  We study a class of games with a continuum of players for which 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.
