Lect07_Slides

Lect07_Slides - Mixed Strategies Keep `em guessing Mixed Strategy Nash Equilibrium A mixed strategy is one in which a player plays his available

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Mixed Strategies Keep ‘em guessing
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Mixed Strategy Nash Equilibrium A mixed strategy is one in which a player plays his available pure strategies with certain probabilities. Mixed strategies are best understood in the context of repeated games, where each player’s aim is to keep the other player(s) guessing, for example: Rock, Scissors Paper. If each player in an n-player game has a finite number of pure strategies, then there exists at least one equilibrium in (possibly) mixed strategies. (Nash proved this). If there are no pure strategy equilibria, there must be a unique mixed strategy equilibrium. However, it is possible for pure strategy and mixed strategy Nash equilibria to coexist, as in the Stag Hunt and Chicken games.
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Example 1: Tennis Let p be the probability that Serena chooses DL, so that 1-p is the probability she chooses CC. Let q be the probability that Venus positions herself for DL, so that 1-q is the probability she positions herself for CC. To find mixed strategies, we add the p-mix and q-mix strategies to the payoff matrix. Venus Williams DL CC q-mix Serena Williams DL 50 , 50 80 , 20 50q+80(1-q) 50q+20(1-q) CC 90 , 10 20 , 80 90q+20(1-q) 10q+80(1-q) p-mix 50p+90(1-p) 50p+10(1-p) 80p+20(1-p) 20p+80(1-p)
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Row Player’s Optimal Choice of p Chose p so as to equalize the payoff your opponent receives from playing either of her pure strategies. This requires understanding how your opponent’s payoff varies with your own choice of p . Graphically, in the Tennis example: For Serena’s choice of p, Venus’s expected payoff from playing DL is: 50p+10(1-p) and from playing CC is: 20p+80(1-p) Venus is made indifferent if Serena chooses p=.70
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Algebraically Serena solves for the value of p that equates Venus’s payoff from positioning herself for DL or CC: 50p+10(1-p) = 20p+80(1-p), or 50p+10-10p = 20p+80-80p, or 40p+10 = 80-60p, or 100p = 70, so p = 70/100 = .70. If Serena plays DL with probability p=.70 and CC with probability 1-p=.30, then Venus’s success rate from DL=50(.70)+10(.30)=38%=Venus’s success rate from CC=20(.70)+80(.30)=38%. Since this is a constant sum game, Serena’s success rate is 100%- Venus’s success rate = 100-38=62%.
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Column Player’s Optimal Choice of q Choose q so as to equalize the payoff your opponent receives from playing either pure strategy. This requires understanding how your opponent’s payoff varies with your choice of q . Graphically, in our example: For Venus’s choice of q, Serena’s expected payoff from playing DL is: 50q+80(1-q) and from playing CC is: 90q+20(1-q) Serena is made indifferent if Venus chooses q=.60
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Algebraically Venus solves for the value of q that equates Serena’s payoff from playing DL or CC: 50q+80(1-q) = 90q+20(1-q), or 50q+80-80q = 90q+20-20q, or 80-30q = 70q+20, or 60 = 100q, so q = 60/100 = .60. If Venus positions herself for DL with probability q=.60
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This note was uploaded on 03/04/2012 for the course ECON 1200 taught by Professor Staff during the Fall '08 term at Pittsburgh.

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Lect07_Slides - Mixed Strategies Keep `em guessing Mixed Strategy Nash Equilibrium A mixed strategy is one in which a player plays his available

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