lecture7 notes - 6.254 Game Theory with Engineering...

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6.254 : Game Theory with Engineering Applications Lecture 7: Supermodular Games Asu Ozdaglar MIT February 25, 2010 1
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Game Theory: Lecture 7 Introduction Outline Uniqueness of a Pure Nash Equilibrium for Continuous Games Supermodular Games Reading: Rosen J.B., “Existence and uniqueness of equilibrium points for concave N -person games,” Econometrica , vol. 33, no. 3, 1965. Fudenberg and Tirole, Section 12.3. 2
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Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Uniqueness of a Pure Strategy Nash Equilibrium in Continuous Games We have shown in the previous lecture the following result: Given a strategic form game �I , ( S i ) , ( u i ) , assume that the strategy sets S i are nonempty, convex, and compact sets, u i ( s ) is continuous in s , and u i ( s i , s i ) is quasiconcave in s i . Then the game �I , ( S i ) , ( u i ) has a pure strategy Nash equilibrium. We have seen an example that shows that even under strict convexity assumptions, there may be infinitely many pure strategy Nash equilibria. 3
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Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Uniqueness of a Pure Strategy Nash Equilibrium We will next establish conditions that guarantee that a strategic form game has a unique pure strategy Nash equilibrium, following the classical paper [Rosen 65]. Notation : Given a scalar-valued function f : R n R , we use the notation �→ f ( x ) to denote the gradient vector of f at point x , i.e., T f ( x ) = f ( x ) , . . . , f ( x ) . x 1 x n Given a scalar-valued function u : I i = 1 R m i �→ R , we use the notation i u ( x ) to denote the gradient vector of u with respect to x i at point x , i.e., T i u ( x ) = u x ( i x 1 ) , . . . , u x ( i m x i ) . (1) 4
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Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Optimality Conditions for Nonlinear Optimization Problems Theorem (3) (Karush-Kuhn-Tucker conditions) Let x be an optimal solution of the optimization problem maximize f ( x ) subject to g j ( x ) 0, j = 1, . . . , r , where the cost function f : R n �→ R and the constraint functions g j : R n �→ R are continuously differentiable. Denote the set of active constraints at x as A ( x ) = { j = 1, . . . , r g j ( x ) = 0 } . Assume that the active constraint gradients, g j ( x ) , j | A ( x ) , are linearly independent vectors. Then, there exists a nonnegative vector λ R r (Lagrange multiplier vector) such that r f ( x ) + λ j g j ( x ) = 0, j = 1 λ j g j ( x ) = 0, j = 1, . . . , r . (2) 5
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Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Optimality Conditions for Nonlinear Optimization Problems For convex optimization problems (i.e., minimizing a convex function (or maximizing a concave function) over a convex constraint set), we can provide necessary and sufficient conditions for the optimality of a feasible solution: Theorem (4) Consider the optimization problem maximize f ( x ) subject to g j ( x ) 0, j = 1, . . . , r , where the cost function f : R n �→ R and the constraint functions g j : R n �→ R are concave functions. Assume also that there exists some ¯ x ) > 0 x such that g j ( ¯ for all j = 1, . . . , r. Then a vector x
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