedish 1952 modification, 1.9 vote weight, 4.1 Swedish 2014 modification, 11.4 voter monotonicity, 9.14 swing voters, 9.9 voter number, 4.9, 14.2, 15.2 Switzerland, 11.6, 12.1, 14.5 votes-to-seats averages, 4.14 votes-to-seats ratio, 4.5, 4.7, 4.12, 5.2, 5.7, 5.8, 6.1, 6.3, 10.5, 10.10, 13.5 target seats, 13.4 voting weight, 12.10 threshold, 7.3, 11.6 of exclusion, 11.1, 11.5 of representation, 11.1, 11.5 weight matrix, 15.1 tie, 3.3, 3.9, 4.2, 4.8, 4.9, 5.2, 6.4, 9.3, 11.1, 12.8, 15.4, 15.8 tie-and-transfer algorithm, 15.9 Zurich, 4.9, 14.2, 16.14