lecture7 notes

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 =...
View Full Document

## This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.

Ask a homework question - tutors are online