Voting Paradoxes and How to Deal with Them

Voting Paradoxes and How to Deal with Them

Springer-Verlag Berlin Heidelberg GmbH

Hannu Nurmi Voting Paradoxes and How to Deal with Them With 12 Figures and 64 Tables Springer

Professor Hannu Nurmi University of Turku Department of Political Science Fin-20014 Turku Finland Library of Congress Cataloging-in-Publication Data Die Deutsche Bibliothek - CIP-Einheitsaufnahme Voting paradoxes and how to deal with them: with 64 tables / Hannu Nurmi. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 1999 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permiss ion for use must always be obtained from Springer-Verlag. Violations are Iiable for prosecution under the German Copyright Law. ISBN 978-3-642-08551-2 ISBN 978-3-662-03782-9 (ebook) DOI 10.1007/978-3-662-03782-9 CISpringer-VerlagBerlin Heidelberg 1999 Originally published by Springer-VerlagBerlin Heidelberg New York in 1999. Softcover reprint of the hardcover Ist edition 1999 The use of general descriptive names, registered names, trademarks, 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. Hardcover-Design: Erich Kirchner, Heidelberg SPIN 10738184 42/2202-5 4 3 2 1 0 - Printed on acid-free paper

For Irmeli

Preface Voting paradoxes are unpleasant surprises encountered in voting. Typically they suggest that something is wrong with the way individual opinions are being expressed or processed in voting. The outcomes are bizarre, unfair or otherwise implausible, given the expressed opinions of voters. Voting paradoxes have an important role in the history of social choice theory. The founding fathers of the theory, Marquis de Condorcet and Jean-Charles de Borda, were keenly aware of some of them. Indeed, much of the work of these and other forerunners of the modern social choice theory dealt with ways of avoiding paradoxes related to voting. One of the early paradoxes, viz. that bearing the name of Condorcet, has subsequently gained such a prominent place in the literature that it is sometimes called the paradox of voting. One of the aims of the present work is to show that Condorcet's is but one of many paradoxes of voting. Some of these are pretty closely interrelated making it meaningful to classify them. This is the second main aim of this book. The third objective is to suggest ways of dealing with paradoxes. Since voting is and has always been an essential instrument of democratic rule, it is of some interest to find out how voting paradoxes are being dealt with by past and present methods of voting. Of even greater interest is to find ways of minimizing the probability of occurrence of various paradoxes. By their very nature some paradoxes are unavoidable. Thus, minimizing their probability is the best one can do in some cases. Parts of this book have been presented at various scientific

VIII Voting Paradoxes and How to Deal with Them meetings, seminars and symposia. I would hereby like to express my gratitude to the organizers and participants of Trento'98 and the 4'th WG-7 meetings as well as of seminars held at Department of Political Science, University of Minnesota and at Department of Philosophy, University of Miami. My special thanks are due to my hosts in these events: Mario Fedrizzi, Mika Widgren, Robert Holt and Risto Hilpinen, respectively. The main part of the work on this book was done during my most pleasant and intellectually rewarding stay as Government of Finland/David and Nancy Speer Visiting Professor of Finnish Studies at University of Minnesota during the fall quarter of 1998. Especially the weekly seminar at Political Science Department was a very important source of inspiration. I am very grateful especially to John Freeman, Robert Holt and W. Phillips Shively for their scholarly advice and warm hospitality. I would also like to thank Manfred J. Holler for encouragement in this project. Executive Economics Editor Werner A. Muller of Springer-Verlag provided me with much needed professional advice during the final stages of the manuscript preparation. I am very grateful to him as well as to an anonymous referee for the numerous comments, suggestions and corrections that led to many improvements of the text. The financial support of the donors of Visiting Professorship in Finnish Studies as well as of the Academy of Finland is gratefully acknowledged. Hannu Nurmi University of Turku Finland May, 1999

Contents Preface vii 1 Introduction 1 2 Basic Concepts and Tools 4 2.1 Alternatives and Opinions 4 2.2 Preference Profile 6 2.3 Pairwise Comparison and Tournament Matrix 7 2.4 McGarvey's Theorem 8 3 Paradoxes of the Enlightenment Era 11 3.1 Borda's Paradox 11 3.2 Condorcet's Paradox 15 3.3 Borda and Condorcet Compared 22 3.4 How Frequent Are the Paradoxes? 25 4 The Geometry of Voting 31 4.1 The Saari Triangles 31 4.2 The Conditions of Paradoxes 34 5 The Paradoxical Act of Voting 41 5.1 The Cost Calculus 41 5.2 The No-Show Paradox 49 6 Monotonicity Paradoxes 55 6.1 Additional Support Paradox 55 6.2 Preference Truncation Paradox 62 6.3 How to Deal with Monotonicity Paradoxes? 67

x Voting Paradoxes and How to Deal with Them 7 Compound Majority Paradoxes 70 7.1 Ostrogorski's Paradox 70 7.2 Anscombe's Paradox 73 7.3 The Paradox of Multiple Elections 74 7.4 The Referendum Paradox 76 7.5 Simpson's Paradox 78 7.6 How to Deal With Compound Majority Paradoxes 81 8 Intra-Profile Paradoxes 87 8.1 Pareto Violations 87 8.2 Inconsistency Paradox 89 8.3 Choice Set Variance Paradoxes 94 8.4 The q-rules and Pareto Violations 98 8.5 Tournament Solutions to Voting Paradoxes 101 9 Paradoxes of Representation 109 9.1 The Alabama Paradox 109 9.2 Other Paradoxes of Hamilton Apportionments 112 9.3 Schwartz's Paradox of Representation 115 9.4 How to Deal with Representation Paradoxes 118 10 Classification of Paradoxes 11 Hard and Soft Solutions List of Figures List of Tables Bibliography Index 120 125 129 131 135 145