LectureSimultaneousmovegamesofcompleteNashInthiscourse,wemainlyconsidernon-cooperativegames.Non-cooperativegametheoryispositivetheory.Giventherulesofthegames,oneconsidersthe esofgamesusingvarioussolutionconcepts.Cooperativegametheoryisnormativetheory.Givennormativecriteria,oneconsiderswhetherthereisanysolutionthatsatisfiesthesecriteriaandwhetherthesolutionisuniqueifitexists.NashbargainingsolutionandShapleyvalueareverycommonusedsolutionconceptsfromcooperativegametheory.Non-cooperativegametheorystudiesstrategicinctionsamongyers.Ingames,thepayofftoayernotonlydependsonhis/herownchoice,butalsoonotheryers’choices.Therefore,whenoneconsidershis/herownchoice,he/sheshouldanticipateotheryers’choices.Thisisdifferentfromthecompetitivemarketwithoutexternalities,whereeachagentisassumedtobeapricetaker,andhis/heractiondoesnotdirectlyaffectotheragents.There,whenanagentmakesadecision,he/sheonlyconsiderstheprices,butnototheragents’choicesdirectly.Let’sconsiderthefollowingTwoyersareeachgivenanenvelopecontainingsomemoney.Theamountsofmoneycontainedinthetwoenvelopesaretwoadjacentnumbersinthesequence2,4,8,16,and32.Eachyeronlyknowstheamountofmoneyinhis/herownenvelope,butnotthatintheotheryer’s.Theythensimultaneouslydecidewhethertorequesttheexchangeoftheenvelopesbetweenthetwoyers.Ifbothyersrequesttheexchange,thentheexchangehappens.Otherwise,thereisnoexchange.Question:Supposeyourenvelopecontains8,shouldyouaskfortheWhenthisquestionwasaskedintheclass,about3/5ofthestudentssaidoneshould.Theother2/5saidoneshouldnot.Thosewhosayoneshouldreasonsthatthereis50%percentchancetheotherenvelopecontains18andonewillwin8and50%percentchancetheotherenvelopecontains4andonewillloss4.However,thisargumentimplicitlyassumesthattheotheryerwillalwaysaskfortheexchange.Thisassumptionisnottrue. with16theenvelopewillneveraskfortheexchangebecauseawith32intheenvelopewillneveraskfortheexchange.Therefore,oneshouldnotaskfortheexchangeifhis/herenvelopecontains8.Followingthisargument,nooneshouldaskfortheexchangeunlesshis/herenvelopecontains2.Consequently,theexchangewillnothappen.(TheseargumentsmaynotseemveryrigorousbuttheycanbemadeThefirstlessonwelearnfromthisexampleisthatonemustanticipatetheotherchoice,oneshould“standintheshoesoftheotheryer”andinferwhattheotheryerwilldo.Thisisthemostimportantmessagefromgametheory.Second,theargumentaboveisbasedonverystrongassumptionsabouttheyers.Whenoneholding4makesthedecision,he/sheassumesthattheoneholding8behavesrationally,he/shealsoassumesthattheoneholding8knowstheoneholding16behavesrationally,andthattheoneholding8knowstheoneholding16knowstheoneholding32behavesrationally;thatis,therationalityoftheyersiscommonknowledgeamongthem.Inmostofthiscourse,wewillassumecommonknowledgeofrationality,whichisaverystrongassumption.Apieceofinformationiscommonknowledgeifitannouncedtoallyerstogether.Third,ifotheryersarenotrational,arationalyermaywanttoydifferentlyfromwhatisarguedabove.Forexample,ifayerholding4knows3/5ofpeopleholding8willaskfortheexchange,thentheexpectedpayofffromaskingforthefortheexchange,andthereforetheyerholding4shouldaskfortheexchange.Ifotheryersarenotrational,knowingitandtakingitintoaccountisimportant.Inthiscourse,wewillconsiderthefollowingfourtypesofgameandthecorrespondingequilibriumconcepts:CompleteSimultaneousmove NashSequentialmovegame(dynamic Sub-gamePerfectNashplete BayesianSequential PerfectBayesianWewillalsodiscusssomeapplicationsofgameSimultaneousmoveElementsofasimultaneousmove(1)yers：i ,actionspace(orstrategyspace)ofeachs(s1,...,sI)：astrategy

ui(s)R-payoffTwoyerfinitegamecanberepresentedbyaExample1:Prisoners’（1）prisoners:1and

payoffs:(s1,s2)(u1,u2(C,Nc)(0,(Nc,C)(9,(C,C)(6,

whereC→confess；NC→notMatrix-6,-0,--6,-0,--9,-1,-CisbetterthanNCregardlessoftheotheryers’action.u1(C,s2)u1(NC,s2for

s2S2CstrictlydominatesUnderlyingassumptionsEachprisoneronlycaresaboutThetwoyersythegameonlyTheeffectsofanythirdpartyinfluenceareallreflectedintheG{I,Si,ui;i1,...,

iscalledthenormalformofthe

u(s,s)()u(s',s

forall {s,..,s,

,...,s},thanwe

isistrictly(weakly)i

s',

'isstrictly(weakly)dominated

sisiDEF:Ifyeri’sstrategysi

strictlydominatesallhisotherstrategies,wesay

isastrictlydominantstrategyofyeriAstrategysisadominantstrategy

u(s*,s)u(s,s)forall andisi{s1,..,si1,si1,...,sI}.

Example2:Second-pricesealed-bidBidderssimultaneouslysubmittheirbids.Thehighestbidwinstheauctionbutonlypaysthesecondhighestbid.Elementsofthisn

ifbimaxbjj

ui

ifbip(Vh ifb Claim:Biddinghisownvaluation

isyeri’sweaklydominantFirst,itdoesnotpaytobidhigher

vi.Ifbidding

alreadywinsthethenbiddinghigherdoesnotchangeanything.Ifbidding

doesnotwinthethenvi<hiandwiningtheauctionbringsalosstoyeri.Inthiscase,biddingthan

maybringalosstoyerSecond,itdoesnotpaytobidlower

vi.Ifbidding

doesnotwinthethenbiddinglowerthan

doesnotchangeanything.Ifbidding

winsthethen

isthehighestbidandthepayofffromwinningtheauctionisnon-negative.thiscase,biddinglowerthan

mayleadtothelossofthisnon-negativeItedelimination(ordeletion)ofdominatedAn

Thereisnodominantstrategyinthisgame.yer1hasnodominantstrategy;yer2hasnodominantstrategyeither.ButRstrictlydominatesC,becauseyer2’spayofffromRisalwaysgreaterthanthatfromCregardlessofyer1’saction.Inotherwords,CisstrictlydominatedbyR,orCisastrictlydominatedstrategy.Therefore,yer2willneveryCandyer1knowsyer2willneveryC.WecantheneliminatecolumnCfromthematrixwithoutaffectingthe eofthegame.AftercolumnCiseliminated,MandDarestrictlydominatedbyUandthereforerowsMandDcanbeeliminatedwithoutaffectingthe eofthegame.AftercolumnMandDareeliminated,LisstrictlydominatedbyRandcanbeeliminated.Aftertheseroundsofelimination,onlythestrategyprofile(U,L)DEF:IfonlyonestrategyprofilesurvivesIESDS,wesaythegameissolvablebyIESDS.Andtheresultisthesolutionofthegame.Note1：TheassumptionbehindIESD:ItisCommonKnowledgethattheyersareNote2:Wheneliminatingweaklydominatedstrategies,thesolutionmaynotbewell-definedinthatitmaydependonthesequenceofelimination.Forexample,inthefollowinggame:U→L→MM→R→U2.4Nash {I,Si,ui;i1,...,

S*(s*,...,s*)isNashequilibrium,ifforanyyeri,u(s*,s*)u(s,s*)for sS,wheres*(s*,...,

,s*,...,s*)

Notes：(1)NEisaselfenforcingagreement(stableagreement);ifyersagreeS*(s*,...,s*),noonewantstounilallydeviatefromthe Forany

sisabestresponsetos*(s*,...,

,s*,...,s*)

Rationalexpectation:Ayerrespondsoptimallytohis/herbeliefaboutwhatyerswilldo,andthosebeliefsare

Thereisnodominantordominatedstrategy.ThisgamehasanuniqueNE(U,L).AprofileofdominantstrategiesmustbeaNashIfagameissolvablethroughIESDS,thenthesolutionmustbeaNashStrategiesinaNashequilibriumsurviveIESDS.EliminatingstrictlydominatedstrategiesdoesnotaffecttheNEs.SetofNashEquilibriumstrategiessetofstrategiesthatsurviveIESDS.TofindNashequilibrium,wecanalwayseliminatestrictlydominatedstrategiesfirst,withoutlosinganyequilibrium.RemarkofNash1,NoguaranteeofParetoefficiencybecauseof2,Itispossibleforallyertobenefitfromthereductionofpayoffsinsomef(x),X

f(x.f(x（沒有損失Gametheorypayoff被減少，可能反而雙方都還 困境的例子NC-1，--NC-1，--0，--6，-NC-1，--9，-N>3雙方都選（-1，-1）比原先的選擇（-6，-6）AnExampleofApplication:CournotCournotequilibrium(tycompetition)2producers;identicalproducts.P(Q)a

Ci(Q)The2firms

(Q1,Q2)simultaneouslyand1(Q1,Q2)(aQ1Q2)Q12(Q1,Q2)(aQ1Q2)Q2Q1Q2

Given

aQ2QC QacQ1(Optimalresponse

同理QacQ2聯(lián)立可以得到Q*Q*a Theintersectionofthetwooptimalresponsecurvesgivesusthe Theequilibrium

**

(a9Benark Iftheyerscancooperate:(Pareto=max(a-c-Q

(a4cc(ac)2**(a

Qa-cQ*Q*2(a- Reason:firmidoesn’ttakeaccountoftheeffectof(negativeexternality)

Qionthe

j|pNotes：InCournotduopoly,thetwoproducers’strategiesarestrategicEquilibriumWhentherearemultipleequilibria,canwe/howdowepickone:Case1:Purecoordinationgame:Therearetwoequilibriainthisgame:(U,L)and(D,R).Therefinementmaydependonofficialrulesandinformalsocialnorms.Forexample,drivinginvolvescoordination.Itissafeifalldrivers