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Unformatted text preview: 14.126 GAME THEORY MIHAI MANEA Department of Economics, MIT, 1. Existence and Continuity of Nash Equilibria Follow Muhamet’s slides. We need the following result for future reference. Theorem 1. Suppose that each S i is a convex and compact subset of an Euclidean space and that each u i is continuous in s and quasiconcave in s i . Then there exists a pure strategy Nash equilibrium. 2. Bayesian Games When some players are uncertain about the characteristics or types of others, the game is said to have incomplete information . Most often a player’s type is simply defined by his payoff function. More generally, types may embody any private information that is relevant to players’ decision making. This may include, in addition to the player’s payoff function, his beliefs about other players’ payoff functions, his beliefs about what other players believe his beliefs are, and so on. The idea that a situation in which players are unsure about each other’s payoffs and beliefs can be modeled as a Bayesian game, in which a player’s type encapsulates all his uncertainty, is due to Harsanyi (1967, 1968) and has been formalized by Mertens and Zamir (1985). For simplicity, we assume that a player’s type is his own payoff and the type captures all the private information. A Bayesian game is a list B = ( N, S, Θ , u, p ) with • N = { 1 , 2 ,...,n } is a finite set of players • S i is the set of pure strategies of player i ; S = S 1 × ... × S n • Θ i is the set of types of player i ; Θ = Θ 1 × ... × Θ n Date : February 14, 2010. 2 MIHAI MANEA • u i : Θ × S → R is the payoff function of player i ; u = ( u 1 ,...,u n ) • p ∈ Δ(Θ) is a common prior (we can relax this assumption). We often assume that Θ is finite and the marginal p i ( θ i ) is positive for each type θ i . Example 1 (First Price Auction with I.I.D. Private Values) . One object is up for sale. Suppose that the value θ i of player i ∈ N for the object is uniformly distributed in Θ i = [0 , 1] and that the values are independent across players. This means that if θ ˜ i ∈ [0 , 1] , ∀ i then p ( θ i ≤ θ ˜ i , ∀ i ) = i θ ˜ i . Each player i submits a bid s i ∈ S i = [0 , ∞ ) . The player with the highest bid wins the object and pays his bid. Ties are broken randomly. Hence the payoffs are given by u i ( θ, s ) = ⎧ ⎪ ⎨ ⎪ ⎩ θ i − s i if s i ≥ s j , ∀ j ∈ N { j ∈ N  s i = s j } otherwise. Example 2 (An exchange game) . Each player i = 1 , 2 receives a ticket on which there is a number in some finite set Θ i ⊂ [0 , 1] . The number on a player’s ticket represents the size of a prize he may receive. The two prizes are independently distributed, with the value on i ’s ticket distributed according to F i . Each player is asked independently and simultaneously whether he wants to exchange his prize for the other player’s prize, hence S i = { agree, disagree } ....
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 Spring '05
 MuhammadYildiz
 Game Theory, player, Nash, MIHAI MANEA

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