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# To see this suppose that s corresponds to the global

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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.

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