# Deep Fictitious Play for Stochastic Differential Games

arXiv: https://arxiv.org/abs/1903.09376 · HTML: https://mlsanity.com/paper/1903.09376/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Mar 2019.

## Built on by (most cited first, 13 of 13)

- [Reinforcement Learning with Dynamic Convex Risk Measures](https://mlsanity.com/paper/2112.13414/) (2021)
- [Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence](https://mlsanity.com/paper/2204.11924/) (2022)
- [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021)
- [DeepHAM: A Global Solution Method for Heterogeneous Agent Models with Aggregate Shocks](https://mlsanity.com/paper/2112.14377/) (2021)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [Linear convergence of a policy gradient method for some finite horizon continuous time control problems](https://mlsanity.com/paper/2203.11758/) (2022)
- [A Class of Dimension-free Metrics for the Convergence of Empirical Measures](https://mlsanity.com/paper/2104.12036/) (2021)
- [Deep Learning for Systemic Risk Measures](https://mlsanity.com/paper/2207.00739/) (2022)
- [Recurrent Neural Networks for Stochastic Control Problems with Delay](https://mlsanity.com/paper/2101.01385/) (2021)
- [Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures](https://mlsanity.com/paper/2509.12484/) (2025)
- [CMAD: Cooperative Multi-Agent Diffusion via Stochastic Optimal Control](https://mlsanity.com/paper/2602.10933/) (2026)
- [Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms](https://mlsanity.com/paper/2307.06450/) (2023)
- [Deep Fictitious Play-Based Potential Differential Games for Learning Human-Like Interaction at Unsignalized Intersections](https://mlsanity.com/paper/2506.12283/) (2025)

## Abstract (arXiv)

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, it i.e., using GPU parallelization, it can be applied to any N-player stochastic differential game with different symmetries and heterogeneities (it 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 under appropriate assumptions, 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.