Proportional Representation

Friedrich Pukelsheim

Proportional RepresentationApportionment Methodsand Their Applications

Second Edition

With a Foreword by Andrew Duff MEP

123

Friedrich PukelsheimInstitut fRur MathematikUniversität AugsburgAugsburg, 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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The just, then, is a species of theproportionate : : :. For proportion is equalityof 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 asthat between the other pair; for there is asimilar distinction between the persons andbetween the things. As the term A, then, is toB, so will C be to D, and therefore,alternando, as A is to C, B will be to D : : :.

This, then, is what the just is—theproportional; the unjust is what violates theproportion. Hence one term becomes toogreat, the other too small, as indeed happensin practice; for the man who acts unjustly hastoo much, and the man who is unjustlytreated 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 UniversityPress.

Foreword to the First Edition

The virtue of parliamentary democracy rests on the representative capability ofits institutions. Even mature democratic states cannot take the strength of itsrepresentative institutions for granted. Newer democracies seek practicable waysand means on which to build lasting structures of governance which will commandthe affinity of the people they are set up to serve. The debate about the structuralreform of parliamentary democracies is never far away. Nor should it be. The powersand composition of parliamentary chambers, their rules and working methods, theorganization 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 andshape of constituencies—all these and more are rightly subject to continual appraisaland are liable to be reformed.

Electoral reform is a delicate business: handled well, it can be the basis on whichnew 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 ofpower, and sow the seeds of destruction of liberty. Electoral systems are centralto the debate in emerging democracies, and the relatively new practice of electionobservation by third parties highlights the need for elections to be run not onlyfairly but also transparently. Voting and counting should be simple, comprehensible,and open to scrutiny—qualities which are too often lacking even in old establisheddemocracies.

Electoral reform is also very difficult to achieve. Those who must legislate forit are those very same people who have a vested interest in the status quo. Thatturkeys don’t vote for Christmas is amply demonstrated in the UK, where reform ofthe House of Lords has been a lost cause for over a century. Advocates of reformneed to stack up their arguments well, be persistent, and enjoy long lives.

Friedrich Pukelsheim has written a definitive work on electoral reform. He takesas his starting point the simple premise that seats won in a parliamentary chambermust represent as closely as possible the balance of the votes cast in the ballotbox. Rigorous in his methodology, the author knows that there is no single perfectelectoral system: indeed, in their quirky details, every system affects the exactoutcome of an election. We are fortunate indeed that this professor of mathematics

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is a profound democrat. He ably brings to the service of politicians the science ofthe mathematician.

Dr Pukelsheim was an indispensable participant at the meeting in Cambridgein 2011, chaired by Geoffrey Grimmett, which devised “CamCom”—the bestconsensual solution to the problem of how to apportion seats in the EuropeanParliament. As the Parliament’s rapporteur for electoral procedure, I am happy thatour ideas are now taken forward in this publication.

The European Parliament

The European Parliament presents unusual challenges both to the scientist andpractitioner. It is one chamber of the legislature of the European Union with a lotof power but little recognition. It reflects a giant historical compromise betweenthe 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 partybut also nationality. The European Parliament is the forum of the political singlemarket where the different political cultures and constitutional practices of the 28member states meet up. MEPs are constitutionally representatives of the Union’scitizens, but they are elected not by a uniform electoral procedure but by differentprocedures under which separate national political parties and candidates fight it out,largely untroubled by their formal affiliation to European political parties.1 Effortsto make more uniform the election of the world’s first multinational parliament tobe directly elected by universal suffrage have been frustrated.

Voter turnout, as we know, has declined at each election to the EuropeanParliament from 62% in 1979 to 43% in 2009, although these overall figures disguisesharp contrasts among the states and between elections. The long financial andeconomic crisis since 2008 has brought to a head a crisis of legitimacy for theEuropean Parliament. If the euro is to be salvaged, and the EU as a whole is toemerge strengthened from its time of trial, transnational democracy needs to workbetter. Banking union and fiscal union need the installation of federal government.That federal government must be fully accountable to a parliament which connectsdirectly to the citizen and with which the citizen identifies. The parliament mustbe composed in a fair and logical way best achieved in accordance with a settledarithmetical formula and not as a result of unseemly political bartering whichborders on gerrymandering and sparks controversy.

It is probable that in spring 2015 there will be a new round of EU constitutionalchange. This will take the form of a Convention in which heads of government andthe European Commission will talk things through with members of the Europeanand 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 forthe apportionment of state seats alongside an ambitious proposal for the creation ofa pan-European constituency for which a certain number of MEPs would be electedfrom transnational party lists.2

There is no reason to doubt that the notion of degressive proportionality, whichstrikes mathematicians as odd, will survive these negotiations because it expressesquite well the broadly understood belief that in a federal polity, the smallerneed to be protected from subordination to the larger. CamCom copes logicallywith degressive proportionality in a way which should satisfy even the austererequirements of the Bundesverfassungsgericht at Karlsruhe.

Nevertheless, as Friedrich Pukelsheim recognizes, fully fledged CamCom meansradical 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 legislatureare balanced by changes to the electoral system in the other. Here, the JagiellonianCompromise, which uses the square root as the basis for weighing the votes of themember states in the Council, deserves a good hearing.

In June 2013, the Council and European Parliament eventually agreed that thenew member state of Croatia should have 11 MEPs in the Parliament which wereelected in May 2014. We worked hard to ensure that the reapportionment of seatswould not contradict the logic of CamCom. There is a first, albeit clumsy, legaldefinition of degressive proportionality. More importantly, the European Union hasnow formally decided to pursue the objective of a formulaic approach to the futuredistribution of seats in the Parliament, coupled with a commitment to revisit thematter 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 electionswhich in the future will make it possible, before each fresh election to the EuropeanParliament, to allocate the seats between Member States in an objective, fair,durable, and transparent way, translating the principle of degressive proportionalityas laid down in Article 1, taking account of any change in their number anddemographic trends in their population, as duly ascertained thus respecting theoverall balance of the institutional system as laid down in the Treaties.

So perhaps CamCom and JagCom are destined to surface together in the next EUtreaty. Legislators who care to understand the maths should start with this book.

Cambridge, UK Andrew Duff MEPSeptember 2013

2For a full exposition of this proposal, see Spinelli Group, A Fundamental Law of the EuropeanUnion, Bertelsmann Stiftung 2013.

Preface to the Second Edition

In this second edition, the text of the first edition has been completely revisedand expanded. Many empirical election results that serve as examples have beenupdated. 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 thepractice of double proportionality, while Chap. 15 explains the required theory.

Chapter 16 is entirely new. It assembles biographical sketches and authoritativequotes from individuals who coined the development of apportionment methodol-ogy.

The author is grateful to the attentive readers of the first edition who providedfeedback and valuable criticisms.

Particular thanks are due to a team of expert scholars and devoted friends whoworked through most or all of the draft for this second edition. Their criticaladvice and numerous suggestions were vital to improve and clarify the exposition.I would like to thank Wolfgang Bischof, Hochschule Rosenheim (DE); XavierMora, Universitat Autònoma de Barcelona (ES); Grégoire Nicollier, Haute Ecoled’Ingénierie, Sion (CH); Antonio Palomares, Universidad de Granada (ES); and BenTorsney, University of Glasgow (UK).

Augsburg, Germany Friedrich PukelsheimJune 2017

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Preface to the First Edition

Proportional representation systems determine how the political views of individualcitizens, 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 berepresented in a committee of a size much smaller than the Parliament itself. Thereare many similar examples all showing that proportional representation inevitablyculminates in the task of translating numbers into numbers—large numbers of thoseto be represented into small numbers of those serving as representatives. The task issolved by procedures called apportionment methods. Apportionment methods andtheir applications are the theme of this work. A more detailed Outline of the Bookfollows the Table of Contents.

By profession a mathematician rather than a politician, I have had the privilegeof getting involved in several proportional representation reform projects in recentyears. These include the introduction of a double-proportional electoral systemin several Swiss cantons since 2006, the amendment of the German FederalElection Law during 2008–2013, and the discussion of the future composition ofthe European Parliament. The practical challenges and the teaching experienceof many lectures and seminars on the subject of proportional representation andapportionment 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 asociety, its constitutional framework, its political culture, its identity. On occasion,the complexities are due to partisan interests of the legislators responsible. Thismélange turns the topic into a truly interdisciplinary project. It draws on such fieldsas constitutional law, European law, political sciences, medieval history, modernhistory, discrete mathematics, stochastics, and computational algorithms, to namebut a few. I became increasingly fascinated by the interaction of so many disciplines.My fascination grew when I had the pleasure of conducting student seminarsjointly with colleagues from the humanities on topics of common interest. Theseexperiences made me realize that proportional representation and apportionmentmethods are a wonderful example to illustrate the universitas litterarum, the unityof arts and sciences.

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In retrospect, I find it much easier to conduct an interdisciplinary seminar than toauthor an interdisciplinary textbook. Nevertheless, I hope that this book may provea useful reference work for apportionment methods, for scholars of constitutionallaw and political sciences as well as for other electoral system designers. The manyapportionment methods studied span a wide range of alternatives in Germany, theEuropean Union, and elsewhere. The book not only describes the mechanics of eachmethod but also lists the method’s properties: biasedness in favor of stronger partiesat the expense of weaker parties, preferential treatments of groups of stronger partiesat 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, voteratios, size of the party system, and so on. These properties are rigorously provedand, whenever possible, substantiated by appropriate formulae.

Since the text developed from notes that I compiled for lectures and seminars, Iam rather confident that it can be utilized for these purposes. The material certainlysuffices for a lecture course or a student seminar in a curriculum of mathematics,quantitative economics, computational social choice, or electoral system design inthe political sciences. I have used parts of the text with particular success in classesfor students who are going to be high-school teachers. The chapters presupposereaders with an appreciation for rigorous derivations and with a readiness to acceptarguments from scientific fields other than their own. Most chapters can then bemastered with a minimum knowledge of basic arithmetic. Three chapters involvemore technically advanced approaches. Chapters 6 and 7 use some stochasticreasoning and Chap. 14 discrete optimization and computer algorithms.

The subject of the book is restricted to the quantitative and procedural rules thatmust be employed when a proportional representation system is implemented; as aconsequence, the book does not explicate the qualitative and normative foundationsthat would be called for when developing a comprehensive theory of proportionalrepresentation. As in all sciences, the classification of quantitative procedures startswith 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. Thistask is resolved by divisor methods or by quota methods. Later, geographicalsubdivisions of the electoral region come into play, as do guarantees for smallunits to obtain representation no matter how small they are, as do restrictions forstronger groups to limit their representation lest they unduly dominate their weakerpartners. In order to respond to these requirements, the basic methods are modifiedinto variants that may achieve an impressive degree of complexity.

When teaching the topic, I soon became convinced that its intricacies can beappreciated only by contemplating real data, that is, data from actual elections inthe real world, rather than imaginary data from contrived elections in the academicivory tower. My Augsburg students responded enthusiastically and set out to devisean appropriate piece of software, BAZI. BAZI has grown considerably since 2000and has proved an indispensable tool for carrying out practical calculations andtheoretical investigations. I would like to encourage readers of this book to use theprogram to retrace the examples and to form their own judgment. BAZI is freelyavailable from the website www.uni-augsburg.de/bazi.

www.uni-augsburg.de/bazi

Preface to the First Edition xv

Acknowledgments

My introduction to the proportional representation problem was the monograph ofMichel 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 toestablish a Theory of Apportionment. This seminal source was soon complementedby the treatise of Klaus Kopfermann (1991) who adds the European dimension tothe proportional representation heritage. Svante Janson’s (2012) typescript provedinvaluable for specific mathematical questions. These books provide the foundationson which the results of the present work are based.

Several colleagues and friends read parts or all of the initial drafts of thisbook and proposed improvements. I have benefited tremendously from the criticalcomments and helpful suggestions of Paul Campbell, Rudy Fara, Martin Fehndrich,Dan Felsenthal, Svante Janson, Jan Lanke, and Daniel Lübbert.

Throughout the project, I had the privilege to rely on the advice and inspirationof my colleagues Karl Heinz Borgwardt, Lothar Heinrich, and Antony Unwin inAugsburg University Institute for Mathematics. A special word of thanks is dueto my nonmathematical Augsburg colleagues who helped me in mastering theinterdisciplinary aspects of the topic. I wish to thank Günter Hägele (MedievalHistory, University Library), Thomas Krüger (Medieval History), Johannes Masing(Constitutional Law, now with the University of Freiburg im Breisgau), MatthiasRossi (Constitutional Law), and Rainer-Olaf Schultze (Political Sciences).

The largest debt of gratitude is due to the current and former members of myworkgroup at Augsburg University. Many of them contributed substantially to thiswork 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 toOlga Birkmeier, Johanna Fleckenstein, Christoph Gietl, Max Happacher, ThomasKlein, Sebastian Maier, Kai-Friederike Oelbermann, Fabian Reffel, and GerlindeWolsleben.

I would like to thank Andrew Duff, MEP, for graciously consenting to contributethe foreword to this book. Finally, I wish to acknowledge the support of the DeutscheForschungsgemeinschaft.

Augsburg, Germany Friedrich PukelsheimOctober 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; hhtii 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)ŒŒt��; Œt� 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 Œ�1I 1�

(Sect. 3.11)h 2 N House size (Sect. 4.1)` 2 f2; 3; : : :g Number of parties entering the apportionment calculations

(Sect. 4.1)k 2 f2; 3; : : :g Number of districts (Sect. 15.1)v D .v1; : : : ; v`/ Vector of vote weights vj 2 Œ0I 1/ for parties j � `

(Sect. 4.1)vC D v1 C � � � C v` 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)nC D fn; n C 1g Upward tie, increment option (Sect. 4.8)n� D fn � 1; ng Downward tie, decrement option (Sect. 4.8)vC=h Votes-to-seats ratio, also known as Hare-quota (Sect. 5.2)

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wj h Ideal share of seats for party j � ` (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 inpractice. Chapter 1 reviews the European Parliament elections of 2014, providinga rich source of examples. Chapter 2 deals with the German Bundestag electionof 2009; emphasis is on the interplay between procedural steps and constitutionalrequirements. 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 methodologythat is developed in the sequels.

Chapters 3–5: Divisor Methods and Quota Methods

A rigorous approach to apportionment needs to appeal to rounding functions androunding rules. They are introduced in Chap. 3. Chapters 4 and 5 discuss the twodominant 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 flexibledivisor for the first step and a specific rounding rule for the second. Quota methodsemploy a formulaic divisor—called quota—for the first step and a flexible roundingrule for the second.

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Chapters 6–8: Deviations from Proportionality

Many apportionment methods deviate from perfect proportionality in a systematicfashion. Chapters 6 and 7 investigate seat biases. A seat bias is the average ofthe deviations between actual seat numbers and the ideal share of seats, assumingthat all vote shares are equally likely or that they follow an absolutely continuousdistribution. Chapter 8 compares two apportionment methods by means of themajorization ordering. The majorization relation allows comparison of two appor-tionment methods. The relation indicates whether one method is more beneficial togroups of stronger parties—and hence more disadvantageous to the complementarygroup 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 ofit is fair, too. This requirement is captured by the notion of coherence. Divisormethods 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 therole of input and output. Given a seat number, the range of vote shares leading tothe given seat number is determined. The results elucidate situations when a straightmajority of votes fails to lead to a straight majority of seats. As a corrective, manyelectoral laws include an extra majority preservation clause. Three majority clausesare 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 showshow to handle minimum requirements as well as maximum cappings. The practicalrelevance of restrictions is shown by examples, such as the allocation of the seatsof the European Parliament between the Member States of the Union. Chapter 13discusses the 2013 amendment of the German Federal Election Law. The systemrealizes practical equality of the success values of all voters in the whole country.Mild deviations from proportionality, due to direct-seat restrictions, are incurredwhen 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 divisionof the electorate as well as of the political division of the voters. The methodsachieve this two-way fairness by apportioning seats to districts proportionally topopulation figures and to parties proportionally to vote counts. The core is thesub-apportionment of seats by district and party in such a way that, for everydistrict, the seats sum to the given district magnitude and, for every party, the seatssum to their overall proportionate due. To this end, two sets of electoral keys arerequired: district divisors and party divisors. While it is laborious to determinethe electoral keys, their publication makes it rather easy to verify the double-proportional seat apportionment. Chapter 14 explains double-proportional divisormethods 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 . . . . . . . . 11.1 The 28 Member States of the 2014 Union . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Austria, Belgium, Bulgaria; Electoral Keys . . . . . . . . . . . . . . . . . . . . . . . 31.3 Cyprus, Czech Republic, Germany; Table Design . . . . . . . . . . . . . . . . 71.4 Denmark, Estonia, Greece; Alliances and Indeps . . . . . . . . . . . . . . . . . 111.5 Spain, Finland, France, Croatia; Vote Patterns and Vote

Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Hungary, Ireland, Italy; Quotas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7 Lithuania, Luxembourg, Latvia; Residual Fits . . . . . . . . . . . . . . . . . . . . 241.8 Malta, the Netherlands, Poland; Nested Stages. . . . . . . . . . . . . . . . . . . . 271.9 Portugal, Romania, Sweden; Method Overview . . . . . . . . . . . . . . . . . . 281.10 Slovenia, Slovakia, United Kingdom; Local Representation . . . . . 331.11 Diversity Versus Uniformity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 Imposing Constitutionality: The 2009 Bundestag Election . . . . . . . . . . . . 412.1 The German Federal Election Law.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.2 Countrywide Super-Apportionment 2009 . . . . . . . . . . . . . . . . . . . . . . . . . 442.3 Per-Party Sub-Apportionments 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.4 Negative Voting Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.5 Direct and Universal Suffrage .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6 Free, Equal, and Secret Ballots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Equality of the Voters’ Success Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.8 Equality of Representative Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.9 Satisfaction of the Parties’ Ideal Shares of Seats . . . . . . . . . . . . . . . . . . 552.10 Continuous Fits Versus Discrete Apportionments .. . . . . . . . . . . . . . . . 56

3 From Reals to Integers: Rounding Functionsand Rounding Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.1 Rounding Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2 Floor Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.3 Ties and the Need for Rules of Rounding . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Rule of Downward Rounding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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3.5 Ceiling Function and Rule of Upward Rounding . . . . . . . . . . . . . . . . . 623.6 Commercial Rounding Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.7 Rule of Standard Rounding.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.8 Signpost Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.9 Rounding Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.10 Stationary Signposts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.11 Power-Mean Signposts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.12 Simple Rounding Does Not Suffice! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4 Divisor Methods of Apportionment: Divide and Round . . . . . . . . . . . . . . . 714.1 House Size, Vote Weights, and Seat Numbers . . . . . . . . . . . . . . . . . . . . . 714.2 Apportionment Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.3 The Five Organizing Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Apportionment Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.5 Divisor Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.6 Max-Min Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.7 Jump-and-Step Procedure and the Select Divisor . . . . . . . . . . . . . . . . . 824.8 Uniqueness, Multiplicities, and Ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.9 Tie Resolution Provisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.10 Primal Algorithms and Dual Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 864.11 Adjusted Initialization for Stationary Divisor Methods .. . . . . . . . . . 874.12 Universal Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.13 Bad Initialization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.14 Highest Comparative Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.15 Authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5 Quota Methods of Apportionment: Divide and Rank . . . . . . . . . . . . . . . . . . 955.1 Quota Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2 Hare-Quota Method with Residual Fit

by Greatest Remainders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.3 Greatest Remainders and the Select Split . . . . . . . . . . . . . . . . . . . . . . . . . . 975.4 Shift-Quota Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.5 Max-Min Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.6 Shift-Quota Methods and Stationary Divisor Methods . . . . . . . . . . . 1005.7 Authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.8 Quota Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.9 Residual Fit Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.10 Quota Method Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Targeting the House Size: Discrepancy Distribution . . . . . . . . . . . . . . . . . . . 1076.1 Seat Total and Discrepancy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.2 Universal Divisor Initialization.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 Recommended Divisor Initialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1096.4 Distributional Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.5 Seat-Total Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.6 Hagenbach-Bischoff Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

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6.7 Discrepancy Probabilities: Formulas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.8 Discrepancy Probabilities: Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.9 Discrepancy Representation by Means

of Rounding Residuals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.10 Invariance Principle for Rounding Residuals . . . . . . . . . . . . . . . . . . . . . . 1226.11 Discrepancy Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7 Favoring Some at the Expense of Others: Seat Biases . . . . . . . . . . . . . . . . . . 1277.1 Seat Excess of a Party . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Rank-Order of Parties by Vote Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.3 Vote Share Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.4 Seat Bias Formula.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.5 Biasedness Versus Unbiasedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307.6 House Size Recommendation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.7 Cumulative Seat Biases by Subdivisions into Districts . . . . . . . . . . . 1337.8 Total Positive Bias: The Stronger Third, the Weaker

Two-Thirds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.9 Alliances of Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357.10 Proof of the Seat Bias Formula .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.11 Proof of the Seat Bias Formula for List Alliances. . . . . . . . . . . . . . . . . 1447.12 Seat Biases of Shift-Quota Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

8 Preferring Stronger Parties to Weaker Parties: Majorization . . . . . . . . 1498.1 Bipartitions by Vote Strengths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498.2 Majorization of Two Seat Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1508.3 A Sufficient Condition via Pairwise Comparisons . . . . . . . . . . . . . . . . 1518.4 Majorization of Two Apportionment Methods . . . . . . . . . . . . . . . . . . . . 1518.5 Majorization of Divisor Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.6 Majorization-Increasing Parameterizations .. . . . . . . . . . . . . . . . . . . . . . . 1538.7 Majorization Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1548.8 Majorization of Shift-Quota Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

9 Securing System Consistency: Coherence and Paradoxes . . . . . . . . . . . . . 1599.1 The Whole and Its Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599.2 The Sixth Organizing Principle: Coherence . . . . . . . . . . . . . . . . . . . . . . . 1609.3 Coherence and Completeness of Divisor Methods .. . . . . . . . . . . . . . . 1619.4 Coherence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1629.5 House Size Monotonicity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1639.6 Vote Ratio Monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1659.7 Retrieval of an Underlying Signpost Sequence .. . . . . . . . . . . . . . . . . . . 1679.8 Proof of the Coherence Theorem in Sect. 9.4 . . . . . . . . . . . . . . . . . . . . . . 1689.9 Coherence and Stationary Divisor Methods . . . . . . . . . . . . . . . . . . . . . . . 1729.10 Coherence and the Divisor Method with Standard Rounding . . . . 1759.11 Violation of Coherence: New States Paradox . . . . . . . . . . . . . . . . . . . . . 1779.12 Violation of House Size Monotonicity: Alabama Paradox . . . . . . . 179

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9.13 Violation of Vote Ratio Monotonicity: Population Paradox . . . . . . 1819.14 Violation of Voter Monotonicity: No-Show Paradox.. . . . . . . . . . . . . 183

10 Appraising Electoral Equality: Goodness-of-Fit Criteria . . . . . . . . . . . . . . 18510.1 Optimization of Goodness-of-Fit Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 18510.2 Voter Orientation: DivStd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18610.3 Curtailment of Overrepresentation: DivDwn .. . . . . . . . . . . . . . . . . . . . . 18910.4 Alleviation of Underrepresentation: DivUpw .. . . . . . . . . . . . . . . . . . . . 19110.5 Parliamentary Orientation: DivGeo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19310.6 Party Orientation: HaQgrR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19510.7 Stabilization of Disparity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19810.8 Success-Value Stability: DivStd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19910.9 Representative-Weight Stability: DivHar . . . . . . . . . . . . . . . . . . . . . . . . . . 20010.10 Unworkable Disparity Functions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20110.11 Ideal-Share Stability: DivStd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20210.12 Ideal Share of Seats Versus Exact Quota of Seats . . . . . . . . . . . . . . . . . 203

11 Tracing Peculiarities: Vote Thresholds and Majority Clauses . . . . . . . . . 20711.1 Vote Share Variation for a Given Seat Number .. . . . . . . . . . . . . . . . . . . 20711.2 Vote Share Bounds: General Divisor Methods . . . . . . . . . . . . . . . . . . . . 20811.3 Vote Share Bounds: Stationary Divisor Methods .. . . . . . . . . . . . . . . . . 20911.4 Vote Share Bounds: Modified Divisor Methods . . . . . . . . . . . . . . . . . . . 21111.5 Vote Share Bounds: Shift-Quota Methods . . . . . . . . . . . . . . . . . . . . . . . . . 21211.6 Overview of Vote Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21311.7 Preservation of a Straight Majority and Majority Clauses . . . . . . . . 21511.8 House-Size Augmentation Clause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21711.9 Majority-Minority Partition Clause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21711.10 The 2002 German Conference Committee Dilemma . . . . . . . . . . . . . 21911.11 Residual-Seat Redirection Clause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22011.12 Divisor Methods and Ideal Regions of Seats . . . . . . . . . . . . . . . . . . . . . . 221

12 Truncating Seat Ranges: Minimum-Maximum Restrictions . . . . . . . . . . 22512.1 Minimum Representation for Electoral Districts . . . . . . . . . . . . . . . . . . 22512.2 Quota Method Ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22612.3 Minimum-Maximum Restricted Variants of Divisor Methods . . . 22712.4 Direct-Seat Restricted Variant of DivDwn. . . . . . . . . . . . . . . . . . . . . . . . . 22812.5 Proportionality Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23212.6 Direct-Seat Restricted Variant of DivStd . . . . . . . . . . . . . . . . . . . . . . . . . . 23212.7 Composition of the EP: Legal Requirements . . . . . . . . . . . . . . . . . . . . . . 23412.8 Cambridge Compromise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23612.9 Power Compromise .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23812.10 Jagiellonian Compromise.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

13 Proportionality and Personalization: BWG 2013 . . . . . . . . . . . . . . . . . . . . . . . 24713.1 The 2013 Amendment of the Federal Election Law . . . . . . . . . . . . . . 24713.2 Apportionment of Seats Among Parties . . . . . . . . . . . . . . . . . . . . . . . . . . . 24813.3 Assignment of Candidates to Seats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

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13.4 Initial Adjustment of the Bundestag Size . . . . . . . . . . . . . . . . . . . . . . . . . . 25113.5 Alternative House Size Adjustment Strategies . . . . . . . . . . . . . . . . . . . . 252

14 Representing Districts and Parties: Double Proportionality . . . . . . . . . . 25914.1 Double Proportionality: Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25914.2 The 2016 Parliament Election in the Canton of Schaffhausen .. . . 26014.3 Discordant Seat Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26414.4 Winner-Take-One Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26514.5 Double Proportionality in Swiss Cantons: The Reality . . . . . . . . . . . 26514.6 Double Proportionality in the European Union: A Vision . . . . . . . . 26614.7 Degressive Compositions and Separate District Evaluations . . . . . 26714.8 Compositional Proportionality for Unionwide Lists . . . . . . . . . . . . . . 268

15 Double-Proportional Divisor Methods: Technicalities . . . . . . . . . . . . . . . . . 27515.1 Row and Column Marginals, Weight and Seat Matrices . . . . . . . . . . 27515.2 Double-Proportional Divisor Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27615.3 Cycles of Seat Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27815.4 Uniqueness of Double-Proportional Seat Matrices. . . . . . . . . . . . . . . . 27915.5 Characterization of Double-Proportional Seat Matrices . . . . . . . . . . 28115.6 Existence of Double-Proportional Seat Matrices . . . . . . . . . . . . . . . . . . 28515.7 A Dual View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28715.8 Alternating Scaling Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28815.9 Tie-and-Transfer Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

16 Biographical Digest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29716.1 Thomas Jefferson 1743–1826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29716.2 Alexander Hamilton 1755–1804 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29816.3 Daniel Webster 1782–1852 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29916.4 Thomas Hare 1806–1891 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30016.5 Henry Richmond Droop 1832–1884 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30116.6 Eduard Hagenbach-Bischoff 1833–1910 . . . . . . . . . . . . . . . . . . . . . . . . . . 30216.7 Victor D’Hondt 1841–1901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30516.8 Joseph Adna Hill 1860–1938.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30616.9 Walter Francis Willcox 1861–1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30816.10 Ladislaus von Bortkiewicz 1868–1931 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30916.11 Edward Vermilye Huntington 1874–1952 . . . . . . . . . . . . . . . . . . . . . . . . . 31016.12 Siegfried Geyerhahn 1879–1960 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31216.13 Jean-André Sainte-Laguë 1882–1950.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31316.14 George Pólya 1887–1985 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31516.15 Horst Friedrich Niemeyer 1931–2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

Foreword to the First EditionThe European Parliament

Preface to the Second EditionPreface to the First EditionAcknowledgments

NotationsOutline of the BookChapters 1 and 2: Apportionment Methods in PracticeChapters 3–5: Divisor Methods and Quota MethodsChapters 6–8: Deviations from ProportionalityChapters 9–11: Coherence, Optimality, and Vote RangesChapters 12 and 13: Practical ImplementationsChapters 14 and 15: Double ProportionalityChapter 16: Biographical Digest

Contents