05 Nash equilibrium bad fashion and bank runsWord格式.docx
《05 Nash equilibrium bad fashion and bank runsWord格式.docx》由会员分享,可在线阅读,更多相关《05 Nash equilibrium bad fashion and bank runsWord格式.docx(18页珍藏版)》请在冰点文库上搜索。
![05 Nash equilibrium bad fashion and bank runsWord格式.docx](https://file1.bingdoc.com/fileroot1/2023-5/10/3540deea-fffd-48b7-8431-04455741ecab/3540deea-fffd-48b7-8431-04455741ecab1.gif)
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.
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