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12005(2005/12/29)1(20points)ThefollowingisaninterpretationoftherivalrybetweentheUnitedStatesandtheSovietUnionforgeopoliticalinfluenceinthe1970sand1980s.Eachsidehasthechoiceoftwostrategies:AggressiveandRestrained.TheSovietUnionwantstoachieveworlddomination,sobeingAggressiveisitsdominantstrategy.TheUnitedStateswantstopreventtheSovietUnionfromachievingworlddomination;itwillmatchSovietaggressivenesswithaggressiveness,andrestraintwithrestraint.Specifically,thepayofftableis:SovietUnionRestrainedAggressiveRestrained4,31,4UnitedStatesAggressive3,12,2Foreachplayer,4isbestand1isworst.(a)Considerthisgamewhenthetwocountriesmovesimultaneously.FindtheNashequilibrium.(5points)(b)Nextconsiderthreedifferentandalternativewaysinwhichthegamecouldbeplayedwithsequentialmoves:(i)TheUnitedStatesmovesfirstandtheSovietUnionmovessecond.(ii)TheSovietUnionmovesfirstandtheUnitedStatesmovessecond.(iii)TheSovietUnionmovesfirstandtheUnitedStatesmovessecond,buttheSovietUnionhasafurthermoveinwhichitcanchangeitsfirstmove.Foreachcase,drawthegametreeandfindthesubgame-perfectequilibrium.(3pointseachfor(i)and(ii);5pointsfor(iii)(c)Whatarethekeystrategicmatters(commitment,credibility,andsoon)forthetwocountries?(4points)(a)SovietUnionhasadominantstrategyofAggressiveandtheUniqueNashequilibriumis(Aggressive,Aggressive),withpayoffs(2,2).(b)Seetheattachedfigure.(c)CommitmentfortheSU;orapromisefortheSU“RifR”.UShavenothingtodo.Hecannotcommittomovefirst(thentheSUwillchoosehisdominantstrategyofAggressive,theoutcomeremainstobethestatusquo);ornotbenecessarytothreatorpromise(sinceiftheSUmovesfirst,hemustchoosetoRestrain,andtheUSneedonlytofollowitsbestchoice.)2(20points)Considerthefollowinggame.Aneutralrefereerunsthegame.Therearetwoplayers,RowandColumn.Therefereegivestwocardstoeach:2and7toRowand4and8toColumn.Thisiscommonknowledge.Then,playingsimultaneouslyandindependently,eachplayerisaskedtohandovertotherefereeeitherhishighcardorhislowcard.Therefereehandsoutpayoffs–whichcomefromacentralkitty,notfromtheplayers–thataremeasuredindollarsanddependonthecardsthathecollects.IfRowchooseshisLowcard2,thenRowgets$2;ifhechooseshis2Highcard7,thenColumngets$7.IfColumnchooseshisLowcard4,thenColumngets$4;ifhechooseshisHighcard8,thenRowgets$8.(a)Drawoutthepayofftable.(4points)(b)WhatistheNashequilibrium?Isitaprisoners’dilemma?(4points)Nowsupposethatthegamehasthefollowingstages.Therefereehandsoutcardsasbefore;whogetswhatcardsiscommonknowledge.ThenatstageI,eachplayer,outofhisownpocket,canhandoverasumofmoney,whichtherefereeistoholdinanescrowaccount.Thisamountcanbezerobutnotnegative.Therulesforthetreatmentofthesesumsareasfollowsandarealsocommonknowledge.IfColumnchooseshisHighcard,thentherefereehandsovertoColumnthesumputbyRowintheescrowaccount;ifColumnchoosesLow,thenRow’ssumrevertsbacktoRow.ThedispositionofthesumdepositedbyColumndependssimilarlyonwhatRowdoes.AtstageII,eachplayerhandsoveroneofhiscards,HighorLow,tothereferee.Thentherefereehandsoutpayoffsfromthecentralkitty,accordingtothepayofftableinsection(a),anddisposesofthesums(ifany)intheescrowaccount,accordingtotherulesjustgiven.(c)Findtherollback(subgame-perfect)equilibriumofthistwo-stagegame.Doesitresolvetheprisoners’dilemma?Whatistheroleoftheescrowaccount?(12points)(a)COLUMNLowHighLow2,410,0ROWHigh0,118,7(b)(Low,Low).Itisaprisoners’dilemma.(c)Thesecondstagegamecanbeshownasthefollowingtable(2points):COLUMNLowHighLow2,410-x,xROWHighy,11-y8-x+y,7+x-y(xisRow’ssumdepositedintheescrowaccount,yisColumn’s.)Ifx4,Column’sdominantstrategyisLow;Ify2,Row’sdominantstrategyisLow.Thus(Low,Low)willbethefinalsecond-stageequilibrium,withpayoffs(2,4).Similarly,ifx4,andy2,theequilibriumwouldbe(Low,High),withpayoffs(10-x,x).Ifsuchaequilibriumisreallywantedbyboth.Rowwilldecreasexasmuchaspossible,sox=4+ε.(ε0andcanbeassmallaspossible)Thepayoffsthusare6-εand4+εrespectively.Ifx4andy2,theequilibriumwouldbe(High,Low),withoptimalpayoffs(2+ε,9-ε),wherey=2+ε.Ifx4andy2,theequilibriumwouldbe(high,high),withpayoffs(6,9),wherex=4+ε,y=2+ε.(4points)Sothepayofftableforthefirststageis(2points),COLUMN2242,42+ε,9-εROW46-ε,4+ε6,93Eachplayerhasadominantstrategyof“4”and“2”.TheonlyNashequilibriumis(“4”,“2”),withpayoffs(6,9)(ornumbersverycloseto6and9).(2points)Yes,itsolvestheprisoners’dilemma.TheescrowaccountisusedtorewardonewhoplaysHigh.(2points)3(20points)AliandBabaarebargainingtosplitatotalthatstartsoutat$100.Alimakesthefirstoffer,statinghowthe$100willbedividedbetweenthem.IfBabaacceptsthisoffer,thegameisover.IfBabarejectsit,adollariswithdrawnfromthetotal,andsoitisnowonly$99.ThenBabagetsthesecondturntomakeanofferofadivision.Theturnsalternateinthisway,adollarbeingremovedfromthetotalaftereachrejection.Ali’sBATNAis$2.25andBaba’sBATNAis$3.50.Whatistherollbackequilibriumoutcomeofthegame?(Pleasewritedownyourprocess,otherwiseyoucanonlygethalfofthepoints.)Sincethesumoftwoparties’BATNAis2.25+3.5=5.75,isbiggerthan5andlessthan6.Thebargaininggamewillendatround95,wherev=6.Inround96,thebargainingwillbreakdownandeachtakeshisBATNA.Inround95,v=6.Amakestheofferandwillgive