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# E f x f x t f x x1 xn given a scalar valued

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Unformatted text preview: 1] for i = 1, 2, and the payoﬀs u1 (s1 , s2 ) = s1 s2 − 2 s1 , 2 2 s2 . 2 Note that ui (s1 , s2 ) is strictly concave in si . It can be seen in this example that the best response correspondences (which are unique-valued) are given by B1 (s2 ) = s2 , B2 (s1 ) = s1 . u2 (s1 , s2 ) = s1 s2 − Plotting the best response curves shows that any pure strategy proﬁle (s1 , s2 ) = (x , x ) for x ∈ [0, 1] is a pure strategy Nash equilibrium. 21 Game Theory: Lecture 6 Continuous Games 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 : Rn �→ R, we use the notation �f (x ) to denote the gradient vector of f at point x , i.e., � � ∂f (x ) ∂f (x ) T �f (x ) = ,..., . ∂ x1 ∂ xn Given a scalar-valued function u : ∏I =1 Rmi �→ R, we use the i notation �i u (x ) to denote the gradient vector of u with respect to xi at point x , i.e., � �T ∂u (x ) ∂u (x ) �i u (x ) = ,..., . (2) ∂ximi ∂xi1 22 Game Theory: Lecture 6 Continuous Games 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 gj (x ) ≥ 0, j = 1, . . . , r , where the cost function f : Rn �→ R and the constraint functions gj : Rn �→ R are continuously diﬀerentiable. Denote the set of active constraints at x ∗ as A(x ∗ ) = {j = 1, . . . , r | gj (x ∗ ) = 0}. Assume that the active constraint gradients, �gj (x ∗ ), j ∈ A(x ∗ ), are linearly independent vectors. Then, there exists a nonnegative vector λ∗ ∈ Rr (Lagrange multiplier vector) such that �f (x ∗ ) + r ∑ λj∗ �gj (x ∗ ) = 0, j =1 ∗ λj gj (x ∗ ) = 0, ∀ j = 1, . . . , r . (3) 23 Game Theory: Lecture 6 Continuous Games Optimality Conditions for Nonlinear Optimization Problems For convex optimization problems (i.e., mini...
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