Deep fictitious play for stochastic differential games

In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop learning theory algorithms for computing Nash equilibrium asymmetric $N$-player non-zero-sum stochastic differential games, which refer as \emph{deep play}, a multi-stage process. Specifically at each stage, propose strategy letting individual player optimize her own payoff subject other players' previous actions, equivalent solve $N$ decoupled control optimization problems, are approximated by DNNs. Therefore, leads structure consisting DNNs, only communicate end stage. The resulted algorithm based on is scalable, parallel model-free, {\it i.e.}, using GPU parallelization, it can be applied any game with different symmetries heterogeneities ({\it e.g.}, existence major players). We illustrate performance comparing closed-form solution linear quadratic game. Moreover, prove convergence under appropriate assumptions, verify that convergent limit forms an open-loop equilibrium. also discuss extensions strategies designed upon closed-loop in end.