Studies in Choice and Welfare

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1 Studies in Choice and Welfare Editors-in-Chief M. Fleurbaey, USA M. Salles, France Series Editors B. Dutta, United Kingdom W. Gaertner, Germany C. Herrero Blanco, Spain B. Klaus, Switzerland P.K. Pattanaik, USA K. Suzumura, Japan W. Thomson, USA For further volumes:

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3 Andranik Tangian Mathematical Theory of Democracy

4 Andranik Tangian WSI Hans-Böckler-Foundation Düsseldorf Germany Studies in Choice and Welfare ISSN ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 To Olga Tangian, who encouraged me to write this book

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7 Preface I have come to the conclusion that politics is too serious a matter to be left to the politicians. Charles De Gaulle ( ) During the last decades, democracy has significantly advanced throughout the world, expanding from the level of single countries to that of world community. Now democracy has gained the status of obligatory password to fruitful international relations, facilitating political dialogue and collaboration between countries with very different economical and cultural backgrounds. Building democratic governance seems to be the only way to reduce the negative aspects and consequences of globalization. Throughout history, the notion of democracy has undergone radical transformations. Democracy originated in Ancient Greece, where its main features were the systematic participation of all citizens in political life and their active work in legislative, executive, and juridical bodies. All important decisions were made by the People s Assembly, and about 600 out of 700 Athenian magistrates were distributed by sortition or rotation. Election was considered appropriate only if the electors shared common values, which was the case for the aristocracy and oligarchy but not for the heterogeneous demos. Therefore, election was regarded as aristocratic or oligarchical, and democracy was associated with the allocation of magistrates by lot. There is a principal difference between voting on proposals and voting on candidates for office. Voting on proposals is democratic, because it is a form of the execution of powers by the people. But voting on candidates for office is different, because it renders the powers to just a few executives, reducvii

8 viii Preface ing the people s access to political participation. Therefore, Athenians used elections only if the elected required special skills and served common goals. Correspondingly, they elected only few high-qualified professionals: military generals, those responsible for water supply, and treasurers. The Roman Republic and the medieval republics in Italy combined elements of monarchy, aristocracy and democracy. For over two thousand years these three forms of power were classified according to the way the power was acquired and executed. Monarchy was identified with hereditary power, aristocracy/oligarchy with elected governors, and democracy with selection of officials by lot. The situation changed radically after the American and French Revolutions. At the outset, both the American and French systems were explicitly opposed to democracy. They practiced no selection of magistrates by lot, and participation by the people was restricted to election of representatives. Nonetheless, in the course of extending the right to vote to all adult citizens, the system of representative government was redefined as democratic. This redefinition produces a number of inconsistencies. Since the will of the electorate is not always adequately represented, certain policies result in a discrepancy between the government and the people. Moreover, aristocratic election, when viewed in a broad democratic context, shows its limits in providing civil consent and political stability. The problem is not occasional failures of particular electoral methods but the general inability of the actual representative democracy to involve people into political life and make them satisfied as citizens. In restricting participation by the people to elections, representative democracy suffers from the so-called democratic deficit recently recognized at all political levels up to the United Nations and the European Union. 1 The link between universal suffrage and democracy also perplexed scholars who attempted to design universal election rules. They encountered voting paradoxes showing that democratic choice cannot be always realized through the election of representatives. In other words, voting, proved to be democratic in the accepting/rejecting of proposals, can constrain the actual election-based democracy. It seems that the intermediation of the people s will must be based on some more advanced methods than on voting for candidates by name. (The reference to paradoxes indicates that voting is be- 1 Democratic deficit (2013). Wikipedia. deficit (cited 28 Mar 2013).

9 Preface ix lieved to be a universal instrument of democratic decision making. However, it does not follow that a tool appropriate for certain tasks can be used elsewhere. Even if a hammer is great for driving nails, its failure in repairing a watch should not be considered paradoxical.) Given these premises, the mathematical theory of democracy analyzes the abilities of single representatives (president, deputy) and representative bodies (parliament, parliament coalition, cabinet of ministers) to express public opinion. The methods developed in this book are applied to study direct and representative democracies and to back up a special election procedure aimed at making representative democracy more representative and, thereby, more democratic. Additionally, the book describes several societal and non-societal applications for which the idea of representation is relevant. It should be emphasized that democracy is not regarded from the viewpoint of the quality of decisions whether they are good or bad should be judged by history. Instead, we evaluate the quality of the representation of public opinion by single representatives and representative bodies as agents of the society. 2 The quality of representation is estimated with regard to the percentage of the population represented and with regard to the frequency of representing a majority. This book consists of three main parts: History, Theory and Applications. The fourth part, Appendix, contains computational formulas and statistical tables. Part I, History, contains five chapters that focus on important steps in the development of democracy and its philosophical comprehension by political thinkers. These chapters are devoted to Ancient Athens, the Roman Republic, medieval republics in Italy, the Age of Enlightenment, and the modernity up to the present. Each chapter outlines the contemporary mathematical findings relevant to democracy. Part II, Theory, also contains five chapters that in a sense mirror the historical ones. They deal with modeling direct democracy of Athenian type, analyzing the relationships between democracy and dictatorship like that in Rome, revealing the bottlenecks of republican representative democracy, developing a probabilistic approach to representatives that dates back to the Age of Enlightenment, and discussing a method to improve the performance 2 We do not study representation in the sense of John Adams ( ) and Mirabeau ( ) as a miniature portrait of the society, which would lead to the problem of proportional representation not considered here.

10 x Preface of representative democracy, responding to the current political agenda. For illustration, the method discussed is applied to hypothetically redistribute the seats in the German Bundestag, achieving a considerable gain in its representativeness. Four chapters of Part III, Applications, among other things, elaborate on the idea that from the mathematical standpoint, neither the society nor representatives are necessarily human, so some objects can represent the behavior of other objects. In particular, American stock prices represent in advance the fluctuations of German stock prices, so that American representatives of German stocks can be used for financial predictions. Similarly, the traffic situations at one street s intersections represent in advance the situations at other intersections, which can be used to anticipate traffic jams and prevent them by switching on the green wave of traffic lights in the appropriate direction. The non-societal applications demonstrate that the mathematical theory of democracy is based on quite general principles beyond the societal specificity. Consequently, democracy itself can be regarded as a rather universal approach to social organization over and above national, cultural, ethnical or religious particularities. Düsseldorf, June 12, 2013 Andranik Tangian

11 Acknowledgements A good critic should not judge a writer by what he leaves unsaid, but from what he says. Polybius (ca. 200 BC 118 BC) This book summarizes the author s works, which were encouraged, influenced and supported by several people. In 1977 Boris Aleksandrovich Efimov, the director of my thesis at the Central Economical-Mathematical Institute of the USSR Academy of Sciences (CEMI), Moscow, suggested to analyze the alternate disappearance and appearance of Arrow s dictators in a model with infinite voters considered a few years earlier by Fishburn and Kirman and Sondermann. It turns out that the infinite model highlights latent decision making hierarchies, which are not easily seen in the finite model where they merge with dictators. As coalition structures, the hierarchies are preserved under the model isomorphisms, which is not always the case for their top levels dictators. The prime role of hierarchies and secondary role of dictators implied an alternative interpretation of Arrow s Impossibility Theorem, as described in Section Viktor Ivanovich Danilov-Daniljan from the Institute of Systems Research of the USSR Academy of Sciences (VNIISI), Moscow, encouraged me to present all of these as a PhD thesis and to defend it at his institute. He also invited to the thesis defense Yuri Ivanovich Manin, who drew an illuminating analogy from axiomatic decision rules implying a spatial hierarchical structure to general legislation principles, further implying a specific governmental organization. xi

12 xii Acknowledgements Since Arrow s dictator shares each opinion with a certain coalition from the decision making hierarchy, he can be regarded as its representative. This prompted an opportunity to measure the dictator s representativeness by the average size of the coalition represented. It was proved that there always exist good Arrow s dictators, who on average represent a majority of the society. When the related findings were presented at the 1988 Leningrad game theory seminar, its chairman Nikolaj Nikolaevich Vorobjev conceptualized them with a metaphor mathematical theory of democracy, which channeled the further studies in this direction. Gorbachev s Perestroika intensified East West scientific contacts. In 1989 Josef Gruber invited about 20 Soviet researchers including myself to the Second International Conference on Econometric Decision Models, at the University of Hagen, Germany. My paper was positively evaluated by Wulf Gaertner, the co-editor of the journal Social Choice and Welfare, and the following year, Gruber arranged my visit to the University of Hagen to prepare a camera-ready monograph, Aggregation and Representation of Preferences, for Springer. It included the evaluation of representative bodies, like parliament and cabinet of ministers (magistrate), and in 1991 it was published with the subtitle Introduction to Mathematical Theory of Democracy. From I worked at the University of Hagen on the project Constructing objective functions initiated by Gruber. Thanks to this project I became acquainted with Bernard Grofman, whom I visited in 1995 at the University of California, Irvine, where I learned about his conjecture on the similarity of outcomes from the Condorcet and Borda voting methods. This conjecture is proved true; see Section 4.8. In 2002 Timothée Ngakoutou, UNESCO, ordered a report on the mathematical theory of democracy. He asked for it to be written in a popular way, illustrated with examples and enhanced policy implications. Correspondingly, the report contained a political introduction (influenced by Bertrand Russell s History of Western Philosophy and Bernard Manin s Principles of Representative Government), so the model was reformulated in terms of simple indices of representativeness, and an application to the 2002 German Bundestag election was developed. This experience reoriented the research toward more political and practical issues, as reflected by Parts I and III. Since 2003 the work has continued at the Institute for Economic and Social Research (WSI) in the Hans-Böckler-Stiftung (political foundation of the DGB German Confederation of Trade Unions), Düsseldorf. I am thankful

13 Acknowledgements xiii to the Foundation head, Nikolaus Simons, and the institute directors, Heide Pfarr and Brigitte Unger, who favored mathematical studies in democracy and generously supported this project. The presentation of the material acquired its current shape while teaching a course organized on the initiative of Susanne Fuchs-Seliger and Clemens Puppe at the Karlsruhe Institute of Technology (formerly University of Karlsruhe). The communication with students required further clarifications, examples, and relevance to practice. One of the students, Matthias Hölzlein, collected internet data for the example in Chapter 8. Finally, I acknowledge the great help of Mary Beth Robinson, who checked the style and suggested numerous editorial improvements. I express my sincere gratitude to all the persons mentioned in this short note. This book has been made possible due to their stimulating influence and organizational support. Düsseldorf, June 12, 2013 Andranik Tangian

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15 Contents Part I History 1 Athenian Democracy Introduction PathtoDemocracy EstablishingDemocracy CulminationofDemocracy Philosophy of Democracy Aristotle s Mathematical Model of Two Types of Governance Summary References Echoes of Democracy in Ancient Rome Introduction Typology of States and the Theory of Mixed Government RotationofGovernmentTypes TheRomanRepublic Pliny slogicalanalysisofasenatehearing Summary References Revival of Democracy in Italian Medieval City-Republics Introduction xv

16 xvi Contents 3.2 TheVenetianRepublic TheFlorentineRepublic TheoryofMixedGovernmentRevitalized FirstMathematicalApproachestoElections Summary References Enlightenment and the End of Traditional Democracy Introduction MontesquieuandtheSeparationofPowers Rousseau and the General Will TheBordaMethodofElection Laplace s Justification of the Borda Method ProblemsPosedbytheBordaMethod Condorcet s Social Choice Theory The Condorcet Method Versus the Borda Method Summary Proofs ProofofTheorem ProofofJuryTheorem ProofofLemma References Modernity and Schism in Understanding Democracy Introduction American and French Revolutionary Constitutions Principles of Representation Tocqueville s Understanding of Democracy Evolution of Representative Government UniversalSuffrageandDemocracy GeneralCommitmenttoDemocracy DemocraticPerspectives Voting Theory in the Age of Representative Government From Egalitarianism to Hierarchies...206

17 Contents xvii 5.11Summary Proofs ProofofTheorem ProofofTheorem References Part II Theory 6 Direct Democracy Introduction Example: Evaluating Athenian Politicians in 462 BC Indices of Representativeness Geometric Interpretation of the Indices DecisiveBodiesSelectedfromtheSociety Ambiguous Effects of the Enlargement of Decisive Bodies Magistrate Versus Parliament with Regard to Their Functions InefficiencyofDemocracyinanUnstableSociety Summary Proofs ProofofTheorems6.1and ProofofTheorem ProofofTheorem ProofofTheorem References Dictatorship and Democracy Introduction Measuring Representativeness for one Preference Profile Indices of Representativeness for the Simplest Model Indices of Representativeness for the Combinatorial Model LeadersVersusDictators FindingLeadersinaLargeSociety DiscussionofDefinitions Summary...306

18 xviii Contents 7.9 Proofs ProofofTheorem ProofofTheorem ProofofTheorem References Representative Democracy Introduction DataStructureforAnalysis Evaluation of Parties and Bundestag Evaluation of Party Coalitions Universality Versus Popularity Summary ProofofTheorem InternetLinkstoPublicOpinionPolls References Statistically Testing the Representative Capacity Introduction Geometric Method AlgebraicMethod Probabilistic Method Interpolation Methods Example: Significance of Representativeness of Parties and Coalitions Summary References Concluding Discussion: Bridging Representative and Direct Democracies DemocracyandVoting ShortcomingsofElection ReconsiderationofElectionbyName ArchitectureofElectionProcedure...431

19 Contents xix 10.5 Example: Enhancing the Representativeness of the Bundestag ElectionCalculusVersusAxiomaticStudies TowardTrulyDemocraticElections References Part III Applications 11 Simple Applications Planning a Public Opinion Poll with no Cyclic Majorities Planning a Representative Public Opinion Poll Invitation to a Round Table SufficiencyofLimitedInformationforDecisionMaking References Application to Collective Multicriteria Decisions Introduction SpecifyingTravelAlternativesforanOuting SelectingOneTravelAlternative SelectingSeveralTravelAlternatives Measuring the Individual Satisfaction Summary References Application to Stock Exchange Predictions Introduction Dow Jones Stocks as Representatives of DAX Stocks TestingStand-AlonePredictors TestingtheBestPredictoroutofMany TestingaPredictorSelectedbyLot Groups of Dow Jones Stocks as Collective Predictors Totality of Dow Jones Stocks as an Aggregate Predictor Summary ProofofTheorem References...502

20 xx Contents 14 Application to Traffic Control Introduction Traffic Forecasts Using the Model of Representatives Testing a Stand-Alone Forecaster Testing the Best Forecaster out of Many Testing a Forecaster Selected by Lot Groups of Intersections as Collective Forecasters Totality of Intersections as an Aggregate Forecaster Summary References Part IV Appendix A Computational Formulas A.1 Chebyshev s Inequality and the Central Limit Theorem A.2 Beta Distribution and Incomplete Beta Function A.3 Multinomial Sums and Falling Factorials A.4 Proofs References B Probabilities of Unequal Choices by Vote and by Candidate Scores HowtoUseB-Tables Probability Tables C Statistical Significance of Representative Capacity HowtoUseC-Tables Significance of Representativeness of a Single Candidate (k = 1) Significance of Representativeness of a Two-Member Coalition (k = 2) Significance of Representativeness of a Three-Member Coalition (k = 3) Index...571