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# M 1 2 m is the set of resources si is the set of

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Unformatted text preview: dition s l �= s k for every 0 ≤ l �= k ≤ N − 1. Theorem A game G is an exact potential game if and only if for all ﬁnite simple closed paths, γ, I (γ) = 0. Moreover, it is suﬃcient to check simple closed paths of length 4. Intuition: Let I (γ) �= 0. If potential existed then it would increase when the cycle is completed. 20 Game Theory: Lecture 8 Potential Games Inﬁnite Potential Games Proposition Let G be a continuous potential game with compact strategy sets. Then G has at least one pure strategy Nash equilibrium. Proposition Let G be a game such that Si ⊆ R and the payoﬀ functions ui : S → R are continuously diﬀerentiable. Let Φ : S → R be a function. Then, Φ is a potential for G if and only if Φ is continuously diﬀerentiable and ∂ ui ( s ) ∂ Φ (s ) = ∂si ∂ si for all i ∈ I and all s ∈ S . 21 Game Theory: Lecture 8 Potential Games Congestion Games Congestion Model: C = �I , M, (Si )i ∈I , (c j )j ∈M � where: I = {1, 2, · · · , I } is the set of players. M = {1, 2, · · · , m} is the set of resources. Si...
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