POLITICAL ECONOMY UNIVERSITY OF

W.ALLENWALLISInstituteofPOLITICALECONOMYUNIVERSITYOFROCHESTERABayesianModelofVotinginJuriesJohnDuggan,andCesarMartinelliWorkingPaperNo.18November1998ABayesianModelofVotinginJuriesJohnDugganDepartmentofPoliticalScienceandDepartmentofEconomicsUniversityofRochesterRochester,NY14627C¶esarMartinelliCentrodeInvestigaci¶onEcon¶omicaInstitutoTecnol¶ogicoAut¶onomodeM¶exicoSantaTeresa930,M¶exico,D.F.10700November9,1998AbstractWetakeagame-theoreticapproachtotheanalysisofjuriesbymodellingvotingasagameofincompleteinformation.Ratherthantheusualassumptionoftwopossiblesignals(oneindicatingguilt,theotherinnocence),weallowjurorstoperceiveafullspectrumofsignals.Weo®erthreemainresults.First,givenanyvotingrulerequiringa¯xedfractionofvotestoconvict,wecharacterizetheuniquesymmetricequilibriumofthegame.Second,weobtainaconditionunderwhichunanimityruleexhibitsabiastowardconvictingtheinnocent.Third,weprovea\jurytheoremforthecontinuoussignalmodel:asthesizeofthejuryincreases,theprobabilityofmakingamistakenjudgmentgoestozeroforeveryvotingrule,exceptunanimityrule;forunanimityrule,theprobabilityofamistakeisboundedstrictlyabovezero.1IntroductionConsideragroupofdecision-makerswhomustchooseoneoftwoalterna-tives.Votersagreeontheoverallobjective,but,onthebasisofdi®erentialinformation,theymaydisagreeonwhichalternativebestachievesthatgoal.Someexamplesare:²Ajurydecidingwhethertoconvictoracquitadefendant.Jurorsagreeaboutthedesirabilityofacquittinganinnocentandconvictingaguiltydefendant,buttheyhavedi®erentopinionsaboutwhetherthedefen-dantisinnocentorguilty.²Theboardofdirectorsofacompanydecidingwhethertoapproveanewinvestmentproject.Allmembersoftheboardagreeabouttheobjectiveofmaximizingpro¯ts,buttheydisagreeintheirestimatesofthepro¯tabilityoftheproject.²Anacademicdepartmentdecidingwhethertohireajobcandidate.Thecommonobjectiveistohireaproductiveresearcher,butthereisdisagreementabouttheacademicpotentialofthecandidate.Althoughourresultsapplyequallywelltoallsituationsexhibitingthisstructure,wefollowanoldliteratureoninformationaggregationinelectionsbyfocusingonthejuryexample.TheliteraturetracesbacktoCondorcet's(1785)jurytheorem,whichassertsthat,undermajorityvoting,largeelec-toratesshouldreachcorrectdecisionswithveryhighprobability(cf.Miller(1986),GrofmanandFeld(1988),Young(1988),Ladha(1992)).Itistra-ditionallyassumedthateachvotersimplybehaves\naively,i.e.,asifthevoterweredecidingtheoutcomealone,butAusten-SmithandBanks(1996)1observethat,givennaivebehavioronthepartofjurors,somemayhaveanincentivetovote\strategically.Inotherwords,naivebehaviordoesnotgenerallyconstituteanequilibrium.Wefollowsubsequentpapersinanalyz-ingvotinginjuriesasaBayesiangameinwhichthejurors'opinionsofguiltorinnocence,i.e.,their\signals,areprivateinformation.Severalversionsofthejurytheoremunderstrategicvotinghavebeeno®ered.FeddersenandPesendorfer(1997)assumediscretedistributionsofsignalsandacontinuumofstates.Myerson(1997)introducesuncertaintyaboutthesizeoftheelectorateandconsidersacountablesetofplayers'types(signals),wherethenumberofvotersreceivinganygivensignalisdrawnfromaPoissondistribution,themeanofwhichdependsonguiltorinnocence.Mc-Clennan(1998)provesthatthereisatleastoneequilibriumthatmaximizestheexantepayo®sofjurorsovertheclassofsymmetricstrategypro¯les.FeddersenandPesendorfer(1998)analyzeasimplemodelinwhichtherearetwopossiblesignals,oneindicatingguiltandtheotherinnocence.Givenanyvotingrulerequiringa¯xedfractionofvotestoconvict,theyareabletoexplicitlysolvefortheuniquesymmetric,responsiveBayesianequilibriumofthevotinggame.Theyshowthatajurytheoremholdsforallvotingrulesotherthanunanimity:asthesizeofthejuryincreases,theprobabilityofmakingamistakenjudgmentgoestozeroforallvotingrules,exceptuna-nimity;inthatcase,theprobabilityofamistakeisboundedstrictlyabovezero.FeddersenandPesendorfer(1998)alsogiveanexamplecomparingdif-ferentvotingrulesfora¯xedjurysize:there,theprobabilityofconvictinganinnocentdefendantunderunanimityruleisgreaterthantheprobabilityun-dermajorityoranyothersupermajorityrule.McKelveyandPalfrey(1998)o®erexperimentalresultsonthebinarysignalmodelroughlyconsistentwith2theequilibriumpredictions.Wedepartfromthepreviousliteratureonjuriesbyassumingthatthesignalsrepresentingthejurors'opinionsofguiltorinnocencearedrawnfromcontinuous,ratherthandiscrete(usuallybinary),distributions.Thisismeanttocapturethefactthatajuror'sopinionoftheevidenceagainstthedefendant,thecasemadebytheprosecutor,etc.,mayre°ectaveryrichspectrumofpossibilities|possibilitiesthatcannotbesummarizedbyadichotomoussignalmerelyindicatingguiltorinnocence.Weimposefewrestrictionsonthedistributionsofsignals,andweareactuallyabletoob-tainacontinuousanalogueofthe\binarysignalmodelasaspecialcase.UnlikeFeddersenandPesendorfer(1997),wecon¯neourattentiontothecaseinwhichtheobjectivesofthejurorsareperfectlyaligned.Withinthisframework,weanalyzevotinginjuriesasagameofincompleteinformation.Weo®erthreemainresults.First,weestablishtheexistenceofasym-metric,responsiveequilibriumcharacterizedbyacuto®signal:jurorswhogetsignalsindicatingahigherl