研究生课程 博弈论 英文课件3

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1、MicroeconomicAnalysisLectureintheGraduatePrograminInternationalEconomicsandFinance———–OTTO-VON-GUERICKE-UNIVERSITYMAGDEBURGPart3Dr.MagnusHoffmannmagnus.hoffmann@ovgu.de1/24Chapter2.2-Two-PlayerZero-SumGamesDefinition2.1(Two-playerzero-sumgames)Two-playerzero-sumgamesisaspecialclas

2、sofnormalformgames,satisfyingu2(a1,a2)=−u1(a1,a2),(2.1)∀a1∈A1and∀a2∈A2.AnexampleforthiskindofgameisthefollowingExample2.1712a12a22a32a110,01,−17,−7a214,−42,−23,−3a319,−90,01,−1A1={a11,a21,a31},A2={a12,a22,a32}2/24IFirstargumentofplayer1:Player2movessecond1.Sinceplayer2isintelli

3、gent,hewillpredictanyactiona1∈A1thatplayer1maychoose.2.Sinceplayer2isrational,hewillchoosetheactiona2thatmaximizeshispayoff(or,equivalently,minimizesthepayoffofplayer2).P1:a1⇒P2:a1⇒u=0121minP1:a2⇒P2:a2⇒u=2121minP1:a3⇒P2:a2⇒u=0121min3.Sinceplayer1isintelligent,hewillpredictthereact

4、ionofplayer2andfigureouttheexpectedpayoffm1(a1)fromplayinga1:m1(a1)=minu1(a1,a2),a2∈A2={u(a1,a1),u(a2,a2),u(a3,a2)}112112112∀a1∈A1.3/2412a12a22a32m1(a1)a110,01,−17,−70a214,−42,−23,−32a319,−90,01,−104.Sinceplayer1isrational,hewillchoosetheactiona1thatmaximizesthepayoffm1(a1).5.Ther

5、efore,theexpectedpayoffofplayer1willbe·¸m1=maxminu1(a1,a2)a1∈A1a2∈A2©¡¢¡¢¡¢ªm=maxua1,a1,ua2,a2,ua3,a21112112112a1∈A1¡¢=ua2,a2112=2.4/24IFirstargumentofplayer2:Player1movessecond1.Sinceplayer1isintelligent,hewillpredictanyactiona2∈A2thatplayer2maychoose.2.Sinceplayer1isrational,he

6、willchoosetheactiona1thatmaximizeshispayoff(or,equivalently,minimizesthepayoffofplayer2).P2:a1⇒P1:a3⇒u=−9212minP2:a2⇒P1:a2⇒u=−2212minP2:a3⇒P1:a1⇒u=−7212min3.Sinceplayer2isintelligent,hewillpredictthereactionofplayer1andfigureouttheexpectedpayoffm2(a2)fromplayinga2:m2(a2)=minu1(a1,a2

7、)a1∈A1=min{u(a3,a1),u(a2,a2),u(a1,a3)},212212212a1∈A1∀a1∈A1.5/2412a12a22a32a110,01,−17,−7a214,−42,−23,−3a319,−90,01,−1m2(a2)−9−2−74.Sinceplayer2isrational,hewillchoosetheactiona2thatmaximizesthepayoffm2(a2).5.Therefore,theexpectedpayoffofplayer2willbe·¸m2=maxminu2(a1,a2)a2∈A2a1∈A

8、1©ª=maxu(a3,a1),u(a2,a2),u(a1,a3)212212212a2∈A2