167hw5 - Game Theory Steven Heilman Please provide complete...

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Game Theory Steven Heilman Please provide complete and well-written solutions to the following exercises. Due February 16th, in the discussion section. Homework 5 Exercise 1. Suppose we have a two-person zero-sum game with ( n + 1) × ( n + 1) payoff matrix A such that at least one entry of A is nonzero. Let x, y Δ n +1 . Write x = ( x 1 , . . . , x n , 1 - n i =1 x i ), y = ( x n +1 , x n +2 , . . . , x 2 n , 1 - 2 n i = n +1 x i ). Consider the function f : R 2 n R defined by f ( x 1 , . . . , x 2 n ) = x T Ay . Show that the Hessian of f has at least one positive eigenvalue, and at least one negative eigenvalue. Conclude that any critical point of f is a saddle point. That is, if we find a critical point of f (as we do when we look for the value of the game), then this critical point is a saddle point of f . In this sense, the minimax value occurs at a saddle point of f . (Hint: Write f in the form f ( x 1 , . . . , x 2 n ) = 2 n i =1 b i x i + 1 i n, n +1 j 2 n c ij x i x j , where b i , c ij R . From here, it should follow that there exists a nonzero matrix C such that the Hessian of f , i.e. the matrix of second order partial derivatives of f , should be of the form 0 C C T 0 .
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