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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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 Spring '10
 AsuOzdaglar

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