8-raffinements-e-article

# 8-raffinements-e-article - F Koessler Equilibrium Renement...

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F. Koessler / November 20, 2007 Equilibrium Refinement and Signaling Games 1/ Dynamic Games of Incomplete Information Equilibrium Refinement and Signaling Games Outline (November 20, 2007) Introductory Examples Sequential Rationality and Perfect Bayesian Equilibrium Strong Belief Consistency and Sequential Equilibrium Signaling Games Application: Spence’s (1973) Model of Education 2/ In games with imperfect information, subgame perfection is not always strong enough to eliminate “irrational decisions” or “incredible threats” off the equilibrium path Example. c (1 , 4) a b 1 D (0 , 0) G (3 , 2) D (0 , 0) G (2 , 3) 2 ( c, D ) is a (SP)NE but D is not an optimal decision at player 2’s information set Sequential rationality generalization of backward induction Require rational decisions even at information sets off the equilibrium path (even if they are not singleton information sets) Player 2 plays G Player 1 plays a

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F. Koessler / November 20, 2007 Equilibrium Refinement and Signaling Games 3/ Example. ( Selten ’s (1975) “horse”) L 1 1 R 1 L 2 2 R 2 1 , 1 , 1 3 R 3 3 , 2 , 2 L 3 0 , 0 , 0 R 3 4 , 4 , 0 L 3 0 , 0 , 1 2 pure strategy (SP)NE: ( R 1 , R 2 , L 3 ) and ( L 1 , R 2 , R 3 ) But in ( L 1 , R 2 , R 3 ) the action R 2 of player 2 is not sequentially rational given that player 3 plays R 3 ( 4 > 1 ) 4/ In the previous examples we have eliminated SPNE in which the action of some player is never optimal, whatever his belief about past play Modification of the first example: c (1 , 4) a b 1 D (0 , 3 ) G (3 , 2) D (0 , 0) G (2 , 3) 2 If player 1 plays c , sequential rationality of player 2 is not well defined (playing G or playing D ?) The strategy profile is usually not sufficient to define sequential rationality The solution concept is not only characterized by a strategy profile but also by a belief system
F. Koessler / November 20, 2007 Equilibrium Refinement and Signaling Games 5/ Belief System 0 1/4 1/2 1 1/4 2 1/3 2/3 0 Bayes’ rule can be applied: μ 2 = ( 1 3 , 2 3 , 0) 6/ ? ? ? 0 0 0 1 1 2 Bayes’ rule cannot be applied: μ 2 = ? (divide by zero) Belief system : collection of probability distributions on decision nodes, one distribution for each information set trivial in perfect information games (probability 1 at every node)

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F. Koessler / November 20, 2007 Equilibrium Refinement and Signaling Games 7/ A pair ( σ, μ ) , where σ is a profile of behavioral strategies and μ a belief system, is a weak sequential equilibrium , or perfect Bayesian equilibrium (PBE), if Sequential Rationality. For every player i and every information set of player i , the local strategy of player i at this information set maximizes his expected utility given his belief at this information set and the strategies of the other players Weak Belief Consistency. In every subgame (along and off the equilibrium path), beliefs are computed by Bayes’ rule according to σ when it is possible. When Bayes’ rule cannot be applied, beliefs can be chosen arbitrarily 8/ Example. ( d, G ) is a perfect Bayesian equilibrium (PBE) 3/4 1/4 0 c a b 1 d (2 , 0) D (0 , 0) G (0 , 1) D (0 , 0) G (0 , 1) D (0 , 1) G (0 , 0) 2 Remark. Many other belief systems are possible (
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