Potential Games

Consider a symmetric oligopoly Cournot competition with linear cost functions ci (qi ) = cqi , 1 ≤ i ≤ n. The inverse demand function, F(Q), Q > 0, is a positive function (no monotonicity, continuity, or differentiability assumptions on F are needed). The profit function of Firm i is defined on Rn ++ as 5i (q1, q2, . . . , qn) = F(Q)qi − cqi , where Q =∑nj=1 qj . Define a function P: Rn ++ −→ R: P(q1, q2, . . . , qn) = q1q2 · · · qn(F(Q)− c). For every Firm i , and for every q−i ∈ Rn−1 ++ , 5i (qi , q−i )−5i (xi , q−i ) > 0, iff P(qi , q−i )− P(xi , q−i ) > 0, ∀qi , xi ∈ R++. (1.1)