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Unformatted text preview: physical
systems. It will be useful both for locating pure strategy Nash
equilibria and also for the analysis of “myopic” dynamics. 12 Game Theory: Lecture 8 Potential Games Potential Functions and Games
Let G = �I , (Si ), (ui )� be a strategic form game.
Deﬁnition
A function Φ : S → R is called an ordinal potential function for the game G if for
each i ∈ I and all s−i ∈ S−i ,
ui (x , s−i ) − ui (z , s−i ) > 0 iﬀ Φ(x , s−i ) − Φ(z , s−i ) > 0, for all x , z ∈ Si .
G is called an ordinal potential game if it admits an ordinal potential.
Deﬁnition
A function Φ : S → R is called an (exact) potential function for the game G if
for each i ∈ I and all s−i ∈ S−i ,
ui (x , s−i ) − ui (z , s−i ) = Φ(x , s−i ) − Φ(z , s−i ), for all x , z ∈ Si .
G is called an (exact) potential game if it admits a potential.
13 Game Theory: Lecture 8 Potential Games Example
A potential function assigns a real value for every s ∈ S . Thus, when we represent the game payoﬀs with a matrix (in ﬁnite...
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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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