Coarse correlated equilibria for continuous time mean field games in open loop strategies

In the framework of continuous time symmetric stochastic differential games in open loop strategies, we introduce a generalization of mean field game solution, called coarse correlated solution. This can be seen as the analogue of a coarse correlated equilibrium in the $N$-player game, where a moderator randomly generates a strategy profile and asks the players to pre-commit to such strategies before disclosing them privately to each one of them; such a profile is a coarse correlated equilibrium if no player has an incentive to unilaterally deviate. We justify our definition by showing that a coarse correlated solution for the mean field game induces a sequence of approximate coarse correlated equilibria with vanishing error for the underlying $N$-player games. Existence of coarse correlated solutions for the mean field game is proved by means of a minimax theorem. An example with explicit solutions is discussed as well.