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1. Turnpike properties in linear quadratic Gaussian N-player differential games

   Cohen, Asaf; Jian, Jiamin (June 2026, ESAIM Control optimisation and calculus of variations)

   We consider the long-time behavior of equilibrium strategies and state trajectories in a linear quadratic N-player game with Gaussian initial data. By comparing the finite-horizon game with its ergodic counterpart, we establish exponential convergence estimates between the solutions of the finite-horizon generalized Riccati system and the associated algebraic system arising in the ergodic setting. Building on these results, we prove the convergence of the time-averaged value function and derive a turnpike property for the equilibrium pairs of each player. Importantly, our approach avoids reliance on the mean field game limiting model, allowing for a fully uniform analysis with respect to the number of players N. As a result, we further establish a uniform turnpike property for the equilibrium pairs between the finite-horizon and ergodic games with N players. Numerical experiments are also provided to illustrate and support the theoretical results. 

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2. Synchronization Games

   https://doi.org/10.1287/moor.2024.0416  

   Höfer, Felix; Soner, H Mete (May 2026, Mathematics of Operations Research)

   We propose a new mean-field game model with two states to study synchronization phenomena, and we provide a comprehensive characterization of stationary and dynamic equilibria along with their stability properties. The game undergoes a phase transition with increasing interaction strength. In the subcritical regime, the uniform distribution, representing incoherence, is the unique and stable stationary equilibrium. Above the critical interaction threshold, the uniform equilibrium becomes unstable and there is a multiplicity of stationary equilibria that are self-organizing. Under a discounted cost, dynamic equilibria spiral around the uniform distribution before converging to the self-organizing equilibria. With an ergodic cost, however, unexpected periodic equilibria around the uniform distribution emerge. Funding: This work was supported by the National Science Foundation [Grant DMS 2406762]. 

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3. Soft-Bellman Equilibrium in Affine Markov Games: Forward Solutions and Inverse Learning

   https://doi.org/10.1109/CDC49753.2023.10384097  

   Chen, Shenghui; Yu, Yue; Fridovich-Keil, David; Topcu, Ufuk (December 2023, Conference on Decision and Control)

   Markov games model interactions among multiple players in a stochastic, dynamic environment. Each player in a Markov game maximizes its expected total discounted reward, which depends upon the policies of the other players. We formulate a class of Markov games, termed affine Markov games, where an affine reward function couples the players’ actions. We introduce a novel solution concept, the soft-Bellman equilibrium, where each player is boundedly rational and chooses a soft-Bellman policy rather than a purely rational policy as in the well-known Nash equilibrium concept. We provide conditions for the existence and uniqueness of the soft-Bellman equilibrium and propose a nonlinear least-squares algorithm to compute such an equilibrium in the forward problem. We then solve the inverse game problem of inferring the players’ reward parameters from observed state-action trajectories via a projected-gradient algorithm. Experiments in a predator-prey OpenAI Gym environment show that the reward parameters inferred by the proposed algorithm outper- form those inferred by a baseline algorithm: they reduce the Kullback-Leibler divergence between the equilibrium policies and observed policies by at least two orders of magnitude. 

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4. The Dyson and Coulomb games

   Carmona, Rene; Cerenzia, Mark; Palmer, Aaron Zeff (October 2019, ArXivorg)

   We introduce and investigate certain N player dynamic games on the line and in the plane that admit Coulomb gas dynamics as a Nash equilibrium. Most significantly, we find that the universal local limit of the equilibrium is sensitive to the chosen model of player information in one dimension but not in two dimensions. We also find that players can achieve game theoretic symmetry through selfish behavior despite non-exchangeability of states, which allows us to establish strong localized convergence of the N-Nash systems to the expected mean field equations against locally optimal player ensembles, i.e., those exhibiting the same local limit as the Nash-optimal ensemble. In one dimension, this convergence notably features a nonlocal-to-local transition in the population dependence of the N-Nash system. 

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5. Finite State Mean Field Games with Wright-Fisher Common Noise as Limits of $$N$$-Player Weighted Games

   https://doi.org/10.1287/moor.2021.1230  

   Bayraktar, Erhan; Cecchin, Alekos; Cohen, Asaf; Delarue, François (March 2022, Mathematics of Operations Research)

   Forcing finite state mean field games by a relevant form of common noise is a subtle issue, which has been addressed only recently. Among others, one possible way is to subject the simplex valued dynamics of an equilibrium by a so-called Wright–Fisher noise, very much in the spirit of stochastic models in population genetics. A key feature is that such a random forcing preserves the structure of the simplex, which is nothing but, in this setting, the probability space over the state space of the game. The purpose of this article is, hence, to elucidate the finite-player version and, accordingly, prove that N-player equilibria indeed converge toward the solution of such a kind of Wright–Fisher mean field game. Whereas part of the analysis is made easier by the fact that the corresponding master equation has already been proved to be uniquely solvable under the presence of the common noise, it becomes however more subtle than in the standard setting because the mean field interaction between the players now occurs through a weighted empirical measure. In other words, each player carries its own weight, which, hence, may differ from [Formula: see text] and which, most of all, evolves with the common noise. 

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