19 Subgame perfect equilibrium matchmaking and strategic.docx
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19 Subgame perfect equilibrium matchmaking and strategic.docx
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19Subgameperfectequilibriummatchmakingandstrategic
GameTheory:
Lecture19Transcript
Overview:
Weanalyzethreegamesusingournewsolutionconcept,subgameperfectequilibrium(SPE).Thefirstgameinvolvesplayers'trustingthatotherswillnotmakemistakes.IthasthreeNashequilibriabutonlyoneisconsistentwithbackwardinduction.WeshowtheothertwoNashequilibriaarenotsubgameperfect:
eachfailstoinduceNashinasubgame.Thesecondgameinvolvesamatchmakersendingacoupleonadate.TherearethreeNashequilibriainthedatingsubgame.Weconstructthreecorrespondingsubgameperfectequilibriaofthewholegamebyrollingbackeachoftheequilibriumpayoffsfromthesubgame.Finally,weanalyzeagameinwhichafirmhastodecidewhethertoinvestinamachinethatwillreduceitscostsofproduction.Welearnthatthestrategiceffectsofthisdecision--itseffectonthechoicesofothercompetingfirms--canbelarge,andifweignorethemwewillmakemistakes.
November12,2007
< ProfessorBenPolak: Solasttimewecoveredawholebunchofnewideas,anditwasreallyquitealotofideasforoneclass.Here'ssomeoftheideaswecovered.Wetalkedaboutinformationsets,andthesewerewaystoallowustomodelimperfectinformation.Sowhat'simperfectinformation? It'sawayofbeingabletocapturebothsimultaneityinmovesandsequentialmovesinthesamegame.Soit'sawaythat'sgoingtoallowustomeldthelessonsfrombeforethemid-termandafterthemid-term. Thenwetalkedaboutwhatstrategiesmeantinthiscontext,andthebasicideaisstrategiesareinstructions--strategiesforeachplayer--givethemaninstructionateachoftheirinformationsets.Thenwetalkedaboutwhatsub-gameswere,and,leavingasidetechnicalities,sub-gameswerejustgameswithingames.Andfinallyweintroducedtheideaofsub-gameperfectionwhichisournewsolutionconceptthatrefinestheideaofNashequilibrium.Whatsub-gameperfectionisgoingtodoisit'sgoingtoinstructtheplayerstoplayaNashequilibriumineverysub-game.Anotherwayofsayingitis,asub-gameequilibriumisaNashequilibriuminthewholegame,butineachsub-gameitinducesNashplayaswell. Now,we'regoingtoseetodayexamples.IfwehavetimeI'llgothroughthreedifferentexamples,andI'lltellyouattheendofeachexamplewhatitisI'mhopingtobeabletotakeawayfromthatexample.So,lasttimewasalotofformalstuff.Todayisgoingtobealotofexamples.Okay,that'souragenda.Here'sagame.Here'sourfirstexample.AndIcallthisexample,Icallthisgame,"don'tscrewup,"forreasonswe'llseeinaminute. SothisisagameinwhichPlayer1hastochoosebetweenUpandDown.IfPlayer1choosesUpthenPlayer2getstomoveandchoosesbetweenleftandright.AndifPlayer2choosesleftthenPlayer1getstomoveagainandPlayer1choosesbetweenupordown.Everyonelookingatthatgame? Sowhydon'tweplaythisgamesincewehaven'tplayedagameforawhile.We'llplayacoupleofgamestoday. SowhatI'mgoingtodoislet'sdividetheclassintwo.SoifIjustdrawalinedownthemiddleoftheclass,everybodytomyleft(toyourright),everybodyonthissideoftheclassisaPlayer1.Okayyou'reallPlayer1's.Andeveryoneonthissideofclassyou'rePlayer2,includingyouguyshidingfromthecamera,you'rePlayer2's.Okay,solet'sfigureoutwhatwe'regoingtodo.Everyonehadthetimetolookatthegame? SoPlayer1'syougettomovefirst,thoseofyouwhoaregoingtochooseDownraiseyourhandnow.Raiseyourhand.Waveitintheair.Keepitupsothecameracanseeyou.AndthoseofyouwhoaregoingtochooseUpraiseyourhands. LotsmoreUps.ThoseofyouwhochoseUpwhydon'tyouallstandup.Idon'twanttodoalltheexercisehere,soallthosewhochoseUp,standup.Soyoucanseethatchoosingdownendsthegame,sothismanypeoplearestillplayingthegame.Everyonewhoisstillsittingdown,everyonewhosatdownherehasexited.Allright,Player2'syougettomovenow.SoPlayer2's,thoseofyouwhochooseright,includingthepeopleonthisaisle,thosepeoplewhochooserightraiseyourhandnow--onerightoverthere.Thoseofyouwhochooseleftraiseyourhands.Whydon'tyouguysallstandup,justtogetyouawakeonaMondaymorning,everyone'ssleepyotherwise. Let'sgobacktoPlayer2's--sorryPlayer1--thoseofyouwhoarestillinthegame.SothoseofyouwhochoseUpthefirsttime,howmanyofyounowchoosedown? Raiseyourhandifyouchoosedown;andraiseyourhandifyouchooseup.Justtogetasampleofthis,let'sgetthe2'stositdownagainsopeoplecanseethem.So2'ssitdown.ThoseofyouPlayer1'swhoarestillinthegamewhowerechoosingupraiseyourhandnow.Ithinkthat'severybody,isthatcorrect? Okay,youcanallsitdown.Solet'sjusttalkaboutthisgameforawhileandthenwe'llanalyzeit. Now,thisisnotadifficultgamefromthepointofviewofstuffwe'vedonesincethemid-term.It'sprettyclearwhatweshoulddointhisgamebybackwardinduction.Sowhydon'twestartthere.Okay,sobybackwardinduction,wefindthatifPlayer1getstomoveasecondtimethenthey'rechoosingbetween4and3,andthey'regoingtochoose4.Player2,iftheygettomove,knowingthatPlayer1isgoingtochooseuptomorrow,they'regoingtobechoosingbetween3iftheychooseleftor2iftheychooseright.Sothey'regoingtostayinthegameandchooseleft,whichiswhatmostofyoudid.Finally,Player1atthebeginningofthegame,knowsthatPlayer2isgoingtochooseleftwhereuponshe'sgoingtochooseup,soifshechoosesUpshe'sgoingtoendupgetting4,andifshechoosesDownshe'sgoingtoget2,soshe'sgoingtochooseUp. Soit'sclearwhatbackwardinductiondoesinthisgameandthat'swhatmostpeopledidinthegame.Isthatright? However,noteverybodydidit.SomeofthePlayer1'sactually,whydon'tyouraiseyourhand,thosepeoplewhochoseDown--theoneswhochosedown.Thereweremorethanthat.Youcanallstandup.Thoseofyouwhodidn'tstandupjustnowraiseyourhands.Peoplearehidingnow,butthat'sokay.ThosepeoplewhochoseDown,theymayhavehadareasonforchoosingDown,andtheirreasonforchoosingDownmighthavebeenthattheythought--eventhoughtheycandobackwardinduction--soeventhoughtheyknowthatbybackwardinductionUpgetsthemthebetteranswer--theymightbeworriedthatiftheychooseUp,Player2willscrewupandchooseright.NoticethatifPlayer2choosesrightthenPlayer1onlygets1,whereasDownyielded2.SoinsomesenseDownwasthe"safe"thingtodoforPlayer1giventhattheymightbeworriedthatPlayer2mightscrewup. Doesthatroughly--justnodifthisisthecase: forthosepeoplewhochosedownisthatkindofwhatyouwerethinking? Somepeopleareshakingtheirheads,butsomepeoplearenodding.That'sagoodsign.NowwhymightPlayer2infactscrewupandchooseright.BecausePlayer2might,themselves,thinkthatPlayer1mightscrewupatthisstage.IfPlayer1weretoscrewupatthelaststageandchoosedown,thenPlayer2bychoosingleftwouldonlyget1,andforhimthesafeoptionthereforeisrightwhichyields2.So,togetthebackwardinductionanswerhere--whichmostofusdid--togetthebackwardinductionanswerherereliesonPlayer1trustingPlayer2toplaybackwardinduction,andthatrequiresPlayer1tohavetrustinPlayer2trustingPlayer1nottoscrewupinthelaststage.Sosayitagain,Player1needsPlayer2nottoscrewup,andthatmeansPlayer1needstotrustthatPlayer2willtrusthernottoscrewup. Everyoneseethegame? Okay,solet'stryandanalyzethisgameusingwhatwelearnedlasttimeandseewhatwefind.Sothefirstthingtodoislet'slookatstrategiesinthisgame.SoPlayer2justhastwostrategies,leftandright,becausePlayer2onlyhasoneinformationset--andnoticethisgameisactuallyagameofperfectinformationsoit'sgoingtobeveryeasy.Player1hastwoinformationsets,thisinformationsetandthatinformationset.AteachofthemPlayer1hastwochoicessoshemusthavefourstrategiesinall.Sothisgamewhenweputitinitsmatrixformisgoingtobea4x2game. Hereitis.AndthestrategiesforPlayer1are(Up,up),(Up,down),(Down,up),and(Down,down).AndthestrategiesforPlayer2arejustleftandright.Andnowwecanputthepayoffsin.So((Up,up),left)getsus(4,3).((Up,up),right)getsus(1,2).((Up,down),left)getsus(3,1).((Up,up,right)getsus(1,2)again: weendupexitingthegamehere.(Down,up)iseasybecauseit'sjustexitingthegameatthefirststage,soallofthesearegoingtobejust(2,1).Everyonehappywiththat? SowhatI'vedoneistranslatedthegameintoitsmatrixform. Andlet'slookforNashequilibriainthisgame.Letmedoitattheboardsinceit'squiteeasyatthisstage.SotolookforNashequilibria,let'sjustworryaboutpurestrategyequilibriafornow.SoifPlayer2waschoosingleftthenPlayer1'sbestresponseisthe(Up,up)strategy.AndifPlayer2ischoosingright,thenPlayer1'sbestresponseiseither(Down,up)or(Down,down).That'sexactlytheconversationwejusthad.IfPlayer2wasgoingto"screwup"andchooserightthenPlayer1wantstogetoutofthegameimmediately.Conversely,ifPlayer1ischoosing(Up,up)thenPlayer2ishappyandisgoingtochooseleft,trustingPlayer1.IfPlayer1wasgoingtochoose(Up,down),however,that'sPlayer1screwingupatthesecondstage.SointhatcasePlayer2wantstogetoutofthegameandchooseright.IfPlayer1ischoosing(Down,up)thenPlayer2isactually--itdoesn'tmatterthey'
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