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SocialChoiceandtheMathematicsofManipulation
Honestyinvoting,itturnsout,isnotalwaysthebestpolicy.Indeed,intheearly1970s,
AllanGibbardandMarkSatterthwaite,buildingontheseminalworkofNobel
LaureateKennethArrow,provedthatwiththreeormorealternativesthereisno
reasonablevotingsystemthatisnon-manipulable;voterswillalwayshavean
opportunitytobenefitbysubmittingadisingenuousballot.Theensuingdecades
producedanumberoftheoremsofstrikingmathematicalnaturalitythatdealtwiththe
manipulabilityofvotingsystems.Thisbookpresentsmanyoftheseresultsfromthe
lastquarterofthetwentiethcentury–especiallythecontributionsofeconomistsand
philosophers–fromamathematicalpointofview,withmanynewproofs.The
presentationisalmostcompletelyself-containedandrequiresnoprerequisitesexcepta
willingnesstofollowrigorousmathematicalarguments.
alan d. tayloristheMarieLouiseBaileyProfessorofMathematicsatUnion
College,wherehehasbeensincereceivinghisPh.D.fromDartmouthCollegein1975.
Hisresearchinterestshaveincludedlogicandsettheory,finiteandinfinitary
combinatorics,simplegames,andsocialchoicetheory.HeistheauthorofMathematics
andPolitics:Strategy,Voting,Power,andProofandcoauthorofFairDivision:From
Cake-CuttingtoDisputeResolutionandTheWin-WinSolution:GuaranteeingFair
SharestoEverybody(bothwithStevenJ.Brams)andSimpleGames:Desirability
Relations,Trading,andPseudoweightings(withWilliamS.Zwicker).
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OUTLOOKS
PUBLISHEDBYCAMBRIDGEUNIVERSITYPRESSAND
THEMATHEMATICALASSOCIATIONOFAMERICA
Mathematical content is not confined to mathematics. Eugene Wigner noted
theunreasonableeffectivenessofmathematicsinthephysicalsciences.Deep
mathematicalstructuresalsoexistinareasasdiverseasgeneticsandart,finance
andmusic.Thediscoveryofthesemathematicalstructureshasinturninspired
newquestionswithinpuremathematics.
IntheOutlooksseries,theinterplaybetweenmathematicsandotherdisci-
plinesisexplored.Authorsrevealmathematicalcontent,limitations,andnew
questionsarisingfromthisinterplay,providingaprovocativeandnovelviewfor
mathematicians,andforothersanadvertisementforthemathematicaloutlook.
ManagingEditor
RonaldL.Graham,UniversityofCalifornia,SanDiego
EditorialBoard
JohnBarrow,UniversityofCambridge
FanChung,UniversityofCalifornia,SanDiego
IngridDaubechies,PrincetonUniversity
PersiDiaconis,StanfordUniversity
DonZagier,MaxPlanckInstitute,Bonn
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Social Choice and the Mathematics
of Manipulation
ALAN D. TAYLOR
UnionCollege
MathematicalAssociationofAmerica
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Contents
Preface page ix
PART ONE
1 AnIntroduction to Social Choice Theory 3
1.1 Some Intuitions, Terminology, and an Example 3
1.2 A Little History 9
1.3 Arrow’s Theorem 13
1.4 Twenty Voting Rules 20
1.5 Exercises 29
2 AnIntroduction to Manipulability 37
2.1 Set Preferences and Manipulability 37
2.2 Specific Examples of Manipulation 44
2.3 Summary of the Main Results 51
2.4 Agenda Manipulability and Transitive Rationality 53
2.5 Exercises 56
3 Resolute Voting Rules 60
3.1 The Gibbard–Satterthwaite Theorem 60
3.2 Ties in the Ballots 68
3.3 The Equivalence of Arrow’s Theorem and the
Gibbard–Satterthwaite Theorem 69
3.4 Reflections on the Proof of the Gibbard–Satterthwaite
Theorem 72
3.5 Exercises 77
PART TWO
4 Non-Resolute Voting Rules 81
4.1 The Duggan–Schwartz Theorem 81
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viii Contents
4.2 Ties in the Ballots 87
4.3 Feldman’s Theorem 88
4.4 Expected Utility Results 95
5 Social Choice Functions 102
5.1 The Barbera´–Kelly Theorem 102
5.2 Ties in the Ballots 109
5.3 Another Barbera´ Theorem 110
5.4 The MacIntyre–Pattanaik Theorem 113
6 Ultrafilters and the Infinite 118
6.1 The Infinite Version of Arrow’s Theorem 118
6.2 Infinite Gibbard–Satterthwaite without Invisible Dictators 122
6.3 Invisible Dictators Resurrected 123
6.4 Infinitely Many Voters and Infinitely Many
Alternatives 125
PART THREE
7 More on Resolute Procedures 133
7.1 Combinatorial Equivalents 133
7.2 Characterization Theorems for Resolute Voting Rules 136
7.3 Characterization Theorems for Resolute Social Choice
Functions 140
7.4 Characterizations for Resolute Social Welfare Functions 142
8 More on Non-Resolute Procedures 147
8.1 Ga¨rdenfors’ Theorem 147
8.2 Characterization Theorems for Non-Resolute Voting Rules 152
8.3 Another Feldman Theorem 154
8.4 Characterization Theorems for Non-Resolute Social
Choice Functions 157
9 Other Election-Theoretic Contexts 160
9.1 Introduction 160
9.2 Ballots That Are Sets: Approval Voting and
Quota Systems 160
9.3 The Barbera´–Sonnenschein–Zhou Theorem 163
9.4 Outcomes That Are Probabilistic Vectors:
Gibbard’s Theorem 164
References 167
Index 173
