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Proposition 2 if g is an approximation to g and is an

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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 α-approximation 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 α-approximation 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.

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