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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