gameL4 - Pearce, "Rationalizable Strategic Behavior...

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Pearce, “Rationalizable Strategic Behavior and the Problem of Perfection,” Econometrica 1984 Rationalizability is a weaker (broader) solution concept than Nash equilibrium. It looks at the implications of common knowledge of rationality, without imposing con- sistency requirements on strategy pro f les. Suppose we are interested in narrowing the set of “rea- sonable” actions we predict a player might choose, with no communication or observation of past play. (A1) Each player forms a subjective probability distribu- tion over the other players’ actions, and the distribution cannot contradict his/her knowledge. (A2) Each player chooses an action that maximizes ex- pected utility, based on the subjective beliefs about oth- ers’ actions. (A3) The structure of the game, and the rationality of players based on (A1) and (A2), is common knowledge.
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Let N = { 1 ,...,n } and let H denote the convex hull of the set H . The procedure to remove mixed strategies that are not rationalizable is the following. Start with H i 4 ( A i ) ,fo r i N . Let H i (0) = H i . Inductively de f ne H i ( t ) by H i ( t )= α i H i ( t 1) : γ × j N H j ( t 1) such that α i is a best response in H i ( t 1) to γ . Thus, retain α i if it is a best response to some conjecture over α i , that have not been removed at an earlier stage.
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De f ne R i ( H 1 ,...,H n )= \ t =1 H i ( t ) .
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gameL4 - Pearce, "Rationalizable Strategic Behavior...

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