This preview shows page 1. Sign up to view the full content.
Unformatted text preview: d
ui : S → R are bounded (measurable) functions.
˜
We deﬁne the distance between the utility function proﬁles u and u as
˜
max sup ui (s ) − ui (s ).
˜
i ∈I s ∈S Consider two strategic form games deﬁned by two proﬁles of utility functions:
˜
G = �I , (Si ), (ui )�,
G = �I , (Si ), (ui )�.
˜
If σ is a mixed strategy Nash equilibrium of G , then σ need not be a mixed
˜
strategy Nash equilibrium of G .
˜
Even if u and u are very close, the equilibria of G and G may be far apart.
˜
For example, assume there is only one player, S1 = [0, 1], u1 (s1 ) = �s1 ,
and u1 (s1 ) = −�s1 , where � > 0 is a suﬃciently small scalar. The
˜
∗
˜
unique equilibrium of G is s1 = 1, and the unique equilibrium of G is
∗ = 0, even if the distance b etween u and u is only 2�.
˜
s1
5 Game Theory: Lecture 6 Continuous Games Closeness of Two Games and �Equilibrium
However, if u and u are very close, there is a sense in which the equilibria of
˜
˜
G are “almost” equilibria of G .
Deﬁnition
(�equilibrium) Given � ≥ 0, a mixed strategy σ ∈ Σ is called an �equilibrium if
for all i ∈ I and si ∈ Si ,
ui (si , σ−i ) ≤ ui (σi , σ−i ) + �.
Obviously, when � = 0, an �equilibrium is a Nash equilibrium in the usual sense. 6 Game Theory: Lecture 6 Continuous Games Continuity Property of �equilibria
Proposition (1)
Let G be a continuous game. Assume that σk → σ, �k → �, and for each k , σk
is an �k equilibrium of G . Then σ is an �equilibrium of G .
Proof:
For all i ∈ I , and all si ∈ Si , we have ui (si , σk i ) ≤ ui (σk ) + �k , −
Taking the limit as k → ∞ in the preceding relation, and using the
continuity of the utility functions (together with the convergence of
probability distributions under weak topology), we obtain,
ui (si , σ−i ) ≤ ui (σ ) + �,
establishing the result.
7 Game Theory: Lecture 6 Continuous Games Closeness of Two Games
We next deﬁne formally the closeness of two strategic form games.
Deﬁnition
Let G and G � be two strategic form games with
G = ...
View
Full
Document
This document was uploaded on 03/19/2014 for the course EECS 6.254 at MIT.
 Spring '10
 AsuOzdaglar

Click to edit the document details