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ProblemSet10SolutionEcon159a/MGT522a,YaleUniversityM.Chenmomotocmx@hotmail.com1.Usingdi erentequilibriatocreateincentives.(a)A:$10,B:$0,revenue=$15-10,0 -10,0 -7 , 12 -10,0 -2.5 , 7.5 -10,0 2 , 3 -10,0 -10,0 1/5 1/2 4/5 1/5 1/2 4/5 Thepure-NEsare(15;45);(12;12);(45;15) .(b)Thenormalmatrixformofthisgameisshownasfollows.0,0 0 ,0 0 ,-10 0,-10 0,-10 3 , 2 0, 0 0 , 0 0 , 0 0,-10 7.5 ,-2.5 0,-10 0, 0 0 , 0 0 , 0 12,-7 0,-10 0,-10 -10,0 -10,0 -7,12 -10,-10 -10,-10 -4, 14 -10,0 -2.5, 7.5 -10,0 -10,-10 5,5 -10,-10 2 , 3 -10,0 -10,0 14 ,-4 -10,-10 -10,-10 $0,1/5 $0,1/2 $0,4/5 $10,1/5 $10,1/2 $10,4/5 $0,1/5 $0,1/2 $0,4/5 $10,1/5 $10,1/2 $10,4/5 A B ItcanbeseenthatthereexistSPEsinwhicheachplayerstartsbyinvesting$0.Theyare[($0;12);($0;12)],[($0;12);($0;45)],[($0;45);($0;12)]and[($0;45);($0;45)].12(c)ThereisnoSPEinwhicheachplayerstartbyinvesting$10.(d)2.A nitelyrepeatedgame.3 , 1 0,0 0,0 5 ,0 0,0 1 , 3 0,0 0,0 0,0 0,0 2 , 2 0,0 0,0 0, 5 0,0 4,4 a b c d A B C D (a)Thepure-NEsofthisgameare(A;a),(B;b)and(C;c).(b)Play(D;d),thenplay(C;c)if(D;d)wasplayed.(reward)play(B;b)ifplayer1deviatedfromD.(Punishmentforplayer1)play(A;a)ifplayer2deviatedfromd.(Punishmentforplayer2)Validation:If(D;d)wasplayedinthe rststage,play(C;c)inthesecondstage.HenceThepayo forbothplayersis4+2=6.Ifplayer1defects,thenthemaximumpayo player1cangetisu1(A;d)+u1(B;b)=6.Therefore,thereisnoincentiveforplayer1todefect.Similarly,itcanbearguedthatplayer2willnotdefect.Alternatively,thetemptation(theadditionalpayo onecanget)forplayer1todefectfromDtoCisu1(A;D) u(D;d)=5 4=1.However,player1willpaythepriceofu1(C;c) u1(B;b)=2 1=1forhisdefect.Sincethetemptationisnotgreaterthanthepriceforthetemptation,player1willnotdefect.3.In nitelyrepeatedgames:anapplication.HeatedCompetition.(a)Thesimpli edpayo matrixisasfollows.Negativesignmeansthenegativepayo whileplussignmeansthepositivepayo .dpistheminimumpricechange,e.g.dp=0:1meanspricehasastepof0:1anditcannotchangefromlike1to1:05.30 ... p p+dp ... 1 ... p p+dp ... 2 ... - - - - - - - - - - - - 0 0 - - - - - - - - - - 0 0 0 - - - - - - - - - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + +(m) +(k) + + 0 0 0 0 0 0 0 0 +(n) +(q) + + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 p=0 0p1 0p+dp1 p=1 1p2 1p+dp2 p=2 p2 Wecanseefromthematrixthatpricep=0isstrictlydominatedbyp=1.Priceintherange(0;1)isweaklydominatedbyp=1.Inthe gureabove,mnandkqwhenp2[1:5;2)(notethat(p 1)Q(p)achieveitsmaximumatp=1:5).Hence,foranypricepintherange(1:5;2),itisweaklydominatedbystrategyp=1:5.TherearetwoNEsinthisgame,(ph;pw)=(1;1)and(ph;pw)=(1+dp;1+dp).(b)4.In nitelyrepeatedgames:practiceatthetheory.(a)Player1:temptationtoday[(Valueofpromisetomorrow) (Valueofdefecttomorrow)]3 2(2+2+22+) 0=21 )13Player2:temptationtoday[(Valueofpromisetomorrow) (Valueofdefecttomorrow))]4 1(1++2+) 0=1 )34Theminimumvalueofis34.(b)4(c)Player1:temptationtoday[(Valueofpromisetomorrow) (Valueofdefecttomorrow)]3 2h(Valueof2forever) (valueof0fromtomorrowforTrperiodsthen2foreverfromperiodTr+1)i3 221 (0+0+02++0Tr 1+2Tr+2Tr+1+)121 2Tr1 )Trlog 12(1)Player2:temptationtoday[(Valueofpromisetomorrow) (Valueofdefecttomorrow)]4 111 Tc1 )Tclog4 3(2)From(1)and(2),wehaveTr;min=log 12Tc;min=log4 3Foranygiven,itcanbeshownthatTr;minTc;min.Sincecolumnplayerhasalargertemptationtodefect,punishmentperiodhastobelongerinordertoachieveanSPE.