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ReprintrequestsshouldbesentbyemailtoGoereeatjg2n@virginia.edu.Thisresearchwasfundedinpartbythe*NationalScienceFoundation(SBR-9617784andSBR-9818683).WewishtothankMelayneMcInnesforahelpfulsuggestion.Stochasticgametheory:forplayinggames,notjustfordoingtheoryJacobK.GoereeandCharlesA.Holt*DepartmentofEconomics,RoussHall,UniversityofVirginia,Charlottesville,VA22903Recenttheoreticaladvanceshavedramaticallyincreasedtherelevanceofgametheoryforpredictinghumanbehaviorininteractivesituations.Byrelaxingtheclassicalassumptionsofperfectrationalityandperfectforesight,weobtainmuchimprovedexplanationsof(i)initialdecisions,(ii)dynamicpatternsoflearningandadjustment,and(iii)equilibriumsteady-statedistributions.IntroductionAboutfiftyyearsago,JohnNashwalkedintotheofficeoftheChairofthePrincetonMathematicsDepartmentwithasolutionconceptforN-persongames,alongwithanexistenceproofthatwassoonpublishedintheseProceedings[1].JohnvonNeumannwasdismissiveandremarkedthat'strivial,youknow.That'sjustafixedpointtheorem[2].ButwordofNash'stheoremspreadquicklyatRANDontheWestCoast,whereresearchersworkingondefensestrategyweredissatisfiedwiththereceivedtheoryofzero-sumgames,sincetheassumptionthatoneplayer'sgainisanother'slossisoflimitedrelevancebeyondsimplecardgames.Twomathematicians,DresherandFlood,designedalaboratoryexperimenttotestNash'sequilibriumconceptthesamedaytheyheardabouthisproof.Theirexperimentimplementedagameinwhichtwoplayershaveunilateralincentivestodefecteventhoughbotharebetteroffwhenbothcooperate.Nash'sthesisadvisor,Tucker,latersawthepayoffsforthisexperimentontheblackboardinsomeone'sofficeanddevisedthefamousstoryoftheprisoner'sdilemma,whichheusedinaseminarforthePsychologyDepartmentatStanfordUniversity[3].Theapplicationsofgametheoryhaveexpandedgreatlysincethen,andwiththeNashequilibriumasitscenterpiece,gametheoryhasfinallygainedthecentralrolefirstenvisionedbyvonNeumannandMorgenstern[4].Ifanything,gametheoryistheleadingcontenderfor2becomingageneraltheoryofsocialscience,withextensiveapplicationsineconomics,politicalscience,psychology,law,andbiology.Indeed,insomeareasofeconomicsvirtuallyallrecenttheoreticaldevelopmentsareapplicationsofgametheory.Thereis,however,widespreadcriticismoftheoriesbasedontheclassicalrationalchoiceassumptionsofperfectdecisionmaking(noerrors)andperfectforesight(nosurprises),especiallywhentheyareappliedtodescribebehaviorincomplexinteractivesituations.Thisskepticismisreinforcedbyevidencefromlaboratoryexperimentswithfinancially-motivatedsubjectswhichoftenproducebehaviorpatternsthataresystematicallybiasedawayfromrationalchoicepredictions.Nashhimselfparticipatedinexperimentsasasubjectandlaterdesignedexperimentsofhisown,butheandhiscoauthorslostwhateverconfidencetheyhadingametheorywhentheysawhowpoorlyitpredictedactualbehavior[2].AndReinhardSelten,whosharedthe1995EconomicsNobelPrize(withNashandHarsanyi),remarkedthatgametheoryisfordoingtheory,notforplayinggames.Likemanyothers,hehasarguedthatdecisionsarestochasticornoisy,wherethenoiseinsubjects'behaviormaybeduetoerrorsinperception,calculation,orrecordingdecisions[5].Alternatively,apparentnoisemayrepresentfullyrationalresponsestofactorslikebenevolence,envy,orotheridiosyncraticfactorsthatarenotmeasuredbytheexperimenter[6].Regardlessofthesourceandinterpretationofthenoise,theeffectwillbethatdifferentplayersencounterdifferenthistoriesofothers'play,andlearninginsuchenvironmentsmayleadtovariationsinindividuals'beliefsanddecisions.Thispaperdescribesthreenewdevelopmentsingametheorythatrelaxtheclassicalassumptionsofperfectrationalityandperfectforesight.Theseapproachestonoisyintrospection(priortoplay),learning(frompreviousplays),andequilibrium(afteralargenumberofplays)providecomplementaryperspectivesforexplainingactualbehaviorinawidevarietyofgames.CoordinationandSocialDilemmaGamesThemodelssummarizedherehavebeenstronglyinfluencedbydatafromexperimentsthatshowdisturbingdifferencesbetweengame-theoreticpredictionsandbehaviorofhumansubjectswhoareearningmoneyincontrolledstrategicsituations.Forexample,GoereeandHolt[7]showthatallthestandardtypesofgamescanbeimplementedinamannerthatyieldspredictionsconsistent3withtheNashequilibriumforsomeparametervalues,andyetineachcasetheobserveddatawillshiftdramaticallyinresponsetoapayoffchangethatdoesnotaltertheNashprediction.Similaranomalousresultshavebeenreportedinmanyotherexperiments,e.g.,matchingpenniesgames,centipedegames,two-stagegames,marketpricinggames,andbargaininggames[8-12].WewillpresentthemainargumentofasocialdilemmagameforwhichNash'stheorypredictsauniqueequilibriumthatisbadforallconcerned,andacoordinationgameinwhichanycommoneffortlevelisanequilibrium,i.e.theNashequilibriummakesnopredictionatall.Thesocialdilemmaisbasedonastoryinwhichtwotravelersloseluggagewithidenticalcontents,andtheairlineofficialpromisestopayanyclaiminanacceptablerangeaslongastheclaimsareequal.Ifnot,thepersonmakingthehigherclaimisassumedtohavelied,andbothwillbereimbursedatthelowerclaim,withareward,R1,beingdeductedfromthereimbursementtothehighclaimantandgiventothelowclaimant.ANasheq