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Econ106G_081_Lecture9

# Econ106G_081_Lecture9 - Econ106G Lecture Note 9 Hong Feng 1...

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Econ106G Lecture Note 9 Hong Feng July 24, 2008 1 Incomplete Information and Bayesian Nash Equilibrium A key assumption in most of the games we have discussed so far was that each player knows the whole structure of the game, i.e. they know their own possi- ble strategies, the strategies of other players, and the payo/s from all possible situations that we want to explain with game theory. In particular, knowledge in competition rarely know each others±costs. A (simpli&ed version of) strategic-form game with Incomplete In- formation is given by A set of players i = 1 ; 2 Types of each players: t i 2 T i with distribution p i ( t i ) ; i = 1 ; 2 : Actions of each player: a i 2 A i ; i = 1 ; 2 Strategies: s i 2 S i : Each pure strategy s i assigns some action a i to every type t i : Payo/s: u 1 ( a 1 ;a 2 ;t 1 ;t 2 ) ; u 2 ( a 1 ;a 2 ;t 1 ;t 2 ) : ( s 1 ( t 1 ) ;s 2 ( t 2 )) is a Bayesian Nash equilibrium if 1 is playing a best reponse to 2 s 1 ( t 1 ) maximizes E t 2 [ u 1 ( a 1 ;s 2 ( t 2 ) ;t 1 ;t 2 )] for all t 1 i:e: E t 2 [ u 1 ( s 1 ( t 1 ) ;s 2 ( t 2 ) ;t 1 ;t 2 )] ± E t 2 [ u 1 ( a 0 1 ;s 2 ( t 2 ) ;t 1 ;t 2 )] for all a 0 1 2 A 1 ; for all t 1 and 2 is playing a best response to 1 s 2 ( t 2 ) maximizes E t 2 [ u 2 ( s 1 ( t 1 ) ;a 2 ;t 1 ;t 2 )] for all t 2 in a BNE, for each type t i of player i , the assigned action s i ( t i ) must be a best response to s j ; i.e. it must generate the highest expected payo/ for t i : 1

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2 Fighting an Unkown Opponent stronger one wins. The strength of player 1 is commonly known to be inter-
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Econ106G_081_Lecture9 - Econ106G Lecture Note 9 Hong Feng 1...

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