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Unformatted text preview: I , (Si ), (ui )�, G � = �I , (Si ), (ui� )�. Then G � is an α−approximation to G if for all i ∈ I and s ∈ S , we have ui (s ) − ui� (s ) ≤ α. 8 Game Theory: Lecture 6 Continuous Games �−equilibria of Close Games
The next proposition relates the �−equilibria of close games.
Proposition (2)
If G � is an αapproximation to G and σ is an �equilibrium of G � , then σ is
an (� + 2α)equilibrium of G .
Proof: For all i ∈ I and all si ∈ Si , we have
ui (si , σ−i ) − ui (σ) = ui (si , σ−i ) − ui� (si , σ−i ) + ui� (si , σ−i ) − ui� (σ) +ui� (σ) − ui (σ)
≤ α+�+α
= � + 2α. 9 Game Theory: Lecture 6 Continuous Games Approximating a Continuous Game with an Essentially
Finite Game
The next proposition shows that we can approximate a continuous
game with an essentially ﬁnite game to an arbitrary degree of
accuracy.
Proposition (3)
For any continuous game G and any α > 0, there exists an “essentially
ﬁnite” game which is an αapproximation to G . 10 Game Theory: Lecture 6 Continuous Games Proof Since S is a compact metric space, the utility functions ui are uniformly
continuous, i.e., for all α > 0, there exists some � > 0 such that
ui ( s ) − ui ( t ) ≤ α for all d (s , t ) ≤ �. Since Si is a compact metric space, it can be covered with ﬁnitely many
open balls Uij , each with radius less than � (assume without loss of
generality that these balls are disjoint and nonempty).
Choose an sij ∈ Uij for each i , j .
Deﬁne the “essentially ﬁnite” game G � with the utility functions ui� deﬁned
as
j
ui� (s ) = ui (s1 , . . . , sIj ), ∀ s ∈ Uj = I ∏ Uk .
j k =1 Then for all s ∈ S and all i ∈ I , we have ui� (s ) − ui (s ) ≤ α,
since d (s , s j ) ≤ � for all j , implying the desired result.
11 Game Theory: Lecture 6 Continuous Games Proof of Glicksberg’s Theorem
We now return to the proof of Glicksberg’s Theorem. Let {αk } be a scalar
sequence with αk ↓ 0.
For each αk , there exists an “essentially ﬁnite” αk approximation G k
of G by Prop...
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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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