Naïve playersdistractorfocuser Chapter1,UsingRules-of-Thumb:ANoteonSophisticatedvs.SimpleMixinginTwo-PlayerRandomlyMatchedGamesChristianDiegoAlcocerArgüelloAugust15,2016AbstractWepostulateanewbehavioralbiasinhowpeopleplaymixedstrategiesbyproposingtheexistenceofsimpleplayerswholackstrategicdepth.Wethemasthosewho,whenindifferentbetweenchoices,followasimplerule-of-thumbandassignapredeterminedprobabilitytoeach.Weshowthatiftheyplay22games,anequilibriumgenerallyfailstoexist.However,underrandommatchingwithinpopulationswithsomeproportionofsimpleplayers,equi-libriumisrestoredandisindistinguishablefromNashequilibriaingameswithunrestrictedstrategychoices,aslongasthepercentageofsimplemixersissmallenough.Assuch,playersareunabletotakeadvantageofthepresenceofsimplemixers,andsimplemixersdonoworsethanmoresophisticatedplayers.Keywords:JEL:C72,D03,D831IntroductionManygamesrequirethatplayersmixamongpurestrategiesinordertoattainanequilibrium.Frequentlysuchmixedstrategyequilibriumareintuitivelyplausible:childrenrecognizetheadvantageofbeingunpredictablewhenplayinggamessuchrock-paper-scissors,andsomedescriptionsofthe`matchingDepartmentofEconomics,MichiganStateUniversity,EastLansing,MI48824,alcocer@msu.edu.1pennies'gameactuallyhaveplayerspennies(ratherthanchoosingheadsortails)inordertodeterminepayouts.Thelogicofwantingtobeunpredictableandwantingtokeepone'srival(s)guessingaboutwhattoexpectreadilyextendstomorecomplexgames.Andyet,ingamesinwhichequilibriumrequiresplayerstomix,wearesometimesatalossofintuitiveexplanationsfortheprescribedequilibrium;particularlysinceitmaybediftodeterminetheoptimalmixingdistribution.Thistaskiscomplicatedbythefactthat,inequilibrium,theplayerisexindifferentbetweenchoosinganyparticularpurestrategythatwouldbeplayedwithpositiveprobability.Ouranalysisismotivatedbytheapparentneedformodelingofoff-equilibriumbehavior.Inmostoftheexperimentalliterature,players'behaviorisdifferentthanthestrategiespredictedbygametheory;inreal-lifescenarios(MisirlisoyandHaggard,2014)andparticularlyinnormal-formgamesinexperiments(NyarkoandSchotter,2002;Stahl,1995).Masiliunasetal.(2004)thatonly7%ofchoicesareconsistentwithNashequilibriumandcannotrejectthehypothesisthatincomplexgames,players'choicesareuniformlydistributed.FailureofNEholdsevenwhentheexperimentersprivatelyrecommendoptimalstrategies(CasonandSharma,2007)orwhenparticipantsareallowedtoexplicitlyinputmixed-strategyprobabilitiesthatareautomaticallyimplementedinsteadofhavingtogothroughtheextracognitivestepofexecutingthemmanually1994).Relatedly,Parkhurstetal.(2015)thatwhenindividualsfaceinformationoverload,theytendtousesimplifyingheuristics.Thecharacterizationofbehindtotally-mixedNashequilibriacanbeapproachedheuristicallyasatwo-stepdecisionprocess.First,aplayermaximizeshisutilityanddetermines,givenhisbeliefsregardinghisopponents'strategies,thathehasnouniquebestresponse.Hethenproceedstoamorestrategicallysophisticateddeliberationthatdetermineswhatsortofmixingensurestheotherplayersareindifferentaswellandpreventsthemfromhavingauniquebestresponse.Inthispaper,weexplorewhathappenswhensomeplayers,ratherthanascertainingequilibriummixingdistributions,neglectthesecondstepdescribedaboveandshootfromthehipbyusingsimplerulesofthumbtochooseamongthepurestrategiesthatleadtothehighestpayoff.Werefertosuchplayersassimplemixers.Incontrast,asophisticatedmixerisaplayerwhocanfreelydetermineandemployanymixedstrategies.Weshowthatin22games,unlesstheruleofthumbthatsimplemixersareguidedbyhappenstocoincidewiththeequilibriummixedstrategyusedbysophisticatedplayers(sothattheirplayisindistin-guishable),anequilibriumfailstoexist.However,underanonymousrandommatchingwithinpopulations2withsimpleandsophisticatedmixers,whenonlythefractionofeachpopulationiscommonknowledge,thenequilibriumisrestorediftheprobabilitythatagivenplayerisasimplemixerisbelowanupperbound.Thisequilibriumisattainedbysophisticatedmixersadjustingtheirstrategiestoaccountforthedistor-tionsinducedbythepresenceandbehaviorofsimplemixersinthepopulation.Asaresult,allplayersŠsophisticatedandsimplealikeŠobtainthesamepayoffsthatprevailintheequilibriumofthestandardgameinwhichtherearenosimplemixers.Animplicationofthisisthatsophisticatedplayersareunabletotakeadvantageofthepresenceofsimplemixerswhodonoworse.Inasense,simplemixersfree-rideoffthesophisticationoftheotherplayers.2LiteratureReviewTheissueofcomplexityinthedeterminationofequilibriahasbeenextensivelyanalyzedinthecompu-tationalliterature.Liptonetal.(2003)discusshowfortwo-playergames,theknownalgorithmshavepolynomialorworse(exponential)runningtimeinthenumberpurestrategies.ThiskeepstheminsomeclassbetweenPandNPsinceitiseasy(i.e.inpolynomialtime)toverifyifagivenstrategyisanequilibrium.Moreover,thecomplexityofsolvingagamecanincreaseconsiderably,dependingontheforminwhichitisrepresented(Kolleretal.,1994)sincetherelationofthenumberofstrategiestothesizeofthetreeformisexponential.Furthermore,fitheincreaseinsizeoftransformingagamedescribedbyasetofrulesintoamatrix-formgameisnotevenboundedingeneral,whichoftenforcesanalyststosolutiontechniquestailoredtogamesfl(Kolleretal.,1994).Becauseofthis,Liptonetal.(2003),forin-stance,considernormalsimplestrategies,strategieswhichareuniformonasmallgenericsupportset.Theyjustifythiswayofthinkingbyconsideringpurestrategiesasresources.Assuch,anequilibriumiscalledfiimpracticalflifaplayerhastorandomizeoveranextensivesetofresources.EversinceNash(1951),therehavebeenattemptstoextendthenotionofNashequilibrium(NE)todealbetterwithempiricaldata.theory1(Harsanyi,1973)andtheconceptofaquantalresponseequilibrium(QRE)2describehowmixed-strategyequilibriacanbeinterpretedasthelimitofpure-strategy1Formally,inthemostgeneralterms,ifplayeri'spayoffgiventhe(pure)strategysisui(s);thenwhatheactuallyobservesisUi(s;qi):=ui(s)+esiqsiwhereesiqsiistheGaussiani.i.d.perturbationscaledbyhistypeandE(esi)=0(seeFudenbergandTirole,1991):Thevectorqicanbeinterpretedasrelatedtolevelsofrationality:ifallitscomponentsarezero,theplayer(perfectly)observesanunperturbedgame.IfVaresi!¥orqsi!¥;heobservesonlynoiseandmixesuniformlyacrossallŒincludingdominatedŒstrategies.2Inthiscase,ifplayeri'spayoffgiventhe(pure)strategysisui(s);thenwhatheactuallyobservesisUi(s):=ui(s)+esiwhereesiisthei.i.d.perturbationE(esi)=0,(McKelveyandPalfrey,1995;McKelveyandPalfrey,1998;Goeree,HoltandPalfrey,3equilibriaingameswhenperturbationstoallpayoffsareassumed.Ex-ante,allpurestrategiesreceiveapositiveprobabilityandprobabilitiesaregreaterforthosestrategiesthatyieldgreaterexpectedpayoffs.AQREcanbeapproximatedandestimatedasalogitequilibrium(McFadden,1973):3ExperimentalevidencesuggestsQREissometimesabetterpredictorthanNEand(ErleiandSchenk-Mathes,2012).Underagivensetofassumptions,aQREcanbeapproximatedandestimatedasalogitequilibrium(McFadden,1973).Cognitivehierarchytheoryorlevel-kthinking(Camereretal.,2004;VanDamme,1991)assumesthattherearemultiplelevelsofplayerswithvaryingdegreesofrationality.Therearenon-strategiclevel-0playerswhoalwaysmixuniformlyamongalltheirpurestrategies.4Level-kplayersbest-respondtolevel-(k-1)populations.Assuch,thispaperattemptstoenhancetheliteratureonoff-equilibriumplay.ThemodelbelowisconceptuallydifferentfromandQREwhereifthevariancesoftheperturbationstendtozero,allstrategiestendtotheNEwhereasasimpleplayerwillnevermixwithanon-degenerateprobabilitydifferentthansrnorwillheplayastrongly-dominatedstrategy.Also,inequilibriumtherelativelikelihoodofoutcomesundercoincidewiththerelativelikelihoodsintheNE.Finally,alevel-0isdistinctfromanaïveplayersincethelateronlymixesuniformlyamongstasubsetofpurestrategies.3ModelWeestablishthemainbymodifyingasimple22game,butitwillbereadilyapparentthattheinsightscarryovertocomplexmulti-playergamesstraightforwardly.Considerageneric22payoffstructureGsuchasthatdepictedinFigure1.Superscriptsdenoteplayersandsubscriptsdenotepurestrategies,withuij;kgivingPlayeri'spayoffwhenPlayer1playss1jandPlayer2playss2k.Weletsi1:=siindicatetheprobabilitywithwhichPlayerichoosessi1,sotheprobabilityofpickingsi2issi2:=1si.Whilemaintainingthestructureofthegameregardingpurestrategiessim,weextendtheanalysistoincludebehavioralplayersŒnamelysimplemixersthatarerestrictedintheirchoiceofmixingamongtheir2002).3Foragivenli0;thelogisticquantalresponseisassji(sjijsi)=expnliUji;io=Kåk=1expnliUki;iowhichistheprobabilitythatiassignstoactionsji(fromsomeactionsetofcardinalityK)giventheotherplayers'strategies,andUji;iistheperturbedpayoffhegetsaftersjiandsi.Theparameterlicanbeinterpretedasameasureofi'srationality.Asl!¥;ifthereexistsauniqueNE(s);asisthecasethroughoutthegamesdescribedbelow,then(s1;:::;sN)convergestos:Asl!0;playersbecomeblindandmixuniformlyacrossallstrategiessuchthattheprobabilitiestendto1K.4Thisisoneofthesimplestpossiblebehaviorsintermsofcomputationalcomplexity(KollerandMegiddo,1996;Koller,MegiddoandVonStengel,1994;Liptonetal.,2003).4Player2s21s2 2s1 1u1 1;1;u2 1;1u1 1;2;u2 1;2s1Player1s12u1 2;1;u22;1u12;2;u22;21s1s21s2Figure1:Generic22GameGpurestrategies.Insteadofbeingallowedtoassignanyprobabilitysimtoanypurestrategysi m(aslongassim2[0;1]andåm2Msi m1whereMisthesetofavailablepurestrategies),simplemixerscanonlyutilizesomepredeterminedmixsrorpurestrategies.Weindexsuchfirule-of-thumbflplayersbyranddenotetheirbehaviorbysr20;sr;1:4AnalysisofSimplePlayAssumethatitiscommonknowledgethateitherisasimplemixerandthatGhasauniqueequilibrium.Lemma1Letx:=s1;s2denoteanequilibriuminthegeneric22gamewithnosimpleplayers.Then,1.thepresenceofasimplemixeridoesnotaffectequilibriumbehavior,ifsi20;sr;1;2.andiftheequilibriumxisuniqueandsi=20;sr;1,thennoequilibriumexists.Lemma1followsdirectlyfromtheofequilibriaandsimplemixing.First,ifsi20;sr;1;thepresenceofasimplemixerdoesnotaffectequilibriumbehaviorsincenoplayerhasanyincentivetodeviatefroms1;s2asexpectedpayoffsareunchanged.Second,toseewhyequilibriacanbelost,supposethatPlayeriisasimplemixerwherei2f1;2gandŸs1;Ÿs2isanequilibriumsuchthatŸsi=20;sr;1:SincetheequilibriumisuniqueandPlayeriisnotallowedtoplayitgiventheofwhichmixedstrategiesareavailabletosimplemixers,thegamehasnoequilibrium.Asanillustration,considertheasymmetricmatchingpenniesexample,withu11;1>1.TheuniqueNEiss1=1=2ands2=11+u11;1:IfPlayer2isasimplemixerthatfollowstheruleofthumbsr=1=2;5then5Itwillbereadilyapparentthat,inmorecomplexgames,themainresultsholdwithanypredeterminedmixingdistribution,aslongasitispublicknowledge.Furthermore,behavioral,experimentalandinsufargumentscanbemadetojustifywhyuniformmixing(sr=12inthe22case)isagoodstartingpoint.5Player2s21s2 2s1 1u1 1;1;00;1s1Player1s120;11;01s1s21s2Figure2:AsymmetricMatchingPenniesthegamehasnoequilibrium,asdepictedinFigure3thatdepictsthelossofconvexityini'sstrategyspace.Figure3:Non-ExistenceofEquilibrium5RandomlyMatchedPlayPreviouslyweassumedthatallplayersweresimplemixers.Wenowgeneralizethegamebyallowingproportionsofbothsimpleandsophisticatedplayers.Thisreconvestrategyspaces.Wewilluseinvertedhatstodistinguishthegeneralizedgamethatincludessimpleandsophisticatedmixersfromthegenericgamethatonlyincludessophisticatedplayers(G)souiandsiwilldenotetheequilibriumpayoffandmixingofPlayeriinthegeneralizedgame.Consideralargepopulationofplayerswhoarerandomlymatched.Foreaseofexposition,wewillfocusonthecasewheretheequilibriumisunique,Player2canbeeithersimpleorsophisticated,and6Player1issophisticatedwithprobabilityone.Below,wediscusshowthemultiple-equilibriaanalysisisnotfundamentallydifferent.Also,thecasewherePlayer1canbesimpleissymmetrictothepropositionbelow.Letl2[0;1]denotethefractionofsimplemixersinthePlayer2population,andassumethatthisfractioniscommonknowledge.Asimpleplayer'smixingisrestrictedtotheprobabilitiesinf0;sr;1g:Proposition2StabilityTheorem:ForeveryequilibriumEofthegeneric22gameG,thereexists¯lG2(0;1]suchthatwheneverl20;¯lG:1.Thereexistsatleastoneequilibriuminthegeneralizedgamewithsimpleandsophisticatedplay-ersthatisoutcome-equivalenttotheequilibriumxinthat(a)Player1andPlayer2,regardlessoftype,obtainthesamepayoffsasinx(b)Thefrequencyinwhichpurestrategiesarechosenisthesame.2.Whenevers2=2f0;sr;1g;sophisticatedPlayer2'sequilibriumstrategys2rdivergesfromhisstrategyinEs2intheoppositedirectionofthedistortionsr:ifsrs2andifsr>s2!s2r0j8l2[0;lG]:u1=u1;u2=u2;s1=s1;s2=s2:WeproceedbyproposinganequilibriumEthatisinlinewiththerestrictionsimposedonsimpleplay-ers.Notethatifthepopulations'expectedstrategiesimpliedbyEcoincidewiththoseofE,thenpayoff-equivalencefollowsdirectly.Thatis,fori=1;2:si=siimpliesui=uiasuiisjustanexpectedvaluethatdependsontheprobabilitiessiandthepayoffsuij;k.First,supposes2=s2:IfPlayer1isplayingapurestrategyinEsuchthats12f0;1g;thenhewillhavenoincentivetodeviate.Similarly,ifhisequilibriumstrategyismixed,Player1'sequilibriumstrategys1isdeterminedbysolvings1u21;1+1s1u22;1=s1u21;2+1s1u22;2;afunctionofu2j;kwhich7isthesameconditionfors1:Thus,whetherPlayer1wasplayingapureoramixedstrategy,s1=s1holdsaslongass2=s2:Now,supposes1=s1:Ifintheequilibriumx,Player2isplayingapurestrategythen,asnotedintheprevioussection,thepresenceofsimplemixersdoesnotaffecthisequilibriumbehavior.Nowsupposes22(0;1):ThesimplePlayer2,beingindifferentbetweenhistwoactions(s21;s22),hass2r=srasabestresponse.Finally,theconvolutionthataverage(expected)equilibriummixingintherestrictedgameofthesophisticatedPlayer2(s2r)isdeterminedbysolvinghsrl+(1l)s2riu11;1+h1srl(1l)s2riu11;2=hsrl+(1l)s2riu12;1+h1srl(1l)s2riu12;2withsolutions2r=srlu11;1+u11;2srlu11;2srlu12;1u12;2+srlu12;2lu11;1u11;1+u11;2lu11;2+u12;1lu12;1u12;2+lu12;2=11l u12;2u11;2u11;1u11;2u12;1+u12;2lsr!whichisdecreasinginsr:Notethatu11;1u12;16=u11;2u12;2sinces22(0;1)isuniquebyassumption.6Theoutcome-equivalencestatedbytheStabilityTheoremisvbycalculatingtheweightedaverageofthemixingofbothpopulations,whichcoincideswithPlayer2'smixinginE:s2=ls2r+(1l)s2r=u12;2u11;2u11;1u11;2u12;1+u12;2=s2:Sinces1=s1ands2=s2;thenu1=u1andu2=u2(forboththesimpleandsophisticatedPlayer2populations):6Ifu11;1u12;1=u11;2u12;2;Player1wouldhaveastrictlydominantstrategy.If,forinstance,bothsidesoftheequationarepositive,thens11strictlydominatess12:Inturn,thiswouldimplythatifs22(0;1);thenu21;1=u21;2sos2isnotunique.Weassumeduniqueness,butitisclearthatpayoffinvarianceholdsifuniquenessfailsasinthecasewhenu21;1=u21;2.Thefrequencyinwhichpurestrategiesarechosenisalsothesamesince,asintherestofthisanalysis,thesophisticatedPlayer2hasnoincentivetodeviatefroms2r=s2ls2r1l:8ThevalueofthethresholdlGwilldependonwhethers2?sr.Sinceprobabilitiesarebounded(inparticulars2r2[0;1]),thecompensationdonebythesophisticatedPlayer2'swhens2sr,thenthesophisticatedPlayer2hastocompensateintheoppositedirectionandlGisthesolutionforlsuchthats2r=1:lG=1sru11;1u12;1u11;1u11;2u12;1+u12;2:Finally,ifs2=sr,tohaveanequilibriumthereisnoneedtohavesophisticatedplayersatall,andlG=1.Thereforeintherestrictedgameonehas:llG=minn1s21sr;s2sro.Theintuitionisthatsinces2=ls2r+(1l)s2r;asophisticatedPlayer2hasnoincentivetodeviatefroms2r=s2ls2r1l=s2Pr(Player2isSimple)s2rPr(Player2isSophisticated)wherethedenominatorisameasureofhowmuchhecompensatesforthebehaviorofhissimplecounterpart.Atthesametime,thesophisticatedPlayer1,sincehehasnosimplecounterpartandexpectsnoequilibriumdeviation,hasnoincentivetodeviateeither.LetD:=s2ls2r;thenumeratorabove.IfD<0orD>1l;thens2r=2[0;1];implyingthereisnoequilibrium.IfD2[0;1l];thenlG=8>>>><>>>>:s2sr;ifs2sr(thesolutiontos2r=1)thuslG2n1;1s21sr;s2sro:Ifthesimplepopulationisgreaterthanthisthreshold(ifl>lG),thenthereisnothingthesophisticatedpopulationcandotorestoreanyequilibriumandthetheoryisatalosswhentryingtopredicttheoutcomeofthisstaticgame.Notethattheequilibrium(s1;s2r;s2r)=(s1;s2lsr1l;sr):=Eisnotunique.ThestrategiesinEarebest-responsesbut,followingouroriginalmotivation,theboundedrationalityofasimpleplayercouldimplythatheonlyplayspurestrategies.(Alternatively,thiscouldbe9seenasthespecialcasewheresr2f0;1g:)Inthiscase,theanalysisdoesnotfundamentallychange.IfEispurelymixedanditiscommonknowledgethatPlayer2willalwaysplaysr=0;thens2r=s21landlG=1s2:Likewise,ifPlayer2willalwaysplaysr=1;thens2 r=s2l1landlG=s2:Thesearethetwocaseswherethedistortioninducedbysimpleplayersisatitsmaximum.Toillustrate,Figure4depictshowtheStabilityTheoremstatesthatforanyparametrizationuij;k;thegameisstableinthesensethatallpayoffsareunchangediftherule-of-thumbpopulationchanges,aslongasitsprevalenceissmallenoughandcommonknowledge.Figure4:RestorationofEquilibriumFollowingthepreviousexample,asshowninFigure4,thetheoremalsopredictsacompensatingbe-haviorbythesophisticatedpopulation.Sinces2<1=2;thens2rlG);therewouldexistnoequilibrium.106ExtensionsFromanevolutionaryperspective,inasymmetric22game(fors;t2f1;2g:uts;s=uts;sanduts;s=uts;s),purestrategysi1risk-dominatessi 2if,andonlyif,E(uijsi 2;si=12)E(uijsi 1;si=12):Itstrictly-dominatesitiftheinequalityisstrict.Inthiscase,thisholdsifu11;1+u1 1;2u1 2;1+u12;2:[?]Itcanbeshownthatthebestresponsetouniformsimplemixingcoincideswithplayingariskdominantstrategy.Assuch,inthespecialcaseofsr=12,arisk-minimizingagentwillhavethesamebehaviorasasimpleplayerandnosimplepopulationwillbeabletoinvadeaNE-mixingpopulationsuccessfully.Notethatifbothplayersareriskminimizers(insteadofutilitymaximizers),eveniftheyaresimplemixers,thenanequilibriumexistsif,andonlyiflisbelowsomeupperboundlG.Player2s21s2 2s1 1u1 1;1;u1 1;1u1 1;2;u1 2;1s1Player1s1 2u1 2;1;u1 1;2u1 2;2;u1 2;21s1s21s2Figure5:Generic22SymmetricGameWecananalyzelearninginrepeatedgameswithsimpleplayers.SupposetwoagentsrepeatedlyplaytheasymmetricmatchingpenniesgameofFigure2asstablepairswithnorematching.lasthe(prior)probabilitythatPlayer2isasimplemixerthatfollowssr=1=2.NowletlT:=Pr(Player2isSimplejs2T;s2T1;:::;s2 1)betheposteriorprobabilitywheres2T:=(Player2'sactionattimeT).Assumenodiscountingandperfectmyopia(nostrategicbehavioracrosstimeperiods).IfthegameisrepeatedTtimes,thereisadecreasinglimitlT+1GlT G;therepeatedgamehasnoequilibrium.IfllT G;therepeatedgameisstableinthatallplayersgetthesameexpectedutilityinthegeneralizedgameasinthegenericgamewithnosimplemixers:u1T=u1 Tandu2 T=u2 T.TheinequalitylT+1G<>:>lTG;ifs2T1=1b;wherebSrepresentstheCNEstrategyandthesameholdsforP2.24Whenweincreasetheprevalenceofbotsto50%asinstage3,thenbothplayertypesareindifferent(and,thus,reachanequilibrium)iftheyplaybS=gS=1:Thisisnot,however,theonlyequilibriumsinceifg=1,thenanyb2[0;1]isabestresponseandg=1isalwaysabestresponse.Additionally,anyexpectedperturbationbyP2wouldmakeP1haveauniquebestresponse:ifP1believesg<1;thenhisactionisb=0:Thisisimportantsince,typically,experimentalpopulationsdonotplaytheNEandthusitwouldoftenbeoptimaltoplayb=0ifP1org=1ifP2:Instage4,thebotbehaviormovesevenfurtherawayfromtheNE.Inthisstage,players'compensationisexpectedtobegreaterthanintheprevioustwostages.,thebotsplay¯bN=¯gN=0whiletheirprevalencerelativetothepreviousstageisunchanged(50%).Inasense,thisistheeasieststagetoplay.Withorwithoutperturbations,thegameisnolongerstrategicinthatnoweachplayerhasastronglydominantstrategy.Inthiscase,playersarenolongerexpectedtomix,andtheirbestresponsesareb=0andg=1.4ResultsAtotalofforty-fourundergraduatestudentsfromMichiganStateUniversityenrolledintheexperiment.10Allparticipantsattendedoneofthetwosortingsessions.11Participantswerethendividedintotwogroups(naïveandsophisticated)consistingof20and24participants,respectively.Despiteeffortstoensurethatparticipantscouldandwouldcometothefollow-uporbotssessions(includingmakingsureallparticipantsindicatedtheycouldattendeitherofthefollow-upsessions,anddelayingpaymentoftherstsessionshow-upfeeuntilthesecondsession),only16ofthenaïveparticipantsand20ofthesophisticatedparticipantsattendedthefollow-upbotssessions.1210Participantsearnedonaverage$24:17inthesortingsessionsand$28:89inthebotsessions.Inanefforttomaximizethecognitiveeffortoftheparticipantsandtoremovepotentialwealtheffects,3(4)roundswererandomlyselectedattheendofthesessionforpaymentinthesorting(bots)sessions.Participantsearnedtokens(0,1or3)worth$5sothemaximumtheycouldearninaroundwas$15:11Itwasvthatplayerswhoself-selectedintobothsessions(bychoosingthedatetheywishedtoparticipate)werenotsystematicallydifferent.Simplestatisticaltestsfoundnoevidenceforapparentdifferencesonbehavior(means),cognitivelevels(testscores)orobservablecharacteristics(questionnaire)betweenbothgroups.12Bycomparingearnings,testscores,andestimatedbehavior,itwasvthatattritionwasnotsystematic.254.1SortingSessionFortheŸGgamesinthesortingsessionswefailtorejectthenullthatparticipantsonaverageplayedtheNE(theNEis12;theaveragemixingwas0:533andthep-valuerelatedtothetestforequalityis0:851).13Assuch,duringtheserounds,itwouldnothavebeenforplayerstodeviate.14Bothpopulationsseemedtounderstandthestandardsymmetricmatchingpenniesgameandplayit,onaverage,asthetheoryinitssimplestformwouldforecast.Ontheotherhandandconsistentwiththeexperimentalliterature,bothplayertypes(P1andP2)donotappeartohaveunderstoodthemorecomplexgameGasclearly.Infact,everyP2wouldhavebydeviatingtoapurestrategyinresponsetoP1'splay.,bothpopulationsplayedamixedstrategyestimatedtobelessthantheNE(P1played‹b=0:601,15P2played‹g=0:707,NE=0:75),butonlytheP1population'splaywasstatisticallydifferent(p-value=0:003)thantheNE.164.1.1SortingProcessTheP1andP2populationsweredividedinhalf;thoserelativelynaïveandthoserelativelysophisticated:Withineachpopulation,eachplayerwasassophisticatediftheirbehaviorwhileplayingGwasrelativelyclosetotheNE,orasnaïveifitwasrelativelyclosetouniformmixing.SinceGisnotsymmetric,itwasnecessarytoperformthesortingseparatelywithineachplayertypepopulation.Theapproachtakenwastogenerateadummy(si)equaltooneifeachindividuali'smixingwasabovethemedianamongstitstype,sothathalfofeachpopulationwasasnaïve(si=0)andtheotherhalfassophisticated(si=1):ThemedianmixingforP1wasplayingDownwithprobability0:588:ThemedianmixingforP2wasplayingrightwithaprobabilityof0:725:13EachchoiceisaBernoullitrialandwefocusonitsparameterp.Assuch,ouranalysisisessentiallynon-parametricandourresultscoincidenumericallywiththosefromtheMann-WhitneyUtest.Thisisbecausetherelevantestimators(i.e.differencesbetweenaverages)areexactlythesameinbothcases.Furthermore,whendoinghypothesistesting,theirassumeddistribution(normal)isalsothesamesincebothtestsinvokethecentrallimittheorem.14More,whenusingOLStoestimatethemodel‹M1;i=d0+d1Typei+d2Mixing1;i+d3TypeiMixing1;i+uiwhereM1;iisindividuali'saverageexperimentalpayoffsinallŸGgamesofthesortingsession,TypeiisadummyequaltooneifiistypeP1andMixing1;iishisaveragebehaviorthroughoutalltheG1games,noevidencewasfoundtorejectthenullhypothesisH0:d2=d3=0:Fordetailsonthismodel,estimatesandp-valuesoftheWaldtests,contacttheauthors.15ThisobservedmixingwasthusintheintervalbetweentheNEandthelevel-0orpurelynaïvemixingthatcoincideswiththesimplestuniformdistributionthatassignsaprobabilityof0:50toeachavailableactionorpurestrategy.ForP1andP2itwasalsovthattheaveragemixingwasstatisticallygreaterthan0:50(thep-valuesare,respectively,0:033and0:000):16Throughoutalltheexperimentsandtreatments,oneoftherobustnesschecksthatwasdonewastoignoretheve(also,whereapplicable,theten)gameroundstoverifyiftakinglearningintoconsiderationfundamentallychangedtheresults.Itneverdid:ifonlythebehaviorduringthelastiterationsofthegamesismeasured,itisstillthecasethatplayers'strategiescoincidedwiththeNashequilibriuminG1butwasstatisticallylowerthantheNashequilibriuminG3.26Wechosetousetherelativelysimplemethoddescribedabovebecauseitisintuitiveandquiterobustwhencomparedtomanyalternativemethods.17Underthesealternativesortingmethods,thedivisionofparticipantsdidnotchangeconsiderably,andthecorrelationbetweensiandthedummiesgeneratedbyalternativesortingmethodswasneverstatisticallynegative.Likewise,whereapplicable,intheregressionmodelingtheprobabilityofbeingassophisticated(describedbelow),theestimatedpartialeffectoftestscoreswasnevernegative.Also,thedifferencesinbehaviorbetweensophisticatedandnaïveplayersduringthebotssessionwerequalitativelyverysimilarwhendistinguishingthembythealternativesortingmethods.Overall,wecannotrejectthehypothesisthatthelabelingwasconsistent:playerssortedaseithersophisticatedornaïveseemedtobehaveassuchthroughout.4.2BotsSession,MainResultsAsdescribedabove,theexperimentevidenceoftheexistenceofstablesetsofrelativelynaïveandsophisticatedplayers.18Naïveplayers'mixingisoftenclosertouniformmixingthanthesophisticatedplayers'mixing;notonlyduringthesortingsession(truebyconstruction)but,ashypothesized,alsointhebotssessionsunderdifferentbotprevalencelevelsandoff-equilibriumbotbehaviors.Sophisticatedplayers'mixingisalsooftenclosertotheCNE.19Furthermore,sophisticatedplayersfrequentlyreactfibetterfltovaryingoff-equilibriumdistortionsinducedbythebots.Table4.2presentsthemainresults.Itincludesthemeanstrategy(averagechoicesarebforP1andgforP2)foreachofthefourstages,foreachofthefourplayertypes:NP1;SP1;NP2andSP2.InStageŸG_25_50behaviorisnotstatisticallydifferentthantheNEof1=2:20IntroducingNE-playingbotsdoesnotseemto,initself,causeplayerstochangetheirbehavior.Duringthelastthreestages,mixingisalwaysstatisticallydifferentthantheCNEforallplayertypes2117Thefulllistofalternativesortingmechanismsinvestigatedincludedsortingbasedon:questionnaireresults,payoffsearnedduringthescreeningsession,payoffswonduringthesecondhalfofthescreeningsession,behaviorduringthescreeningsession(ignoringthe10or20iterationsofG3toaccountforlearning),behaviorduringthescreeningsession(poolingP1andP2playerstogether),behaviorcomparedtotheirbestresponse(asopposedtotheNE)giventheiropponent'sbehaviorduringeitherthewholeorthelast40iterationsofthescreeningsession,anddroppingfromthedatasetallbutthemostsophisticatedandthemostnaïve(accordingtotheoriginalscreeningcriterium)players.18Thatis,evidenceofacontinumofsophistication(ornaiveté)levels.Wefoundnopurelynaïvenorpurelysophisticatedplayers.Forpracticalpurposes,though,wewillrefertothewholepopulationasconsistingofonlytwodistinctsetswithnointersectioneventhoughsubtledistinctionswereobservedonthedata(andthiscouldbethebasisoffurtherresearch).Theresultswererobusttorelaxingtheimpliedassumptions.19Thisdistinctioncomparesbehaviortoequilibria,nottothebestresponsesthatcorrespondtoactualplay.BestresponsesaredescribedinAppendix4.20P-values:0:266;0:551;0:374;0:426.21Thep-valuesare0:000inallcasesexceptNP2instageG_50_0,wherep-value=0:013.27Player1(b)Player2(g)StageTHPCNENaïve(NP1)Sophisticated(SP1)THPCNENaïve(NP2)Sophisticated(SP2)1)ŸG_25_500:500:56ƒ0:53ƒ0:500:55ƒ0:54ƒ2)G_25_500:830:570:620:830:670:763)G_50_500:000:430:431:000:750:824)G_50_00:000:230:111:000:931:00ƒDenotesNaïveandSophisticatedMixingareDifferentatthe5%LevelƒDenotesMixingEqualtoCNEatthe5%Level#NaïvePlayers:16;#SophisticatedPlayers:20TableII.4.2:MainResults,EstimatedMixedStrategieswiththeexceptionoftheSP2playerswhoplayexactlytheCNEinstageG_50_0.EveryoneoftheSP2playersplayedrightinall10roundsofthislaststage.Fromabehavioralperspective,itcouldbearguedthatitisharderforP1tocompensatetob=1thanforP2tocompensatetog=1:G1'sasymmetryimpliesitisP1whohastheresponsibilityofdeterminingthesizeofthecake(sosurplusmaximizationgoesintheoppositewayofcompensation)whereasP2onlydetermineswhogetsthecake.Pursuingthisquestionisbeyondthescopeofthisanalysis,butthedatasuggeststhestrengthoftheeffectsofbeingnaïvedonotcompletelyoverpowerothereffectslikealtruisticpreferences,surplusmaximizationorlossaversion.Thedata,therefore,suggeststhatingamesthatarecomplexenoughsuchasstages2and3(i.e.,thosegameswheretheNEisdifferentthat1=2andwithoutastronglydominantstrategy),thesophisticatedplay-ersdonotcompensateenoughtorestoretheequilibriumgiventheoff-equilibriumbehaviorofbots.Thisis,insomeways,notinconsistentwiththestylizedfactthatparticipantpopulationsdonotplaytheNEevenwithoutbots.Thisfurtherimpliestheexistenceofauniquebestresponseanditisremarkablethattheevidencesuggeststhisbestresponsemightbepredictable.Ifwhatwehereisthenwhencomplex-enoughgamesasdescribedabove,theoptimalstrategyistoassumesimplythatyouwillbematchedagainstaperfectlynaïveplayer.Thatis,ineveryvariationsofGabove;anyplayercouldobtainabove-equilibriumpayoffsbyplayingthemixedstrategyderivedfromassumingtheopponentpopulationwillbemixinguniformly.Inasense,duetothenaïvecomponentoftheirmixing,experimentalpopulationscanbetakenadvantageof.WiththeexceptionoftheP1playersduringstage3(G_50_50)whereNP1andSP1effectivelybehavedthesame,22themixingofthesophisticatedplayersinthemorecomplexstages(2and3)isstatistically22‹bNP1=0:4267;‹bSP1=0:433andthep-valueofH0:‹bNP1=‹bNP1is0:877:28closertotheCNE(andfurtherfrom50/50mixing)thantherespectiveplayoftheirnaïvecounterparts.2324Figures1and2belowillustratethisforbothplayertypes(P1inFigure1andP2inFigure2)acrossstages2and3(botprevalence0:25and0:5)andincludethedatafromthesortingsessions(botprevalenceof0)asareference.Notethat,overall,sophisticatedplayersplayclosertotheCNEandnaïveplayersalwaysmixcloserto1=2(althoughnotstatisticallysoforplayer1types.Theconsistencywithwhichthenaïveplayersinthemorecomplexgamesinourexperimentmixcloserto1=2Œbothacrossthedifferentstagesandthetwo-weekseparatedsortingandbotssessionsŒsupportsthehypothesisoftheexistenceofastablesetofrelativelynaïveplayersinthepopulation.Furthermore,aswillbediscussedinthenextsubsection,thissetofplayersmaybeabletobethroughlowerscoresonquantitativetests.TheeffectsofinteractingwithbotsandthedistinctionbetweennaïveandsophisticatedplayersisillustratedinFigures1-4.FigureII.1:CNEandP1'sBehavior(NaïveandSophisticated)asthePrevalenceoftheBotPopulationIncreases23P-values:0:028;0:021and0:036:24Totestforlearning,thesetwostageswereseparatedintohalvesandanalyzedseparately.Althoughtherewerecertainlystrongdynamicinteractions(seeserialcorrelation,below),theevidenceforlearningisinconclusive:thereappearednoconsistentorsystematicadjustmentofthepopulations'behaviorwhencomparedthettothesecondhalfofthestages.Theadjustments,ifany,weresometimesdoneinoppositedirections.29FigureII.2:CNEandP2'sBehavior(NaïveandSophisticated)asthePrevalenceoftheBotPopulationIncreasesFigureII.3:BestResponsesandP1'sBehavior(NaïveandSophisticated)asthePlayoftheBotPopulationDeviatesfromtheEquilibrium30FigureII.4:BestResponsesandP2'sBehavior(NaïveandSophisticated)asthePlayoftheBotPopulationDeviatesfromtheEquilibriumWenowformallycomparetheimpactofvaryingbotdistortionsonthenaïveversustheimpactonthesophisticatedplayers.First,focusingontheGgameswithbots,wecancontrasttheeffectofanincreaseoftheirpopulationfrom25%to50%(wecallthistreatmentT1)bycomparingnaïveP1'sbehaviorinstage2(‹b=0:57)withstage3(‹b=0:43).Thedifferenceis0:14andthesamecalculationwiththesophisticatedpopulationyieldsadifferenceof0:19:Thedifference-in-differences(DiD)estimateforthetreatmenteffectisthus0:05;implyingthesophisticatedpopulationreactedmoretotreatmentT1.AsseeninTable4.2b,doingthesamederivationsforP2yieldsanestimatedeffectwiththesamesign.Yet,theinterpretationistheoppositesincetheexpectedcompensationwentup:theCNEis0:83instage2and1:00instage3.TheevidencesuggeststhattheP2naïvepopulationreactedmoretoT1.However,allDiDpointestimatesareat10%orhigherlevels.OnlyP1aftertreatmentT2(comparingstages3and4:gameswith50%botsthatgofromtoplayingb=g=0),withanestimatedpartialeffectof0:108,hasarelativelylowp-valueof0:127.ThesignofthiseffectisnegativewhichimpliesthesophisticatedpopulationreactedmoretoT2.31P1P2TreatmentDiDNonlinearDiDDiDNonlinearDiDT10:052(0:386)0:053(0:384)0:016(0:766)0:003(0:948)T20:108(0:127)0:111(0:088)0:001(0:986)0NATableII.4.2b:TreatmentEffectsThereisacaveattothisanalysisanditsintuitionisanalogoustothecaseofdependentbinaryvariables:adesirablefeatureofprobitandlogitmodelsversuslinearregressionsisdiminishingmarginalmagnitudesofthepartialeffects.Toillustrate,considerP2'sreactiontoT1:theaverageadjustmentofNP2(0:08)isoflargersizethanthechangebySP2;0:06:However,whenadjustingawayfromuniformmixingitisprobablybehaviorallyeasierifone'smixingstartsclosetouniformityandharderwhenmixingisalreadyclosetoeitherbound:zeroorone.Thisisanotherwayofsayingthatanyvariablethathasaneffectonplayers'choiceswilllikelyhavediminishingmarginaleffects.Assuch,apositiveadjustmentfromb=0:67(NP2'sbehaviorinstage2)wouldbeeasierthanthesamefromg=0:76(SP2):ThisexplainswhythenonlinearDiDestimationforthetreatmenteffectofT1onP2showninTable4.2bismuchclosertozerothanthelinearDiD.25UnderadifferentinsteadofF;withlocallyfasterdiminishingmarginaleffects,thiscoefwouldturnpositive,inlinewithpreviousresultsthatseemtoshowsophisticatedplayersunderstandthegamesbetterthanthenaïve.ThecoefforP1aftertreatmentsT1andT2arealsoslightlyenhancedbythenonlinearestimation,andtheeffectofT2onP2cannotbeestimatedwiththismethodduetocolinearity.Theestimationoftherateatwhichthesemarginaleffectsdecreaseis,however,beyondthescopeofthisanalysis.4.3ModelingCognitiveHeterogeneityNext,weinvestigateifthereisalinkbetweengeneralanalyticabilitiesandbeinganaïveplayer.Ourmeasureofintellectualskillsisaten-questiontestthatparticipantsansweredimmediatelyaftersession25Withalinearmodel,wewouldhavechoice=a+bs+gTk+dsTk+uwheresi:=0[isophisticated]+1[iisnaïve]andTk:=0[Stage(k+1)]+1[Stage(k+2)].(NotetheestimationofdisnumericallyequivalentwhetherwesorTkasthetreatmentvariable.)WenowspecifyPr(choice=1js;Tk)=F(a+bs+gTk+dsTk):ThesearealldummyvariablesandPuhani(2012)showsthetreatmenteffectisF(a+b+g+d)F(a+b+g)andstandarderrorscanbeconsistentlyestimated.32(twoweeksbeforethebotssession;seeAppendix2).OurresultsthathighscoresrelatetoplayingrelativelyclosetotheCNE:Weinterpretthisasfurtherevidencethattheproposedconceptsofnaivetéandsophisticationareempiricallygrounded:relativelynaïveplayersdoexist.Thisalsosuggests,thatthecognitivelevelsdeterminedinthesortingphasecanbepartiallycapturedbytheproposedanalyticaltest.Second,byconditioningonone'stestscore,thereisnostatisticallyimpactontheprobabilityofbeingnaïveofothercharacteristicssuchasincome,demographicsoracademics.Thehypothesisisthattheprobabilityofbeingsophisticateddependsnon-linearlybutpositivelyoneveryindividuali'scognitiveability(ci).Thatis,8i:Pr(si=1jci;xi)=f(dcci+xidx);wherexiisavectorofobservables,andf()isanincreasingfunction.Thevariablesinxiweregatheredinasurveyafterthetest.WediscussthetestandthequestionnaireinAppendix2.WereporttheseresultsinTables4.2.3a,4.2.3band4.2.3c.Letcibethetotalscoreonthetest.Thesimplestprobitregression,withf(z)=F(z)(thestandardnormalc.d.f.)andwithnoothercovariates,estimates‹dc=0:25(p-valueof0:042).Therelatedmarginaleffectis0:090suggestingthataonepointincreaseontheten-questiontestisassociatedwitha9percentagepointincreaseintheprobabilitythattheplayerissophisticated.Theseresultsarerobusttoseveralotherlogitandlinearregressions,ciassomesubsetofthequestionnaire,oraddingadifferentsetofcontrolsinxi:ModelProbitLogitLinearResponseVariablesssTestScore0:2530:4190:092[MarginalEffect][0:090][0:091][0:092](p-value)(0:042)(0:051)(0:040)(Robust)(0:053)(0:060)(0:029)TableII.4.2.3a:RegressionsofsonTestScore33ModelProbitProbitProbitProbitProbitProbitProbitResponseVariablesssssssTestScore0:2530:3470:2960:2630:2270:2540:296(p-value)(0:042)(0:018)(0:025)(0:037)(0:095)(0:049)(0:027)GenderŒ1:064ŒŒŒŒŒ(0:034)RaceŒŒ0:146ŒŒŒŒ(0:300)AgeŒŒŒ0:161ŒŒŒ(0:524)GPAŒŒŒŒ0:259ŒŒ(0:618)ReportedIncomeŒŒŒŒŒ0:271Œ(0:288)WeeklyExpendituresŒŒŒŒŒŒ0:370(0:213)TableII.4.2.3b:RegressionsofsincludingdifferentsetsofregressorsModelProbitProbitProbitModelProbitResponseVariabless_Alls_L_40ResponseVariablesTestScore0:2530:2320:172GREQuestions0:222(p-value)(0:042)(0:056)(0:150)(p-value)(0:214)TableII.4.2.3c:RegressionsofsincludingdifferentsetsofregressorsWeinterpretthisasevidencethathighcognitiveabilities(andnoneoftheothertestedobservables)areindeedpositivelycorrelatedwithbeingsophisticated.Asimplewaytovisualizethesecorrelationswithouttheaidofmodelingistocomparethehistogramsofthetestscoresforthesi=0(naïve)andsi=1(sophisticated)populations;thesegraphsprovideevidenceforthepositiverelationshipbetween(s)andtestscores,asshowninthebelow.34Figure5:NaïveandSophisticatedPlayersQuantitativeTestResults4.4SerialCorrelationAllplayers'choicesexhibitpositiveserialcorrelationfromoneroundtothenext,measuredacrossthe80roundstheyplayedinthesecondsession.Table4.2.4showstheserialcorrelationestimationsforeachplayerandstage(p-valuesinparenthesis).Eventhoughthegameisasymmetric,asdiscussedbefore,noobviousdistinctioncanbemadebetweentheP1andP2populationsinthisregard;noneexhibitedsystematicallydifferentmeasuresofserialcorrelation.Interestingly,naïveplayers(si=1)playedwithaserialcorrelationclosertozero.Thisisperhapsmoreclearlyseenbylookingattheregressionresultsbelow,andcouldbeinterpretedasanindicationthatsophisticatedplayers'understoodthegamebetterandplayedclosertothecornersolutionsthatmaximizedtheirpayoffs.Also,sinceanegativeserialcorrelationcanbeinterpretedasaneffectoftryingtopreventbeingpredictable,apositiveserialcorrelationisprobablyrelatedtoplayingagainstapopulationwithbots,whenbeingunpredictableisnotconsideredasimportant.35StageP1P21)ŸG_25_500:204(0:006)0:283(0:000)2)G_25_500:378(0:000)0:360(0:000)3)G_50_500:392(0:000)0:476(0:000)4)G_50_00:309(0:000)0:310(0:000)TableII.4.2.4:SerialCorrelation\Choicei=0:409+0:302L:Choicei0:098si+0:194L:Choiceisi(0:000)(0:000)(0:000)(0:000)5ConclusionsInthisarticle,wedescribethedesignandresultsofatwo-sessionexperimentthatincludedcomputer-controlledbots.WetestandtheexistenceofplayerswhosestrategymixingispersistentlyclosertouniformmixingthantopureormixedNE(i.e.thoserelativelynaïve),andthoserelativelysophisticated.Consistentmethodstoidentifybothtypesofplayersweredevelopedandtested.Moreover,theprobabilityofbeingnaïvecanbepartiallypredictedbythescoreonaquantitativetest,takentwoweeksbeforethemaingames.Wealsofoundevidencethatthesophisticatedpopulationreactsbettertooff-equilibriumbehaviorinthetheoretically-predicteddirectionbutnotwiththemagnituderequiredtorestoreequilibria.ThisthestylizedfactthatNEarenotplayedincomplexgamesand,fromthenaïvecomponentofpopulations'mixing,thedirectionofthedeviationscanthusbepredicted.Weemployedthistodesignsimplemechanismstoobtainabove-equilibriumpayoffsundertheseexperimentalconditions.Theseresultsopenupseveralresearchpossibilities.Fromatheoreticalperspective,wecanforeseeap-pliedmodelswherenaïveplayerscanbetakenadvantageofbyplayerswithamoremoderaterationality36bound.Thisway,apolicymakercanpotentiallygenerateaPareto-improvementingameswithourpos-tulatedassumptionsonrationality.Empirically,wecantestthedeterminantsthatmakeaplayerbenaïverelativetohim/herself(asopposedtorelativetotherestofthepopulation)underdifferentexperimentalcon-ditionsincludingvaryinggamecomplexityandtimeconstraints,beliefelucidation,accesstorandomizationdevices,whitenoise,andinformationoverload.Lastly,weconjecturethatthemethodsdescribedabovetoobtainabove-equilibriumpayoffscanbegeneralizedtoothernormalandextensiveformgames.37APPENDICES38APPENDIXA,InstructionsHello,andthankyouforparticipating!Bycominghere,youalreadyearned$10(tenUSdollars).Inafewmoments,youwillbegiventheopportunitytoaddtotheseearnings.Themoneyyouwinwillbepaidtoyouattheendoftheexperiment.Everytokenyouearnwillearnyou$5.00veUSdollars).Youwillbeplayingsimplegameswitharandomlychosenplayerfromthisroom.Thissessionwilllastanestimated60minutes.Itwillconsistof60iterationsofsimplegamesandTHREEofthesewillberandomlychosentodetermineyourtotalearnings.Example:supposeyouearn1,3and0tokensinthethreerandomlychosengames.Inthiscase,yourtotalearningswillbe$30($10plusfourtimes$5).Irrespectiveofyourresults,youwillbeinvitedtoparticipateinasimilarsessionintwoweeks.Thiswillbethescreenyouandtherestoftheparticipantswillbeseeing.Youhavetwooptions:chooseAorB:Likewise,theplayeryouwerepairedwithcanpickeitherCorD:Yourdecisionwillbemadebeforeyouknowwhattheotherplayerdid.SupposeyoupressBandthenitisrevealedthattheotherplayerchoseD:Inthiscaseyouwillearn1tokenandtheotherplayerwillreceive0tokensforthisgame.Noteyourpayoffisrepresentedinredandisthenumberineachcell,andtheotherplayer'spayoffisthesecondnumber,ingreen.Furtherinstructionswillappearhere.Pleasekeepyourattentiononyourowncomputerscreenandstaysilentthroughoutthisexperiment.Ifyouhaveanyquestions,pleaseraiseyourhandandasktheexperimentadministrators.PRESSOKTOSTARTASAMPLEROUND.39APPENDIXB,QuestionnaireandTestBelowisthequestionnaireandthetestthattheparticipantsansweredafterthesortingsession.Therewasanemphasison.Playerswerethroughloginnamestheycreated.Thedemographicvariables(1-9)followstandardliterature.OneexampleisFryerandLevitt(2005).Thesocioeconomicvariables(10-13)weremeanttobeaproxyforincome.Questions(1;2)fromthetestaretheLindaParadoxandtheWasonSelectionTask,standardintheliter-ature,usedforinstanceinCharnessandSutter(2013).26Questions(3;4)aretheCNEmixing(lastequationofthesolution)inGandŸGrespectively.Questions(5;10)arebasicallythesame,andtheansweristrivial;someoftheinformationtheyprovideistobedisregarded,givenhowthesetupisworded.Thepurposewastomeasureifthereisacontradictorybehaviorinthesetwoanswers(answeringcorrectlywhenplay-ingagainstfibotsflinvideogamesbutincorrectlywhenfacingarealopponentwithstrategicthinking)thatsimilarbehaviorwhenplayingagainstcomputerbots.(Noevidenceforthiskindofcontradictorybehaviorwasfound.)Finally,questions(69)weretakenfromthePracticeBookforthePaper-basedGRERevisedGeneralTest,SecondEdition,byETS(2012),availableonline.FromthetwoQuantitativeReasoningsampleexams,thequestionswiththehighestpercentageofexamineeswhoansweredcorrectly,notcountingquestionsin-volvinggraphs,werechosen.High-percentage(theirrangewasin82%Œ88%)questionswerepreferredsincethepopulationtestedtogetthesepercentageswasmostlystudentstryingtogetintograduateschoolwhohavepracticedfortheexam,whereastheparticipantsoftheseexperimentswerestudentswhohadnotyettheirundergraduateeducation.Moreover,theaimwastochooseGRE-typeques-tionsthatwouldbecorrectlyansweredabout50%ofthetime(tomaximizethescorevariance).Graphquestionsweredisregardedbecausetheyareprobablylessrelatedthanothertypesofquestionstotheabili-tieswewantedtomeasureand,moreimportantly,thesesectionstypicallyinvolvevequestionsonthesamegraph(s).Questionnaire:Thankyouforparticipating!26OthersimilaroptionsincludetheCognitiveTest(CRT)byFrederick(2005).40Pleaseoutthissurveyandanswerthetestatthebackofthispage.Asithasalreadybeenexplained,thecashyouhavealreadyearneddependedonlyonyourgameresults.Also,irrespectiveofthese,youwillbeinvitedtoafollow-upsessiononeortwoweeksfromtodaywithahigherexpectedpayoff.Youwillhave20minutes.PleaserememberthatalltheinformationyouprovidewillremainYoudonotneedtowriteyourname.Thesequestionsfollowstandardlabor,education,developmentandhealtheconomicsliteratureandaremeanttoidentifyyoursocioeconomicanddemographiccharacteristics.1)Gender:MaleFemale2)Race(selectoneormore):AmericanIndianorAlaskanNativeAsianBlackorAfricanAmericanHispanicorLatinoNativeHawaiianWhiteOther3)Age:_______4)Education(GPA):_______5)Education(Major):_________6)Education(CurrentSemester):_______7)EducationLevel(Father):_____8)EducationLevel(Mother):_________9)Citizenship:U.S.CitizenOther(Specify):________________10)Howwouldyouclassifyyourparentsregardingincome(lower,lower-middle,middle,upper-middle,orupperclass)?LowerLowerMiddleMiddleUpperMiddleUpper11)Doyouownacar?YesNo12)Doyoulivealoneorshare?AloneShare13)HaveyoutraveledoutofMichiganinthelastsixmonths?YesNo14)Howmuchmoneydoyouusuallyspendeveryweek?US$_______Writeanyname,word,numberorcombinationofcharactersthatwillallowustoanonymouslyidentifyyouinthenextsession(pleasedon'tforgetit!).Examples:fiGandalffl,fi789fl,fiGreenSpartanfl,etc._________________________________________Test41Youhave20minutestothistest.Pleaseselectorwritethebestanswer.1)Lindais31yearsold,single,outspoken,andverybright.Shemajoredinphilosophy.Asastudent,shewasdeeplyconcernedwithissuesofdiscriminationandsocialjustice,andalsoparticipatedinanti-nucleardemonstrations.Whichismoreprobable?a.Lindaisabankteller.b.Lindaisabanktellerandisactiveinthefeministmovement.2)Youareshownasetoffourcardsplacedonatable,eachofwhichhasanumberononesideandacoloredpatchontheotherside.Thevisiblefacesofthecardsshow3,8,redandgreen.Whichcard(s)mustyouturnovertotestthetruthofthefollowingproposition?Ifacardshowsanevennumberononeface,thenitsoppositefaceisred.a.Tobecertain,youonlyneedtoturnoverthe3card.b.Tobecertain,youonlyneedtoturnoverthe8card.c.Tobecertain,youonlyneedtoturnovertheredcard.d.Tobecertain,youonlyneedtoturnoverthegreencard.e.Tobecertain,youonlyneedtoturnoverthe8and3cards.f.Tobecertain,youonlyneedtoturnoverthe8andredcards.g.Tobecertain,youonlyneedtoturnoverthe8andgreencards.h.Tobecertain,youneedtoturnoverallcards.3)ThesolutionforPintheequation3P=1ŒPis:P=__0.25__4)Ifx=Q,andx=1ŒQ,then:Q=__0.5__5)Youaregoingtoshootapenaltykick,andifthegoaliedoesnotguessthedirection,youaregoingtoscoreforsure.Youhavethreeoptions,shootingtotheleft,centerorrightofthegoalie.Heisalefty,andyouaresurehewilljumptohisleftwitha35%probability,stayatthecenterwitha35%probabilityandjumptotherightwitha30%42probability.Youcannotshootwithyourleftfootsoitiswellknownyoushootpenaltykickstotheleftofgoalkeepersonly5%ofthetime,5%tothecenterand90%totheirright.Inwhatdirectionshouldyoushoot?a.Totheleftofthegoalieb.Tothecenterc.Totherightofthegoalie6)AtCompanyY,theratioofthenumberoffemaleemployeestothenumberofmaleemployeesis3to2.Ifthereare150femaleemployeesatthecompany,howmanymaleemployeesarethereatthecompany?______100_____maleemployees.7)Thespaceinacertainmarketisrentedfor$15per30squarefeetforoneday.Inthemarket,Alicerentedarectangularspacethatmeasured8feetby15feet,andBettyrentedarectangularspacethatmeasured15feetby20feet.Ifeachwomanrentedherspaceforoneday,howmuchmoredidBettypaythanAlice?a.$27b.$36c.$54d.$90e.$1808)Abusinessownerobtaineda$6,000loanatasimpleannualinterestrateofrpercenttopurchaseacomputer.Afteroneyear,theownermadeasinglepaymentof$6,840torepaytheloan,includingtheinterest.Whatisthevalueofr?a.7.0b.8.4c.12.3d.14.0e.16.89)Workingattheirrespectiveconstantrates,machineImakes240copiesin8minutesandmachineIImakes240copiesin5minutes.Attheserates,howmanymorecopiesdoesmachineIImakein4minutesthanmachineImakesin6minutes?a.1043b.12c.15d.20e.2410)Inthebattleofavideogameyoureallywanttobeat,whenFrankensteinattacks,youcanparry,blockordodge.Damageisonlypreventedbypickingtherightdefense,butheissofastyouhavetochoosewithoutthatinformation.Theorderofhisattacksisrandom,butyouknowthatparryingiscorrectslightlymoreoftenthanblockingandthatblockingiscorrectalittlemoreoftenthandodging.Sofaryouhavebeenparryingmoreoftenthanblockinganddodging,butonlybecauseyoulikedhowitlooked.Whatshouldyoudonexttotrytobeathim?a.Parry.b.Block.c.Dodge.44APPENDIXC,InterfaceCarewastakentodiminishframingandotherexternaleffects.Apartfromtheinclusionofbotsthatpubliclyannouncedtheirprevalenceandbehavior,andgame-orderrandomizationasdescribedabove,thegameswerecodedinzTree(Fischbacher,2007)andhadthefollowingcharacteristics.Beforethe(sorting)session,theplayerswerepresentedwithabrieftutorialdescribingthegames.Afterthetutorial,apracticeroundwasplayed.Thesecond(bots)sessionincludedanewtutorialfeaturingexampleswhichintroducedthecomputerbotsandanotherpracticeround.Attheendofeachsession,allplayerswereshownatablewiththeirresults.RegardlessofwhethertheywereP1orP2;playersalwayssawthegamesasarowplayerplayingagainstacolumnplayer:P2typeplayerssawatransposedversionofthegames.Duringeachround,theplayerspressedoneoftwobuttonslabeledAandBtoselecttheiraction.AnOKbuttonpresswasrequired.Aftereachround,theresultswereshowedusingcolorstohighlighttheplayers'andtheiropponents'actions,includingaverbaldescriptionofthepayoffs.Thebelowshowsascreencaptureoftheinterfacedevelopedforthispaper.Duringtheinvitations,tutorialsandrounds,carewastakentouseneutralandsimplelanguagelikeotherplayerorplayingwithinsteadofopponentorplayingagainst.Carewasalsotakentoavoidtechnicalterms.45FigureII.A:InterfacewithBots46APPENDIXD,BestResponsesTableA6showsthebestresponseofeachpopulationtothestrategyoftherespectiveopponentpopulation(thenaïveplayersfromtheP1population,NP1onlyplayedagainstNP2,forinstance).Itreferstothebestresponseofthispopulationtothereportedbehaviorofitsrelevantopponents.Weconsideritacornersolutioninf0;1gonlyiftheopponentpopulationplayedastrategythatwasstatisticallydifferentthanthecompensatedNE(iftheopponent'smixingwasoutsidethe99%interval).Player1(b)Player2(g)StageNaïve(NP1)Sophisticated(SP1)Naïve(NP2)Sophisticated(SP2)1)ŸG_25_50AnyAnyAnyAny2)G_25_5000113)G_50_5000114)G_50_00011TableA6:BestResponses47REFERENCES48Working paperThe Journal of Behavioral FinanceEconometricaEconometricaCognitive Ability and Anomalous PreferencesJournal of Economic Behavior & OrganizationThe Quarterly Journal of EconomicsThe American Economic ReviewEconomic TheoryThe Journal of Economic PerspectivesJournal of Economic LiteratureThe Review of Economic StudiesJournal of Behavioral and Experimental EconomicsJournal of Risk and UncertaintyTUC Working Papers in EconomicsPractice Book for the Paper-based GRE Revised General TestExperimental EconomicsJournal of Economic PerspectivesNational Bureau of Economic ResearchGame TheoryReview of FinanceJournal of Political EconomyInformation Economics and Policy, Vol. 1The American Economic ReviewJournal of Economic TheoryGames and Economic BehaviorInternational Journal of Game TheoryJournal of Economic LiteratureAmerican Economic ReviewEconometrica: Journal of the Econometric SocietyA Treatise on ProbabilityInternational Journal of Game TheoryProceedings of the Twenty-Sixth Annual ACM Symposium on Theory of ComputingJournal of Economic Behavior & OrganizationProceedings of the 4th ACM Conference on Electronic CommerceMicroeconomic TheoryWorking Paper Series in EconomicsEconometrica: Journal of the Econometric SocietyExperimental EconomicsGames and Economic BehaviorCurrent BiologyGames and Economic BehaviorAnnals of MathematicsJournal of Agricultural and Resource EconomicsJournal of the Econometric SocietyGames and Economic BehaviorGames and Economic BehaviorThe Economic JournalEconomics LettersModeling Bounded RationalityEuropean Economic ReviewThe American Economic ReviewMathematical Methods of Operations ResearchModels of Bounded Rationality: Behavioral Economics and Business Organization, Vol. 2Games and Economic BehaviorStability and Perfection of Nash EquilibriaChapter3,DeterminantsofNaïveversusSophisticatedMixingChristianDiegoAlcocerArgüelloMichiganStateUniversityƒ2016-7-16AbstractAlcocerandJeitschko(2014)postulateabehavioralbiasinhowindividualsplaymixedstrategies.Theynaïveplayersasthosewho,whenindifferent,assignanoff-equilibriumpredeterminedprob-abilitytoeachaction.AlcocerandShupp(2016)evidenceoftheexistenceoftheseplayers:thosewhosestrategymixingisconsistentlyclosertouniformmixingthantheobservedbehavioroftherestofthepopulation.Wenowfocusonindividualresponsestochangingexperimentalconditions.Weevidencethatthereexistdistractors(andfocusers)thatpushplayers'towards(away)naïvemixinginmatchingpenniesgames.Thisallowsformethodstotakeadvantageofthisbiasandattainabove-equilibriumpayoffs.Usingcomputerbots,wealsoisolatealtruisticcomponentsofplayers'strategies.Welookatgameswherewepreviouslyfoundevidenceofsurplus-maximizingbehaviorthatisdifferentfromequilibriummixing.Addingaproportionoftransparentbotsthat(ex-ante)donotincentivizeanychangeinbehaviorbutimplythatsurplusiswastediftheygetanypayoff,behaviorgetsclosertoNashequilibria.1IntroductionWeinvestigateexperimentalconditionsŒdistractorsandfocusersŒthatpushplayers'behaviortowardsandawaywhatAlcocerandJeitschko(AJ,2016)identifyaseitherrelativelynaïveorsophisticatedmixing.Ourmainanalysisfocusesonobservingmeasuresofhowplayers'strategiesrespondtothesecontrolsandmoveawayorclosertoNashequilibria(NE).ThisallowsustoforecastthedirectionofthisdeviationinKeywords:Experimental,Behavioral,BoundedRationality,CompensatedEquilibrium,ComputerBots,Heuristics,MixedEquilibria,NaïveandSophisticatedPlayers.JEL:C72,C91,D03,D83.ƒWewishtothankJonX.Eguia,ThomasD.Jeitschko,ArijitMukherjeeandJeffreyM.Wooldridgeforhelpfulcomments.53gameswithtotally-mixedequilibria.Doingsocanallowonetodesignandtestsimplemechanismsthattakeadvantageofthisoff-equilibriummixingbehavioralbias,andearnabove-equilibriumpayoffs.Theevidencesuggeststhatbest-respondingtonaïvemixingyieldsabove-equilibriumrentsandthattheserentsincrease(decrease)ifthepopulationsaredistracted(focused).Relatedly,wealsoisolatealtruisticcomponentsofplayers'strategiesbyhavingpopulationsrandomlyplayagainstvaryingprevalencesofcomputerplayers(bots)withvaryingpublicly-announcedstrategies.Weshowthatwhenbots'behaviordoesnotdirectlyincentiveanyparticularresponse,itstillpushesplayerstowardsequilibriumincaseswherewehaveevidenceofbehaviorthathasaltruisticcomponentsinthatitmaximizestotal-surplus.Wearguethatliketheationofdistractorsandfocusers,ofaltruismbiasesallowsfortheexistenceofmechanismstotakeadvantageofsomepredictedoff-equilibriummixing.Also,wediscusshowthesehelpexplainsomepuzzlingresultsfromAlcocerandShupp(AS,2016).OurresultsareconsistentwiththestylizedfactthatbehaviorisincreasinglydifferentthanNEwhendealingwithincreasinglycomplexgames,inparticularinthosewithunique,totally-mixedNE.Thisstylizedfacthasbeenoneoftheprimarymotivationsforthetheoreticalliteratureonequilibriumextensionsthatallownon-NEstrategiestohavepositiveprobabilities.However,thequestionofwhichmixedorpurestrategiesindividualsactuallyfollowremainsopen.AJpostulateabehavioralbiasorrationalityboundinhowpeopleplaymixedstrategies.Asanillus-tration,onecanimagineanagentthatcanchoosebetweentwoactions,leftorright,knowingthatgivenothertheplayers'strategies,hisexpectedpayoffisthesameineithercase.AccordingtoneoclassicalNEtheory,theprobabilitywithwhichtheywillchooseeitherofthetwooptionsdependsonwhattheyknowabouttheotherplayers'payoffs.Theirpaperrelaxesthetypicalassumptionsonrationalityandproposesthetheoreticalexistenceofnaïveplayerswholackanystrategicdepthandthus,followingtheprincipleofindifference(e.g.,Keynes(1921);itisalsocalledprincipleofinsufreasoninprobabilitytheory),willalwaysacoinwhenindifferent.ThemainresultofAJ'spaperisthat,inegames,ifsomeproportionoftheplayersisnaïve(andfollowsomeoff-equilibriummixingwhenindifferent),thenexpectedpayoffsforallplayersarethesameasinthestandardperfectrationalitycase.Theonlyassumptionisthatthepercentageofnaïveplayersissmallenoughotherwiseequilibriadonotgenerallyexist.Thesophisticatedplayersessentiallyplayoff-equilibriuminresponsetothenaïveplayersandthecombinedresultistobringusbacktopayoffsequivalent54totheNEpayoffsonaverage.Thiscompensationisintheoppositedirectionofthedistortioninducedbythenaïvemixingbias.ThismeansthegeneralizedgameisanisomorphicsettingtotheNEsincetheconvolutionresultingfromthelinearcombinationofthemixingofnaïveandtherespondingsophisticatedplayerscoincideswiththemixingdistributionofthestandardperfect-rationalitycase.ASuseasetoflaboratoryexperimentstotest(andtheexistenceofsomenaïvelikeplayerswhoconsistentlyacoin,ormixcloseto50%,indifferentsettings.Theydevelopconsistentmodelstoidentifythemwithinapopulationandalsoshowthattheprobabilityofbeingnaïvecanbepartiallypredictedbyasimplequantitativetest.Theysortparticipants(withouttheirknowledge)intotwogroupsbasedonbehaviorinasetofsimplenormalform22gameswithauniqueandtotallymixedNE(asymmetricmatchingpenniesgames):naïveandtheirsophisticatedcounterparts.Theythenhaveeachgroupseparatelyplayagainstchangingproportionsofautomatedplayers(bots)thatfollowvaryingoff-equilibriummixedstrategies.Overall,theyevidenceoftheexistenceofplayersthatarerelativelynaïveandthatthesophisticatedplayersreactinthedirectionpredictedbyAJ.Also,theiranalysissuggeststhattheprobabilityofbeingnaïvecanbepartiallypredictedbyasimplequantitativetest.ThispaperbuildsonAS'initialworkandusessimilarexperimentstodetermine;1)ifgivenexperimentalconditionscanmakeindividualsmoreorlessnaïverelativetothemselves(analogoustowithin-estimation,weinvestigatewhatmakesplayersmoreorlessnaïverelativetothemselves;asopposedtoanalysisbetweenpopulations,asinAS,whoinvestigatehowtoidentifyplayerswhoarenaïverelativetoapopulation),and2)ifthereisasocialcomponentinutilitiesthatcanbedwiththeaidofbots.Thehypothesisbehind(1)isthatplayerssometimesbehaverelativelyclosetocoinunderanexperimentalcontrolwelabeldistractorandconsistsofaddingweaklydominatedstrategiestomatchingpenniesgames,thanwhentheyfaceadifferentcontrolwedenominatefocuserwhichconsistsofmonotonicallyincreasingpayoffssuchthatequilibriaarenotThehypothesisbehind(2)isthatwhenthereissomechanceplayersarematchedagainstabot,altruisticeffects(utilitygainedbytotalsurplusmaximizing,evenifitisanotherplayerwhogetsit)decreaseandbehaviormovestowardswhatispredictedbyutilitymaximization.2LiteratureReviewMixedstrategiesareprevalentineverydaylife.Anillustrativeexamplefromsportsoccurswheninbaseballabattertriestooutguesswhatwillthepitcherthrownext.Theyarealsorelevantincrimepreventionsituations55wherethelocationofanattackcannotbeperfectlypredictedbecauseagentsaremixingtheirstrategies.InanepistemicgametheoryapproachaboutNEwithmixedstrategies,playershaveunobservablepriorbeliefsabouttheirownstrategies,aboutplayers'beliefs,andsoon(hierarchiesofbeliefs).1Intwoseminalpapers,Aumann(1987)andAumannandBrandenburger(1995),showthatmutualbeliefinrationalityandcommonknowledgeofthegame'smixedstrategiesandpayofffunctionsaresufconditionsforNE.Mixedstrategiesarenotconsciousrandomizations,butingconjecturesastowhatotherplayerswilldo.Consequently,ifaNEfailstobeobserved,oneoftheseassumptionsfailedtoo.Healy(2011)linkstheseepistemicfoundationswiththeexperimentalevidencethatpopulationstendnottoplayNE.Byelicitingsubjects'beliefs,hethesourcesoffailuretoplaytheNEinveclassic22games:asymmetricmatchingpennies,dominancesolvablegame(wheretheNEisnotPareto-dominated),prisoners'dilemma,symmetriccoordination(battleofthesexes)andasymmetriccoordination.HethattheAumannandBrandenburger(1995)assumptionthatgenerallyfailsisthatplayershaveimperfectbeliefsaboutothers'payoffs,evenwhenthegameclearlythem.Thisresultisclosetothefundamentalassumptionsoftheory(Harsanyi,1973),quantalresponseequilibrium(McKelveyandPalfrey,1995;McKelveyandPalfrey,1998;Goeree,HoltandPalfrey,2002)andlogitequilibrium(McFadden,1973)wheremixed-strategyequilibriacanbeinterpretedasthelimitofpure-strategyequilibriaingameswhenperturbationsareaddedtopayoffs.2Goeree,HoltandPalfrey(2003)andSeltenandChmura(2008)showthatquantalresponseequilibriumisabetterpredictorthanNEinseveralvariantsof22gameswithunique,mixedNEthattheycallasymmetricmatchingpennies.Inmostoftheexperimentalliterature,players'behaviorisydifferentthanthestrategiespredictedbygametheory(SeltenandChmura,2008).Experimentalevidencesuggeststhatinformationincreasesarecorrelatedtodecisionvariance(SchramandSonnemans,2011).Theimpactofexcessiveinformationandcognitiveloadsvariesacrossindividuals(Swansonetal.,2011)and,forassetchoices,isgreaterforthosewithlessbackgroundin(AgnewandSzykman,2005).Relatedly,CamererandLovallo(1999)runexperimentsofagameofsimultaneousentrywithcongestionwheretheNE1Theproblembehindbeliefformationofotherplayers'strategiesinmixedequilibria(letaloneotherplayers'beliefs)isknowntobenon-trivial.Feldman(1959)ranaseriesofexperimentswhereindividualswereshownsequencesofzerosandones.Eventhoughthesequenceswererandom,agentswouldtrytocomeupwiththeoriesandheuristicstopredictthem;actuallyseeingpatternswheretheydonotexist.2Analternatefamilyoftheoriesoftenusedtomodeloff-equilibriumbehaviorislevel-kthinkingorcognitivehierarchytheory(Camerer,HoandChong,2004;VanDamme,1991).56istoenterwithanondegenerateprobability(i.e.,itisamixed-strategyNE).TheyevidenceonovexcessentrycomparedtotheNE.Asymmetricorgeneralizedmatchingpennies,or,moregenerally,22gameswithunique,mixedNEareextensivelyanalyzedintheliterature.Awell-knownexamplethatisveryrelevanttothediscussioninthenextsectionsisGoereeandHolt(2001).TheythatpopulationsdonotplaytheNEbut,interestinglyandopposedtothetheory(butperhapsnotsurprisingly),theyndthatchangesinoneplayer'spayoffsinoneoutcomeincreasetheprobabilitythattherelatedstrategyisplayed.(WhereasinamixedNEachangeinone'spayoffsonlyaffectstheotherplayer'sequilibriumstrategy.)ShachatandSwarthout(2004)haveindividualsplayasymmetricmatchingpenniesgamesagainstcomputerbotsandwhileitisthatindividualsgenerallydetectdeviationsfromNashequilibriumandhaveanintuitionofhowtoexploitthem,consistentlywiththerestoftheliterature,playersdonotperfectlyfollowtheresultingbestresponses.GillandProwse(2014)alsoacorrelationbetweencognitiveability(measuredbyaRaventest),andeconomicrationality(measuredinlevel-kterms).Parkhurstetal.(2015)thatwhenindividualsfaceinformationoverload,theytendtousesimplifyingheuristics(GretherandWilde,1983).Whenplayershavetwooptionsandareoverloadedwithinformation,theytendtomixuniformlymixinginanexperimentalsetting.Similarly,DuffyaandSmith(2014)thatwhenplayingaprisoner'sdilemma,playerswhoareheavilydistractedbyhavingtomemorizea7digitnumberplayworsethanthosefacingthesmallercognitiveloadofrememberinga2digitnumber.3ExperimentalDesignandProceduresWeimplementanexperimentconsistingofthreetreatmentsdesignedtotestifplayers'behaviorvariespre-dictablywhenplayingingamesthatincludewhatwedenominatedistractors(Treatment1)andfocusers(Treatment2),andagainstcomputer-simulatedplayers(bots)withvaryingbehaviorandprevalence(Treat-ment3).Attheendofeachsession,participantsanswerasimplequantitativetestidenticaltothatusedinAS.3Eachtreatmentinvolvedtwosessions(seeTable3fordetails).3Thetestanditsdiscussionisincludedinanappendix.57TreatmentNameRoundsShowUp$PerTokenPayingRounds1Focuser:MonotonicIncrease5010522Distractor:BorderedGame5010323Bots:AltruismMeasure751053TableIII.3:ExperimentalTreatmentsAcrossthethreetreatments,allsessionsinvolvethesamebasicsetting.Afterarrivingandloggingontothecomputer,halftheplayersarerandomlyassignedasP1(firowflplayers)andhalfweredesignatedasP2(ficolumnflplayers).WhilewedifferentiaterowandcolumnplayersinourdiscussionandalthoughthegameisdesignedsuchthattheNEisnumericallythesameforboth,theasymmetricdistinctionbetweenbothisfundamentalfortheanalysisbelow.Thatsaid,allplayersviewedthegameasarowplayeronthecomputerwhethertheywerearowplayerornot.Playersmaintainedtheirtypethroughoutasessionandplayedmultipleroundsofthefollowingmatchingpenniesgamealongwiththe25to50roundsofthegamevariantsdescribedbelow.Notethat,followingAS(2016),inthecontextofthisgamenaïvemixingisasoccurringwhenplayersassignaprobabilitystatisticallycloseto50%toeachavailableaction.Similarly,sophisticatedornon-naïvemixingisasoccurringifplayisclosertotheNE:b=g=3=4.Playerswereanonymousandwererandomlyre-matchedaftereachround.Eachsessionincludedatutorialandpracticeround.58P2LeftRightP1Up3;00;3(1b)Down0;11;0(b)(1g)(g)NE:b=g=3=4FigureIII.3:BaseGame3.1TreatmentDescriptions3.1.1Focuser:MonotonicIncreaseTheFocuserTreatmentsessionsinclude25roundsofthebasegameand25roundsofthemonotonicallyincreasedgame(seebelow)inrandomorder.Notethatunderthiscontrol,payoffsaremultipliedbythreesuchthattheNEremainsthesameb=g=34:ThehypothesisisthatplayerswillconsistentlyplayclosertotheNE(andawayfromsimple50-50mixing)whentheyfaceamonotonicallyexpandedgame.Theideaisstraightforward:whenthereismoreatstake,playersconcentratemoreontheirchoicesandtheirbehaviorgetsclosertotheonepredictedbyneoclassicaltheory(whichassumeshighrationalitylevels)thantonaïvemixingorrandomnoise.4LeftRightUp9;00;3(1b)Down0;93;0(b)(1g)(g)NE:b=g=34FigureIII.3.1.1:MonotonicallyIncreasedGame3.1.2Distractor:BorderedGameAsintheFocuserTreatment,theDistractorTreatmentsessionsinclude25roundsofthebasegameand25roundsofamborderedgameinrandomorder.Theborderedgameaddsaweakly-dominatedpure4NotethatinaBernoullitrial,varianceismaximizedwhenp=1=2:59strategytothebasegamesuchthattheNEofthisnewgamecoincideswiththeNEofthebasegame.Asinpreviousresearchthisstrategyaddition,whiledominated,isexpectedtoaddalayerofcomplexitytothechoiceandthuspotentiallythiscognitiveloadwillpushbehaviorawayfromtheNE.Thehypothesisisthatwhenplayingaborderedversionofthebasegame,behaviorwillmovetowardsnaïvemixing.Tobeclear,ourhypothesesdonotonlyimplythatdistractorspushmixingawayfromtheNE.Throughoutallexperiments,wealsocannotstatisticallyrejectthehypothesisthatthedirectionofthisdis-tortionispredictablebyAJ'sresults.LeftRightz'Up3;00;30;0(1ab)Down0;11;00;0(b)z0;00;00;0(a)(1gd)(g)(d)NE:b=g=3=4;a=d=0FigureIII.3.1.2:BorderedGame3.2BotsTreatmentDuringthetwosessionsoftheBotsTreatment,participantsplaythebasegameduring75rounds.Twentyveoftheseroundsincludenobotswhile50roundsdo.Inthe50roundswithbots,thebotsplayedtheNEduring25roundsandwerecoin-(simulatingperfectlynaïve)fortheother25rounds.Notethattheorderinwhichthe75roundswereplayedwasrandomandthatthebotstrategiesandtheirprevalence(eachplayerhada25%probabilityofbeingmatchedagainstabotduringthebotrounds)wereannouncedbeforeeachroundandwerepublicinformation.GameTypeNumberBotBotRandomizationNEorCNEofRoundsPrevalenceIfP1IfP2IfP1IfP2Base250%NANAb=0:75g=0:7522525%bB=0:75gB=0:75b=0:75g=0:7532525%bB=0:50gB=0:50b=0:83g=0:83TableIII.3.2:BotsTreatment60Theoff-equilibriumstrategythebotsusedduringthe25roundsimplyadistortionallowingustoobserveseveralmeasuresofhowdifferentplayersreacttoit.,thesebotshelpustoisolateapotentialsocialcomponentofutilitiesthatmayimpactplay.Theintuitionbehindthisisthatthebasegameisasymmetricso,fromabehavioralperspective,itisharderforP1tocompensateandreacttothedistortionscreatedthanforP2.ThisisbecauseitisP1whohastheresponsibilityofdeterminingthesizeofthecake,whereasP2onlydetermineswhogetsthecake.Inthiscase,forP1;surplusmaximizationandlossaversion(playingUpoften)gointheoppositewayofcompensation(playingDownoften).3.2.1CompensatedNashEquilibriumPredictionsFollowingAJ,Table3.2alsoshowsthe`CompensatedNashEquilibrium'(CNE)fortheroundswithoff-equilibriumbots.TheCNEis,inessence,theNashequilibriumaftertakingintoaccountthepresenceofplayersthatarefollowingpubliclyknownoff-equilibriumstrategies.Duringallstages,itconsistsofuniqueandinterior(purelymixed)strategies,implyingthatanydeviationcanbetakenadvantageofandcollapsestheopponent'sbest-responserelationintoadegeneratedistribution.Fortype2gamesinwhichthebotsareplayingtheNE,theirpresenceshouldhavenoimpactontheCNE.Assuch,theCNEfortype2gamescoincideswiththeNEofthebasegame.Forthetype3game,thebotpopulationis(i.e.,theyplayŸb=Ÿg=1=2).Sincetheirprevalenceis25%,theuniqueCNEistoplaybS=gS=0:83.TheCNEhasthecharacteristicthatitpredictsbehaviorthatcompensatesbymovingintheoppositedirectionofthedistortion.Thatis,sincehumanplayersknowsomeautomatedplayerswillmixwithaprobabilitythatislessthantheequilibriumprobability(ŸbWE;whereWMandWBareeachpopulation'sstrategiesinthebaseandborderedgames.Table4.2showstheresults.Thesearenotasclear-cutasthosefromtheFocuserTreatment.Focusingoncolumns2and3,astraightforwardt-testyieldsnostatisticaldifferenceinbehaviorbetweenborderedandnot-borderedgames.ForbothP1andP2;estimatedstrategiesareessentiallythesamewhencomparingthebaseandtheborderedgames.Therespectivep-valuesare0.502and0.431.99Poolingthedatafromboththerowandthecolumnplayersdidnothelp:p-value=0:306:64P1MixingP2MixingŸP1MixingŸP2MixingStage(¯b)(¯g)(Ÿb)(Ÿg)Base0.5950.7040.6020.741(Std.Err.)(0.024)(0.022)(0.024)(0.023)Bordered0.6190.7290.6070.693(Std.Err.)(0.026)(0.024)(0.025)(0.025)Difference0.0240.0260.005-0.048(Std.Err.)(0.017)(0.016)(0.016)(0.015)TableIII.4.2:DistractorGameResultsHowever,lookingatthedatamoreclosely,wethattherearetwooutlierplayers(outof34)thateitherdidnotunderstandthegameorwerenottryingtomaximizetheirpayoffs.Thebehaviorofthesetwooutliersdidnotshowanyapparentpattern.Oneofthemplayedthedominatedactionsixtimes(outof25)whereastheotherplayeditvetimes.OneofthemwastypeP1andtheotherP2.Ifweremovethetwooutliers,behaviorforP1doesnotchange(p-value=0:8875)butforP2wehavesomeevidencethatitdoes,andintherightdirection(p-value=0:0789).Thisprovidessomeevidencethatplayersgetdistractedwhenweaklydominatedstrategiesareaddedtoagameandthatthisdistractionpushestheirbehaviortowardsuniformmixing.AswedowiththeresultsoftheFocuserTreatment,weinterpretthisas(albeitweak)thatunderadistractorcontrol,behaviorcanbepushedawayfromtheNE.Thisistobeexpectedandcoincideswiththeexperimentalliterature.Ourcontributionisevidencethatwithoutthesecontrols,behaviorisdistortedinthedirectionofnaïvemixing(uniformmixingamongstallpurestrategiesincludedintheNE).Thistreatmentalsotestswhethernaïvemixingcanbetterforecastbehaviorthantheequilibriumconceptsof,quantalresponseequilibria,andlevel-kthinkingwhichallplacepositiveprobabilityevenondominatedstrategies.Assuch,theseconceptspredictthatthepurestrategieszandz'willbeplayedwithsomestrictlypositiveprobabilitywhereasanaïveplayerisassumedtobeableto(always)avoida(conspicuously)dominatedactionandplayeitherzorz'withzeroprobability.Theevidenceisinconclusive.Wewanttoverifyifweaklydominatedstrategiesareplayedwithanystrictlypositiveprobabilityasthisneverhappensundernaïvemixing.Aftertheremovalofthetwoout-65liers,theestimatedprobabilitythatthedominatedactionisplayedis1.33%.10Thisisevidenceagainstthebehavioralconceptofnaïvemixingatthepopulationlevel.Still,itmustbenotedthatitwas7(outof34)playersthatplayed(z)or(z')onceveofthem)ortwice(twoplayers).Thisimplies79.4%oftheplayershaveanestimatedbehaviorofexactly¯a=0(ifP1)or¯d=0(ifP2):So,afterlookingatthemindividually,wecannotrejectthatthesearepartiallynaïveplayerseventhoughtheother20.6%ofthepopulationarenot.4.3BotsTreatmentResultsWhenfacingthebasegame,ASthat,eventhoughtheNEmixingismathematicallythesameforbothplayertypes(b=g=3=4),P2typeplayers'actualestimatedbehavior(¯g)isclosetotheNEwhereasP1typeplayersmixedclosertonaïvemixing(1=2):Theyalsothatthisbehavioraldistinctionholdswhenplayinggamesthatincludeoff-equilibriumbots.Whydothetwoplayertypes'behaviordiffer?Weconjecturethatwhenplayingthebasegame,P1facesaproblemthatisbehaviorallyhardersince,inasense,hehasthectingresponsibilityofkeepinghisopponentindifferent(heuristically,thisistheconditiontocalculatetotally-mixedequilibria)andmaximizingtotalsurplus.IfhechoosesUp,totalsurplusisthreetimesgreaterthanifheplaysDownso,irrespectiveofwhathisopponentdoes,hemighthaveanextrabehavioralincentivetoplayUporb=0.Ontheotherhand,P2typesactiononlydetermineswhogetsthissurplus.Ifforanygivenroundthereisapublicly-knownprobabilitythatoneismatchedwithabotŒwhogainsnoutility(i.e.,willnotearnanymoney)Œthenanyaltruisticandsurplus-maximizingincentivesarereducedinfavorofstraightforwardutilitymaximization(i.e.,keepinghisopponentindifferent).Thatis,theexpectedaltruisticutilitydiminishesiftheprobabilityofbeingmatchedwithahumanisreduced.Moreover,ifthesebotsareplayingtheNE,theyaretransparentinthattheirpresencedoesnotimplyadistortionthatneedstobecompensatedifanequilibriumistoberestored.Ontheotherhand,AJshowthatbotslikethoseincludedinTreatment3areexpectedtoinduceacompensationandtheuniqueCNEistoplaybS=gS=0:83.Ifaltruistic/totalsurplusmaximizingincentivesplayaroleinthedifferencebetweenP1andP2behavior,thenoneshouldexpectwhencomparingtreatments1(no-bots)and2(transparent-bots)tothatP1'sbehaviormovesclosertotheNEandourresultssupportthis.Denoting¯bkasP1'sestimatedbehaviorwhen10Nostatisticaltestisneededsincethenullisthatthemixingisexactlyzero.66playingStagekasdescribedinTable2,weget:¯b1=0:57;¯b2=0:64and¯b3=0:52:11Thecrucialrelationisthat¯b1<¯b2andthattheyarestatisticallydifferent.Thep-valueoftheone-sidedt-testthatcomparesbehaviorwithnobotsandwithNEbotsis:0:030allowingtherejectionofthenullhypothesisb1=b2:Ifwerestrictourselvestogame-theoreticaltoolslikeequilibriumconceptsorutility-maximizationcri-teria,themagnitudeofthedifference(¯b2¯b1>0)isnotaseasytointerpretasitssign.Thisisbecause,inthesegames,players'mixingisonlyexpectedtoadjusttotheiropponents'payoffs,nottotheirownpayoffssinceinaninternalequilibriumtheyareindifferentamongsttheirpurestrategies.Westillinterprettheofb1