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This paper provides a mathematical study of the well-posedness of master equation on finite state space involving terms modelling common noise. In this setting, the solution of the master equation depends on an additional variable modelling…

Analysis of PDEs · Mathematics 2024-03-06 Charles Bertucci , Charles Meynard

The goal of the paper is to introduce a formulation of the mean field game with major and minor players as a fixed point on a space of controls. This approach emphasizes naturally the role played by McKean-Vlasov dynamics in some of the…

Probability · Mathematics 2016-10-19 Rene Carmona , Peiqi Wang

This paper establishes unique solvability of a class of Graphon Mean Field Game equations. The special case of a constant graphon yields the result for the Mean Field Game equations.

Optimization and Control · Mathematics 2022-03-14 Peter E. Caines , Daniel W. C. HO , Minyi Huang , Jiamin Jian , Qingshuo Song

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

The formulation of Mean Field Games (MFG) typically requires continuous differentiability of the Hamiltonian in order to determine the advective term in the Kolmogorov--Fokker--Planck equation for the density of players. However, in many…

Numerical Analysis · Mathematics 2024-04-03 Yohance A. P. Osborne , Iain Smears

We develop value iteration-based algorithms to solve in a unified manner different classes of combinatorial zero-sum games with mean-payoff type rewards. These algorithms rely on an oracle, evaluating the dynamic programming operator up to…

Computer Science and Game Theory · Computer Science 2024-11-12 Xavier Allamigeon , Stéphane Gaubert , Ricardo D. Katz , Mateusz Skomra

This paper concerns value functions of time-dependent tug-of-war games. We first prove the existence and uniqueness of value functions and verify that these game values satisfy a dynamic programming principle. Using the arguments in the…

Analysis of PDEs · Mathematics 2021-04-06 Jeongmin Han

We study a mean-field game of optimal stopping and investigate the existence of strong solutions via a connection with the Bank-El Karoui's representation problem. Under certain continuity assumptions, where the common noise is generated by…

Optimization and Control · Mathematics 2025-07-28 Giorgio Ferrari , Anna Pajola

We consider the problem of payoff division in indivisible coalitional games, where the value of the grand coalition is a natural number. This number represents a certain quantity of indivisible objects, such as parliamentary seats, kidney…

Computer Science and Game Theory · Computer Science 2026-04-02 Mikołaj Czarnecki , Michał Korniak , Oskar Skibski , Piotr Skowron

We study a class of zero-sum stochastic games between a stopper and a singular-controller, previously considered in [Bovo and De Angelis (2025)]. The underlying singularly-controlled dynamics takes values in…

Optimization and Control · Mathematics 2025-06-25 Andrea Bovo , Alessandro Milazzo

In this paper, we study fully coupled nonlocal second order quasilinear forward-backward partial differential equations (FBPDEs), which arise from solution of the mean field game (MFG) suggested by Lasry and Lions [Japan. J. Math. 2 (2007),…

Probability · Mathematics 2022-12-13 Ziyu Huang , Shanjian Tang

We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar…

Probability · Mathematics 2018-04-11 Francois Delarue

Mean Field Games with state constraints are differential games with infinitely many agents, each agent facing a constraint on his state. The aim of this paper is to provide a meaning of the PDE system associated with these games, the…

Optimization and Control · Mathematics 2019-01-01 Piermarco Cannarsa , Rossana Capuani , Pierre Cardaliaguet

This paper considers mean field games in a multi-agent Markov decision process (MDP) framework. Each player has a continuum state and binary action, and benefits from the improvement of the condition of the overall population. Based on an…

Optimization and Control · Mathematics 2021-01-05 Minyi Huang , Yan Ma

In the present work, we study deterministic mean field games (MFGs) with finite time horizon in which the dynamics of a generic agent is controlled by the acceleration. They are described by a system of PDEs coupling a continuity equation…

Analysis of PDEs · Mathematics 2020-07-29 Yves Achdou , Paola Mannucci , Claudio Marchi , Nicoletta Tchou

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the…

Optimization and Control · Mathematics 2020-03-03 Jin Ma , Eunjung Noh

The presence of artificial intelligence (AI) in our society is increasing, which brings with it the need to understand the behavior of AI mechanisms, including machine learning predictive algorithms fed with tabular data, text or images,…

Machine Learning · Statistics 2025-06-06 Pedro Delicado , Cristian Pachón-García

We discuss and compare two methods of investigations for the asymptotic regime of stochastic differential games with a finite number of players as the number of players tends to the infinity. These two methods differ in the order in which…

Probability · Mathematics 2012-10-23 Rene Carmona , Francois Delarue , Aime Lachapelle

In this paper, we present a model of a game among teams. Each team consists of a homogeneous population of agents. Agents within a team are cooperative while the teams compete with other teams. The dynamics and the costs are coupled through…

Computer Science and Game Theory · Computer Science 2023-10-20 Jayakumar Subramanian , Akshat Kumar , Aditya Mahajan

Mean Field Game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to $0$.…

Optimization and Control · Mathematics 2019-03-13 Pierre Cardaliaguet , Alessio Porretta