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# Suppose now that p q a bq and costs ci qi are

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Unformatted text preview: h equilibrium. Hence s Note, however, that there may also be other pure strategy Nash equilibria corresponding to local maxima. 15 Game Theory: Lecture 8 Potential Games Examples of Ordinal Potential Games Example: Cournot competition. I ﬁrms choose quantity qi ∈ (0, ∞) The payoﬀ function for player i given by ui (qi , q−i ) = qi (P (Q ) − c ). � � We deﬁne the function Φ(q1 , · · · , qI ) = ∏I =1 qi (P (Q ) − c ). i Note that for all i and all q−i > 0, ui (qi , q−i ) − ui (qi� , q−i ) > 0 iﬀ Φ(qi , q−i ) − Φ(qi� , q−i ) > 0, ∀ qi , qi� > 0. Φ is therefore an ordinal potential function for this game. 16 Game Theory: Lecture 8 Potential Games Examples of Exact Potential Games Example: Cournot competition (again). Suppose now that P (Q ) = a − bQ and costs ci (qi ) are arbitrary. We deﬁne the function I I i =1 i =1 Φ∗ (q1 , · · · , qn ) = a ∑ qi − b ∑ qi2 − b I ∑ 1≤i <l ≤I I qi ql − ∑ ci (qi ). i =1 It can be show...
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