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GameTheoryDr.ZouYashengLecture7wMixedStrategyEquilibriumZouYashengGameTheory&BusinessStrategy21.AZero-SumGame:AnyoneforTennis?1a.TheMinimaxTheorem2.ANon-Zero-SumGame3.ChoosetheRightMix4.WhatifthePayoffsChange?5.UniqueSituationsZouYashengGameTheory&BusinessStrategy3UnpredictabilityAcriticalelementofstrategywheneveronesidelikesacoincidenceofactionswhiletheotherwishestoavoidit.•TheATOwantstoaudittaxevaders;taxcheatershopetoavoidanaudit.•Theeldersisterwantstoridherselfoftheyoungerbrother,whowantstobeincluded.•Theinvaderswantchoiceoftheplaceofattacktosurprise,thedefenderswanttoconcentratetheforcesontheplaceofattack.•Thebeautifulpeoplewantexclusivity,thehoipolloiwanttobeupwiththelatesttrends.(AsYogiBerrasaid,“Thatnightclubissocrowded,no-onegoesthereanymore.”)•Whatisthebestamountofafinegivenafrequencyofdetection?ZouYashengGameTheory&BusinessStrategy4ChoosingtheLevelofUnpredictabilityWhilethetaxman’sortheattackers’decisiononanyoccasionmaybeunpredictable,thereareruleswhichgoverntheselection.Thecorrectamountofunpredictabilityshouldnotbelefttochance.Theoddsofchoosingonemoveoveranothercanbepreciselydeterminedfromtheparticularsofthegame.ZouYashengGameTheory&BusinessStrategy51.AZero-SumGame:AnyoneforTennis?Theserver,Stefan,wantstominimisetheprobabilitythatthereceiver,Rod,canreturnserve,andRodwantstomaximisethisprobability.It’sazero-sumgame:Stefan’swinisRod’sloss.IfRodcananticipateStefan’saim(toRod’sforehandorbackhand)thenRodwillmoveappropriately(forehandorbackhand)toincreasetheprobabilityofasuccessfulreturn.StefanwilltrytodisguiseormisleadRoduntilthelastsecond,hopingtocatchRodoffguardandwrong-footed.ZouYashengGameTheory&BusinessStrategy6TennisServe&ReturnA2×2payoffmatrixwhichsetsoutthepercentagesofRod’ssuccessfullyreturningserver:Stefan:theServer;Rod:theReceiver.wTABLE1.ThepercentageoftimesRodsuccessfullyreturns.wAnon-cooperative,zero-sumgame.(Rod,Stefan).wNoNashequilibriuminpurestrategies.ZouYashengGameTheory&BusinessStrategy7Stefan’staskStefanwantstokeepthesuccessfulreturnpercentageaslowaspossible;Rodhastheexactoppositeinterest:ashighaspossible.Ifthetwoplayersdecideontheirstrategiesbeforethematch,knowingtheaboveprobabilities,whatshouldtheirstrategiesbe?Tohelpanswerthisquestion,wenowplot:thepercentageoftimesRodreturnsserveagainsttheprobabilityofStefanaimingtoRod’sforehand.ZouYashengGameTheory&BusinessStrategy8IfSplayed0.5:0.5F:B,whatshouldRdo?Stefanwantstokeepthesuccessfulreturnpercentageaslowaspossible,alongtheredline.Rod:theexactoppositeinterest,ashighaspossible,alongthegreenline.ZouYashengGameTheory&BusinessStrategy9MixingstrategiesByplottingthetwostraightlines,we’reconsideringthepossibilitythatStefan(andRod)canmixtheirmoves,usingprobability:Stefan:“ifIalwaysservetotheforehand,thentheservewillbereturned90%ofthetime,butifIalwaysservetothebackhand,thepercentagefallsto60%.Inbothcases,Rodlearnstocorrectlyanticipatewhatmy(unchangingorpure)strategyis.“WhatifImixmyshotsandservehalftotheforehandandhalftothebackhandatrandom?ThenRodwillbekeptguessing,andwon’tbeabletoanticipatecorrectlyallthetime.”•IfRodanticipatesforehand,hewillberightwithprobabilityhalf(andreturn90%ofthetime)andwillbewrongwithprobabilityhalf(andreturnonly20%ofthetime).Thepercentageofsuccessfulreturnswillbe90/2+20/2=55%.•IfRodanticipatesbackhand,thepercentageofsuccesswillfallto60/2+30/2=45%.ZouYashengGameTheory&BusinessStrategy10ThebestmixRod(upperenvelope)willbebetteroff(55%success)ifhemovestowardsStefan’sforehand.(theupperline)ForStefan(lowerenvelope),areturnpercentageof55%isbetterthanthe90%or60%ofunchangingserving.(Remember:StefanwantstominimisethepercentageofsuccessfulreturnsbyRod.)Fromthediagram,Stefan’sbestmixistoservetotheforehandwithprobabilityof0.4,resultinginasuccessfulrateofreturnof48%,thebest(lowest)Stefancanachieve.Atthismix,Rodisindifferentbetweenmovingtoforehandormovingtobackhand:Rodcannotimprovethesuccessrateof48%.Theexactproportionsofthemixfollowfromthefouroutcomesofthebasicinteraction.Ifthesenumberschange,sowillthebestmixedstrategy.ZouYashengGameTheory&BusinessStrategy11Rod’stask„FromRod’spointofview,wegetadifferentchart:wANashequilibriumatRMF:0.3,SAF:0.4.wSAF:StefanaimsatforehandwSAB:StefanaimsatbackhandZouYashengGameTheory&BusinessStrategy12Asymmetry:Oneline(SAF)correspondstoStefanaimingtoforehand,one(SAB)tobackhand.Thepercentageofsuccessfulreturnsdependsonbothplayer’smoves,fromthepayoffmatrix.AsRod’sprobabilityofforehandreturnsincreases,above0.3,therateofhissuccessfallsto20%,becauseSadjuststoR’splay;below0.3forehand,theratealsofalls,to30%.Ditto.At0.3forehand,therateofsuccessfulreturnsis48%.Stefanrespondsappropriately.Note:eachplayerreachesthesamerateofasuccessfulreturn:48%.UsinghisbestmixStefanisabletokeepRoddowntothis,thebestRodisabletoachieveusinghisbestmix.ZouYashengGameTheory&BusinessStrategy131a.TheMinimaxTheoremThispropertyofzero-sumgamesistheMinimaxTheorem:When,inzero-sumgames,oneplayerattemptstominimiseheropponent’smaximumpayoff,whileheropponentattemptstomaximisehisownminimumpayoff,thesurprisingconclusionisthattheminimumofthemaximu