gameL9 - Perfect Bayesian Equilibrium For an important...

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Perfect Bayesian Equilibrium For an important class of extensive games, a solution concept is available that is simpler than sequential equi- librium, but with similar properties. In a Ba y e s iane x t en s i v egam ew i thob s e r vab l ea c - tions ,naturemoves f rst and independently selects a type for each player. Afterwards, the actions chosen by players are observed by all.
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De f nition 231.1: A Ba y e s iane x t en s i v egam ew i th observable actions is a tuple, h Γ , ( Θ i ) , ( p i ) , ( u i ) i where 1. Γ = h N,H,P i is an extensive game form with perfect information and (possibly) simultaneous moves 2. The set of types for player i , Θ i ,isa f nite set, and we denote Θ = × i N Θ i 3. p i is a probability measure on Θ i ,whe re p i ( θ i ) > 0 for all θ i Θ i , and these measures are independent 4. u i : Θ × Z R is a von Neumann-Morgenstern utility function.
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The set of histories is { } ( Θ × H ) The information set for player i P ( h ) is of the form I ( θ i ,h )= { (( θ i 0 i ) ): θ 0 i Θ i } To solve the game, we will be looking for a pro f le of be- havioral strategies in Γ (for each player and each type) (( σ i ( θ i )) and a belief system µ i ( h ) that speci f es a com- mon belief, after the history h , held by all players other than i about player i ’s type. Let s be a pro f le of behavioral strategies in Γ .D e f ne O h ( s ) to be the probability measure on terminal histories of Γ generated by s ,g iventheh isto ry h . De f ne O ( σ i ,s i i | h ) to be the probability measure on terminal histories of Γ , given i uses s i , other players use type-dependent behavioral strategies σ i ,theh istory reached is h , and beliefs are given by µ i .
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De f nition 232.1: Let h Γ , ( Θ i ) , ( p i ) , ( u i ) i be a Bayesian extensive game with observable actions, with Γ = h N,H,P i . Apair (( σ i ) , ( µ i )) = (( σ i ( θ i )) i N,θ i Θ i , ( µ i ( h )) i N,h H \ Z ) is a perfect Bayesian equilibrium if the following con- ditions are satis f ed 1. Sequential Rationality : For every non-terminal his- tory h H \ Z ,eve ryp laye r i P ( h ) , and every type θ i Θ i , the probability measure O ( σ i i ( θ i ) i | h ) is weakly preferred by type θ i to O ( σ i ,s i i | h ) for any strategy s i of player i in Γ . 2. Correct initial beliefs : µ i ( )= p i for each i N . 3. Action-determined beliefs :I f i/ P ( h ) and a A ( h ) ,then µ i ( h, a µ i ( h ) .I f i P ( h ) , a A ( h ) , a 0 A ( h ) ,and a i = a 0 i µ i ( h, a µ i ( h, a 0 ) .
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4. Bayesian updating :I f i P ( h ) and a i is in the support of σ i ( θ i )( h ) for some θ i in the support of µ i ( h ) , then for any θ 0 i Θ i ,wehave µ i ( h, a )( θ 0 i )= σ i ( θ 0 i )( h )( a i )[ µ i ( h )( θ 0 i )] P θ i Θ i σ i ( θ i )( h )( a i )[ µ i ( h )( θ i )] . Interpreting conditions 3 and 4: Action-determined beliefs—since types are independent, beliefs about player i ’s type cannot be in f uenced by the actions of the other players. You can’t signal what you don’t know. If player i does not make a move after the history h , then the action pro F le after history h does not change µ i .I f p l a y e r i does move after the history h ,
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This note was uploaded on 07/17/2008 for the course ECON 817 taught by Professor Peck during the Fall '07 term at Ohio State.

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gameL9 - Perfect Bayesian Equilibrium For an important...

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