國(guó)外博弈論課件lecture(1)-展示頁(yè)
2024-10-28 13:57本頁(yè)面
【正文】 mixed strategies ((r*, 1r*), (q*, 1q*)) is a Nash equilibrium if and only if v1((r*, 1r*), (q*, 1q*)) ? EU1(s11, (q*, 1q*)) v1((r*, 1r*), (q*, 1q*)) ? EU1(s12, (q*, 1q*)) v2((r*, 1r*), (q*, 1q*)) ? EU2(s21, (r*, 1r*)) v2((r*, 1r*), (q*, 1q*)) ? EU2(s22, (r*, 1r*)) Player 2 s21 ( q ) s22 ( 1 q ) Player 1 s11 ( r ) u1(s11, s21), u2(s11, s21) u1(s11, s22), u2(s11, s22) s12 (1 r ) u1(s12, s21), u2(s12, s21) u1(s12, s22), u2(s12, s22) May 29, 2021 73347 Game TheoryLecture 8 12 Theorem 1: illustration ? Player 1: ? EU1(H, (, )) = (1) + 1=0 ? EU1(T, (, )) = 1 + (1)=0 ? v1((, ), (, ))=?0+?0=0 ? Player 2: ? EU2(H, (, )) = 1+ (1) =0 ? EU2(T, (, )) = (1)+ 1 = 0 ? v2((, ), (, ))= 0+ 0=0 Matching pennies Player 2 H () T () Player 1 H () 1 , 1 1 , 1 T () 1 , 1 1 , 1 May 29, 2021 73347 Game TheoryLecture 8 13 Theorem 1: illustration ? Player 1: ? v1((, ), (, )) ? EU1(H, (, )) ? v1((, ), (, )) ? EU1(T, (, )) ? Player 2: ? v2((, ), (, )) ? EU2(H, (, )) ? v2((, ), (, )) ? EU2(T, (, )) ? Hence, ((, ), (, )) is a mixed strategy Nash equilibrium by Theorem 1. Matching pennies Player 2 H () T () Player 1 H () 1 , 1 1 , 1 T () 1 , 1 1 , 1 May 29, 2021 73347 Game TheoryLecture 8 14 ? Employee’s expected payoff of playing “work” ? EU1(Work, (, )) = 50 + 50=50 ? Employee’s expected payoff of playing “shirk” ? EU1(Shirk, (, )) = 0 + 100=50 ? Employee’s expected payoff of her mixed strategy ? v1((, ), (, ))=?50+?50=50 Theorem 1: illustration Employee Monitoring Manager Monitor () Not Monitor () Employee Work () 50 , 90 50 , 100 Shirk () 0 , 10 100 , 100 May 29, 2021 73347 Game TheoryLecture 8 15 ? Manager’s expected payoff of playing “Monitor” ? EU2(Monitor, (, )) = 90+ (10) =80 ? Manager’s expected payoff of playing “Not” ? EU2(Not, (, )) = 100+ (100) = 80 ? Manager’s expected payoff of her mixed strategy ? v2((, ), (, ))= 80+ 80=80 Theorem 1: illustration Employee Monitoring Manager Monitor () Not Monitor () Employee Work () 50 , 90 50 , 100 Shirk () 0 , 10 100 , 100 May 29, 2021 73347 Game TheoryLecture 8 16 ? Employee ? v1((, ), (, )) ? EU1(Work, (, )) ? v1((, ), (, )) ? EU1(Shirk, (,
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