# Signatured Deep Fictitious Play for Mean Field Games with Common Noise

arXiv: https://arxiv.org/abs/2106.03272 · HTML: https://mlsanity.com/paper/2106.03272/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 56 references that are in the map.
- First posted on arXiv 6 Jun 2021.

## Builds on (most cited first, 27 of 27)

- [A Machine Learning Framework for Solving High-Dimensional Mean Field Game and Mean Field Control Problems](https://mlsanity.com/paper/1912.01825/) (2019)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Fictitious Play for Mean Field Games: Continuous Time Analysis and Applications](https://mlsanity.com/paper/2007.03458/) (2020)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [Deep Learning Approximation for Stochastic Control Problems](https://mlsanity.com/paper/1611.07422/) (2016)
- [Unified Reinforcement Q-Learning for Mean Field Game and Control Problems](https://mlsanity.com/paper/2006.13912/) (2020)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU](https://mlsanity.com/paper/2001.00706/) (2020)
- [Learning in Mean Field Games: the Fictitious Play](https://mlsanity.com/paper/1507.06280/) (2015)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Mean Field Game of Controls and An Application To Trade Crowding](https://mlsanity.com/paper/1610.09904/) (2016)
- [Deep Fictitious Play for Finding Markovian Nash Equilibrium in Multi-Agent Games](https://mlsanity.com/paper/1912.01809/) (2019)
- [Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/1903.09376/) (2019)
- [Convergence of Deep Fictitious Play for Stochastic Differential Games](https://mlsanity.com/paper/2008.05519/) (2020)
- [Efficiency of the Price Formation Process in Presence of High Frequency Participants: a Mean Field Game analysis](https://mlsanity.com/paper/1305.6323/) (2013)
- [Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource](https://mlsanity.com/paper/1607.02130/) (2016)
- [Optimal Policies for a Pandemic: A Stochastic Game Approach and a Deep Learning Algorithm](https://mlsanity.com/paper/2012.06745/) (2020)
- [Reinforcement Learning for Mean Field Game](https://mlsanity.com/paper/1905.13357/) (2019)
- [Contact rate epidemic control of COVID-19: an equilibrium view](https://mlsanity.com/paper/2004.08221/) (2020)
- [The Master Equation in Mean Field Theory](https://mlsanity.com/paper/1404.4150/) (2014)
- [Many-player games of optimal consumption and investment under relative performance criteria](https://mlsanity.com/paper/1905.11782/) (2019)
- [Convolutional Signature for Sequential Data](https://mlsanity.com/paper/2009.06719/) (2020)
- [N-player and Mean-field Games in Ito-diffusion Markets with Competitive or Homophilous Interaction](https://mlsanity.com/paper/2106.00581/) (2021)
- [Wellposedness of Mean Field Games with Common Noise Under a Weak Monotonicity Condition](https://mlsanity.com/paper/1406.7028/) (2014)
- [Translation invariant mean field games with common noise](https://mlsanity.com/paper/1409.7345/) (2014)
- [Stable solutions in potential mean field game systems](https://mlsanity.com/paper/1612.01877/) (2016)

## Built on by (most cited first, 12 of 12)

- [Recent Developments in Machine Learning Methods for Stochastic Control and Games](https://mlsanity.com/paper/2303.10257/) (2023)
- [Global universal approximation of functional input maps on weighted spaces](https://mlsanity.com/paper/2306.03303/) (2023)
- [Reinforcement Learning for Mean Field Games, with Applications to Economics](https://mlsanity.com/paper/2106.13755/) (2021)
- [Learning High-Dimensional McKean-Vlasov Forward-Backward Stochastic Differential Equations with General Distribution Dependence](https://mlsanity.com/paper/2204.11924/) (2022)
- [A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guarantee](https://mlsanity.com/paper/2302.05816/) (2023)
- [Generative models for decision-making under distributional shift](https://mlsanity.com/paper/2604.04342/) (2026)
- [Deep Learning for Systemic Risk Measures](https://mlsanity.com/paper/2207.00739/) (2022)
- [Learning Mean Field Games on Sparse Graphs: A Hybrid Graphex Approach](https://mlsanity.com/paper/2401.12686/) (2024)
- [Global Solutions to Master Equations for Continuous Time Heterogeneous Agent Macroeconomic Models](https://mlsanity.com/paper/2406.13726/) (2024)
- [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026)
- [Multivariate Systemic Risk Measures and Computation by Deep Learning Algorithms](https://mlsanity.com/paper/2302.10183/) (2023)
- [Directed Chain Generative Adversarial Networks](https://mlsanity.com/paper/2304.13131/) (2023)

## Abstract (arXiv)

Existing deep learning methods for solving mean-field games (MFGs) with common noise fix the sampling common noise paths and then solve the corresponding MFGs. This leads to a nested-loop structure with millions of simulations of common noise paths in order to produce accurate solutions, which results in prohibitive computational cost and limits the applications to a large extent. In this paper, based on the rough path theory, we propose a novel single-loop algorithm, named signatured deep fictitious play, by which we can work with the unfixed common noise setup to avoid the nested-loop structure and reduce the computational complexity significantly. The proposed algorithm can accurately capture the effect of common uncertainty changes on mean-field equilibria without further training of neural networks, as previously needed in the existing machine learning algorithms. The efficiency is supported by three applications, including linear-quadratic MFGs, mean-field portfolio game, and mean-field game of optimal consumption and investment. Overall, we provide a new point of view from the rough path theory to solve MFGs with common noise with significantly improved efficiency and an extensive range of applications. In addition, we report the first deep learning work to deal with extended MFGs (a mean-field interaction via both the states and controls) with common noise.