Disposing dictators, demystifying voting paradoxes

257 Pages·2008·1.224 MB·English

by  Saari D.G.

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cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
DISPOSINGDICTATORS,DEMYSTIFYINGVOTINGPARADOXES
Wedecidebyelections,butdoweelectwhothevotersreallywant?Thean-
swer,aswehavelearnedoverthepasttwocenturies,is“notnecessarily.”That
isanegative,frighteningassertionaboutaprincipaltoolofdemocracy!This
negativism has been supported by two hundred years of published results
showinghowbadthesituationcanbe.Thisexpository,largelynontechnical
book is the first to find positive results showing that the situation is not
nearly as dire and negative as we have been led to believe. Instead there
aresurprisinglysimpleexplanationsforthenegativeassertions,andpositive
conclusionscanbeobtained.
DonaldG.SaariisDistinguishedProfessorofMathematicsandEconomics
andHonoraryProfessorofLogicandPhilosophyofScienceattheUniversity
ofCaliforniaatIrvine,whereheisDirectoroftheInstituteforMathematical
Behavioral Sciences. He previously served on the faculty of Northwestern
University from 1968 to 2000, where he held the Pancoe Professorship of
Mathematics.AmemberoftheU.S.NationalAcademyofSciences,Fellow
oftheAmericanAcademyofArtsandSciences,andFellowoftheAmerican
Association for the Advancement of Science, Professor Saari is the former
ChiefEditoroftheBulletinoftheAmericanMathematicalSociety.Theauthor
of more than 170 published papers, he has also written numerous books,
includingBasicGeometryofVoting(1995),DecisionsandElections:Explain-
ingtheUnexpected(CambridgeUniversityPress,2001),ChaoticElection!A
MathematicianLooksatVoting(2001),TheWayItWas:Mathematicsfromthe
EarlyYearsoftheBulleting(2003),andCollisions,Rings,andOtherNewtonian
N-BodyProblems(2005).
cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
Disposing Dictators, Demystifying
Voting Paradoxes
Social Choice Analysis
DONALD G. SAARI
UniversityofCalifornia,Irvine
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521516051
© Donald G. Saari 2008
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2008
ISBN-13 978-0-511-42893-7 eBook (EBL)
ISBN-13 978-0-521-51605-1 hardback
ISBN-13 978-0-521-73160-7 paperback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
Formysister
KatriSaari,
acreativeartistanddedicatededucatorwhohasinfluencedmany.
Amongourmanysharedmemories,severalareassociatedwith
growingupintheforested,northernmostreachesoftheUnitedStates.
cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
cuus338-Sarri cuus338/Saari ISBN:9780521516051 July26,2008 14:7
Contents
Preface pagexi
1 SubtleComplexityofSocialChoice 1
1.1 DoesEverythingGoWrong? 1
1.2 AndtheProudFatherIs... 3
1.3 Enemies? 7
1.4 CurseofDimensionality 13
1.5 Outline 15
1.5.1 DethroningDictators,andThenParadoxes 16
1.5.2 The“WilloftheVoters”:WhatIsIt? 18
2 DethroningDictators 20
2.1 MajorNegativeConclusions 21
2.1.1 Arrow’sTheorem 21
2.1.2 Sen’sSeminalResult 23
2.1.3 TopologicalDictators 26
2.1.4 “ParadoxofVoting”andCondorcet’sTriplets 28
2.1.5 List’sLists 29
2.1.6 Anscombe 30
2.1.7 AStandardRequirement 31
2.2 Commonality 31
2.2.1 Condorcet’sIdeasDominate 32
2.2.2 CanWeTrusttheMajorityVotingoverPairs? 40
2.2.3 ACommonExplanation 42
vii
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viii Contents
2.3 WhyDoTheseNegativeResultsOccur? 43
2.3.1 Arrow’sTheorem 44
2.3.2 Sen’sResult 47
2.3.3 TopologicalDictatorsandBeachParties 51
2.4 PositiveReplacements 58
2.4.1 Arrow’sResult 58
2.4.2 Sen’sResult 64
2.4.3 TopologicalDictatorsandMoreEffectiveAgents 70
2.5 FinalThoughts 72
3 VotingDictionaries 74
3.1 WhatGoesWrong? 76
3.1.1 AxiomaticApproachversusParadoxes 77
3.1.2 Dictionaries 77
3.1.3 AggregationRules 80
3.2 LassieandtheAxiomaticApproach 81
3.3 Dictionaries 84
3.3.1 ALittleChaos 86
3.3.2 ChaoswithinVotingTheory 90
3.3.3 DictionaryListingsforPositionalRules 91
3.4 UsingtheDictionaries 96
3.4.1 VarietyComingfromVarieties 99
3.4.2 OtherDictionaries 104
3.5 ComparingOutcomesoveraSetofCandidates 107
3.5.1 GeneralResults 107
3.5.2 LetElementaryGeometryDotheWork 109
3.5.3 ProcedureLines 111
3.5.4 OtherRules,SuchAsApprovalVoting 113
3.5.5 AWorkingToolforActualElections 116
3.5.6 BacktotheOriginalProblem:CreatingExamples 118
4 ExplainingAllVotingParadoxes 123
4.1 ProfileCoordinates 124
4.1.1 CoffeeReflections 124
4.1.2 The“Water”forVotingRules 127
4.1.3 NothingGoesWrong 133