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lecture6 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 6: Continuous and Discontinuous Games Asu Ozdaglar MIT February 23, 2010 1
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Game Theory: Lecture 6 Introduction Outline Continuous Games Existence of a Mixed Nash Equilibrium in Continuous Games (Glicksberg’s Theorem) Existence of a Mixed Nash Equilibrium with Discontinuous Payoffs Construction of a Mixed Nash Equilibrium with Infinite Strategy Sets Uniqueness of a Pure Nash Equilibrium for Continuous Games Reading: Myerson, Chapter 3. Fudenberg and Tirole, Sections 12.2, 12.3. Rosen J.B., “Existence and uniqueness of equilibrium points for concave N -person games,” Econometrica , vol. 33, no. 3, 1965. 2
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Game Theory: Lecture 6 Continuous Games Continuous Games We consider games in which players may have infinitely many pure strategies. Definition A continuous game is a game �I , ( S i ) , ( u i ) where I is a finite set, the S i are nonempty compact metric spaces, and the u i : S R are continuous functions. We next state the analogue of Nash’s Theorem for continuous games. 3
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Game Theory: Lecture 6 Continuous Games Existence of a Mixed Nash Equilibrium Theorem (Glicksberg) Every continuous game has a mixed strategy Nash equilibrium. With continuous strategy spaces, space of mixed strategies infinite dimensional, therefore we need a more powerful fixed point theorem than the version of Kakutani we have used before. Here we adopt an alternative approach to prove Glicksberg’s Theorem, which can be summarized as follows: We approximate the original game with a sequence of finite games, which correspond to successively finer discretization of the original game. We use Nash’s Theorem to produce an equilibrium for each approximation. We use the weak topology and the continuity assumptions to show that these converge to an equilibrium of the original game. 4
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Game Theory: Lecture 6 Continuous Games Closeness of Two Games Let u = ( u 1 , . . . , u I ) and u ˜ = ( u ˜ 1 , . . . , u ˜ I ) be two profiles of utility functions defined on S such that for each i ∈ I , the functions u i : S R and u ˜ i : S R are bounded (measurable) functions. We define the distance between the utility function profiles u and u ˜ as max sup u i ( s ) u ˜ i ( s ) . i ∈I s S | | Consider two strategic form games defined by two profiles of utility functions: G = �I , ( S i ) , ( u i ) , G ˜ = �I , ( S i ) , ( u ˜ i ) . If σ is a mixed strategy Nash equilibrium of G , then σ need not be a mixed strategy Nash equilibrium of G ˜ . Even if u and u ˜ are very close, the equilibria of G and G ˜ may be far apart. For example, assume there is only one player, S 1 = [ 0, 1 ] , u 1 ( s 1 ) = s 1 , and u ˜ 1 ( s 1 ) = s 1 , where > 0 is a sufficiently small scalar. The unique equilibrium of G is s 1 = 1, and the unique equilibrium of G ˜ is s 1 = 0, even if the distance between u and u ˜ is only 2 .
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