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Strategy13Handout

# Strategy13Handout - Mixed Strategies Matching Pennies We...

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10/16/2008 1 Mixed Strategies – Matching Pennies ± We denote the strategies on the game bi-matrix Guildenstern q1 - q Heads Tails ± Rosencrantz’s payoff to H : (1)q + (-1)(1-q) = 2q-1 T : (-1)q + (1)(1-q) = 1-2q Rosencrant zp Heads 1,-1 -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Matching Pennies ± Rosencrantz’s & Guildenstern’s best responses: q 1 The best response curves intersect at the Nash p 01 1/2 1/2 equilibrium p=1/2, q=1/2 Mixed Strategies – Matching Pennies ± Suppose R’s payoff to H increases Guildenstern - q Heads Tails ± Rosencrantz’s payoff to H : (2)q + (-1)(1-q) = 3q-1 T : (-1)q + (1)(1-q) = 1-2q ± Now, 3q-1 = 1-2q requires q=2/5 Rosencrant Heads 2,-1 -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Matching Pennies ± Rosencrantz’s & Guildenstern’s best responses: q 1 The best response curves intersect at the Nash p 1/2 1/2 the Nash equilibrium p=1/2, q=2/5 2/5 Mixed Strategies – Matching Pennies ± Suppose R’s payoff to H increases Rt Guildenstern - q Heads Tails ± Rosencrantz’s payoff to H : (2)q + (-1)(1-q) = 3q-1 T : (-1)q + (1)(1-q) = 1-2q ± Now, 3q-1 = 1-2q requires q=2/5 Rosencrant Heads -1 , 1 1-p Tails -1 , 1 1 , -1 Mixed Strategies – Chicken ± Since we allow for randomization over pure strategies, so we can consider mixed-strategy Nash equilibria even in non-zero-sum games ± Recall the game of Chicken… ± We have two pure-strategy NE Daisy Duke Swerve Don’t Swerve Billy-Jo Bob Swerve 1 , 1 1 , 2 Don’t Swerve 2 , 1 0 , 0

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10/16/2008 2 Daisy Duke q1 -
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Strategy13Handout - Mixed Strategies Matching Pennies We...

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