10 Mixed strategies in baseball dating and paying your taxes.docx
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10 Mixed strategies in baseball dating and paying your taxes.docx
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10Mixedstrategiesinbaseballdatingandpayingyourtaxes
GameTheory:
Lecture10Transcript
Overview:
Wedevelopthreedifferentinterpretationsofmixedstrategiesinvariouscontexts:
sport,anti-terrorismstrategy,dating,payingtaxesandauditingtaxpayers.Oneinterpretationisthatpeopleliterallyrandomizeovertheirchoices.Anotheristhatyourmixedstrategyrepresentsmybeliefaboutwhatyoumightdo.Athirdisthatthemixedstrategyrepresentstheproportionsofpeopleplayingeachpurestrategy.Thenwediscusssomeimplicationsofthemixedequilibriumingames;inparticular,welookhowtheequilibriumchangesinthetax-compliance/auditorgameasweincreasethepenaltyforcheatingonyourtaxes
October8,2007
< ProfessorBenPolak: Allright,solasttimewedidsomethingIthinksubstantiallyharderthananythingwe'vedoneintheclasssofar.Welookedatmixedstrategies,andinparticular,welookedatmixed-strategyequilibria.Therewasabigidealasttime.Thebigideawasifaplayerisplayingamixedstrategyinequilibrium,theneverypurestrategyinthemix--that'stosayeverypurestrategyonwhichtheyplacesomepositiveweight--mustalsobeabestresponsetowhattheothersideisdoing.Thenweusedthattrick.Weuseditinthisgamehere,tohelpusfindNashEquilibriaandthewayitallowedustofindtheNashEquilibriaisweknewthatif,inthiscase,VenusWilliamsismixingbetweenleftandright,itmustbethiscasethatherpayoffisequaltothatofrightandweusethattofindSerena'smix. Conversely,sinceweknewthatSerenaismixingagainbetweenlandr,weknewshemustbeindifferentbetweenlandrandweusedthattofindVenus'mix.SoIwanttogobacktothisexamplejustforafewmomentsjusttomakeonemorepointandthenwe'llmoveon,butwe'llstillbetalkingaboutmixedstrategiesthroughouttoday.Sothiswasthemixthatwefoundbeforewechangedthepayoffs,wefoundthatVenus'equilibriummixwas.7,.3andSerena'sequilibriummixwas.6,.4.Andareasonablequestionatthispointwouldbe,howdoweknowthat'sreallyanequilibrium? Wekindoffounditbutwedidn'tkindofgobackandcheck. SowhatIwanttodonowisactuallydothat,dothatmissingstep.Werusheditabitlasttimebecausewewantedtogetthroughallthematerial.Let'sactuallycheckthatinfactP*isabestresponsetoQ*.SowhatIwanttodoisIwanttocheckthatVenus'mixP*isabestresponseforVenusagainstSerena'smixQ*.ThewayI'mgoingtodothatisI'mgoingtolookatpayoffsthatVenusgetsnowsheknows-orrathernowweknowshe'splayingagainstQ*.Solet'slookatVenus'payoffs..I'mgoingtofigureoutherpayoffsforL,herpayoffsforR,andalsoherpayoffforwhatshe'sactuallydoingP*. SoVenus'payoffs,ifshechoosesLagainstQ*thenshegets--verysimilartowhatwehadontheboardlastweek,butnowI'mgoingtoputinwhatQ*isexplicitly--shegets50times.6..[ThisisQ*andthisis1-Q*.].Soshegets50times.6and80times1minus.6whichis.4,80times.4.Wecanworkthisout,andIworkeditoutathome,butifsomebodyhasacalculatortheycanpleasecheckme.Ithinkthiscomesto.62.Somebodyshouldjustcheckthat.IfVenuschoseR--rememberRheremeansshootingtoSerena'sright,toSerena'sforehand--ifshechoseRthenherpayoffsare90 Q*.So90(.6)plus20(1-Q*)so20(.4),so90(.6)plus20(.4),andagainIworkedthatoutathome,andfortunatelythatalsocomesoutat.62.Sowhat'sVenus'payoffforP*? We'vegotherpayoffforbothherpurestrategies,soherpayofffromactuallychoosingP*iswhat? Well,P*is.7,so.7ofthetimeshewillactuallybeplayingLandwhensheplaysL,she'llgetapayoffof.62,and.3ofthetimeshe'llbeplayingR,andonceagain,she'llbegettingapayoffof.62and--doIhaveacalculator? Sorry,thankyou.SoP*is.7,yes,you'reabsolutelyright,sothisisP*and1-P*,Solet'smakethatclearer.I'llshowyouwhattheequilibriumisbutP*itselfis.7.SowhenVenusplaysLwithprobabilityof.7,then.7ofthetimeshe'llgettheexpectedpayoffof.62and.3ofthetimeshe'llgetapayoffagainof.62andthat'sthekindofmathIdon'thavetodoathome,that'sgoingtocomeoutat.62.Again,assumingmymathiscorrect.SoallI'vereallydonehereisconfirmwhatwedidalreadylasttime. Weknew--weinfactchoseSerena'smixQtomakeVenusindifferentbetweenLandR.Andthat'sexactlywhatwefoundhere,goingleftit's.62,goingrightitgets.62andhenceP*gets.62.ButIclaimwecannowseesomethingalittlebitelse.Wecannowaskthequestion,isP*infactthebestresponse? Well,foritnottobeabestresponse,forthisnottobeanequilibrium,therewouldhavetobesomedeviationthatVenuscouldmakethatwouldmakeherstrictlybetteroff.Letmerepeatthat.Ifthiswerenotanequilibrium,therewouldhavetobesomedeviationforVenus,thatwouldmakeherstrictlybetteroff.ByplayingP*she'sgettingareturnof.62.Soonethingshecoulddeviateto,isplayingLallthetime.IfshedeviatestoplayingLallthetime,herpayoffisstill.62soshe'snotstrictlybetteroff.That'snotastrictlyprofitabledeviation.Anotherthingshecoulddeviateto,isshecoulddeviatetoplayingR.IfshedeviatestoplayingR,herpayoffwillbe.62.Onceagain,she'snotstrictlybetteroff: she'sthesameasshewasbefore,sothat'snotastrictlyprofitabledeviation. SowhathaveIshownsofar? I'veshownthatP*isasgoodasplayingL,andP*isasgoodasplayingR.Infactthat'showweconstructedit.SodeviatingtoLisnotastrictlyprofitabledeviationanddeviatingtoRisnotastrictlyprofitabledeviation.Butatthispoint,somebodymightaskandsay,okay,you'veshownmethatthere'snowaytodeviatetoapurestrategyinastrictlyprofitableway,buthowaboutdeviatingtoanothermixedstrategy? So,sofarwe'veshown--we'veshownjustuphere--wecanseethatVenushasnostrictlyprofitablepure-strategydeviation.Shehasnostrictlyprofitablepure-strategydeviationbecauseeachofherpurestrategiesyieldsthesamepayoffasdidhermixedstrategy,yieldsthesameasP*.Buthowdoweknowthatshedoesn'thaveamixedstrategythatwouldbestrictlybetter? Howdoweknowthat? Anybody? Nohandsgoingup;oh,therewasahandup,good. Student: Anymixbetweenleftandrightwillstillyield.62. ProfessorBenPolak: Good,soanymixthatVenusdeviatesto,willbeamixbetweenLandR,andanymixbetweenLandRwillbeamixbetween.62and.62andhencewillyield.62.Sowe'regoingtouseagain,thisfactwedevelopedlastweek.Thefactwedevelopedlastweekwasthatanymixedstrategyyieldsapayoffthatisaweightedaverageofthepurestrategypayoffs,thepayoffstothepurestrategiesinthemix.Anymixedstrategyyieldsapayoffthatisaweightedaverageofthepayofftothepurestrategiesinthemix.Thatwasourkeyfactlastweek.Sohereifwe'veshownthatthere'snopure-strategydeviationthat'sstrictlyprofitable,thentherecan'tbeanymixedstrategydeviationthat'sstrictlyprofitable. Why? Becausethemixedstrategydeviationsmustyieldpayoffsthatlieamongthepurestrategydeviations.Sothisisagreatfactforus.What'sthelessonhere? Thelessonisweonlyeverhavetocheckforstrictlyprofitablepure-strategydeviations.That'sagoodjob.Why? Becauseifwehadtocheckformixedstrategydeviationsonebyone,we'dbehereallnight,becausethere'saninfinitenumberofpossiblemixed-strategydeviations.Buttherearen'tsomanypurestrategydeviationswehavetocheck.Let'sjustrepeattheidea.Supposethereisn'tanypure-strategydeviationthat'sprofitable,thentherecan'tbeanymixedstrategydeviationthat'sprofitable,becausethehighestexpectedreturnyoucouldevergetfromamixedstrategy,isoneofthepurestrategiesinthemix,andyou'vealreadycheckedthatnoneofthoseareprofitable. Sothissimpleidea,thesimpleideawedevelopedlasttime,notonlyhelpsustofindNashEquilibria,butalsotocheckforNashEquilibria.NowalotofpeopleIgatheredfromfeedbackfromsectionswereleftprettyconfusedlasttime.It'sahardidea.ActuallyIlookedatthetapeovertheweekend,Icouldseewhereitcouldbeconfusing.Butit'sactually,Ithinkwhat'sreallyconfusinghere--itwasn'tsomuch--Ithinkitwasn'tsomuchthatIcouldhavebeenclearerthoughI'msureIcouldhavebeen.It'sthatthisisreallyahardidea,thisideaofmixedstrategies.Sowe'regoingtoworkonitagaintoday,butIthinkoneoftheideasthatgetspeopleconfused,isthefollowingidea. Theysay,lookwefoundVenus'equilibriummixbychoosingaPanda1-PtomakeSerenaindifferent.WefoundSerena'sequilibriummixbyfindingaQanda1-QtomakeVenusindifferentandanaturalquestionyouhearpeopleaskthenis,whyisVenus"tryingtomakeSerenaindifferent? "WhyisSerena"tryingtomakeVenusindifferent? "That'snotreallythepointhere.Itisn'tthatVenusistryingtomakeSerenaindifferent.It'sthatinequilibrium,sheisgoingtomakeSerenaindifferent.Itisn'thergoalinlifetomakeSerenaindifferentbetweenlandr,anditisn'tSerena'sgoalinlifetomakeVenusindifferentbetweenLandR,butinequilibriumitendsupthattheymakeeachotherindifferent.ThewaythatwecanseethatisthatifVenusputs--wesaidlasttimeit'srepeated--ifVenusputstoomuchweight,morethan.7onL,thenSerenajustcheatstotheleftallthetime,andthatcan'tpossiblybeanequilibrium.AndifVenusputstoomuchweightonR,thenSerenacheatstotherightallthetimeandthatcan'tbeanequilibrium.SoithastobethatwhatVe
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