Mean Field Games

Mean-Field Price Formation on Trees with a Network of Relative Performance Concerns ArXiv ID: 2512.21621 “View on arXiv” Authors: Masaaki Fujii Abstract Financial firms and institutional investors are routinely evaluated based on their performance relative to their peers. These relative performance concerns significantly influence risk-taking behavior and market dynamics. While the literature studying Nash equilibrium under such relative performance competitions is extensive, its effect on asset price formation remains largely unexplored. This paper investigates mean-field equilibrium price formation of a single risky stock in a discrete-time market where agents exhibit exponential utility and relative performance concerns. Unlike existing literature that typically treats asset prices as exogenous, we impose a market-clearing condition to determine the price dynamics endogenously within a relative performance equilibrium. Using a binomial tree framework, we establish the existence and uniqueness of the market-clearing mean-field equilibrium in both single- and multi-population settings. Finally, we provide illustrative numerical examples demonstrating the equilibrium price distributions and agents’ optimal position sizes. ...

Deep Learning and Elicitability for McKean-Vlasov FBSDEs With Common Noise ArXiv ID: 2512.14967 “View on arXiv” Authors: Felipe J. P. Antunes, Yuri F. Saporito, Sebastian Jaimungal Abstract We present a novel numerical method for solving McKean-Vlasov forward-backward stochastic differential equations (MV-FBSDEs) with common noise, combining Picard iterations, elicitability and deep learning. The key innovation involves elicitability to derive a path-wise loss function, enabling efficient training of neural networks to approximate both the backward process and the conditional expectations arising from common noise - without requiring computationally expensive nested Monte Carlo simulations. The mean-field interaction term is parameterized via a recurrent neural network trained to minimize an elicitable score, while the backward process is approximated through a feedforward network representing the decoupling field. We validate the algorithm on a systemic risk inter-bank borrowing and lending model, where analytical solutions exist, demonstrating accurate recovery of the true solution. We further extend the model to quantile-mediated interactions, showcasing the flexibility of the elicitability framework beyond conditional means or moments. Finally, we apply the method to a non-stationary Aiyagari–Bewley–Huggett economic growth model with endogenous interest rates, illustrating its applicability to complex mean-field games without closed-form solutions. ...

Optimal hedging of an informed broker facing many traders ArXiv ID: 2506.08992 “View on arXiv” Authors: Philippe Bergault, Pierre Cardaliaguet, Wenbin Yan Abstract This paper investigates the optimal hedging strategies of an informed broker interacting with multiple traders in a financial market. We develop a theoretical framework in which the broker, possessing exclusive information about the drift of the asset’s price, engages with traders whose trading activities impact the market price. Using a mean-field game approach, we derive the equilibrium strategies for both the broker and the traders, illustrating the intricate dynamics of their interactions. The broker’s optimal strategy involves a Stackelberg equilibrium, where the broker leads and the traders follow. Our analysis also addresses the mean field limit of finite-player models and shows the convergence to the mean-field solution as the number of traders becomes large. ...

Mean Field Game of Optimal Tracking Portfolio ArXiv ID: 2505.01858 “View on arXiv” Authors: Lijun Bo, Yijie Huang, Xiang Yu Abstract This paper studies the mean field game (MFG) problem arising from a large population competition in fund management, featuring a new type of relative performance via the benchmark tracking constraint. In the n-agent model, each agent can strategically inject capital to ensure that the total wealth outperforms the benchmark process, which is modeled as a linear combination of the population’s average wealth process and a market index process. That is, each agent is concerned about the performance of her competitors captured by the floor constraint. With a continuum of agents, we formulate the constrained MFG problem and transform it into an equivalent unconstrained MFG problem with a reflected state process. We establish the existence of the mean field equilibrium (MFE) using the partial differential equation (PDE) approach. Firstly, by applying the dual transform, the best response control of the representative agent can be characterized in analytical form in terms of a dual reflected diffusion process. As a novel contribution, we verify the consistency condition of the MFE in separated domains with the help of the duality relationship and properties of the dual process. ...

Unified continuous-time q-learning for mean-field game and mean-field control problems ArXiv ID: 2407.04521 “View on arXiv” Authors: Unknown Abstract This paper studies the continuous-time q-learning in mean-field jump-diffusion models when the population distribution is not directly observable. We propose the integrated q-function in decoupled form (decoupled Iq-function) from the representative agent’s perspective and establish its martingale characterization, which provides a unified policy evaluation rule for both mean-field game (MFG) and mean-field control (MFC) problems. Moreover, we consider the learning procedure where the representative agent updates the population distribution based on his own state values. Depending on the task to solve the MFG or MFC problem, we can employ the decoupled Iq-function differently to characterize the mean-field equilibrium policy or the mean-field optimal policy respectively. Based on these theoretical findings, we devise a unified q-learning algorithm for both MFG and MFC problems by utilizing test policies and the averaged martingale orthogonality condition. For several financial applications in the jump-diffusion setting, we obtain the exact parameterization of the decoupled Iq-functions and the value functions, and illustrate our q-learning algorithm with satisfactory performance. ...

Mean field equilibrium asset pricing model with habit formation ArXiv ID: 2406.02155 “View on arXiv” Authors: Unknown Abstract This paper presents an asset pricing model in an incomplete market involving a large number of heterogeneous agents based on the mean field game theory. In the model, we incorporate habit formation in consumption preferences, which has been widely used to explain various phenomena in financial economics. In order to characterize the market-clearing equilibrium, we derive a quadratic-growth mean field backward stochastic differential equation (BSDE) and study its well-posedness and asymptotic behavior in the large population limit. Additionally, we introduce an exponential quadratic Gaussian reformulation of the asset pricing model, in which the solution is obtained in a semi-analytic form. ...

A Mean Field Game between Informed Traders and a Broker ArXiv ID: 2401.05257 “View on arXiv” Authors: Unknown Abstract We find closed-form solutions to the stochastic game between a broker and a mean-field of informed traders. In the finite player game, the informed traders observe a common signal and a private signal. The broker, on the other hand, observes the trading speed of each of his clients and provides liquidity to the informed traders. Each player in the game optimises wealth adjusted by inventory penalties. In the mean field version of the game, using a Gâteaux derivative approach, we characterise the solution to the game with a system of forward-backward stochastic differential equations that we solve explicitly. We find that the optimal trading strategy of the broker is linear on his own inventory, on the average inventory among informed traders, and on the common signal or the average trading speed of the informed traders. The Nash equilibrium we find helps informed traders decide how to use private information, and helps brokers decide how much of the order flow they should externalise or internalise when facing a large number of clients. ...