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