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Unformatted text preview: 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 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 , , R 3 4 , 4 , L 3 , , 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 1/4 1/2 1 1/4 2 1/3 2/3 1/4 1/2 1 1/4 2 ➥ Bayes’ rule can be applied: μ 2 = ( 1 3 , 2 3 , 0) 6/ ? ? ? 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) 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. Whenwhen it is possible....
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This note was uploaded on 01/23/2011 for the course ECONOMICS gt512 taught by Professor Breviart during the Spring '10 term at Télécom Paris.
 Spring '10
 breviart
 Game Theory

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