國外博弈論課件lecture(1)-在線瀏覽
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【正文】 )) ? Manager ? v2((, ), (, )) ? EU2(Monitor, (, )) ? v2((, ), (, )) ? EU2(Not, (, )) ? Hence, ((, ), (, )) is a mixed strategy Nash equilibrium by Theorem 1. Theorem 1: illustration Employee Monitoring Manager Monitor () No Monitor () Employee Work () 50 , 90 50 , 100 Shirk () 0 , 10 100 , 100 May 29, 2021 73347 Game TheoryLecture 8 17 ? Use Theorem 1 to check whether ((2/3, 1/3), (1/3, 2/3)) is a mixed strategy Nash equilibrium. Theorem 1: illustration Battle of sexes Pat Opera (1/3) Prize Fight (2/3) Chris Opera (2/3 ) 2 , 1 0 , 0 Prize Fight (1/3) 0 , 0 1 , 2 May 29, 2021 73347 Game TheoryLecture 8 18 Mixed strategy equilibrium: 2player each with two strategies ? Theorem 2 Let ((r*, 1r*), (q*, 1q*)) be a pair of mixed strategies, where 0 r*1, 0q*1. Then ((r*, 1r*), (q*, 1q*)) is a mixed strategy Nash equilibrium if and only if EU1(s11, (q*, 1q*)) = EU1(s12, (q*, 1q*)) EU2(s21, (r*, 1r*)) = EU2(s22, (r*, 1r*)) ? That is, each player is indifferent between her two strategies. 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 19 Use indifference to find mixed Nash equilibrium (2player each with 2 strategies) ? Use Theorem 2 to find mixed strategy Nash equilibria ? Solve EU1(s11, (q*, 1q*)) = EU1(s12, (q*, 1q*)) ? Solve EU2(s21, (r*, 1r*)) = EU2(s22, (r*, 1r*)) May 29, 2021 73347 Game TheoryLecture 8 20 Use Theorem 2 to find mixed strategy Nash equilibrium: illustration ? Player 1 is indifferent between playing Head and Tail. ? EU1(H, (q, 1–q)) = q (1) + (1–q) 1=1–2q ? EU1(T, (q, 1–q)) = q 1 + (1–q) (1)=2q–1 ? EU1(H, (q, 1–q)) = EU1(T, (q, 1–q)) 1–2q = 2q–1 4q = 2 This give us q = 1/2 Matching pennies Player 2 H ( q ) T ( 1–q ) Player 1 H ( r ) 1 , 1 1 , 1 T ( 1–r ) 1 , 1 1 , 1 May 29, 2021 73347 Game TheoryLecture 8 21 Use Theorem 2 to find mixed strategy Nash equilibrium: illustration ? Player 2 is indifferent between playing Head and Tail. ? EU2(H, (r, 1–r)) = r 1+(1–r) (1) =2r – 1 ? EU2(T, (r, 1–r)) = r (
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