微觀經(jīng)濟(jì)學(xué)（趙耀輝）第28章(文件)
【正文】 3 220 34? ? ? ? ? ? ? ? .(0,4) U, D, L, R, 43 415352(1,2) 9/20 3/20 (0,5) (3,2) 6/20 2/20 Mixed Strategies Player B Player A A’s expected Nash equilibrium payoff is 1 920 0 320 0 620 3 220 34? ? ? ? ? ? ? ? .B’s expected Nash equilibrium payoff is 2 920 4 320 5 620 2 220 165? ? ? ? ? ? ? ? .(0,4) U, D, L, R, 43 415352(1,2) 9/20 3/20 (0,5) (3,2) 6/20 2/20 Mixed Strategies ?For games with multiple pure strategies, there also exists mixed strategies. ?Example: Chicken game. The probability that each player plays straight is 189。 . p pL L? ?3 1( )(1,2) (0,4) (0,5) (3,2) L,pL R,1pL U, D, 5352Player B Mixed Strategies Player A So for there to exist a Nash equilibrium, A must be indifferent between playing Up or Down。 that is, as playing purely one or the other. ? The strategies presented so far are players’ pure strategies （ 純粹策略） . ? Consequently, equilibria are pure strategy Nash equilibria. ? Must every game have at least one pure strategy Nash equilibrium? Pure Strategies Player B Player A Here is a new game. Are there any pure strategy Nash equilibria? (1,2) (0,4) (0,5) (3,2) U D L R Pure Strategies Player B Player A Is (U,L) a Nash equilibrium? (1,2) (0,4) (0,5) (3,2) U D L R Pure Strategies Player B Player A Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? (1,2) (0,4) (0,5) (3,2) U D L R Pure Strategies Player B Player A Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? No. Is (D,L) a Nash equilibrium? (1,2) (0,4) (0,5) (3,2) U D L R Pure Strategies Player B Player A Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? No. Is (D,L) a Nash equilibrium? No. Is (D,R) a Nash equilibrium? (1,2) (0,4) (0,5) (3,2) U D L R Pure Strategies Player B Player A Is (U,L) a Nash equilibrium? No. Is (U,R) a Nash equilibrium? No. Is (D,L) a Nash equilibrium? No. Is (D,R) a Nash equilibrium? No. (1,2) (0,4) (0,5) (3,2) U D L R More Examples Matching Pennies Player B Player A (1,1) (1, 1) (1,1) (1, 1) H T H T More Examples 點(diǎn)球 進(jìn)攻球員 守門員 (1,0) (0, 1) (0,1) (1, 0) 左 右 左 右 Pure Strategies Player B Player A So the game has no Nash equilibria in pure strategies. Even so, the game does have a Nash equilibrium, but in mixed strategies (混合策略 ). (1,2) (0,4) (0,5) (3,2) U D L R Mixed Strategies ?Instead of playing purely Up or Down, Player A selects a probability distribution (pU,1pU), meaning that with probability pU Player A will play Up and with probability 1pU will play Down. ?Player A is mixing over the pure strategies Up and Down. ?The probability distribution (pU,1pU) is a mixed strategy for Player A. Mixed Strategies ?Similarly, Player B selects a probability distribution (pL,1pL), meaning that with probability pL Player B will play Left and with probability 1pL will play Right. ?Player B is mixing over the pure strategies Left and Right. ?The probability distribution (pL,1pL) is a mixed strategy for Player B. Mixed Strategies Player A This game has no pure strategy Nash equilibria but it
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