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In this short note, we consider an inverse problem to a mean-field games system where we are interested in reconstructing the state-independent running cost function from observed value-function data. We provide an elementary proof of a…

Analysis of PDEs · Mathematics 2024-08-16 Kui Ren , Nathan Soedjak , Kewei Wang , Hongyu Zhai

This work investigates the ambient potential identification problem in inverse Mean-Field Games (MFGs), where the goal is to recover the unknown potential from the value function at equilibrium. We propose a simple yet effective iterative…

Optimization and Control · Mathematics 2025-10-14 Jiajia Yu , Jian-Guo Liu , Hongkai Zhao

Finite-state mean-field games (MFGs) arise as limits of large interacting particle systems and are governed by an MFG system, a coupled forward-backward differential equation consisting of a forward Kolmogorov-Fokker-Planck (KFP) equation…

Optimization and Control · Mathematics 2026-02-16 William Hofgard , Asaf Cohen , Mathieu Laurière

When controlling multi-agent systems, the trade-off between performance and scalability is a major challenge. Here, we address this difficulty by using mean field games (MFGs), which is a framework that deduces the macroscopic dynamics…

Optimization and Control · Mathematics 2021-08-06 Daisuke Inoue , Yuji Ito , Takahito Kashiwabara , Norikazu Saito , Hiroaki Yoshida

Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform optimal execution or statistical arbitrage, where…

Mathematical Finance · Quantitative Finance 2019-04-02 Philippe Casgrain , Sebastian Jaimungal

By following the study in [24], we consider an inverse boundary problem for the mean field game system where a probability density constraint is enforced on the game agents. That is, we consider the case that reflective boundary conditions…

Analysis of PDEs · Mathematics 2024-02-22 Hongyu Liu , Shen Zhang

In this paper, we propose and study the utilization of the Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions $r, k$ and cost function $F$ in a stationary mean field game (MFG) system. This study features several…

Optimization and Control · Mathematics 2023-08-15 Ming-Hui Ding , Hongyu Liu , Guang-Hui Zheng

Mean field games (MFGs) model interactions in large-population multi-agent systems through population distributions. Traditional learning methods for MFGs are based on fixed-point iteration (FPI), where policy updates and induced population…

Machine Learning · Computer Science 2025-02-17 Chenyu Zhang , Xu Chen , Xuan Di

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. In the $n$-player model, each agent aims to…

Optimization and Control · Mathematics 2026-04-16 Lijun Bo , Yijie Huang , Xiang Yu

In this work, we study the contraction conditions of iterative algorithms for stationary and finite-horizon discrete-time regularized mean-field games (MFGs) with multiple populations, where each population only interacts with the state…

Optimization and Control · Mathematics 2026-05-26 Uğur Aydın , Tamer Başar

Mean field games (MFGs) provide a mathematically tractable framework for modelling large-scale multi-agent systems by leveraging mean field theory to simplify interactions among agents. It enables applying inverse reinforcement learning…

Machine Learning · Computer Science 2025-12-02 Yang Chen , Libo Zhang , Jiamou Liu , Michael Witbrock

In this paper, we investigate the interaction of two populations with a large number of indistinguishable agents. The problem consists in two levels: the interaction between agents of a same population, and the interaction between the two…

Optimization and Control · Mathematics 2018-10-30 Alain Bensoussan , Tao Huang , Mathieu Laurière

We propose a control-theoretic framework for evolutionary clustering based on Mean Field Games (MFG). Moving beyond static or heuristic approaches, we formulate the problem as a population dynamics game governed by a coupled…

Numerical Analysis · Mathematics 2026-03-31 Alessio Basti , Fabio Camilli , Adriano Festa

Here, we develop numerical methods for finite-state mean-field games (MFGs) that satisfy a monotonicity condition. MFGs are determined by a system of differential equations with initial and terminal boundary conditions. These non-standard…

Numerical Analysis · Mathematics 2017-05-02 Diogo Gomes , Joao Saude

We consider the problem of representing collective behavior of large populations and predicting the evolution of a population distribution over a discrete state space. A discrete time mean field game (MFG) is motivated as an interpretable…

Machine Learning · Computer Science 2018-04-24 Jiachen Yang , Xiaojing Ye , Rakshit Trivedi , Huan Xu , Hongyuan Zha

The inverse problem method is tested for a class of mean field statistical mechanics models representing a mixture of particles of different species. The robustness of the inversion is investigated for different values of the physical…

Mathematical Physics · Physics 2015-06-12 M. Fedele , C. Vernia , P. Contucci

Mean-field games (MFGs) are a modeling framework for systems with a large number of interacting agents. They have applications in economics, finance, and game theory. Normalizing flows (NFs) are a family of deep generative models that…

Optimization and Control · Mathematics 2023-05-24 Han Huang , Jiajia Yu , Jie Chen , Rongjie Lai

We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows…

Optimization and Control · Mathematics 2023-03-07 J. Frédéric Bonnans , Pierre Lavigne , Laurent Pfeiffer

Mean field games (MFG) and mean field control (MFC) problems have been introduced to study large populations of strategic players. They correspond respectively to non-cooperative or cooperative scenarios, where the aim is to find the Nash…

Computer Science and Game Theory · Computer Science 2023-12-19 Rene Carmona , Gokce Dayanikli , Francois Delarue , Mathieu Lauriere

This paper proposes a novel Mean-Field Game (MFG) framework for large-scale attacker-defender systems aimed at protecting one or multiple High-Value Units (HVUs). Motivated by classical agent-wise attrition models, we introduce a…

Analysis of PDEs · Mathematics 2026-04-03 Avetik Arakelyan , Tigran Bakaryan , Davit Alaverdyan , Naira Hovakimyan , Isaac Kaminer