A Primer in Social Choice Theory Revised Edition LSE Perspectives in Economic Analysis Series editors Timothy Besley and Frank Cowell The LSE Perspectives in Economic Analysis series provides concise and original insight into a wide range of topics in economics Each book is accessibly written but scholarly to appeal to advanced students of economics, and academics and professionals wishing to expand their knowledge outside their own particular field Books in the series A Primer in Social Choice Theory byWulf Gaertner A Primer in Social Choice Theory, Revised Edition byWulf Gaertner Strategy and Dynamics in Contests by Kai A Konrad Forthcoming books Econometric Analysis of Panel Data by Vassilis Hajivassiliou Measuring Inequality by Frank Powell A Primer in Social Choice Theory Revised Edition Wulf Gaertner Great Clarendon Street, Oxford 0x2 6DP Oxford University Press is a department of the University of Oxford It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Hong Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toronto With of fices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore South Korea Switzerland Thailand Turkey Ukraine Vietnam Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © Wulf Gaertner, 2009 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2006 All rights reserved No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose the same condition on any acquirer British Library Cataloguing in Publication Data Data available Library of Congress Cataloging in Publication Data Data available Typeset by SPI Publisher Services, Pondicherry, India Printed in Great Britain on acid-free paper by MPG Biddles Ltd., King’s Lynn, Norfolk ISBN 978-0-19-956530-6 10 987654321 To the memory of my parents and to my family PREFACE TO THE REVISED EDITION In her otherwise very positive review of the first edition of this Primer in the journal Social Choice and Welfare (Vol 30,2008), Antoinette Baujard deplored the absence of exercises This criticism was more than justified, since a primer is written for beginners It is especially important that they find out for them-selves whether they have correctly understood the concepts introduced in the text, and furthermore some of the logical inferences may be uncommon to them In this revised edition, the reader will find 8-10 exercises at the end of each of Chapters 1-9 Some of the exercises are very easy, and are just intended to make sure that the reader has thoroughly understood what is discussed in the text, others are a bit ‘trickier’ Work with pencil and paper can be very illuminating and rewarding Hints toward a solution of some of the exercises are gathered at the end of the book I am grateful to Nick Baigent and John Weymark for allowing me to take a look at some of the exercises that they devised for their own course in social choice theory Otherwise, this new edition sees a few minor additions and amendments that are meant to lead to greater clarity I wish to thank both teachers and students of collective choice theory for the comments and suggestions that they made on the first edition These were very encouraging for me In particular, I wish to thank Greg Fried for his observations in relation to the Arrow—Sen proof discussed in section 2.2 Finally, I am much indebted to OUP, especially to Sarah Caro, for making this new edition possible Osnabrück and London March 2009 Wulf Gaertner PREFACE AND ACKNOWLEDGEMENTS TO THE FIRST EDITION This book is meant to be an introductory text into the theory of social choice It is not a book for readers who have already acquired a basic knowledge of social choice theory and now wish to tackle more specialized issues There exist some very fine advanced textbooks on collective choice and related questions This primer is written for undergraduates and first year graduates Prerequisites are very small: some knowledge of elementary set theory and some basic knowledge of mappings in IRn The main aim is to attract readers to an area which revolves around the problem of aggregating individual preferences These questions are interesting and highly relevant both for small communities and large societies It would be nice if, while going through the various chapters of this primer, the reader were to develop an interest and curiosity for more There is so much more which is not covered in this book As said above, there are very good books that will guide the reader beyond what is being discussed in the present text This primer in social choice theory is based on various courses that the author has taught at different places over the years Long and very fruitful discussions with Nick Baigent, Prasanta Pattanaik, Maurice Salles, Amartya Sen, Kotaro Suzumura, John Weymark, and Yongsheng Xu are gratefully acknowledged Without the gentle advice and guidance of these and other eminent scholars, this book would never have been written I am deeply indebted to all of them I am very grateful to Constanze Binder at Groningen University who took pains to read most parts of the text I received a lot of interesting and very helpful comments from her I also wish to thank two referees for their constructive criticism I am grateful to Brigitte Arnold who helped me tremendously to turn the various versions of my manuscript into a readable text I also wish to thank Christian Aumann who did a fine job in producing the figures for this primer We hope that these graphs will enhance the understanding on the part of the reader Tim Besley and Frank Cowell from STICERD at the London School of Economics were kind enough to accept this primer as the first book in a new series Special thanks to them Last but not least I wish to thank Sarah Caro and her collaborators at Oxford University Press for the production of this book Dutta 90 Farquharson 3, 75–77, 85, 89 Fishburn 41 Fleurbaey 67 Foley 190 Gaertner 4, 52, 64, 67–70, 76, 90, 107, 172, 173, 177 Gans 90 Gardenfors 104 Gauthier 147, 155–157 Geanakoplos 25, 80 Gehrlein 43 Gevers 30, 122, 126, 128–130 Gibbard 15, 59–69, 76, 77, 80, 85, 92, 97 Groves 78 Guilbaud 43 Hammond 126 Harsanyi 10, 123, 134–136, 145, 148, 149 Heal 185, 186 Heinecke 52 Helvétius 121 Hicks 199 Hurwicz 78, 79 Hutcheson 121 Imai 155 Inada 50–52 Jackson 80 Jehle 25 Jungeilges 173 Kalai 139, 151–155, 157, 158, 163–165, 167 Kaldor 199 Kelly 45, 60, 77, 99, 107, 110 Kemeny 112–114, 182 Klamler 114 Klemisch—Ahlert 157 Kolm 190 Konow 178 Kreps 193 Krüger 70 Lagerspetz Lancaster 194 Leininger 114–116 Lensberg 158 London 3–5 Luce 144, 145, 147, 151 Lull 3, 4, 199 Mas-Colell 99, 101 Maskin 52 May 40, 41, 52 McLean 3-5 Merlin 113 Michaud 111 Mill 57 Mishan Mongin 136 Morgenstern 7, 134, 141 Moulin 15, 86-89, 91 Muller 80 Nash 65, 66, 140, 142,144-148,150-152, 155, 157, 163-165 Ng Nitzan 113 Nozick 13, 67, 193 Nurmi 111-113,116 Pattanaik 46, 67-70, 89, 193-196, 198, 199 Pliny the Younger 3, 4, 77 Polybios Pufendorf Puppe 198, 199 Raiffa 144, 145, 147, 151 Ratliff 114 Rawls 121-124, 127, 128,134, 146, 147, 163, 164, 171-173 Reny 25, 80 Roberts 90 Roemer 145, 152 Roth 154 Rubinstein 150 Saari 103, 105-107, 113, 184 Samuelson 19, 34 Saporiti 91 Saposnik 51, 52 Satterthwaite 15, 77, 80, 85, 89, 92, 97 Schokkaert 164 Scitovsky Sen 7-9, 13, 15, 21, 29, 38, 46, 49, 52, 57-62,66-69,97, 98,125, 126,192-197 Sengupta 90 Smart 90 Smorodinsky 151-155,157, 163-165, 167 Sonnenschein 99, 101 Sprumont 188-190 Strasnick 126 Suzumura 5, 67-70 Thomson 152, 157, 158, 188,191 Tohmé 91 von Neumann 7, 134, 141 Weymark 28, 129,134 Xu 193-196,198, 199 Yaari 163-165,167-169, 171,174, 177 Young 102-105,110, 111 Zeckhauser 89 Zeuthen 148-150,155, 156 SUBJECT INDEX absolute majority rule 38, 39, 41 accountability principle 178 acyclicity 6-8, 58, 59, 98 alienable rights 59-61, 76 alternating concessions 148-150, 155-157 anonymity condition 37, 38, 40, 41, 53, 88, 99, 102, 103, 105, 125, 128, 130, 131, 133, 135, 183-187, 189-191 antiplurality rule 103, 105 Arrovian social welfare function 8, 22, 49, 57, 80, 98 Arrow’s impossibility result 14-16, 19,21, 38, 41, 42, 80 averaging rule 184 bargaining from equal split 164 Bargaining from zero 165, 167 Bargaining over the strong Pareto set 165, 169 Bargaining theory 14, 29, 123, 140, 141, 145,146, 151, 152, 156, 163 Bayesian concept of rationality 134 Bergson social welfare function Best element 6-9, 50 Betweenness 45, 46 Borda rank-order rule 3, 4, 10, 12, 15, 77, 89, 102-110, 182 Borda score 103, 104, 106-108, 113 Borda winner 105, 106, 113, 114 broad Borda rule 107, 108 cancellation property 104, 105, 107 cardinal measurability 141 chi-squared test 170, 176 choice function 7-9, 58, 60, 75, 76, 81, 84, 97, 98 choice set 7, 8, 60, 61, 63, 65, 66, 69 choice space 184 citizens’ sovereignty 21, 37 clockwise cycle 51, 52 Cobb-Douglas utility function 78 Collective choice rule 8, 10, 58, 61, 62, 64 competitive allocation 78 composition condition 198 conditional preferences 62-64 Condorcet loser 105, 106, 112-114 Condorcet paradox 42, 50, 51, 77 Condorcet principle Condorcet winner 4, 50, 85-88, 90, 91, 105, 106, 111-114, 116, 117, 182 Condordet winner 106 consistency 8, 9, 64, 102, 108, 110, 115, 144, 178 consistency of sets of voters 102-105, 110 contagion property 14, 21, 24, 25 continuity 28, 33, 181-183, 185 continuous aggregation rules 182, 185 contractibility 185, 186 contractible space 185, 186 contraction consistency 9, 108, 110, 144 cooperative Nash solution 145, 163 copeland method 110 counter-clockwise cycle 51, 52 cyclical balance 51, 52 cyclical majorities 4, 111 decisiveness 21-25, 33, 57-60, 62, 64, 66, 68-70, 99, 100 demand revealing mechanisms 78 dichotomous preferences 50-52 dictator 24, 28, 32, 33, 37, 57, 81, 84, 97, 99, 100 difference principle 121, 124, 134, 171, 172 disagreement point 139, 141, 154, 158 distance function 111, 113, 186 distance minimization 111, 114, 182 Dodgson ranking 114 Dodgson rule 4, 111, 114 dominant strategy 66, 70, 78 dominant strategy equilibrium 70 egalitarian rule 191 egalitarian solution 157-159 envy-freeness 190-192 equal split 169 equiprobability model 135 equity axiom 126, 128, 130, 132, 171-173, 175, 176 equity requirement 157 essential alternative 199 ethical observer 10, 123 exclusionary power 60, 66, 70 expansion consistency 9, 108, 110 expected utility 11, 134,136, 141, 149 extended orderings 172, 173 fair equality of opportunity 121 fairness 121, 164 faithful 104 free disposal of utility 154 freedom of choice 181, 192–199 game form 15, 65–67, 70 general possibility theorem 21 generalized Condorcet winner 88 generalized equity axiom 157 generalized median voter scheme 15, 88, 91, 92 generalized single—peaked preferences 92 generalized utilitarian rules 133, 135 Gibbard’s conditional preferences 62 Gibbard’s libertarian claim 61 Gibbard—Satterthwaite impossibility theorem 15, 80, 81,85 Harsanyi’s aggregation theorem 134 I—monotonicity 197 i—variants 62, 68, 76 ideal point 151, 152,154–156 impartial observer 135, 136 implementation theory 80 impossibility of a Paretian liberal 15, 57, 58 incentive compatibility 78, 79 inclusionary power 99 independence condition 12, 15, 20, 22, 25, 29, 37, 38,41, 80,100, 108, 110, 111, 125, 128, 144, 194, 195,197 independent prospects 135 indifference between no—choice situations 194, 196 individual rationality from equal division 191 inequity axiom 126, 130, 132 informational aspects 12, 14–16, 20, 128 informational constraints 12 informational structure 122, 123 instrumental aspect 193, 199 instrumentalist view 192 interpersonal comparisons of utility 122, 123, 126, 130, 135, 141, 146, 157, 159 intrinsic value of freedom 193, 194 intrinsic view 192 inversion metric 113 irrelevance of dominated alternatives 197 justice as fairness 121 K—dimensional boxes 92 Kalai—Smorodinsky solution 152–155, 157, 163–165 Kemeny metric 113, 114, 182 Kemeny ranking 112–114, 117 Kemeny rule 112–114 latin square 42, 45, 77, 109 legislative body 88 lexicographic difference principle 124 lexicographic egalitarian solution 158 lexicographic maximum solution in relative utility gains 155 leximaxprinciple 127, 128, 132 leximin principle 15, 124, 127–129, 132 liberal paradox 58 liberalism 58, 59, 62, 67–69 local decisiveness 57 locally integrable vector field 183 lottery 134, 135, 140, 141, 148, 157 majority rule 4, 10, 38, 42, 46, 47, 49, 50, 52, 53, 85, 116 majority rule with run-off 116 majority winner 50, 85 manipulability 75–78, 90 manipulation 3, 4, 76, 84, 91, 107 market failure 155 matching pennies 64, 65 maximal element 6, 7, 152 maximal set 6, maximin principle 15, 68, 89, 122, 124, 127, 128, 134, 165, 167–169, 171 mechanism design 80 median peak 86, 87, 90 median voter 15, 49, 86, 88, 90–92 method of majority decision 15, 43, 46, 49, 52, 102, 104 metric on preferences 182, 186, 187 minimal equity axiom 128 minimal liberalism 58 mixed alternative 140, 141 mixed—strategy equilibrium 65, 69 monotonicity 85, 147, 151, 152, 154, 155, 158 multi—dimensional choice space 181 multi—dimensional social choice 92 narrow Borda rule 107, 108 Nash bargaining solution 15, 140, 144, 146, 148, 150, 164, 165 Nash equilibrium 65, 66 Nash formula 145, 147 Nash product 150 Nash’s independence condition 144,147, 151, 155 neutrality condition 29, 38,40-42,102-104, 125,129 non-cooperative bargaining 155 non-cooperative games 148 non-utility information 12, 13, 29, 129 nondictatorship 20, 37,99, 101,184 observer-dependence 136 one-sided resource-monotonicity 191 opportunity set 193, 197, 198 paradox of voting 25, 42, 45, 48, 98,108, 109 Paretian property Pareto indifference 29,134, 135, 192 Pareto principle 5, 19, 20,22, 25,27, 29, 30, 33, 37, 38, 42, 53, 59-61, 64,66, 70, 80, 81, 83-85, 98, 99, 101,125, 128-135,146, 148, 153, 154, 158, 178, 184, 189,192 Pareto-extension rule 97-99, 101 Parliamentary vote 114, 116 phantom voters 88, 89, 91 Pigou-Dalton principle 178 pivotal mechanism 78 plurality rule 103, 104, 106, 112, 113,116 plurality winner 105 political spectrum 90 positional dictatorship 126,127, 129, 132 positional ranking procedure 3, 12 positive responsiveness 39-42, 99, 101 preference reversal 111, 113 primary goods 121, 122 prisoners’ dilemma 64, 66 private goods 78 proportional rule 191 proportional solution 191 proximity of preferences 183 proximity preservation 182, 186, 187 public goods 78 public opinion 164 pure exchange 78 pure prospects 134, 140 quasi-transitivity 6, 98-102 rank dictatorship 126, 129 rational choice 7, rationalizability rationing scheme 188 Rawlsian approach 14 Rawlsianism 122, 128, 163 reflective equilibrium 164 resource-monotonicity 191 restricted Pareto optimality 90 rights-system 58, 61 xrights-waiving solution 61 riskaversion 89, 134 risk limit 149 robustness 52 roll-call 115 scoring function 102, 103, 108 scoring rule 103, 107 separability of unconcerned individuals 126, 128,129, 132 set contraction 144 side payments 78 simple majority rule 3, 10, 12, 15, 37-43, 45, 49, 51-53, 77, 85-87, 89, 102, 109, 110, 125 single-cavedness 47 single-crossing 90, 91 single-peaked preferences 4, 15, 43-50, 85-92, 181,187, 188, 191,192 single-plateaued preferences 192 single-troughedness 47-49 social choice function 80, 81, 84 social decision function 8, 58-60, 97-102 social evaluation functional 29, 30, 123, 128 social indifference curves social welfare function 1, 5, 19-22, 29, 37, 41, 45, 49, 50, 52, 57, 80, 85, 99, 139 stability ofa voting situation 85, 89, 90 status quo 11,15, 38, 39, 140-148,152, 155-157, 163 step-by-step negotiation condition 158 strategic misrepresentation 75, 85 strategic voting 75, 77, 90 strategy-proof voting rules 76, 88 strategy-proofness 76, 79, 80, 84, 85, 87-89, 91,92, 189-192 strict monotonicity 194 strict Pareto principle 125, 128, 130-133,135, 158 strong monotonicity 158 strong neutrality 29, 38 surjective 84, 85 symmetry 143, 153, 154, 158,165 tenability 163, 164, 167, 168 threat advantage 140, 147 top cycle 89 topological domain restriction 185 transfer principle 178 transitive closure method 109, 110, 114 transitivity 5, 6, 9, 22, 25, 27, 32, 42, 46, 47, 50–53, 80, 89–91, 98, 99, 101, 130, 131, 133 unanimity 19, 113, 183–187 unconditional preferences 62–64, 66 uniform allocation rule 189–191 uniform rule 187, 190–192 unique Condorcet winner 85–87, 90 unit—comparable utilities 124, 135 unrestricted domain of preferences 20, 29, 37, 40, 41, 58, 59, 76, 77, 80, 81 utilitarianism 10, 15, 121, 122, 124, 126, 128, 129, 133–135, 163, 165, 170, 171, 173 value-restricted preferences 49 veil ofignorance 123, 147, 198 veto power 97, 99 von Neumann—Morgenstern axiom 134 von Neumann—Morgenstern utility function 134, 141 voters’ sovereignty 76, 84 voting cycle 89 waiver set 61 Walrasian mechanism 78, 79, 191 weak dictatorship 99–101 weak Pareto principle 19,20, 22, 25,29, 30, 37, 42, 58,61, 64, 104, 110, 132, 142, 144, 158,184 welfarism 12,13, 29, 30,122, 129, 130, 140, 143,159, 170 willingness to pay 78, 170,171 Zeuthen’s Principle 148–150 The internet address is http://nts4.oec.uni-osnabrueck.de/mikro/darp.pdf All in all, we had given six different situations to the students All these situations are fully reproduced in Gaertner and Jungeilges (2002) We should mention that in Osnabrück, we had two versions of our questionnaire, a technical and a non-technical version (the technical version is reproduced here and on the internet) The non-technical version did not use the specification in terms of extended orderings but provided a somewhat lengthier verbal description of the same ‘facts’ instead Of course, each student only saw one version Since in Israel and the Baltics, we used the non-technical version, table 9.7 gives the results from this version only The Osnabrück results for the two versions did not show any difference on the basis of a two-sample non-parametric test, given an error probability of 5% ... Strategy and Dynamics in Contests by Kai A Konrad Forthcoming books Econometric Analysis of Panel Data by Vassilis Hajivassiliou Measuring Inequality by Frank Powell A Primer in Social Choice Theory. .. concise and original insight into a wide range of topics in economics Each book is accessibly written but scholarly to appeal to advanced students of economics, and academics and professionals wishing... Germany He is one of the managing editors of the journal Social Choice and Welfare In the past, he has been visiting scholar at Harvard University and the London School of Economics He was awarded