博弈论基础练习1

1.TheHawk-DoveGame.Thisproblemisbasedonanexampledevelopedbythebiologist

JohnMaynardSmithtoillustratetheusesofgametheoryinthetheoryofevolution.Malesofacertainspeciesfrequentlycomeintoconflictwithothermalesovertheopportunitytomatewithfemales.Ifamalerunsintoasituationofconflict,hehastwoalternative“strategies”.Amalescanplay“Hawk”inwhichcasehewillfighttheothermaleuntilheeitherwinsorisbadlyhurt.Orhecanplay“Dove”,inwhichcasehemakesadisplayofbraverybutretreatsifhisopponentstartstofight.IfananimalplaysHawkandmeetsanothermalewhoisplayingHawk,theybothareseriouslyinjuredinbattle.IfheisplayingHawkandmeetsananimalwhoisplayingDove,theHawkgetstomatewiththefemaleandtheDoveslinksofftocelibatecontemplation.IfananimalisplayingDoveandmeetsanotherDove,thebothstrutaroundforawhile.Eventuallythefemaleeitherchoosesoneofthemorgetsboredandwandersoff.Theexpectedpayoffstoeachoftwomalesinasingleencounterdependonwhichstrategyeachadopts.Thesepayoffsaredepictedintheboxbelow.

AnimalBHawkDove

AnimalAHawk−5;−510;0Dove0;104;4Nowwhilewanderingthroughtheforest,amalewillencountermanyconflictsituationsof

thistype.SupposethathecannottellinadvancewhetheranotheranimalthathemeetswillbehavelikeaHawkorlikeaDove.ThepayofftoadoptingeitherstrategyoneselfdependsontheproportionoftheotherguysthatisHawksandtheproportionthatisDoves.(a)Ifstrategiesthataremoreprofitabletendtobechosenoverstrategiesthatareless

profitable,explainwhytherecannotbeanequilibriuminwhichallmalesactlikeDovesorallactlikeHawks.FindtheNashequilibriaoftheHawk-DoveGameinpurestrategies.(b)SupposethatthefractionofalargemalepopulationthatchoosestheHawkstrategyis

p.ThenifoneactslikeaHawk,thefractionofones’sencountersinwhichhemeetsanotherHawkisaboutpandthefractionofone’sencountersinwhichhemeetsaDoveisabout1−p.

•FindtheNashequilibriaoftheHawk-DoveGameinmixedstrategies.

•Ifthemoreprofitablestrategytendstobeadoptedmorefrequentlyinfutureplays,thenifthestrategyproportionsareoutofequilibrium,willchangestendtomovetheproportionsbacktowardequilibriumorfurtherawayfromequilibrium?

2.Agroupofnstudentsgotoarestaurant.Itiscommonknowledgethateachstudentwill

simultaneouslychoosehis/herownmeal,butallstudentswillsharethetotalbillequally.Ifastudentgetsamealofpricepandcontributesxtowardspayingthebill,his/herpayoff

√

willbep−x.

(a)ComputetheNashequilibriumofthisgame.(b)Discussthelimitingcases:n=1andn→∞.

IntroductiontoGameTheory-WasedaUniversityR´obertVeszteg

Exercises1

3.Considerthefollowingextensiveformgame:

Player 1TBXPlayer 2(2;2)Player 2RRL(3;1)(0;0)(5;0)L(0;1)(a)Findtheplayers’strategysets.(b)Findthenormal-formrepresentationofthisgame.(c)Findallpure-strategyNashequilibriaofthisgame.(d)Whichoftheseequilibriaaresubgameperfect?

IntroductiontoGameTheory-WasedaUniversityR´obertVeszteg