Social choice and the mathematics of manipulation PDF
Preview Social choice and the mathematics of manipulation
P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 This page intentionally left blank ii P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 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). i P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 ii P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 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 iii P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 iv P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:20 Social Choice and the Mathematics of Manipulation ALAN D. TAYLOR UnionCollege MathematicalAssociationofAmerica v ccaammbbrriiddggee uunniivveerrssiittyy pprreessss CCaammbbrriiddggee,,NNeewwYYoorrkk,,MMeellbboouurrnnee,,MMaaddrriidd,,CCaappeeTToowwnn,,SSiinnggaappoorree,,SSããooPPaauulloo CCaammbbrriiddggee UUnniivveerrssiittyy PPrreessss TThheeEEddiinnbbuurrgghhBBuuiillddiinngg,,CCaammbbrriiddggeeccbb2222rruu,,UUKK PPuubblliisshheeddiinntthheeUUnniitteeddSSttaatteessooffAAmmeerriiccaabbyyCCaammbbrriiddggeeUUnniivveerrssiittyyPPrreessss,,NNeewwYYoorrkk wwwwww..ccaammbbrriiddggee..oorrgg IInnffoorrmmaattiioonnoonntthhiissttiittllee::wwwwww..ccaammbbrriiddggee..oorrgg//99778800552211881100552244 ©©AAllaannDD..TTaayylloorr22000055 TThhiissbbooookkiissiinnccooppyyrriigghhtt..SSuubbjjeeccttttoossttaattuuttoorryyeexxcceeppttiioonnaannddttootthheepprroovviissiioonnooff rreelleevvaannttccoolllleeccttiivveelliicceennssiinnggaaggrreeeemmeennttss,,nnoorreepprroodduuccttiioonnooffaannyyppaarrttmmaayyttaakkeeppllaaccee wwiitthhoouutttthheewwrriitttteennppeerrmmiissssiioonnooffCCaammbbrriiddggeeUUnniivveerrssiittyyPPrreessss.. 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MMaatthheemmaattiiccaall AAssssoocciiaattiioonn ooff AAmmeerriiccaa 11552299EEiigghhtteeeenntthhSStt..NNWW,,WWaasshhiinnggttoonn,,DDCC2200003366,,UUSSAA P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:27 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 vii P1:JZY/KDF CB799-FM.xml CB799B/Taylor 0521810523 March31,2005 9:27 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
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