Lecture 13_Bayesian Games 2010 6

Lecture 13_Bayesian Games 2010 6 - ECON 4109H LECTURE 13...

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Unformatted text preview: ECON 4109H LECTURE 13 Bayesian Games December 16, 2010 ECON 4109H LECTURE 13 Incomplete Information Until now, we have assumed that all aspects of the game are common knowledge. In many situations, there is uncertainty about certain aspects of the game or other players preferences. In an auction, a bidder may be uncertain about the valuations of other bidders. Bidders may be uncertain about the true value of the object they are bidding for. Firms may be uncertain about the other firms cost functions. Voters may be uncertain about preferences of other voters. ECON 4109H LECTURE 13 Battle of the Sexes Consider a variant of battle of the sexes in which player 1 is uncertain whether player 2 would like to go out with him. Player 2 is knows player 1s preferences: B S B 2, 1 0, 0 S 0, 0 1, 2 B S B 2, 0 0, 2 S 0, 1 1, 0 Player 2 knows the game. Player 1 thinks that with equal probability the two games are played. ECON 4109H LECTURE 13 Battle of the Sexes B S B 2, 1 0, 0 S 0, 0 1, 2 B S B 2, 0 0, 2 S 0, 1 1, 0 A strategy for player 1 is simply an action B or S . Player 2 can be of two types , type t 1 who prefers to go out with player 1, and a type t 2 who prefers to avoid player 1. Each type occurs with equal probability. A strategy for player 2 is a function from player 2s type to { B , S } . ECON 4109H LECTURE 13 Battle of the Sexes A Nash equilibrium is a pair of strategies which are best responses to one another. But remember that a strategy for player 2 is a function from his type to his action. ( B , B ) ( B , S ) ( S , B ) ( S , S ) B 2, 1 2 1, 3 2 1, 0 0, 1 S 0, 1 2 1 2 , 0 1 2 , 3 2 1, 1 ECON 4109H LECTURE 13 Battle of the Sexes ( B , B ) ( B , S ) ( S , B ) ( S , S ) B 2 , 1 2 1 , 3 2 1 , 0 0, 1 S 0, 1 2 1 2 , 0 1 2 , 3 2 1 , 1 ( B , ( B , S )) is the unique pure strategy Nash Equilibrium. Suppose that player 1 chooses S , then player 2 will match with player 1 half the time. But given that player 2 will match player 1 half the time, player 1 prefers to match when player 2 chooses B , so there is no equilibrium in which player 1 chooses S . ECON 4109H LECTURE 13 Battle of the Sexes Now suppose neither player knows whether the other player wants to go out with them: B S B 2, 1 0, 0 S 0, 0 1, 2 B S B 2, 0 0, 2 S 0, 1 1, 0 B S B 0, 1 2, 0 S 1, 0 0, 2 B S B 0, 0 2, 2 S 1, 1 0, 0 Player 1 knows his type (i.e., whether we are in the top or bottom row). Player 2 knows his type (i.e, whether we are in the right or in the left). ECON 4109H LECTURE 13 Battle of the Sexes Theorem The only pure strategy equilibria are (( B , B ) , ( B , S )) and (( S , B ) , ( S , S )) . Proof. First note that if player 1 selects ( S , S ) , then player 2 will choose ( S , B ) . But then player 1 will prefer ( B , B ) . So there cannot be an equilibrium in which player 1 plays ( S , S ) . Likewise, there is no equilibrium in which player 2 chooses ( B , B ) . If player 1 plays ( B , S ) , then player 2 will play ( S , S ) , but then player 1 will want to play (...
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Lecture 13_Bayesian Games 2010 6 - ECON 4109H LECTURE 13...

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