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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 Spring '10
 AsuOzdaglar

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