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10 game theory lecture 7 uniqueness of a pure strategy

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Unformatted text preview: ( ¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x ) x ¯ ¯ (10) ∗ ∗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 ). ¯ ¯ 10 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium 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. (7), 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. ¯ ¯ Combining the preceding two relations with the relation in (10) yields a contradiction, thus concluding the proof. 11 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Sufficient Condition for Diagonal Strict Concavity Let U (x ) denote the Jacobian of �u (x ) [see Eq. (5)]. In particular, if the xi are all 1-dimensional, then U (x ) is given by ⎛ ∂2 u (x ) ∂2 u (x ) ⎞ 1 1 ··· 2 ∂x1 ∂x2 ⎜ ∂x1 ⎟ ⎜2 ⎟ .. U (x ) = ⎜ ∂ u2 (x ) ⎟. . ⎝ ∂x2 ∂x1 ⎠ . . . Proposition For all i ∈ I , assume that the strategy sets Si are given by Eq. (3), where hi is a concave function. Assume that the symmetric matrix (U (x ) + U T (x )) is negative definite for all x ∈ S , i.e., for all x ∈ S , we have y T (U (x ) + U T (x ))y < 0, ∀ y �= 0. Then, the payoff functions (u1 , . . . , uI ) are diagonally strictly concave for x ∈ S . 12 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Proof Let x ∗ , x ∈ S . Consider the vector ¯ x ( λ ) = λx ∗ + (1 − λ )x , ¯ for some λ ∈ [0, 1]. Since S is a convex set, x (λ) ∈ S . Because U (x ) is the Jacobian of �u (x ), we have d �u (x (λ)) dλ = or �1 0 dx (λ) d (λ) U (x (λ))(x ∗ − x ), ¯ = U (x (λ)) U (x (λ))(x ∗ − x )d λ = �u (x ∗ ) − �u (x ). ¯ ¯ 13 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Proof Multiplying the preceding by (x − x ∗ )T yields ¯ ( ¯ − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x ) x ¯ ¯ �1 1 =− (x ∗ − x )T [U (x (λ)) + U T (x (λ))](x ∗ − x )d λ ¯ ¯ 20 > 0, where to get the strict inequality we used the assumption that the symmetric matrix (U (x ) + U T (x )) is negative definite for all x ∈ S . 14 Game Theory: Lecture 7 Supermodular Games Supermodular Games Supermodular games are those characterized by strategic complementarities Informally, this means that the marginal utility of increasing a player’s strategy raises with increases in the other players’ strategies. Implication ⇒ best response of a player is a nondecreasing f...
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