Unformatted text preview: 1] for i = 1, 2, and the payoﬀs
u1 (s1 , s2 ) = s1 s2 − 2
s1
,
2 2
s2
.
2
Note that ui (s1 , s2 ) is strictly concave in si . It can be seen in this example
that the best response correspondences (which are uniquevalued) are
given by
B1 (s2 ) = s2 ,
B2 (s1 ) = s1 . u2 (s1 , s2 ) = s1 s2 − Plotting the best response curves shows that any pure strategy proﬁle
(s1 , s2 ) = (x , x ) for x ∈ [0, 1] is a pure strategy Nash equilibrium.
21 Game Theory: Lecture 6 Continuous Games Uniqueness of a Pure Strategy Nash Equilibrium
We will next establish conditions that guarantee that a strategic form
game has a unique pure strategy Nash equilibrium, following the
classical paper [Rosen 65].
Notation:
Given a scalarvalued function f : Rn �→ R, we use the notation
�f (x ) to denote the gradient vector of f at point x , i.e.,
�
�
∂f (x )
∂f (x ) T
�f (x ) =
,...,
.
∂ x1
∂ xn
Given a scalarvalued function u : ∏I =1 Rmi �→ R, we use the
i
notation �i u (x ) to denote the gradient vector of u with respect to xi
at point x , i.e.,
�
�T
∂u (x )
∂u (x )
�i u (x ) =
,...,
.
(2)
∂ximi
∂xi1
22 Game Theory: Lecture 6 Continuous Games Optimality Conditions for Nonlinear Optimization Problems
Theorem (3)
(KarushKuhnTucker conditions) Let x ∗ be an optimal solution of 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 continuously diﬀerentiable. Denote the set of active constraints at x ∗ as
A(x ∗ ) = {j = 1, . . . , r  gj (x ∗ ) = 0}. Assume that the active constraint
gradients, �gj (x ∗ ), j ∈ A(x ∗ ), are linearly independent vectors. Then, 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 . (3)
23 Game Theory: Lecture 6 Continuous Games Optimality Conditions for Nonlinear Optimization Problems
For convex optimization problems (i.e., mini...
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 Spring '10
 AsuOzdaglar
 Game Theory, Nash, continuous games

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