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This work investigates continuous time stochastic differential games with alarge number of players, whose costs and dynamics interact through theempirical distribution of both their states and their controls. The controlprocesses are assumed to be open-loop. We give regularity conditionsguaranteeing that if the finite-player game admits a Nash equilibrium, thenboth the sequence of equilibria and the corresponding states processes satisfya Sanov-type large deviation principle. The result requires existence of aLipschitz continuous solution of the master equation of the corresponding meanfield game, and is based on concentration inequalities for Lipschitz FBSDEs.The result carries over to cooperative (i.e. central planner) games. We studythe linear-quadratic case of such games in details.

Laplace principle for large population games with control interaction