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
