Lecture13 - Mixed Strategies Matching Pennies We denote the...

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3/3/2009 1 Mixed Strategies – Matching Pennies We denote the strategies on the game bi-matrix Guildenstern q 1-q Heads Tails Rosencrantz’s payoff to H : (1)q + (-1)(1-q) = 2q-1 T : (-1)q + (1)(1-q) = 1-2q Rosencrantz p 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 0 1 1/2 1/2 the Nash equilibrium p=1/2, q=1/2 Mixed Strategies – Matching Pennies Suppose R’s payoff to H increases Guildenstern q 1-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 Rosencrantz p 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 0 1 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 Guildenstern q 1-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 Rosencrantz p Heads 2 , -1 -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
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