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Unformatted text preview: Game Theory May 1, 2009 1 Mixed Strategies Sometimes players randomize over their pure strategies: for instance a player can decide to play a strategy with 50% probability and another strategy with 50% probability. Example Left Right Up 1 , 2 4 , 5 Down 2 , 1 3 , There are two pure strategy NE: ( Down,Left ) and ( Up,Right ). But there is another NE in mixed strategies where player 1 plays Up with probability 25% and player 2 plays Left with probability 50%. A game might not have any NE in Pure Strategies, but (under very mild conditions) it ALWAYS has at least one equilibrium in Mixed Strategies (Nash Theorem). Example Left Right Up 1 , 1 1 , 1 Down 1 , 1 1 , 1 In this game there a no Pure Strategy NE, but there is a Mixed Strategy NE where every player plays one of the two strategies with probability 50%. 1.1 Expected Payoffs Before analyzing the equilibrium in mixed strategy I will introduce the con cept of expected payoff. When the environment is uncertain the payoff that 1 a agent receives becomes random: before the uncertainty is resolved we can compute an expectation of the payoff. The expected payoff is given by the sum of the individual payoffs multi plied for the probability of them happening. Example: Suppose you enter this bet: you pay 3$, then you roll a dice and receive an amount of money equal to 1 dollar times the number obtained. What is the expected payoff of this gambling game? The possible outcomes from rolling the dice are { 1 , 2 , 3 , 4 , 5 , 6 } each with equal probability p = 1 / 6. Then the expected payoff is E [ π ] = 3 + 1 1 6 + 2 1 6 + 3 1 6 + 4 1 6 + 5 1 6 + 6 1 6 = 3 + 3 . 5 = 0 . 5 Example: Consider the game Left Right Up 1 , 2 4 , 5 Down 2 , 1 3 , Suppose player 2 is playing Left with probability 50% (and Right with probability 50% of course). What is the expected payoff for player 1 from choosing Up ? he gets 1 with probability 0.5 and 4 with probability 0.5. So E [ π 1  Up ] = 1 1 2 + 4 1 2 = 2 . 5 What is the expected payoff that player 1 receives by playing Down ? he gets 2 with probability 0.5 and 3 with probability 0.5. So E [ π 1  Down ] = 2 1 2 + 3 1 2 = 2 . 5 If player 2 plays each strategy with probability 50% then player 1 becomes indifferent between Up and Down: this is not a case, actually in order to have a mixed strategy equilibrium everybody has to be indifferent....
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This note was uploaded on 05/09/2009 for the course ECON 200 taught by Professor Junnie during the Spring '08 term at NYU.
 Spring '08
 JunNie
 Game Theory

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