Deep Fictitious Play for Stochastic Differential Games
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In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric N-player non-zero-sum stochastic differential games, for which we refer as deep fictitious play, a multi-stage learning process. Specifically at each stage, we propose the strategy of letting individual player optimize her own payoff subject to the other players' previous actions, equivalent to solve N decoupled stochastic control optimization problems, which are 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., 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 over a small time horizon, and verify that the convergent limit forms an open-loop Nash equilibrium. We also discuss the extensions to other strategies designed upon fictitious play and closed-loop Nash equilibrium in the end.
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