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（b）Thereisoneinitialnode（I）forHanselmakingthefirstmove;

fourdecisionnodes（D）includingtheinitialnode,whichrepresentthenodeswhereHanselorGretelmakeadecision;

andnineterminalnodes（T）.

（c）Thereisoneinitialnode（I）forHanselmakingthefirstmove;

fivedecisionnodes（D）includingtheinitialnode,whichrepresentthenodeswhereHanselorGretelmakeadecision;

andeightterminalnodes（T）.

S2.Forthisquestion,rememberthatactionswiththesamelabel,iftakenatdifferentnodes,aredifferentcomponentsofastrategy.Toclarifytheanswers,thenodesonthetreesarelabeled1,2,etc.（inadditiontoshowingthenameoftheplayeractingthere）.ActionsinastrategyaredesignatedasN1（meaningNatnode1）,etc.ThetreesarebelowinthesolutionstoExerciseS3.Numberingofnodesbeginsatthefarleftandproceedstotheright,withnodesequidistanttotherightoftheinitialnodeandnumberedfromtoptobottom.

（a）Scarecrowhastwostrategies:

（1）Nor

（2）S.Tinmanhastwostrategies:

（1）tifScarecrowplaysN,or

（2）bifScarecrowplaysN.

（b）Scarecrowhastwoactionsatthreedifferentnodes,soScarecrowhaseightstrategies,2•2•2=8.Todescribethestrategiesaccurately,wemustspecifyaplayer’sactionateachdecisionnode.Scarecrowdecidesatnodes1,3,and5,sowewilllabelastrategybylistingtheactionandthenodenumber.Forexample,todescribeScarecrowchoosingNateachnode,wewrite（N1,N3,N5）.Accordingly,theeightstrategiesforScarecroware:

（N1,N3,N5）,（N1,N3,S5）,（N1,S3,N5）,（S1,N3,N5）,（N1,S3,S5）,（S1,N3,S5）,（S1,S3,N5）,and（S1,S3,S5）.

Tinmanhastwoactionsatthreedifferentnodes,soTinmanalsohaseightstrategies,2•2•2=8.Tinman’sstrategiesare:

（n2,n4,n6）,（n2,n4,s6）,（n2,s4,n6）,（s2,n4,n6）,（n2,s4,s6）,（s2,n4,s6）,（s2,s4,n6）,and（s2,s4,s6）.

（c）Scarecrowhastwoactionsatthreedecisionnodes,soScarecrowhaseightstrategies:

2•2•2=8.Scarecrow’sstrategiesare:

（N1,N4,N5）,（N1,N4,S5）,（N1,S4,N5）,（S1,N4,N5）,（N1,S4,S5）,（S1,N4,S5）,（S1,S4,N5）,and（S1,S4,S5）.Tinmanhastwostrategies:

（t2）and（b2）.Lionhastwostrategies:

（u2）and（d2）.

S3.（a）BeginningwithTinman,weseethatTinmanprefersapayoffof2over1,soTinmanchoosest.WithTinmanchoosingt,Scarecrowreceivesapayoffof0forNand1forS,soScarecrowchoosesS.Thus,therollbackequilibriumisScarecrow’schoosingSandTinman’schoosingt（eventhoughhewon’thaveachancetoplayit）.Tinman’sactiondoesnotaffecttherollbackequilibrium,becauseScarecrowexpectsTinmantochooset,soScarecrowbestrespondsbychoosingS.

（b）ThegraphbelowindicateswhichactionScarecrowandTinmanchooseateachnode.Scarecrow’sequilibriumstrategyisS1,S3,N5,andTinman’sisn2,n4,s6.yieldingtheequilibriumpayoff（4,5）.

（c）ThegraphbelowindicateswhichactionScarecrow,Tinman,andLionchooseateachnode.Scarecrow’sequilibriumstrategyisN1,N4,N5;

Tinman’sisb;

andLion’sisd,yieldingthepayoff（2,3,2）.

S4.Thegametreeisshownbelow.

Boeingprefers$300milliontolosing$100million,soBoeingwillpeacefullyaccommodateAirbus’sentryintothemarket.AirbusexpectsBoeingtoaccommodateitsentrypeacefully,soitcanmake$300millionbyentering,ornothingbynotentering,soAirbuswillenterthemarket.Thus,therollbackequilibriumisAirbus’senteringthemarketandBoeing’speacefullyaccommodating,withapayofftoeachfirmof$300millioninprofit.

S5.（a）ForBarneytowinthegame,hemustremovethelastmatchstick,whichmeansthatifheleavesFred1to4matches,FredcanremoveallofthemandBarneywouldlose,soBarneymustleavemorethan4matchsticks.Becausethereare6matchsticksandBarneymusttakeatleast1,Barneyshouldremoveonly1matchstick,whichwillleaveFredwith5matchsticks.NomatterwhatFreddoes,Barney,onhisnextturn,willbeabletoremovealltheremainingmatchstickstowinthegame.Moreprecisely,Barneyshouldtake1matchstickonhisfirstturn.IfFredtakesfmatchsticks,Barneyshouldtake（5–f）matchsticks.

（b）Frompart（a）,weknowthatwhomeverisleftwith5matchstickswilllosethegame,soBarneyshouldremoveenoughmatchstickstoleaveFredonly5.IfBarneyleavesFred6to9matches,thenFredwillleaveBarneywith5,andBarneywilllose,soBarneymustleaveFredwithmorethan9matches.Also,ifBarneyleaves11matches,Fredcanensureheisleftwith6to9matchesbychoosingonly1match,leavingBarneywith10matchesandnowaytokeepFredfromhaving6to9matches.Thus,Barneymusttake2matches,leavingFredwith10,andmustchoose（5–f）matchesoneachsubsequentturn.

Anotherwaytoviewthisproblemisthataplayerwillloseifhisturnbeginswith5matches.Thus,eachplayeralwayswantstoremovematchstickstoleavehisopponentwith5.Sinceweknowwhathappenswhen5matchsticksremain,wecandividethenumberofremainingmatchsticksintounitsof5.Forexample,12matchstickscanbedividedintotwounitsof5with2extra.BarneywantstoforceFredtohavesomemultipleof5,soBarneyremoves2matchesatfirst,andthen（5–f）matchesineachofhissubsequentturns.

（c）Thefullgamehas21matchsticks,andFredbegins.Asdescribedin（b）,FredwantstoleaveBarneywithsomemultipleof5,and21is4unitsof5with1extra.SoFredshouldremove1matchstickonhisfirstturn,andthen（5–b）matchsticksonhissubsequentturns,wherebisthenumberofsticksthatBarneyhasjustremoved.Withoptimalplay,Fredwillwineverytime.

（d）Eachplayerwantstoleavetheotherplayeramultipleof5matchsticks.Sooneachturn,theplayershoulddividetheremainingmatchsticksby5andremovetheremainder.Iftheremainderis0andmorethan4matchsticksremain,thentheplayerisstuckwithamultipleof5.Sothatplayershouldrandomlychoose1to4matchsticks,hopingthattheopponentwillmakeamistakeonasubsequentturn.If4orfewermatchsticksremain,thentheplayershouldremoveallofthemtowinthegame.

S6.（a）Thegametreeis:

（b）Thegraphinpart（a）indicatesthefourrollbackequilibria,whichcanbedescribedasFred’staking1to4matchsticks,andthenBarney’sremovingallremainingmatchsticks.LettingthefirstnumberrepresentthenumberofmatchsticksremovedbyFredandthesecondbyBarney,thefourrollbackequilibriamaybedescribedas:

（1,4）,（2,3）,（3,2）,and（4,1）.

（c）With5matchsticksatthebeginningofthegame,thereisasecond-moveradvantage,becausenomatterwhatquantitythefirstmoverremoves,thesecondmovercanremoveallremainingmatchstickstowinthegame.

（d）Thereismorethanonerollbackequilibrium,becausesolongasBarneyplaysoptimally,anyofFred’sfouractionsattheinitialnodeleadstothesamepayoff.Thusinequilibrium,heisindifferentamongthosefouractionsatthatnode.

S7.（a）Thegametreeisshownbelow.

（b）TherollbackequilibriumisLion1eatstheslave,Lion2doesnoteatLion1,andLion3wouldeatLion2ifgiventheopportunity（whichheisnotinequilibrium）.

（c）Thereisafirst-moveradvantage,becauseLion3willalwayseatLion2ifable,soLion2hasanincentivetonoteatLion1inordertoprotecthimselffromLion3.

（d）Eachlionhastwoactionsatasinglenode（eat,don’teat）,soeachhastwocompletestrategies.

S8.（a）Thegametreeisgivenbelow.

（b）Rollbackpruningisillustratedbyarrowsonbranchesofthetree.TheequilibriumentailsFrieda’schoosingRural;

BigGiant’salwayschoosingUrban（UR,orUifUandUifR）;

andTitan’schoosingUrbanunlessbothFrieda’sandTitanhavechosenRural（UUUR）.Theequilibriumpayoffsare（2,5,5）tothestoresinorderoftheirmoves.

S9.（a）Thegametreeisshownbelow.

（b）TheProposerhasonenodewith11actions;

thustheProposerhas11completestrategies.Wecanlisttheseasthesplitproposed,withthefirstnumberindicatingtheportionfortheProposerandthesecondfortheResponder.The11completestrategiesare:

0/10,1/9,2/8,3/7,4/6,5/5,6/4,7/3,8/2,9/1,and10/0.

TheResponderhas11nodeswithactionsAcceptorRejectateachnode;

thustheResponderhas211,or2,048completestrategies.Someexamplesofpossiblestrategiesincludeacceptingonly5/5,acceptingonly10,acceptingonlyoddnumbers,andrejectingalloffers.

（c）AssumingthattheplayerscareonlyabouttheircashpayoffsmeansthattheResponderwilldefinitelyacceptanypositiveofferandwillbeindifferentbetweenAcceptingandRejectingwhenofferednothing.IftheProposerassumesthattheResponderwillacceptanofferof$0whenindifferent,thentherollbackequilibriumistooffer$0andforittobeaccepted.However,althoughtheProposermaybeunsureoftheResponder’sactioninthefaceofindifference,hecanexpecttheRespondertoacceptanofferof$1.IfthereisuncertaintyabouttheResponder’sactionwhenshe’sindifferent,therollbackequilibriumoccurswheretheProposeroffers$1andthatofferisaccepted.

（d）BecausePeteknowsRachelwillacceptanyofferof$3ormore,Petecanmaximizehispayoffbyofferingonly$3.

（e）TherearemanypossibleutilitiesthatmayrepresentRachel’sutility.Onecommonutilityisfairness,inwhichRachelreceivesautilityequaltothedollaramountiftheofferiswithin40%to60%ofthetotalam