09 Mixed strategies in theory and tennis.docx
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09 Mixed strategies in theory and tennis.docx
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09Mixedstrategiesintheoryandtennis
GameTheory:
Lecture9Transcript
Overview:
Wecontinueourdiscussionofmixedstrategies.Firstwediscussthepayofftoamixedstrategy,pointingoutthatitmustbeaweighedaverageofthepayoffstothepurestrategiesusedinthemix.Wenoteaconsequenceofthis:
ifamixedstrategyisabestresponse,thenallthepurestrategiesinthemixmustthemselvesbebestresponsesandhenceindifferent.Weusethisideatofindmixed-strategyNashequilibriainagamewithinagameoftennis.
October3,2007
< ProfessorBenPolak: Solasttimewesawthis,wesawanexampleofamixedstrategywhichwastoplay1/3,1/3,1/3inourrock,paper,scissorsgame.Today,we'regoingtobeformal,we'regoingtodefinemixedstrategiesandwe'regoingtotalkaboutthem,andit'sgoingtotakeawhile.Solet'sstartwithaformaldefinition: amixedstrategy(andI'lldevelopnotationasI'mgoingalong,soletmecallitPi,ibeingthepersonwho'splayingit)Piisarandomizationoveri'spurestrategies.Soinparticular,we'regoingtousethenotationPi(si)tobetheprobabilitythatPlayeriplayssigiventhathe'smixingusingPi.SoPi(si)istheprobabilitythatPiassignstothepurestrategysi. Let'simmediatelyreferthatbacktoourexample.Soforexample,ifI'mplaying1/3,1/3,1/3inrock,paper,scissorsthenPiis1/3,1/3,1/3andPiofrock--soPi(R)--isa1/3.Sowithoutbelaboringit,that'sallI'mdoinghere,isdevelopingsomenotation.Let'simmediatelyencountertwothingsyoumighthavequestionsabout.Sothefirstis,thatinprinciplePi(si)couldbezero.JustbecauseI'mplayingamixedstrategy,itdoesn'tmeanIhavetoinvolveallofmystrategies.Icouldbeplayingamixedstrategyontwoofmystrategiesandleavetheotheronewithzeroprobability.So,forexample,againinrock,paper,scissors,wecouldthinkofthestrategy1/2,1/2,0.InthisstrategyIassign--Iplayrockhalfthetime,Iplaypaperhalfthetime,butIneverplayscissors. Soeveryoneunderstandthat? Andwhilewe'reherelet'slookattheotherextreme.Theprobabilityassignedbymymixedstrategytoaparticularsicouldbeone.ItcouldbethatIassignalloftheprobabilitytoaparticularstrategy.Whatwouldwecallamixedstrategythatassignsprobability1tooneofthepurestrategies? What'sagoodnameforthat? That'sa"purestrategy."Sonoticethatwecanthinkofpurestrategiesasthespecialcaseofamixedstrategythatassignsalltheweighttoaparticularpurestrategy.So,forexample,ifPi(R)was1,that'sequivalenttosayingthatI'mplayingthepurestrategyrock,i.e.apurestrategy. Sothere'snothinghere.I'mjustbeingalittlebitnerdyaboutdevelopingnotationandmakingsurethateverythingisinplace,andjusttopointoutagain,oneconsequenceofthisiswe'venowgotourpurestrategiesembeddedinourmixedstrategies.WhenI'vegotamixedstrategyIreallyamincludinginthoseallofthepurestrategies.Solet'sproceed.I'mgoingtopushthatupalittlehigh,sorry.SonowIwanttothinkaboutwhatarethepayoffsthatIgetfrommixedstrategies,andagain,I'mgoingtogoalittleslowlybecauseit'salittletrickyatfirstandwe'llgetusedtothis,don'tpanic,we'llgetusedtothisaswegoonandasyouseetheminhomeworkassignmentsandinclass. Solet'stalkaboutthepayoffsfromamixedstrategy.Inparticular,whatwe'regoingtoworryaboutareexpectedpayoffs.SotheexpectedpayoffofthemixedstrategyP,let'sbeconsistentandcallitPi,themixedstrategyPiiswhat? It'stheweightedaverage--it'saweightedaverageoraweightedmixtureifyoulike--oftheexpectedpayoffsofeachofthepurestrategiesinthemix.SothisisalongwayofsayingsomethingagainwhichIthinkisalittlebitobvious,butletmejustsayitagain.Thewayinwhichwefigureouttheexpectedpayoffofamixedstrategyis,wetaketheappropriatelyweightedaverageoftheexpectedpayoffsIwouldgetfromthepurestrategiesoverwhichI'mmixing. Sotomakethatlessabstractlet'simmediatelylookatanexample.Sohere'sanexamplewe'llcomebacktoseveraltimes,butjustoncetoday,andthisagameyou'veseenbefore.HereisthegameBattleoftheSexes,inwhichPlayerAcanchoose--PlayerIcanchooseAandB,andPlayerIIcanchooseaandb,andwhatIwanttodoisIwanttofigureoutthepayofffromparticularstrategies.SosupposethatPisbeingplayedbyPlayerIandPislet'ssay(1/5,4/5).SowhatdoImeanbythat? ImeanthatPlayerIisassigning1/5toplayingAand4/5toplayingB.AndsupposethatQ--soIamgoingtousePandQbecauseit'sconvenienttodosoratherthancallingthemP1andP2.SosupposethatQisthemixturethatPlayerIIischoosingandshe'schoosinga(½,½),soshe'sputtingaprobability1/2onaandaprobability1/2onb.JusttonoticeIswitchednotationonyoualittlebit,forthisexampletokeeplifeeasy,I'mgoingtousePtoberow'smixturesandQtobecolumn'smixtures. AndthequestionIwanttoansweriswhatistheexpectedpayoffinthiscaseofP? WhatisP'sexpectedpayoff? ThewayI'mgoingtodothatis,I'mfirstofallgoingtoaskwhatistheexpectedpayoffofeachofthepurestrategiesthatPinvolves,thepurestrategiesinvolvedinP.Sotostartoff--sothefirststepisaskwhatistheexpectedpayoffforPlayerIofplayingAagainstQandwhatistheexpectedpayoffforPlayerIofplayingBagainstQ? Thatwillbeourfirstquestionandwe'llcomebackandconstructthepayoffforP.SothesearethingswecandoIthink. SotheexpectedpayoffofAagainstQiswhat? Well,halfthetimeifyouplayAyou'regoingtofindyouropponentisplayinga,inwhichcaseyou'llget2,andhalfthetimewhenyouplayAyou'llfindyouropponentisplayingbinwhichcaseyou'llget0.Solet'sjustwritethatup.SoI'mgoingtoget2withprobability1/2plus0withprobability1/2.Everyonehappywiththat? Thatgivesme1.Pleasecorrectmymathinthis.It'sveryeasyattheboardtomakemistakes,butIthinkthatoneisright. Conversely,whatifIplayedB? What'stheexpectedpayofffortherowplayerofplayingBagainstQ,whereQis1/2,1/2? SohalfthetimewhenIplayB,I'llmeetaPlayerIIplayingaandI'llget0andhalfthetimeI'llfindPlayerIIisplayingbandI'llget1.Solet'swritethatup.SoI'llget0halfthetimeandI'llget1halfthetimeforanaverageof1/2.That'sthefirstthingIask.Andnowtofinishthejob,InowwanttofigureoutwhatistheexpectedpayoffforPlayerIofusingPagainstQ? ThatwasthequestionIreallywantedtostartoffwith.What'sthewaytothinkaboutthis? WellPis1/5ofthetime--accordingtoP,1/5ofthetimePlayerIisplayingAand4/5ofthetimePlayerIisplayingB,isthatright? Sotoworkouttheexpectedpayoffwhatwe'regoingtodoiswe'regoingtotake1/5ofthetime,andatwhichcasehe'splayingAandhe'llgettheexpectedpayoffhewouldhavegotfromplayingAagainstQ,and4/5ofthetimehe'sgoingtobeplayingBinwhichcasehe'llgettheexpectedpayofffromplayingBagainstQ. Nowjustplugginginsomenumberstothatfromabove,sowe'vegot1/5ofthetimehe'sdoingtheexpectedpayofffromAagainstQandthat'sthisnumberweworkedoutalready.Sothisnumberherecancomedownhere,1.And4/5ofthetimehe'splayingBagainstQ,inwhichcasehisexpectedpayoffwas1/2,sothis1/2comesinhere.Everyoneokaysofar,howIconstructeditsofar? Isthispodiuminthewayofyouguys,areyouokay? Letmepushitslightly.Sothetotalhereiswhat? It'sgoingtobe1/5of1plus4/5of½.4/5of1/2is2/5,soI'vegotatotalof3/5.Sothetotalhereis3/5.EveryoneunderstandhowIdidthat? Nowwhileit'sherelet'snoticesomething.WhenIplayedP,someofthetimeIplayedAandsomeofthetimeIplayedB.AndwhenIendedupplayingA,IgotA'sexpectedpayoff.AndwhenIplayedB,IgotB'sexpectedpayoff.SothenumberIendedupwith3/5mustliebetweenthepayoffIwouldhavegotfromAwhichis1,andthepayoffIwouldhavegotfromBwhichis1/2. Isthatright? So3/5liesbetween1/2and1.Everyoneokaywiththat? Nowthat'sasimplebutverygeneralandveryusefulideaitturnsout.TheideahereisthatthepayoffI'mgoingtogetmustliebetweentheexpectedpayoffsIwouldhavegotfromthepurestrategies.Letmesayitagain.Ingeneral,whenIplayamixedstrategytheexpectedpayoffIget,isaweightedaverageoftheexpectedpayoffsofeachofthepurestrategiesinthemix,andweightedaveragesalwayslieinsidethepayoffsthatareinvolvedinthemix.Soletmetryandpushthatsimpleideaalittleharder.SupposeIwasgoingtotaketheaverageheightintheclass--averageheightinthisclass.Soletmejust,ratherthanusetheclass,letmejustusesomeT.A.'shere. SoletmegetthesethreeT.A.'stostandupasecond.SupposeIwanttofigureouttheaverageheightofthesethreeT.A.'s.SostandupclosetogethersoIcanatleastseewhat'sgoingonhere.SoIthink,fromwhereI'mstanding,I'vegotthatAleisthetallestandMyrtoisthesmallest,isthatright? SoIdon'tknowinstantaneouslywhatthisaveragewouldbe,butIclaimthatanyweightedaverageoftheirthreeheights,isgoingtogivemeanumberthat'ssomewherebetweenthesmallestheightofthethree,whichisMyrto'sheight,andthetallestheightofthethree,whichisAle'sheight,isthatright? Isthatcorrect? Sothat'saprettygeneralidea.ThanksguysI'llcomebacktoyouinasecond. Let'sthinkaboutthissomewhereelse,let'sthinkaboutthebattingaverageofateam.Theteambattingaverageinbaseball,let'suse
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