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Unformatted text preview: 14.126 Lecture Notes on Rationalizability Muhamet Yildiz April 9, 2010 When we de f ne a game we implicitly assume that the structure (i.e. the set of play ers, their strategy sets and the fact that they try to maximize the expected value of the vonNeumann and Morgenstern utility functions) described by the game is common knowledge. The exact implications of this implicit assumption is captured by rational izability. In this lecture, I will formally demonstrate this fact. I will further extend rationalizability to incomplete information games. Of course, every incompleteinformation game can be represented as a complete information game, and the rationalizability is already de f ned for the latter game. That solution concept is called exante rationalizability . It turns out that that notion is more restrictive and im poses some stronger assumptions than what is intended in incomplete information game. To capture the exact implications of the assumptions in the incompleteinformation game, I will introduce another solution concept, called interim correlated rationalizabil ity , which is related to the rationalizability applied to the interim representation of the game, in which types are considered as players. Along the way, I will introduce a formulation of the Bayesian games that will be used in the remainder of the course. 1 Rationalizability in CompleteInformation Games Consider a completeinformation game ( N, S, u ) , where N is the set of players, with generic elements i, j ∈ N , S = Q i ∈ N S i is the set of strategy pro f les, and u : S R N → is the pro f le of payo f functions u i : S R . A game ( N, S, u ) is said to be f nite if N → and S are f nite. Implicit in the de f nition of the game game that player i maximizes 1 the expected value of u i with respect to a belief about the other players’ strategies. I will next formalize this idea. 1.1 Belief, Rationality, and Dominance De f nition 1 For any player i , a (correlated) belief of i about the other players’ strate Q gies is a probability distribution μ on S − i = j = i S j . − i 6 De f nition 2 The expected payo f from a strategy s i against a belief μ − i is Z u i ¡ s i , μ − i ¢ = E μ i [ u i ( s i , s − i )] ≡ u i ( s i , s − i ) dμ − i ( s − i ) ¡ ¢ P Note that in a f nite game u i s i , μ − i = s − i ∈ S − i u i ( s i , s − i ) μ − i ( s − i ) . De f nition 3 For any player i , a strategy s ∗ i is a best response to a belief μ − i if and only if u i ( s ∗ i , μ − i ) ≥ u i ( s i , μ − i ) ( ∀ s i ∈ S i ) . Here I use the notion of a weak best reply , requiring that there is no other strategy that yields a strictly higher payo f against the belief. A notion of strict best reply would require that s ∗ yields a strictly higher expected payo f than any other strategy....
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 Spring '05
 MuhammadYildiz
 Game Theory, player, common knowledge, rationalizability

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