  We consider a large population dynamic game in discrete time. The peculiarity
of the game is that players are characterized by time-evolving types, and so
reasonably their actions should not anticipate the future values of their
types. When interactions between players are of mean-field kind, we relate Nash
equilibria for such games to an asymptotic notion of dynamic Cournot-Nash
equilibria. Inspired by the works of Blanchet and Carlier for the static
situation, we interpret dynamic Cournot-Nash equilibria in the light of causal
optimal transport theory. Further specializing to games of potential type, we
establish existence, uniqueness and characterization of equilibria. Moreover we
develop, for the first time, a numerical scheme for causal optimal transport,
which is then leveraged in order to compute dynamic Cournot-Nash equilibria.
This is illustrated in a detailed case study of a congestion game.
