section4Myerson-and-Weber(博弈论-哈佛大学)

GOV2005:GameTheorySection4:[e-mail]•Mainideas•Keyterms•Analysis–Themodelofmulticandidateelections–Votingequilibria•Example1:Comparisonofvotingsystems•Example2:Effectofasmallminority–CandidatepositioninggameMainIdeas•Differentelectoralsystemshavedifferentvotingequilibria,evenwhenvoterspreferencesarefixed•Pluralityruleandapprovalvotingcanleadtodifferencesincandidatepositioning•Thereisadifferencebetweentwotypesofcampaignactivities–Thoseintendedtopresentinformationandhelpvotersdeterminetheirpreferences–Thoseintendedtoinfluencetheselectionofanequilibriumoutcomebymanipulatingperceptionsofcandidateviability(viafocalarbiters)PluralityRuleLessRestrictivethanApprovalVotingKeyTerms•Approvalvoting:cangive1/0votestoeachcandidate•Pluralityrule:cangive1votetoasinglecandidate•Bordavoting:gives0votestoonecandidate,1votetoanother,and2votestotheremainingcandidate•Ballot:(v1,v2,…,vk)forcandidates1…k–Forexample,aballotinabordacount,k=3mightbe(2,0,1)•Preferences:(u1,u2,…,uk)forcandidates1…k,whereuiisthepayoffthevoterreceivesifiwins–Forexample,onesetofpreferencesmightbe(10,4,5)–VotersareofthesametypeiftheysharethesamepreferencesPivotProbability:SameforAllVoters•Pivotprobability:pij,theprobability(perceivedbyavoter)thatcandidatesi&jwilltiefor1stplace(pij=pji)•Avoterperceivestheprobabilityhisvotewillbreakatiebetweeni&j,andelectitobe:max[(pij)(vi–vj),0]–Inaplurality-election,r’sballotmightbevr(1,0,0).Ifhispivotprobabilitythat1&2willtiefor1stis½,thenhisperceptionoftheprobabilityhisvotewillelectcandidate#1is½.•Notethatiftheballotwasvr(0,1,0),thereisnowaythatr’sballotwouldelectcandidate#1.Therefore,max[(pij)(vi–vj),0]=0.–Inaborda-count,r’sballotmightbevr(2,1,0),andifhispivotprobabilitythat1and2willtiefor1stis½,hisperceptionoftheprobabilitythathisvotewillelectcandidate#1remains½–Instillanother(approval-votingelection)voterr’sballotmightbevr(1,1,0),andifhispivotprobabilitythat1and2willtiefor1stis½,thenhisperceptionoftheprobabilitythathisvotewillelectcandidate#1iszero.AnticipatingaUtilityGain•Recallthatmax[(pij)(vi–vj),0]istheperceptionoftheprobabilitythattheballotwillbreakatieinfavorofi•G(p,v,u):utilitygainexpectedbyavoteroftypeufromsubmittingballotvwhenpishisvectorofperceivedpivotprobabilities:3-candidateexample,G(p,v,u)=(p12)(u1–u2)(v1–v2)+(p13)(u1–u3)(v1–v3)+(p21)(u2–u1)(v2–v1)+(p23)(u2–u3)(v2–v3)+(p31)(u3–u1)(v3–v1)+(p32)(u3–u2)(v3–v2).But:Either(pij)(ui–uj)(vi–vj)or(pji)(uj–ui)(vj–vi)goesin,butnotboth.(p12)(u1–u2)(v1–v2)=expectedvalueofbreakingatieinfavorofcandidate1(p21)(u2–u1)(v2–v1)=expectedvalueofbreakingatieinfavorofcandidate2Votermayonlydoonethingortheother(orgivebotha‘1’,so(vi–vj)=0)Nomatterwhat,youonlyhavetocountoneoftheseterms--theonethat’s0ProspectiveRating•So,ina3-candidaterace,G(p,v,u)=(p12)(u1–u2)(v1–v2)+(p13)(u1–u3)(v1–v3)+(p21)(u2–u1)(v2–v1)+(p23)(u2–u3)(v2–v3)+(p31)(u3–u1)(v3–v1)+(p32)(u3–u2)(v3–v2).Whatshouldavoterdo?WhatballotmaximizesG?Ask:“Howdoesvotingforcandidateieffectmyexpectedutility--consideringthecasesinwhichit’sacloseracebtwiandj,andalsowhereit’saclosebtwiandh”Answer:checktheprospectiverating:e.g.,theprospectiveratingofcandidate1is(p12)(u1–u2)+(p13)(u1–u3).•Inapluralityelection,youvotefor1candidate–(1,0,0),(0,1,0),(0,0,1)areyouronlyoptions•Inanapprovalelection,youhave7options–(1,0,0),(0,1,0),(0,0,1),(1,1,0),(1,0,1),(0,1,1),(1,1,1)Summary:ProspectiveRating•Theelectiontakesplaceinaninstitutionalsetting–Eitherplurality,bordacount,orapprovalvoting•Eachtypeofvoterhasparticularpreferences•Thereareexpectationsorbeliefsknownaspivotprobabilitiesthataresharedamongtheelectorate–Pivotprobabilities(pij)=chanceofatiebetweeniandj•Givenpivotprobabilities,andthevotingsystem,eachtypeofvotersubmitsaballotthatmaximizesG,theanticipatedgainfromvoting.(Hey!Wakeup!Nextlinesarekey!)–Inapluralityelection,eachtypeofvotersubmitsasinglevoteforthecandidateassociatedwiththehighestprospectiverating–Inapproval-votingelections,eachtypeofvotersubmitsavoteforanycandidateassociatedwithapositiveprospectiverating–Inabordacount,Gismaximizedbyrankingcandidatesinorderoftheirprospectiveratings.MoreKeyTerms•Electionresult:Probabilitydistributionμthatsummarizesthevotingbehaviorofvotersofeachtype–μ(v,u)isthefractionofelectorateoftypeucastingballotv–μ(v,T)isthefractionofelectoratecastingballotv•Predictedscoreofcandidatej:Sj(μ)=Σ(vj)[μ(v,T)].Ex:with3ballots(2,1,0),(1,2,0),and(0,1,2)splitevenlyamongtheelectorate,thepredictedscoreofcandidate1is(1/3)*(2)+(1/3)*1+(1/3)*0=1Likelywinner:candidatewithamaximalpredictedscoreVoterresponseset:setofallelectionresultsinwhichallvoterscastballotsthatmaximizetheirexpectedgains–Ifvotingbehaviorμinagivenelection,withagivenpivot-probabilityvector),isconsistentwithutilitymaximization,thenthatpivot-probabilityvectorissaidtojustifyμv∈VTakeaStepBackVotingEquilibria•Testyourunderstanding:–Ifpivotprobabilitieswere=0,votingdoesn’tmatter•Definition:Orderingconditionforε,withrespecttoμ(forε,1ε0)•Rememberthisorderingcondition,itwillbeimportantlater–Ifjgetsmorevotesthani,thenthep