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1、ECON205MicheleBiavatiAnswersforPS2Fall20131)PenaltyShotsRevisited.Player1hastotakeasoccerpenaltyshottodecidethegame.ShecanshootLeft,Middle,orRight.Player2isthegoalie.Hecandivetotheleft,middle,orright.Actionsarechosensimultaneously.Thepayoffs(whichherearetheprobabilitiesintenthsofw
2、inning)areasfollows.2lmr631L4794741M636136R974a)Foreachplayer,isanystrategydominatedbyanother(pure)strategy?No.b)Forwhatbeliefsaboutplayer1’sstrategyismabestresponseforplayer2?Forwhatbeliefsaboutplayer2’sstrategyisMabestresponseforplayer1?Ifplayer1plays‘L’,thebestresponseforplayer
3、2is‘l’.Ifplayer1plays‘M’,thebestresponseforplayer2is‘m’.Ifplayer1plays‘R’,thebestresponseforplayer2is‘r’.So‘m’isabestresponseto‘M’.Ifplayer2plays‘l’,thebestresponseforplayer1is‘R’.Ifplayer2plays‘m’,thebestresponseforplayer1iseither‘L’or‘R’.Ifplayer2plays‘r’,thebestresponseforplaye
4、r1is‘L’.So,‘M’isneverabestresponse.c)Supposeplayer2“putshimselfinplayer1’sshoes”andassumesthatplayer1,whateverisherbelief,willalwayschooseabest-responsetothatbelief.Shouldplayer2everchoosem?Player2willchoose‘m’onlyifhebelievesthatplayer1willplay‘M’.So,thequestionbasicallyasksifthe
5、rearebeliefsaboutplayer2’sstrategythatwouldmake‘M’abestresponseforplayer1.Thatis,ifweletpdenoteplayer2’sprobabilityofchoosing‘l’andqdenotehisprobabilityofchoosing‘r’,thendoesthereexistbelief(p,1–p–q,q)suchthat:u1(M;p,q)–u1(L;p,q)≥0(1)andu1(M;p,q)–u1(R;p,q)≥0(2)Ifsuchabeliefexisted
6、thenplayer2could“rationalize”choosing‘m’sinceifplayer1heldsuchabelief,‘M’wouldbeabestresponse.Thenitwouldberationalfor2tobelieveplayer1waschoosing‘M’whichinturnmakesitrationalfor2tochoose‘m’.Inordertosolvethis,weneedtowriteformulaeforu1(M;p,q),u1(L;p,q)andu1(R;p,q).u1(L;p,q)=4p+7(
7、1–p–q)+9q=-3p+2q+7u1(M;p,q)=6p+3(1–p–q)+6q=3p+3q+3u1(R;p,q)=9p+7(1–p–q)+4q=2p–3q+7So,u1(M;p,q)–u1(R;p,q)=3p+3q+3–(2p–3q+7)=p+6q–4≥0orp+6q≥4u1(M;p,q)–u1(L;p,q)=3p+3q+3–(-3p+2q+7)=6p+q–4≥0or6p+q≥4Summingyieldsp+q≥8/7,butweknowthatp+qcanatmostbeequalto1(why?).So,therearenobeliefsplay
8、er1couldholdaboutplayer2’sstrateg
