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Lect07and08_Slides

# Lect07and08_Slides - Mixed MixedStrategies Keepem guessing...

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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 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 strategies, then there exists at least one equilibrium in (possibly) mixed strategies. (Nash proved this). If there are nopure 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 Chicken game.
Example 1: Tennis Let p be the probability that Serena chooses DL, so 1 p is the probability that she chooses CC. Let q be the probability that Venus positions herself for DL so 1 q is Let q be the probability that Venus positions herself for DL, so 1 q is the probability that she positions herself for CC. To find mixed strategies, we add the p mix and q mix options. 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) i 50 90(1 ) 80 20(1 ) 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 pure strategy. This requires understanding how your opponent’s payoff varies with your choice of p . Graphically, in the Tennis example: For Serena’s choice of p, Venus’s expected payoff from playing DL is: Venus i d 50p+10(1 p) and from playing CC is: is made indifferent if Serena chooses 20p+80(1 p) p=.70
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 0/100 0 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 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 Column Player s Optimal Choice of Choose q so as to equalize the payoff your opponent receives from playing either pure strategy. h d d h ff h 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 Serena s expected payoff from playing DL is: Serena is made 50q+80(1 q) and from playing CC is: 90q+20(1 q) indifferent if Venus chooses 60 q=.60
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

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Lect07and08_Slides - Mixed MixedStrategies Keepem guessing...

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