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Lecture6 - 6.254 Game Theory with Engineering Applications...

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6.254: Game Theory with Engineering Applications February 23, 2010 Lecture 6: Continuous and Discontinuous Games Lecturer: Asu Ozdaglar 1 Introduction In this lecture, we will focus on: Existence of a mixed strategy Nash equilibrium for continuous games (Glicksberg’s theorem). Finding mixed strategy Nash equilibria in games with infinite strategy sets. Uniqueness of a pure strategy Nash equilibrium for continuous games. 2 Continuous Games In this section, we consider games in which players may have infinitely many pure strategies. In particular, we want to include the possibility that the pure strategy set of a player may be a bounded interval on the real line, such as [0,1]. We will follow the development of Myerson [3], pp. 140-148. Definition 1 A continuous game is a game hI , ( S i ) , ( u i ) i where I is a finite set, the S i are nonempty compact metric spaces, and the u i : S 7→ R are continuous functions. A compact metric space is a general mathematical structure for representing infinite sets that can be well approximated by large finite sets. One important fact is that, in a compact metric space, any infinite sequence has a convergent subsequence. Any closed bounded subset of a finite-dimensional Euclidean space is an example of a compact metric space. More specifically, any closed bounded interval of the real line is an example of a compact metric space, where the distance between two points x and y is given by | x - y | . In 6-1
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our treatment, we will not need to refer to any examples more complicated than these (see the Appendix for basic definitions of a metric space and convergence notions for probability measures). We next state the analogue of Nash’s Theorem for continuous games. Theorem 1 (Glicksberg) Every continuous game has a mixed strategy Nash equilibrium. With continuous strategy spaces, the space of mixed strategies Σ is 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. 2.1 Closeness of Two Games and -Equilibrium 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 7→ R and ˜ u i : S 7→ R are bounded measurable functions. We may define the distance between the utility function profiles u and ˜ u as max i ∈I sup s S | u i ( s ) - ˜ u i ( s ) | .
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