lecture6 notes

Since the hi are concave functions we have hi xi hi xi

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Unformatted text preview: ume also that the payoff functions ˜ (u1 , . . . , uI ) are diagonally strictly concave for x ∈ S . Then the game has a unique pure strategy Nash equilibrium. 26 Game Theory: Lecture 6 Continuous Games Proof Assume that there are two distinct pure strategy Nash equilibria. Since for each i ∈ I , both xi∗ and xi must be an optimal solution for an ¯ optimization problem of the form (5), Theorem 4 implies the existence of ∗ ∗ ¯ ¯ ¯ nonnegative vectors λ∗ = [λ1 , . . . , λI ]T and λ = [λ1 , . . . , λI ]T such that for all i ∈ I , we have �i ui (x ∗ ) + λi∗ �hi (xi∗ ) = 0, (7) λi∗ hi (xi∗ ) = 0, (8) ¯ �i ui (x ) + λi �hi (xi ) = 0, ¯ ¯ ¯ i hi (xi ) = 0. λ ¯ (9) and (10) 27 Game Theory: Lecture 6 Continuous Games Proof Multiplying Eqs. (7) and (9) by (xi − xi∗ )T and (xi∗ − xi )T respectively, and ¯ ¯ adding over all i ∈ I , we obtain 0 = ( ¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x ) x ¯ ¯ (11) ∗ ∗T ∗ T∗ ¯ + ∑ λ �hi (x ) (xi − x ) + ∑ λi �hi (xi ) (x − xi ) ¯ ¯ ¯ i i i i ∈I > i i ∈I ¯ ¯ ¯ ¯ ∑ λi∗ �hi (xi∗ )T (xi − xi∗ ) + ∑ λi �hi (xi )T (xi∗ − xi ), i ∈I i ∈I where to get the strict inequality, we used the assumption that the payoff functions are diagonally strictly concave for x ∈ S . Since the hi are concave functions, we have hi (xi∗ ) + �hi (xi∗ )T (xi − xi∗ ) ≥ hi (xi ). ¯ ¯ 28 Game Theory: Lecture 6 Continuous Games Proof Using the preceding together with λi∗ > 0, we obtain for all i , λi∗ �hi (xi∗ )T (xi − xi∗ ) ¯ ≥ λi∗ (hi (xi ) − hi (xi∗ )) ¯ = λi∗ hi (xi ) ¯ ≥ 0, where to get the equality we used Eq. (8), and to get the last inequality, we used the facts λi∗ > 0 and hi (xi ) ≥ 0. ¯ Similarly, we have ¯ λi �hi (xi )T (xi∗ − xi ) ≥ 0. ¯ ¯ Combi...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

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