# Mean Field Game of Controls and An Application To Trade Crowding

arXiv: https://arxiv.org/abs/1610.09904 · HTML: https://mlsanity.com/paper/1610.09904/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2016.

## Built on by (most cited first, 23 of 23)

- [Generalization in Mean Field Games by Learning Master Policies](https://mlsanity.com/paper/2109.09717/) (2021)
- [Deep Learning for Mean Field Games and Mean Field Control with Applications to Finance](https://mlsanity.com/paper/2107.04568/) (2021)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Reinforcement Learning for Mean Field Games, with Applications to Economics](https://mlsanity.com/paper/2106.13755/) (2021)
- [On the Statistical Efficiency of Mean-Field Reinforcement Learning with General Function Approximation](https://mlsanity.com/paper/2305.11283/) (2023)
- [Signatured Deep Fictitious Play for Mean Field Games with Common Noise](https://mlsanity.com/paper/2106.03272/) (2021)
- [Individual-Level Inverse Reinforcement Learning for Mean Field Games](https://mlsanity.com/paper/2202.06401/) (2022)
- [Model-Based RL for Mean-Field Games is not Statistically Harder than Single-Agent RL](https://mlsanity.com/paper/2402.05724/) (2024)
- [Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics](https://mlsanity.com/paper/2109.11403/) (2021)
- [Deep Learning for Principal-Agent Mean Field Games](https://mlsanity.com/paper/2110.01127/) (2021)
- [Independent RL for Cooperative-Competitive Agents: A Mean-Field Perspective](https://mlsanity.com/paper/2403.11345/) (2024)
- [Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problems](https://mlsanity.com/paper/2502.11506/) (2025)
- [Solving Continuous Mean Field Games: Deep Reinforcement Learning for Non-Stationary Dynamics](https://mlsanity.com/paper/2510.22158/) (2025)
- [Deep Learning for Mean Field Games with non-separable Hamiltonians](https://mlsanity.com/paper/2301.02877/) (2023)
- [Bench-MFG: A Benchmark Suite for Learning in Stationary Mean Field Games](https://mlsanity.com/paper/2602.12517/) (2026)
- [Learning Macroeconomic Policies through Dynamic Stackelberg Mean-Field Games](https://mlsanity.com/paper/2403.12093/) (2024)
- [Mean-field Chaos Diffusion Models](https://mlsanity.com/paper/2406.05396/) (2024)
- [Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization](https://mlsanity.com/paper/2407.02765/) (2024)
- [Consensus Based Stochastic Control](https://mlsanity.com/paper/2501.17801/) (2025)
- [Recurrent Structural Policy Gradient for Partially Observable Mean Field Games](https://mlsanity.com/paper/2602.20141/) (2026)
- [Recursive Maximum Likelihood Estimation for Interacting Particle Systems using Virtual Particles](https://mlsanity.com/paper/2605.00786/) (2026)
- [Towards Model-Free Learning in Dynamic Population Games: An Application to Karma Economies](https://mlsanity.com/paper/2605.11042/) (2026)
- [Neural Parameter Calibration for Finite-State Mean Field Games](https://mlsanity.com/paper/2606.23155/) (2026)

## Abstract (arXiv)

In this paper we formulate the now classical problem of optimal liquidation (or optimal trading) inside a Mean Field Game (MFG). This is a noticeable change since usually mathematical frameworks focus on one large trader in front of a "background noise" (or "mean field"). In standard frameworks, the interactions between the large trader and the price are a temporary and a permanent market impact terms, the latter influencing the public price. In this paper the trader faces the uncertainty of fair price changes too but not only. He has to deal with price changes generated by other similar market participants, impacting the prices permanently too, and acting strategically. Our MFG formulation of this problem belongs to the class of "extended MFG", we hence provide generic results to address these "MFG of controls", before solving the one generated by the cost function of optimal trading. We provide a closed form formula of its solution, and address the case of "heterogenous preferences" (when each participant has a different risk aversion). Last but not least we give conditions under which participants do not need to instantaneously know the state of the whole system, but can "learn" it day after day, observing others' behaviors.