notes_12 - Game Theory May 1 2009 1 Mixed Strategies...

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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.

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notes_12 - Game Theory May 1 2009 1 Mixed Strategies...

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