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Mean field type games (MFTGs) describe Nash equilibria between large coalitions: each coalition consists of a continuum of cooperative agents who maximize the average reward of their coalition while interacting non-cooperatively with a…

Computer Science and Game Theory · Computer Science 2025-07-29 Kai Shao , Jiacheng Shen , Mathieu Laurière

The paper studies the convergence, as $N$ tends to infinity, of a system of $N$ coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. We describe the limit problem in terms of the so-called…

Analysis of PDEs · Mathematics 2015-09-09 Pierre Cardaliaguet , François Delarue , Jean-Michel Lasry , Pierre-Louis Lions

In this tutorial, we provide an introduction to machine learning methods for finding Nash equilibria in games with large number of agents. These types of problems are important for the operations research community because of their…

Optimization and Control · Mathematics 2024-06-18 Gokce Dayanikli , Mathieu Lauriere

Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient-flow method based on the variational characterization of certain MFG. The second one uses…

Numerical Analysis · Mathematics 2015-11-23 Noha Almulla , Rita Ferreira , Diogo Gomes

We consider the variational structure of a time-fractional second order Mean Field Games (MFG) system with local coupling. The MFG system consists of time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations. In such a situation…

Analysis of PDEs · Mathematics 2019-07-03 Tang Qing , Fabio Camilli

This paper studies mean field games for multi-agent systems with control-dependent multiplicative noises. For the general systems with nonuniform agents, we obtain a set of decentralized strategies by solving an auxiliary limiting optimal…

Optimization and Control · Mathematics 2019-06-10 Bing-Chang Wang , Yuan-Hua Ni , Huanshui Zhang

This paper presents a comprehensive study of linear-quadratic (LQ) mean field games (MFGs) in Hilbert spaces, generalizing the classic LQ MFG theory to scenarios involving $N$ agents with dynamics governed by infinite-dimensional stochastic…

Optimization and Control · Mathematics 2025-08-12 Hanchao Liu , Dena Firoozi

In this paper we study a mean-field games system with Dirichlet boundary conditions in a closed domain and in a mean-field of control setting, that is in which the dynamics of each agent is affected not only by the average position of the…

Optimization and Control · Mathematics 2023-06-21 Mattia Bongini , Francesco Salvarani

In this paper, we introduce a natural learning rule for mean field games with finite state and action space, the so-called myopic adjustment process. The main motivation for these considerations are the complex computations necessary to…

Optimization and Control · Mathematics 2020-09-01 Berenice Anne Neumann

This paper is interested in the problem of optimal stopping in a mean field game context. The notion of mixed solution is introduced to solve the system of partial differential equations which models this kind of problem. This notion…

Analysis of PDEs · Mathematics 2017-06-14 C. Bertucci

Mean field games (MFGs) offer a versatile framework for modeling large-scale interactive systems across multiple domains. This paper builds upon a previous work, by developing a state-of-the-art unified approach to decode or design the…

Analysis of PDEs · Mathematics 2025-01-22 Hongyu Liu , Catharine W. K. Lo

The partially observed major minor LQG and nonlinear mean field game (PO MM LQG MFG) systems where it is assumed the major agent's state is partially observed by each minor agent, and the major agent completely observes its own state have…

Optimization and Control · Mathematics 2020-09-11 Dena Firoozi , Peter E. Caines

This paper establishes the existence of equilibria result of a class of mean field games with singular controls. The interaction takes place through both states and controls. A relaxed solution approach is used. To circumvent the tightness…

Optimization and Control · Mathematics 2022-05-10 Guanxing Fu

We formulate a stochastic game of mean field type where the agents solve optimal stopping problems and interact through the proportion of players that have already stopped. Working with a continuum of agents, typical equilibria become…

Optimization and Control · Mathematics 2017-12-01 Marcel Nutz

Mean field games are concerned with the limit of large-population stochastic differential games where the agents interact through their empirical distribution. In the classical setting, the number of players is large but fixed throughout…

Optimization and Control · Mathematics 2019-12-30 Julien Claisse , Zhenjie Ren , Xiaolu Tan

We introduce a new mean field kinetic model for systems of rational agents interacting in a game theoretical framework. This model is inspired from non-cooperative anonymous games with a continuum of players and Mean-Field Games. The large…

Mathematical Physics · Physics 2012-12-27 Pierre Degond , Jian-Guo Liu , Christian Ringhofer

In Mean Field Games of Controls, the dynamics of the single agent is influenced not only by the distribution of the agents, as in the classical theory, but also by the distribution of their optimal strategies. In this paper, we study…

Analysis of PDEs · Mathematics 2023-02-01 Fabio Camilli , Claudio Marchi

First, we study the existence of solutions for a class of first order mean field games systems \begin{equation*} \left\{\begin{aligned} &H(x,u,Du)=F(x,m(t)),\quad &&x\in M,\ \forall\ t\in[0,T],\\ &\partial_t…

Analysis of PDEs · Mathematics 2025-07-15 Xiaotian Hu

In this manuscript we derive a new nonlinear transport equation written on the space of probability measures that allows to study a class of deterministic mean field games and master equations, where the interaction of the agents happens…

Analysis of PDEs · Mathematics 2024-03-25 P. Jameson Graber , Alpár R. Mészáros

We study mean-field games of optimal stopping (OS-MFGs) and introduce an entropy-regularized framework to enable learning-based solution methods. By utilizing randomized stopping times, we reformulate the OS-MFG as a mean-field game of…

Optimization and Control · Mathematics 2025-09-24 Jodi Dianetti , Roxana Dumitrescu , Giorgio Ferrari , Renyuan Xu