Mixed Strategies混合的略

1、mixed strategiesmixed strategiesplayer 1headtailplayer 2head1, -1-1, 1tail-1, 11, -1mixed strategiesplayer 1headtailplayer 2head1, -1-1, 1tail-1, 11, -1mixed strategiesdefinition: a mixed strategy of a player in a simultaneous move game is a probability distribution over the players actions in match

2、ing pennies a mixed strategy will be ai = (ai(h), ai(t), where 0 ai(.) 1mixed strategiesplayer 1headtailplayer 2qhead1, -1-1, 11 - qtail-1, 11, -1where 0 q 1mixed strategiesplayer 1headtailplayer 2qhead1, -1-1, 11 - qtail-1, 11, -1expected payoff2q - 1 1 2qmixed strategiesplayer 21 2q1, -1-1, 1tail2

3、q - 1 -1, 11, -1head pplayer 1 1-pexpected payofftailhead1 - qqexpected 1 - 2p 2p-1 payoffmixed strategies 1 2q 2q 1 if and only if q player 1s best pure-strategy response is:- tail if q - indifferent between h and t if q = mixed strategiesplayer 1p1 pheadtailplayer 2qhead1, -1-1, 11 - qtail-1, 11,

4、-1where 0 p 1 mixed strategiese1(payoff) = pq*1 + p(1 - q)*(-1) + (1 p)q*(-1) + (1 p)(1 q) * 1= (1 2q) + p(4q 2)maximize e1(payoff) choosing p.if 4q 2 0 q 0 q p = 1 (head) is best responseif 4q 2 = 0 q = any p in 0, 1 is a best responsemixed strategiese2(payoff) = pq*(-1) + p(1 - q)*1 + (1 p)q*1 + (

5、1 p)(1 q) *(-1)= (2p - 1) + q(2 4p)maximize e2(payoff) choosing q.if 2 - 4p q = 0 (tail) is best responseif 2 - 4p 0 p q = 2/3americans expected payoff from entering and staying out must be the same:-50p +100(1-p) = 0p + 0(1-p) - p = 2/3symmetric expected payoffs are thus:-50(2/3)(2/3) +100(2/3)(1/3

6、) + 0(1/3)(2/3)+0(1/3)(1/3) = 0note that equalizing the conditional expected payoffs gives you the interior solution (if it exists) while maximizing the unconditional expected payoffs will give you all ne.all ne are thus (1,0),(0,1); (0,1),(1,0); (2/3,1/3),(2/3,1/3) mixed strategiesproposition: ever

7、y simultaneous-move game with vnm preferences and a finite number of players in which each player has finitely many actions has at least one nash equilibrium, possibly involving mixed strategies. mixed strategiesasymmetric game0, 00, 100stay out1 p150, 0-50, -50enterpunitedstay outenter1 - qqamerica

8、nasymmetric united/american solutionconsider the unconditional expected payoff of united:euu = -50pq + 150p(1-q) + 0(1-p)q + 0(1-p)(1-q) = -200pq + 150p = p(150-200q)so uniteds best response correspondence is:if 150-200q 0 q p=1.if 150-200q 0 q 3/4 = p=0.if 150-200q = 0 q = 3/4 = p 0,1.consider the

9、unconditional expected payoff of american:eua = -50pq + 100q(1-p) + 0(1-q)p + 0(1-p)(1-q) = -150pq + 100q = q(100-150p)so americans best response correspondence is:if 100-150p 0 p p=1.if 100-150p 0 p 2/3 = p=0.if 100-150p = 0 p = 2/3 = p 0,1.graph the br correspondences (in p,q space) to find all ne

10、. mixed strategiesasymmetric game pure-strategy nash equilibrium:(enter, stay out)(stay out, enter) mixed-strategy nash equilibrium: (au, aa) = (2/3,1/3), (3/4,1/4)mixed strategiesdefinition: in a strategic game with vnm preferences, player is mixed strategy ai strictly dominates her action ai ifui(

11、ai, a-i) ui(ai, a-i) for every a-i mixed strategies3, .0, .b0, .4, .m1, .1, .trldoes this game have any dominated pure strategies? no, but if the row player mixes equally between m and b, then if the column player plays l, row gets 4(1/2)+0(1/2) = 2 if she mixes while just 1 if she plays t. if colum

12、n plays r, row gets 0(1/2)+3(1/2) = 3/2 if she mixes, while again just 1 by playing t.thus t is strictly dominated by a mixed strategy.mixed strategies20, 1515, 103, 25b15, 10 10, 1010, 15m25, 320, 105, 5trclwhat are the ne (pure and mixed) of this game?method of finding all mixed-strategy nash equi

13、librium for each player i, choose a subset si of her set ai of actions. check whether there exists a mixed strategy profile a such that (1) the set of actions to which each strategy ai assigns positive probability is si and (2) a satisfies the conditions in proposition 116.2 in osborne. repeat the a

14、nalysis for every collection of subsets of the players sets of actionsmixed strategies1, 32, 40, 0s0,10, 04, 2bxsbmixed strategies potential types of equilibria: 1) player one plays 1 strategy, player two plays 1 strategy. these are pure strategy ne. 2) player one plays 1 strategy, player two plays 2 strategies. one plays a pure strategy, two mixes on bs, bx, or sx 3) player one plays 1 strategy, player two plays 3 strategies. one plays a pure strategy, two mixes on bsx 4) player one plays 2 strategies, player two plays 1 strategy. one mixes on