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Lecture7-new

# Lecture7-new - 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
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 hI , ( S i ) , ( u i ) 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 hI , ( S i ) , ( u i ) 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 7→ R , we use the notation f ( x ) to denote the gradient vector of f at point x , i.e., f ( x ) = f ( x ) x 1 , . . . , f ( x ) x n T . Given a scalar-valued function u : I i = 1 R m i 7→ 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., i u ( x ) = u ( x ) x 1 i , . . . , u ( x ) x m i i T . (1) 4
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 7→ R and the constraint functions g j : R n 7→ 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 f ( x * ) + r j = 1 λ * j g j ( x * ) = 0, 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 7→ R and the constraint functions g j : R n 7→ R are concave functions. Assume also that there exists some ¯ x such that g j ( ¯ x ) > 0 for all j = 1, . . . , r. Then a vector x * R n is an optimal solution of the preceding problem if and only if g j ( x * ) 0 for all j
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Lecture7-new - 6.254 Game Theory with Engineering...

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