05 Nash equilibrium bad fashion and bank runs.docx
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05 Nash equilibrium bad fashion and bank runs.docx
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05Nashequilibriumbadfashionandbankruns
GameTheory:
Lecture5Transcript
Overview:
Wefirstdefineformallythenewconceptfromlasttime:
Nashequilibrium.ThenwediscusswhywemightbeinterestedinNashequilibriumandhowwemightfindNashequilibriuminvariousgames.Asanexample,weplayaclassinvestmentgametoillustratethattherecanbemanyequilibriainsocialsettings,andthatsocietiescanfailtocoordinateatallormaycoordinateonabadequilibrium.Wearguethatcoordinationproblemsarecommonintherealworld.Finally,wediscusswhyinsuchcoordinationproblems--unlikeinprisoners'dilemmas--simplycommunicatingmaybearemedy.
September19,2007
< ProfessorBenPolak: Okay,solasttimewecameacrossanewidea,althoughitwasn'tverynewforalotofyou,andthatwastheideaofNashEquilibrium. WhatIwanttodotodayisdiscussNashEquilibrium,seehowwefindthatequilibriumintorathersimpleexamples.AndtheninthesecondhalfofthedayIwanttolookatanapplicationwhereweactuallyhavesomefunandplayagame.AtleastIhopeit'sfun. Butlet'sstartbyputtingdownaformaldefinition.Weonlyusedaratherinformalonelastweek,sohere'saformalone.Astrategyprofile--rememberaprofileisonestrategyforeachplayer,soit'sgoingtobeS1*,S2*,allthewayuptoSM*ifthereareMplayersplayingthegame--sothisprofileisaNashEquilibrium(andI'mjustgoingtowriteNEinthisclassforNashEquilibriumfromnowon)if,foreachi–soforeachplayeri,herchoice--soherchoicehereisSi*,iispartofthatprofileisabestresponsetotheotherplayers'choices.Ofcourse,theotherplayers'choiceshereareS--i*soeveryoneisplayingabestresponsetoeveryoneelse. Now,thisisbyfarthemostcommonlyusedsolutionconceptinGameTheory.SothoseofyouwhoareinterviewingforMcKenzieorsomething,you'regoingtofindthatthey'regoingtoexpectyoutoknowwhatthisis.Soonereasonforknowingwhatitis,isbecauseit'sinthetextbooks,it'sgoingtobeusedinlotsofapplications,it'sgoingtobeusedinyourMcKenzieinterview.That'snotaverygoodreasonandIcertainlydon'twantyoutojumptotheconclusionthatnowwe'vegottoNashEquilibriumeverythingwe'vedoneuptoknowisinsomesenseirrelevant.That'snotthecase. It'snotalwaysgoingtobethecasethatpeoplealwaysplayaNashEquilibrium.Forexample,whenweplayedthenumbersgame,thegamewhenyouchoseanumber,we'vealreadydiscussedlastweekorlasttime,thattheequilibriuminthatgameisforeveryonetochooseone,butwhenweactuallyplayedthegame,theaveragewasmuchhigherthanthat: theaveragewasabout13.Itistruethatwhenweplayeditrepeatedly,itseemedtoconvergetowards1,buttheplayofthegamewhenweplayeditjustoneshotfirsttime,wasn'taNashEquilibrium.Soweshouldn'tformthemistakeofthinkingpeoplealwaysplayNashEquilibriumorpeople,"ifthey'rerational,"playNashEquilibrium.Neitherofthosestatementsaretrue. Nevertheless,therearesomegoodreasonsforthinkingaboutNashEquilibriumotherthanthefactit'susedbyotherpeople,andlet'stalkaboutthoseabit.SoIwanttoputdownsomemotivationshere--thefirstmotivationwealreadydiscussedlasttime.Infact,somebodyintheaudiencementionedit,andit'stheideaof"noregrets."Sowhatisthisidea? Itsays,supposewe'relookingataNashEquilibrium.Ifweholdthestrategiesofeveryoneelsefixed,noindividualihasanincentivetodeviate,tomoveaway.Alright,I'llsayitagain.Holdingeveryoneelse'sactionsfixed,noindividualhasanyincentivetomoveaway.Letmebealittlemorecarefulhere;noindividualhasanystrictincentivetomoveaway.We'llseeifthatactuallymatters.Sonoindividualcandostrictlybetterbymovingaway.Noindividualcandostrictlybetterbydeviating,holdingeveryoneelse'sactions. SowhyIcallthat"noregret"? Itmeans,havingplayedthegame,supposeyoudidinfactplayaNashEquilibriumandthenyoulookedbackatwhatyouhaddone,andnowyouknowwhateveryoneelsehasdoneandyousay,"DoIregretmyactions? "Andtheansweris,"No,Idon'tregretmyactionsbecauseIdidthebestIcouldgivenwhattheydid."SothatseemslikeafairlyimportantsortofcentralideaforwhyweshouldcareaboutNashEquilibrium. Here'sasecondidea,andwe'llseeothersariseinthecourseoftoday.AsecondideaisthataNashEquilibriumcanbethoughtofasself-fulfillingbeliefs.So,inthelastweekorsowe'vetalkedafairamountaboutbeliefs.IfIbelievethegoalkeeper'sgoingtodivethiswayIshouldshootthatwayandsoon.Butofcoursewedidn'ttalkaboutanybeliefsinparticular.Thesebeliefs,ifIbelievethat--ifeveryoneinthegamebelievesthateveryoneelseisgoingtoplaytheirpartofaparticularNashEquilibriumtheneveryone,willinfact,playtheirpartofthatNashEquilibrium.Now,why? WhyisitthecaseifeveryonebelievesthateveryoneelseisplayingtheirpartofthisparticularNashEquilibriumthatthat'ssofulfillingandpeopleactuallywillplaythatway? Whyisthatthecase? Anybody? Canwegetthisguyinred? Student: BecauseyourNashEquilibriummatchesthebestresponseagainstboth. ProfessorBenPolak: Exactly,soit'sreally--it'salmostarepeatofthefirstthing.IfIthinkeveryoneelseisgoingtoplaytheirparticular--ifIthinkplayers2throughNaregoingtoplayS2*throughSN*--thenbydefinitionmybestresponseistoplayS1*soIwillinfactplaymypartintheNashEquilibrium.Good,soaspartofthedefinition,wecanseetheseareself-fulfillingbeliefs.Let'sjustremindourselveshowthataroseintheexamplewelookedattheendlasttime.I'mnotgoingtogobackandre-analyzeit,butIjustwanttosortofmakesurethatwefollowedit. So,wehadthispicturelasttimeinthepartnershipgameinwhichpeoplewerechoosingeffortlevels.AndthislinewasthebestresponseforPlayer1asafunctionofPlayer2'schoice.AndthislinewasthebestresponseofPlayer2asafunctionofPlayer1'schoice.Thisisthepicturewesawlasttime.Andlet'sjustlookathowthose--it'snosecretherewhattheNashEquilibriumis: theNashEquilibriumiswherethelinescross--butlet'sjustseehowitmapsouttothosetwomotivationswejustsaid. So,howaboutself-fulfillingbeliefs? Well,ifPlayer--sorry,Iput1,thatshouldbe2--ifPlayer1believesthatPlayer2isgoingtochoosethisstrategy,thenPlayer1shouldchoosethisstrategy.IfPlayer1thinksPlayer2shouldtakethisstrategy,thenPlayer1shouldchoosethisstrategy.IfPlayer1thinksPlayer2ischoosingthisstrategy,thenPlayerIshouldchoosethisstrategyandsoon;that'swhatitmeanstobebestresponse.ButifPlayer1thinksthatPlayer2isplayingexactlyherNashstrategythenPlayer1'sbestresponseistorespondbyplayinghisNashstrategy.Andconversely,ifPlayer2thinksPlayer1isplayinghisNashstrategy,thenPlayer2'sbestresponseindeedistoplayherNashstrategy.So,youcanthat'saself-fulfillingbelief.Ifbothpeoplethinkthat'swhat'sgoingtohappen,thatisindeedwhat'sgoingtohappen. Howabouttheideaofnoregrets? Sohere'sPlayer1;shewakesupthenextmorning--ohI'msorryitwasahewasn'tit? Hewakesupthenextmorningandhesays,"IchoseS1*,doIregretthis? "Well,nowheknowswhatPlayer2chose;Player2choseS2*andhesays,"nothat'sthebestIcouldhavedone.GiventhatPlayer2didinfactchooseS2*,IhavenoregretsaboutchoosingS1*;thatinfactwasmybestresponse."Noticethatwouldn'tbetrueattheotheroutcome.So,forexample,ifPlayer1hadchosenS1*butPlayer2hadchosensomeotherstrategy,let'ssayS2prime,thenPlayerIwouldhaveregrets.PlayerIwouldwakeupthenextmorningandsay,"ohIthoughtPlayer1wasgoingtoplayS2*;infact,shechoseS2prime.IregrethavingchosenS1*;IwouldhaveratherchosenS1prime.So,onlyattheNashEquilibriumaretherenoregrets.Everyoneokaywiththat? Thisisjustrevisitingreallywhatwedidlasttimeandunderliningthesepoints. So,IwanttospendquitealotoftimetodayjustgettingusedtotheideaofNashEquilibriumandtryingtofindNashEquilibrium.(Igottoturnoffthatprojectorthat'sinthewaythere.Isthatgoingtoupsetthelightsalot? )Sookay,sowhatIwanttodoisIwanttolookatsomeverysimplegameswithasmallnumberofplayerstostartwith,andasmallnumberofstrategies,andIwantustogetusedtohowwewouldfindtheNashEquilibriainthosesimplegames.We'llstartslowlyandthenwe'llgetalittlefaster. So,let'sstartwiththisgame,verysimplegamewithtwoplayers.EachplayerhasthreestrategiesandI'mnotgoingtomotivatethisgame.It'sjustsomerandomgame.Player1canchooseup,middle,ordown,andPlayer2canchooseleft,center,andrightandthepayoffsareasfollows: (0,4)(4,0)(5,3)(4,0)(0,4)(5,3)again(3,5)(3,5)and(6,6).So,wecoulddiscuss--ifwehadmoretime,wecouldactuallyplaythisgame--butisn'taveryexcitinggame,solet'sleaveourplayingforlater,andinstead,let'stryandfigureoutwhataretheNashEquilibriainthisgameandhowwe'regoingtogoaboutfindingit.Thewaywe'regoingtogoaboutfindingthemisgoingtomimicwhatwedidlasttime.Lasttime,wehadamorecomplicatedgamewhichwastwoplayerswithacontinuumofstrategies,andwhatwedidwaswefiguredoutPlayer1'sbestresponsesandPlayer 2'sbestresponses: Player1'sbestresponsetowhatPlayer2'sdoingandPlayer2'sbestresponsetowhatPlayer1isdoing;andthenwelookedwheretheycoincidedandthatwastheNashEqui
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