崔志鵬電郵博弈

TheElectronicMailGame:

StrategicBehaviorUnder"AlmostCommonKnowledge"

ArielRubinsteinTheAmericanEconomicReview(Jun.,1989)PresentedbyCuiZhipeng1TheCoordinatedAttackStory

Army1Army2EnemyAttackatdawnIseeIseeyouseeIseeyouseeIseeAttackatdawnornotAttackatdawnornot2Contributionsofthepaper

Themainmessageofthispaperisthatplayers'strategicbehaviorunder"almostcommonknowledge"maybeverydifferentfromthatundercommonknowledge.

“AlmostCommonKnowledge“VS“CommonKnowledge”3DefinitionofthesetwokindknowledgeCommonknowledge:

itiscommonknowledgebetweentwoplayers1and2thatTheplayedgameisG,

ifbothknowthatthegameisG,

1knowsthat2knowsthatthegameisG.And2knowsthat1knowsthatthegameisG,1knowsthat2knowsthat1knowsthatthegameisG.And2knowsthat1knowsthat2knowsthatthegameisG

andsoonandsoon.“Almostcommonknowledge”:notsoonandsoon,the“knows”arefinite.4IntroduceanexampleTwoplayers,1and2,areinvolvedinacoordinationproblem.EachhastochoosebetweentwoactionsAandB.Therearetwopossiblestatesofnature,aandb.Eachofthestatesisassociatedwithapayoffmatrixasfollows:TheGameGa21AABBMM0000-L-LStatea,probability1-pTheGameGb2AABB00MM00-LStateb,probabilityp1-LL>M>0,p<1/25Both1and2knowsaboutwhichstatearetheyin,stateaorstateb.Andthe“knows”aresoonandsoon.So,iftheyareinstatea,theplayers’beststrategicbehaviorsaretocoordinate,bothchooseA.iftheyareinstateb,theplayers’beststrategicbehaviorsaretocoordinate,bothchooseB.CommonKnowledgeintheExample:6AlmostCommonknowledgeintheexample1.Theinformationaboutthestateofnatureisknowninitiallyonlytoplayer1.————useElectronicMailtosetup2.Thetwoplayersarelocatedattwodifferentsitesandtheycommunicateonlybyelectronicmailsignals.

Dueto"technicaldifficulties"thereisa"small"probabilitye>0,thatthemessagedoesnotarriveatitsdestination.3.Theelectronicmailnetworkissetuptosendaconfirmationautomaticallyifanymessagesreceived,includingnotonlytheconfirmationoftheinitialmessagebutaconfirmationoftheconfirmation;andsoon.7DemonstratetheElectronicMailSystem1.itisassumedthat,whenplayer1getstheinformationthatthestateofnatureisb,hiscomputerautomaticallysendsamessagetoplayer2andthenplayer2'scomputerconfirmsthemessageandthenplayer1'scomputerconfirmstheconfirmationandsoon.

2.Ifamessagedoesnotarrive,thenthecommunicationstops.3.Nomessageissentifthestateofnatureisa.4.Attheendofthecommunicationphasethescreendisplaystotheplayerthenumberofmessageshismachinehassent.LetT,beavariableforthenumberofmessagesi'scomputersent(thenumberoni'sscreen).8DemonstratetheElectronicMailSystemIfthestateisa,nomessagewillbesent.T1=0,T2=0Ifthestateisb,111222……9Ifthetwomachinesexchangeaninfinitenumberofmessages,thenwemaysaythatthetwoplayershavecommonknowledgethatthegameisGb.

However,sinceonlyafinitenumberofmessagesaretransferred,theplayersneverhavecommonknowledgethatthegametheyplayisGb。Inthecommonknowledgesituation,beststrategicbehaviorsare(B,B)Whatarebeststrategicbehaviorsunderthe“Almostknowledgesituation??Aretheythesameasthecommonknowledge’sones?“Almostcommonknowledge”situationissetup!10TheAnalysisoftheElectronicMailGame

T,beavariableforthenumberofmessagesi'scomputersent(thenumberoni'sscreen).(T1,T2)=(0,0),(n+1,n)or(n+1,n+1)Defineplayeri'sstrategyintheelectronicmailgame,Sitobeafunctionfromthesetofnaturalnumbers0,1,2,...intotheactionspace(A,B).ThenSi(t)isinterpretedasi'sactionifhismachinesenttmessages.i'sstrategydependsonTi11TheAnalysisoftheElectronicMailGame

PROPOSITION1:ThereisonlyoneNashequilibriuminwhichplayer1playsAinthestateofnaturea.InthisequilibriumtheplayersplayAindependentlyofthenumberofmessagessent.Let(S1,S2)beaNashequilibriumsuchthatS1(0)=A.Wewillprovebyinduction(歸納法）thatS1(t)=S2(t)=Aforallt.12IfT2=0thenplayer2didnotgetamessage.Heknowsthatitmightbebecauseplayer1didnotsendhimamessage(thiscouldoccurwithprobability1-p)(2)amessagewassentbutdidnotarrive(thishappenswithprobabilitype).Inthefirstcase,player1playsA(S1(0)=A).Ifplayer2playsA,then,player2'sexpectedpayoffisatleast;[(I-p)M+pe0]/[(1-p)+pe]ifheplaysBhegetsatmost[-L(1-p)+peM]/[(l-p)+pe].Thereforeitisstrictlyoptimalfor2toplayA,thatisS2(0)=A.TheAnalysisoftheElectronicMailGame13TheAnalysisoftheElectronicMailGame

Assumption:forallTi<t,players1and2playAinequilibrium.AssumeT1=t.Player1isuncertainwhether:T2=t(inthecasewhereplayer2receivedthetthmessagebut2'stthmessagewaslost)p=(1-e)e/[e+(1-e)e]>1/2.T2=t-1(inthecasewhere2didnotreceivethetthmessage).

z=e/[e+(1-e)e]>1/2.

Itismorelikelythatplayer1‘slastmessagedidnotarrivethanthatplayer2gotthemessage.Example,T1=1,how???14TheAnalysisoftheElectronicMailGameBytheinductiveassumption,player1assessesthat,ifT2=t-1,player2willplayA.Ifplayer1choosesB,playerl'sexpectedpayoffisatmostz(-L)+(1-z)MifhechoosesA,thenhisutilityis0.Give