# Convergence of Deep Fictitious Play for Stochastic Differential Games

arXiv: https://arxiv.org/abs/2008.05519 · HTML: https://mlsanity.com/paper/2008.05519/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Aug 2020.

## Built on by (most cited first, 10 of 10)

- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [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)
- [Convergence of the Backward Deep BSDE Method with Applications to Optimal Stopping Problems](https://mlsanity.com/paper/2210.04118/) (2022)
- [A Class of Dimension-free Metrics for the Convergence of Empirical Measures](https://mlsanity.com/paper/2104.12036/) (2021)
- [Recurrent Neural Networks for Stochastic Control Problems with Delay](https://mlsanity.com/paper/2101.01385/) (2021)
- [Deep Graphic FBSDEs for Opinion Dynamics Stochastic Control](https://mlsanity.com/paper/2204.02506/) (2022)
- [Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures](https://mlsanity.com/paper/2509.12484/) (2025)
- [Deep Fictitious Play-Based Potential Differential Games for Learning Human-Like Interaction at Unsignalized Intersections](https://mlsanity.com/paper/2506.12283/) (2025)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)

## Abstract (arXiv)

Stochastic differential games have been used extensively to model agents' competitions in Finance, for instance, in P2P lending platforms from the Fintech industry, the banking system for systemic risk, and insurance markets. The recently proposed machine learning algorithm, deep fictitious play, provides a novel efficient tool for finding Markovian Nash equilibrium of large N-player asymmetric stochastic differential games [J. Han and R. Hu, Mathematical and Scientific Machine Learning Conference, pages 221-245, PMLR, 2020]. By incorporating the idea of fictitious play, the algorithm decouples the game into N sub-optimization problems, and identifies each player's optimal strategy with the deep backward stochastic differential equation (BSDE) method parallelly and repeatedly. In this paper, we prove the convergence of deep fictitious play (DFP) to the true Nash equilibrium. We can also show that the strategy based on DFP forms an eps-Nash equilibrium. We generalize the algorithm by proposing a new approach to decouple the games, and present numerical results of large population games showing the empirical convergence of the algorithm beyond the technical assumptions in the theorems.