國外博弈論課件lecture(2)-文庫吧資料
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【正文】 Lecture 7 7 Solving matching pennies ? Player 2’s best response B2(r): ? Tail (q=0) if r ? Head (q=1) if r ? Any mixed strategy (0?q?1) if r= Player 2 Head Tail Player 1 Head 1 , 1 1 , 1 Tail 1 , 1 1 , 1 q 1q 12q 2q1 Expected payoffs r 1r Expected payoffs 2r1 12r 1 q r 1 1/2 1/2 May 28, 2021 73347 Game TheoryLecture 7 8 1 q r 1 1/2 1/2 Solving matching pennies ? Player 1’s best response B1(q): ? Head (r=1) if q ? Tail (r=0) if q ? Any mixed strategy (0?r?1) if q= ? Player 2’s best response B2(r): ? Tail (q=0) if r ? Head (q=1) if r ? Any mixed strategy (0?q?1) if r= ? Check r = is best response to q= q = is best response to r= Player 2 Head Tail Player 1 Head 1 , 1 1 , 1 Tail 1 , 1 1 , 1 r 1r q 1q Mixed strategy Nash equilibrium May 28, 2021 73347 Game TheoryLecture 7 9 Mixed strategy ? Mixed Strategy: ? A mixed strategy of a player is a probability distribution over player’s (pure) strategies. Definition Let G be a n playe r gam e with strateg y sets S 1 , S 2 ,.., S n . A m ixed str ategy i? f or player i is a proba bility dist r ibution on S i . If S i has a finite n u m ber of p u re strat eg ies, . } . . . , ,{21 iiKiii sssS ? then a m ixed strat eg y is a functi on ???ii S:? such that 1)(1???iKjiji s?. W e write this m ixed str a tegy as ))( ..., ),( ),((21 iiKiiiii sss ???. May 28, 2021 73347 Game TheoryLecture 7 10 Mixed strategy: example ? Matching pennies ? Player 1 has two pure strategies: H and T ( ?1(H)=, ?1(T)= ) is a Mixed strategy. That is, player 1 plays H and T with probabilities and , respectively. ( ?1(H)=, ?1(T)= ) is another Mixed strategy. That is, player 1 plays H and T with probabilities and , respectively. May 28, 2021 73347 Game TheoryLecture 7 11 Mixed strategy: example ? Player 1: ? (3/4, 0, 188。) is a mixed strategy. That is, ?1(T)=3/4, ?1(M)=0 and ?1(B)=1/4. ? Play
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