We study a class of games with a continuum of players for which a Cournot-Nash equilibria can be obtained by the minimisation of some cost related to optimal transport. This cost is not convex in the usual sense, in general, but it turns out to have hidden strict convexity properties in many relevant cases. This enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations, a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria.
Cournot-Nash equilibria; mean field games; optimal transport; externalities; Monge-Ampère equations; convexity along generalised geodesics;
Mathematics of Operations Research, vol. 41, n. 1, July 16, 2015, pp. 125–145