1.3 Mixed Strategy Equilibrium

1.3 Mixed Strategy Equilibrium - AMS 335/ECO 355 Game...

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AMS 335/ECO 355 Game Theory Fall 2010 Chapter 1 Games of Complete Information Zhen Xu Page 1 of 7 1.3 Mixed Strategy Equilibrium A game in strategic form does not always have a Nash equilibrium in which each player deterministically chooses one of his strategies. However, players may instead randomly select from among these pure strategies with certain probabilities. Randomizing one’s own choice in this way is called a mixed strategy. Nash showed in 1951 that any finite strategic-form game has an equilibrium if mixed strategies are allowed. As before, an equilibrium is defined by a (possibly mixed) strategy for each player where no player can gain on average by unilateral deviation. Average (that is, expected ) payoffs must be considered because the outcome of the game may be random. Example: Quality choice Suppose player 1 is an internet service provider and player 2 a potential customer. They consider entering into a contract of service provision for a period of time. The provider can, for himself, decide between two levels of quality of service, High or Low . The level of service cannot be put verifiably into the contract. High-quality service is more valuable than low-quality service to the customer, in fact so much so that the customer would prefer not to buy the service if she knew that the quality was low. Her choices are to buy or not to buy the service. Here, low-quality service provision, even when the customer decides to buy, has the same low payoff 1 to the provider as when the customer does not sign the contract in the first place, since the customer will opt out later. However, the customer still prefers not to buy when the service is Low in order to spare herself the hassle of entering the contract. In a sense, the opt-out clause in the contract has the purpose of convincing the customer that the high-quality service provision is in the provider’s own interest. Table 1 High-low Quality Game between a Service Provider (Player 1) and a Customer (Player 2) Player 2 Buy Not to Buy Player 1 High 2,2 0,1 Low 1,0 1,1 This game has no dominated strategy for either player. The game has two Nash equilibria in which each player chooses his strategy deterministically. One of them is, the strategy combination ( Low, Not to Buy ). This is an equilibrium since Low is the best response (payoff-maximizing strategy) to Not to Buy and vice versa. The second Nash equilibrium is the strategy combination ( High, Buy ). It is an equilibrium since player 1 prefers to provide high-quality service when the customer buys, and conversely, player 2 prefers to buy when the quality is high. This equilibrium has a higher payoff to both players than the former one, and is a more desirable solution.
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AMS 335/ECO 355 Game Theory Fall 2010 Chapter 1 Games of Complete Information Zhen Xu Page 2 of 7 Both Nash equilibria are legitimate recommendations to the two players of how to play the game. Once the players have settled on strategies that form a Nash equilibrium, neither player has incentive to deviate, so that they will rationally stay
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This note was uploaded on 04/27/2011 for the course ECO 355 taught by Professor Xu during the Fall '10 term at SUNY Stony Brook.

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1.3 Mixed Strategy Equilibrium - AMS 335/ECO 355 Game...

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