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1. Finite State Mean Field Games with Major and Minor Players

   Rene Carmona, Peiqi Wang (July 2018, arXiv.org)

   The goal of the paper is to develop the theory of finite state mean field games with major and minor players when the state space of the game is finite. We introduce the finite player games and derive a mean field game formulation in the limit when the number of minor players tends to infinity. In this limit, we prove that the value functions of the optimization problems are viscosity solutions of PIDEs of the HJB type, and we construct the best responses for both types of players. From there, we prove existence of Nash equilibria under reasonable assumptions. Finally we prove that a form of propagation of chaos holds in the present context and use this result to prove existence of approximate Nash equilibria for the finite player games from the solutions of the mean field games. this vindicate our formulation of the mean field game problem. 

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2. CEMRACS2017: NUMERICAL PROBABILISTIC APPROACH TO MFG

   Angiuli, Andrea; Graves, Christy V; Li, Houzhi; Chassagneux, Jean-Francois; Delarue, Francois; Carmona, Rene (February 2019, ESAIM. Proceedings and surveys)

   This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems . Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhame, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems . The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges. 

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3. CEMRACS 2017: Numerical Probabilistic Approach to MFG

   Andrea Angiuli, Christy V. (May 2018, arXiv.org)

   This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, and by Huang, Caines, and Malhame. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges. 

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4. On Time‐Inconsistency in Mean‐Field Games

   https://doi.org/10.1111/mafi.12456  

   Bayraktar, Erhan; Wang, Zhenhua (July 2025, Mathematical Finance)

   ABSTRACT We investigate an infinite‐horizon time‐inconsistent mean‐field game (MFG) in a discrete time setting. We first present a classic equilibrium for the MFG and its associated existence result. This classic equilibrium aligns with the conventional equilibrium concept studied in MFG literature when the context is time‐consistent. Then we demonstrate that while this equilibrium produces an approximate optimal strategy when applied to the related ‐agent games, it does so solely in a precommitment sense. Therefore, it cannot function as a genuinely approximate equilibrium strategy from the perspective of a sophisticated agent within the ‐agent game. To address this limitation, we propose a newconsistentequilibrium concept in both the MFG and the ‐agent game. We show that a consistent equilibrium in the MFG can indeed function as an approximate consistent equilibrium in the ‐agent game. Additionally, we analyze the convergence of consistent equilibria for ‐agent games toward a consistent MFG equilibrium as tends to infinity. 

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5. Nash equilibrium structure of Cox process Hotelling games

   https://doi.org/10.1017/apr.2021.45  

   Anantharam, Venkat; Baccelli, François (June 2022, Advances in Applied Probability)

   Abstract We study anN-player game where a pure action of each player is to select a nonnegative function on a Polish space supporting a finite diffuse measure, subject to a finite constraint on the integral of the function. This function is used to define the intensity of a Poisson point process on the Polish space. The processes are independent over the players, and the value to a player is the measure of the union of her open Voronoi cells in the superposition point process. Under randomized strategies, the process of points of a player is thus a Cox process, and the nature of competition between the players is akin to that in Hotelling competition games. We characterize when such a game admits Nash equilibria and prove that when a Nash equilibrium exists, it is unique and consists of pure strategies that are proportional in the same proportions as the total intensities. We give examples of such games where Nash equilibria do not exist. A better understanding of the criterion for the existence of Nash equilibria remains an intriguing open problem. 

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