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Unformatted text preview: games), we can also represent the potential function as a matrix, each entry corresponding to the vector of strategies from the payoﬀ matrix. Example
The matrix P is a potential for the “Prisoner’s dilemma” game described
below:
�
�
�
�
(1, 1) (9, 0)
43
G=
,
P=
(0, 9) (6, 6)
30 14 Game Theory: Lecture 8 Potential Games Pure Strategy Nash Equilibria in Ordinal Potential Games Theorem
Every ﬁnite ordinal potential game has at least one pure strategy Nash
equilibrium.
Proof: The global maximum of an ordinal potential function is a pure
strategy Nash equilibrium. To see this, suppose that s ∗ corresponds
to the global maximum. Then, for any i ∈ I , we have, by deﬁnition,
∗
∗
Φ(si∗ , s−i ) − Φ(s , s−i ) ≥ 0 for all s ∈ Si . But since Φ is a potential
function, for all i and all s ∈ Si ,
∗
∗
ui (si∗ , s−i ) − ui (s , s−i ) ≥ 0 iﬀ ∗
∗
Φ(si∗ , s−i ) − Φ(s , s−i ) ≥ 0. ∗
∗
Therefore, ui (si∗ , s−i ) − ui (s , s−i ) ≥ 0 for all s ∈ Si and for all i ∈ I .
∗ is a pure strategy Nas...
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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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