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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 von-Neumann 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 incomplete-information 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 ex-ante 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 incomplete-information 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 Complete-Information Games Consider a complete-information 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
- Game Theory, player, common knowledge, rationalizability