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Unformatted text preview: 6.254 : Game Theory with Engineering Applications Lecture 17: Games with Incomplete Information: Bayesian Nash Equilibria Asu Ozdaglar MIT April 15, 2010 1 Game Theory: Lecture 17 Introduction Outline Incomplete information. Bayes rule and Bayesian inference. Bayesian Nash Equilibria. Auctions. Reading: Fudenberg and Tirole, Sections 6.16.5. Krishna, Chapters 15. 2 Game Theory: Lecture 17 Incomplete Information Incomplete Information In many game theoretic situations, one agent is unsure about the payoffs or preferences of others. Incomplete information introduces additional strategic interactions and also raises questions related to learning. Examples: Bargaining (how much the other party is willing to pay is generally unknown to you) Auctions (how much should you bid for an object that you want, knowing that others will also compete against you?) Market competition (firms generally do not know the exact cost of their competitors) Signaling games (how should you infer the information of others from the signals they send) Social learning (how can you leverage the decisions of others in order to make better decisions) 3 Game Theory: Lecture 17 Incomplete Information Example: Incomplete Information Battle of the Sexes Recall the battle of the sexes game, which was a complete information coordination game. Both parties want to meet, but they have different preferences on Ballet and Football. B F B ( 2, 1 ) ( 0, 0 ) F ( 0, 0 ) ( 1, 2 ) In this game there are two pure strategy equilibria (one of them better for player 1 and the other one better for player 2), and a mixed strategy equilibrium. Now imagine that player 1 does not know whether player 2 wishes to meet or wishes to avoid player 1. Therefore, this is a situation of incomplete information also sometimes called asymmetric information . 4 Game Theory: Lecture 17 Incomplete Information Example (continued) We represent this by thinking of player 2 having two different types , one type that wishes to meet player 1 and the other wishes to avoid him. More explicitly, suppose that these two types have probability 1/2 each. Then the game takes the form one of the following two with probability 1/2. B F B ( 2, 1 ) ( 0, 0 ) F ( 0, 0 ) ( 1, 2 ) B F B ( 2, 0 ) ( 0, 2 ) F ( 0, 1 ) ( 1, 0 ) Crucially, player 2 knows which game it is (she knows the state of the world ), but player 1 does not. What are strategies in this game? 5 Game Theory: Lecture 17 Incomplete Information Example (continued) Most importantly, from player 1s point of view, player 2 has two possible types (or equivalently, the world has two possible states each with 1/2 probability and only player 2 knows the actual state)....
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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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