# Deep Learning for Principal-Agent Mean Field Games

arXiv: https://arxiv.org/abs/2110.01127 · HTML: https://mlsanity.com/paper/2110.01127/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 41 references that are in the map.
- First posted on arXiv 3 Oct 2021.

## Builds on (most cited first, 13 of 13)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [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)
- [Deep Learning Methods for Mean Field Control Problems with Delay](https://mlsanity.com/paper/1905.00358/) (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)
- [A tale of a Principal and many many Agents](https://mlsanity.com/paper/1608.05226/) (2016)
- [Dynamic programming approach to principal-agent problems](https://mlsanity.com/paper/1510.07111/) (2015)
- [An integral control formulation of Mean-field game based large scale coordination of loads in smart grids](https://mlsanity.com/paper/1803.00040/) (2018)
- [Moral Hazard in Dynamic Risk Management](https://mlsanity.com/paper/1406.5852/) (2014)
- [Convex Analysis for LQG Systems with Applications to Major Minor LQG Mean-Field Game Systems](https://mlsanity.com/paper/1810.07551/) (2018)
- [On mean field games models for exhaustible commodities trade](https://mlsanity.com/paper/1807.10344/) (2018)

## Built on by (most cited first, 5 of 5)

- [Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer](https://mlsanity.com/paper/2201.13094/) (2022)
- [Reinforcement Learning with Dynamic Convex Risk Measures](https://mlsanity.com/paper/2112.13414/) (2021)
- [Multi-Agent Reinforcement Learning for Greenhouse Gas Offset Credit Markets](https://mlsanity.com/paper/2504.11258/) (2025)
- [Learning Macroeconomic Policies through Dynamic Stackelberg Mean-Field Games](https://mlsanity.com/paper/2403.12093/) (2024)
- [DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems](https://mlsanity.com/paper/2511.04309/) (2025)

## Abstract (arXiv)

Here, we develop a deep learning algorithm for solving Principal-Agent (PA) mean field games with market-clearing conditions -- a class of problems that have thus far not been studied and one that poses difficulties for standard numerical methods. We use an actor-critic approach to optimization, where the agents form a Nash equilibria according to the principal's penalty function, and the principal evaluates the resulting equilibria. The inner problem's Nash equilibria is obtained using a variant of the deep backward stochastic differential equation (BSDE) method modified for McKean-Vlasov forward-backward SDEs that includes dependence on the distribution over both the forward and backward processes. The outer problem's loss is further approximated by a neural net by sampling over the space of penalty functions. We apply our approach to a stylized PA problem arising in Renewable Energy Certificate (REC) markets, where agents may rent clean energy production capacity, trade RECs, and expand their long-term capacity to navigate the market at maximum profit. Our numerical results illustrate the efficacy of the algorithm and lead to interesting insights into the nature of optimal PA interactions in the mean-field limit of these markets.