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Unformatted text preview: mizing a convex function (or
maximizing a concave function) over a convex constraint set), we can provide
necessary and suﬃcient conditions for the optimality of a feasible solution:
Theorem (4)
Consider the optimization problem maximize
f (x ) subject to gj (x ) ≥ 0, j = 1, . . . , r , where the cost function f : Rn �→ R and 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 .
24 Game Theory: Lecture 6 Continuous Games 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}, (4) 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 ) (5) hi (yi ) ≥ 0. We use the notation �u (x ) to denote �u (x ) = [�1 u1 (x ), . . . , �I uI (x )]T . (6)
25 Game Theory: Lecture 6 Continuous Games 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. (4), where hi is a concave function, and there
˜
exists some xi ∈ Rmi such that hi (xi ) > 0. Ass...
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This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

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