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Unformatted text preview: 6.254 : Game Theory with Engineering Applications Lecture 18: Games with Incomplete Information: Bayesian Nash Equilibria and Perfect Bayesian Equilibria Asu Ozdaglar MIT April 22, 2010 1 Game Theory: Lecture 18 Introduction Outline Bayesian Nash Equilibria. Auctions. Extensive form games of incomplete information. Perfect Bayesian (Nash) Equilibria. 2 Game Theory: Lecture 18 Incomplete Information Incomplete Information In the last lecture, we studied incomplete information games where one agent is unsure about the payoffs or preferences of others. Examples abundant: Bargaining, auctions, market competition, signaling games, social learning. We modeled such games as Bayesian games that consist of A set of players I ; A set of actions (pure strategies) for each player i : S i ; A set of types for each player i : θ i ∈ Θ i ; A payoff function for each player i : u i ( s 1 , . . . , s I , θ 1 , . . . , θ I ) ; A (joint) probability distribution p ( θ 1 , . . . , θ I ) over types (or P ( θ 1 , . . . , θ I ) when types are not finite). 3 Game Theory: Lecture 18 Bayesian Games Bayesian Games Importantly, throughout in Bayesian games, the strategy spaces, the payoff functions, possible types, and the prior probability distribution are assumed to be common knowledge . Definition A (pure) strategy for player i is a map s i : Θ i → S i prescribing an action for each possible type of player i. Given p ( θ 1 , . . . , θ I ) , player i can compute the conditional distribution p ( θ i  θ i ) using Bayes rule , where θ i = ( θ 1 , . . . , θ i 1 , θ i + 1 , . . . , θ I ) . Player i knows her own type and evaluates her expected payoffs according to the conditional distribution p ( θ i  θ i ) . 4 Game Theory: Lecture 18 Bayesian Games Bayesian Nash Equilibria Definition (Bayesian Nash Equilibrium) The strategy profile s ( · ) is a (pure strategy) Bayesian Nash equilibrium if for all i ∈ I and for all θ i ∈ Θ i , we have that s i ( θ i ) ∈ arg max s i ∈ S i ∑ θ i p ( θ i  θ i ) u i ( s i , s i ( θ i ) , θ i , θ i ) , or in the nonfinite case, s i ( θ i ) ∈ arg max s i ∈ S i Z u i ( s i , s i ( θ i ) , θ i , θ i ) P ( d θ i  θ i ) . Hence a Bayesian Nash equilibrium is a Nash equilibrium of the “expanded game” in which each player i ’s space of pure strategies is the set of maps from Θ i to S i . 5 Game Theory: Lecture 18 Auctions Auctions A major application of Bayesian games is to auctions . This corresponds to a situation of incomplete information because the valuations of different potential buyers are unknown. We made the distinction between: Private value auctions : valuation of each agent is independent of others’ valuations; Common value auctions : the object has a potentially common value, and each individual’s signal is imperfectly correlated with this common value....
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This note was uploaded on 05/08/2010 for the course CS 6.254 taught by Professor Asuozdaglar during the Spring '10 term at MIT.
 Spring '10
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