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r then a vector x rn is an optimal solution of the

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Unformatted text preview: d the constraint functions gj : Rn �→ R ¯ are concave functions. Assume also that there exists some x such that gj (x ) > 0 ¯ for all j = 1, . . . , r . Then a vector x ∗ ∈ Rn is an optimal solution of the preceding problem if and only if gj (x ∗ ) ≥ 0 for all j = 1, . . . , r , and 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 . 6 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Uniqueness of a Pure Strategy Nash Equilibrium We now return to the analysis of the uniqueness of a pure strategy equilibrium in strategic form games. We assume that for player i ∈ I , the strategy set Si is given by Si = {xi ∈ Rmi | hi (xi ) ≥ 0}, (3) Rmi where hi : �→ R is a concave function. Since hi is concave, it follows that the set Si is a convex set (exercise!). Therefore the set of strategy profiles S = ∏I =1 Si ⊂ ∏I =1 Rmi , being a i i Cartesian product of convex sets, is a convex set. Given these strategy sets, a vector x ∗ ∈ ∏I =1 Rmi is a pure strategy Nash i equilibrium if and only if for all i ∈ I , xi∗ is an optimal solution of maximizeyi ∈Rmi subject to ∗ ui (yi , x−i ) (4) hi (yi ) ≥ 0. We use the notation �u (x ) to denote �u (x ) = [�1 u1 (x ), . . . , �I uI (x )]T . (5) 7 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Uniqueness of a Pure Strategy Nash Equilibrium We introduce the key condition for uniqueness of a pure strategy Nash equilibrium. Definition We say that the payoff functions (u1 , . . . , uI ) are diagonally strictly concave for ¯ x ∈ S , if for every x ∗ , x ∈ S , we have (x − x ∗ )T �u (x ∗ ) + (x ∗ − x )T �u (x ) > 0. ¯ ¯ ¯ Theorem Consider a strategic form game �I , (Si ), (ui )�. For all i ∈ I , assume that the strategy sets Si are given by Eq. (3), where hi is a concave function, and there ˜ exists some xi ∈ Rmi such that hi (xi ) > 0. Assume also that the payoff functions ˜ (u1 , . . . , uI ) are diagonally strictly concave for x ∈ S . Then the game has a unique pure strategy Nash equilibrium. 8 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium 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 (4), Theorem 2 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, (6) λi∗ hi (xi∗ ) = 0, (7) ¯ �i ui (x ) + λi �hi (xi ) = 0, ¯ ¯ ¯ i hi (xi ) = 0. λ ¯ (8) and (9) 9 Game Theory: Lecture 7 Uniqueness of a Pure Strategy Equilibrium Proof Multiplying Eqs. (6) and (8) by (xi − xi∗ )T and (xi∗ − xi )T respectively, and ¯ ¯ adding over all i ∈ I , we obtain 0 =...
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