Mean Field Games

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3 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

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Latest papers 26

Oct 1, 2026math.OC

Mean field games as a tool for AI safety: a worked example from the July 2026 Hugging Face incident

One way to make AI systems safe is to shape what the system is: its objective and dispositions. We take a complementary route: treat the agents' characteristics as partly unknown and ask what structure of interaction ensures that bad collective outcomes are not equilibria. Mean field games suit this when many interchangeable agents are coupled through an aggregate. We introduce a program for using them in AI safety and carry one example through end to end: the July 2026 incident in which about 1,200 agents in an OpenAI evaluation coordinated on an improvised message board and 684 attacked a third party's infrastructure. We model the decision to attack as a mean field game of optimal stopping whose gain is a product: belief that provenance will be audited, times reachability of the record, minus the perceived hazard. The central result is an exact threshold on the belief. No agent attacks unless the population's confidence that provenance is checked exceeds

π^{**} = η/(η+ ψ+ \varepsilon a \overline{M})

, where

η

is the perceived hazard,

ψ

and

\varepsilon a \overline{M}

measure how far one attacker and the collective can alter the record, and

\overline{M}

is the peak population. Below it, no attack is the unique equilibrium for all agent parameters. The threshold survives every enrichment we consider. We then use the per-agent record to discipline the model. Its features, a stable minority attacking for thirty hours and then a pivot in which most of the board joined within a day, motivate each refinement. The account that emerges is heterogeneous belief meeting a sequence of public discoveries, each lowering the belief at which attacking paid. A few coordinating agents made those discoveries, so the model describes the several hundred who responded, not the few who produced them; a major-player version is left to future work.

P. Jameson Graber

Artificial Intelligence Safety Mean Field Games

Sep 30, 2026cs.AI

Meta-Multi-Agent Reinforcement Learning for Fast Adaptation of Interactive Policies with Applications to Autonomous Driving

This paper develops a meta-multi-agent reinforcement learning (meta-MARL) framework to enable fast adaptation of interactive policies in a multi-agent system (MAS). Meta-reinforcement learning (meta-RL) enables agents to rapidly adapt to new tasks/environments using a bi-level optimization mechanism. However, existing meta-RL generally focuses on single-agent systems. Extending these frameworks and algorithms to multi-agent systems poses additional challenges, as tasks are characterized by not only the environment but also agents' strategic interactions. To address these challenges, we model multi-agent reinforcement learning (MARL) problems as Markov games (MGs) and develop a meta-MARL framework for rapid interactive policy adaptation across a distribution of MGs. A new concept, called meta-NE, is defined to describe the desired solution concept in a meta-MARL problem. Sufficient conditions for the equivalence between a meta-NE and a stationary point of the gradient-play-based meta-MARL algorithm are established. Our evaluation on autonomous-driving tasks demonstrates that the proposed meta-MARL method achieves faster adaptation than pretrained MARL baselines, validating the effectiveness of our framework.

Huiwen Yan, Kyriakos G. Vamvoudakis, Mushuang Liu

Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, USA · Daniel Guggenheim School of Aerospace Engineering, Georgia Tech, Atlanta, GA 30332, USA

Multi-Agent Reinforcement Learning Mean Field Games Autonomous Driving Interaction Data

Sep 29, 2026cs.MA

Regularized policy gradient with learned mixtures of Gaussians for games with continuous actions

Most successes of superhuman game-playing algorithms are in games with discrete actions, yet in auctions, robotics, sports, or trading, actions are nearly continuous. Prior techniques either rely on expert-designed discretizations or are sample inefficient. We present a scalable policy-gradient algorithm for large sequential games with continuous or mixed discrete and continuous actions. It combines magnetic mirror descent with a mixture of Gaussians reparametrization, trained via self-play. We show that it approximates equilibrium in games where gradient descent fails. In sequential games, it outperforms neural fictitious self-play and matches or outperforms the final strategies of policy space response oracles with 3.5--5.5

\times

fewer samples. In heads-up no-limit Texas hold'em, it performs on par with Slumbot.

Ondřej Kubíček, Viliam Lisý, Tuomas Sandholm

Czech Technical University in Prague · Carnegie Mellon University · Artificial Intelligence Center

Mean Field Games Mirror Descent Self-Play

Sep 14, 2026cs.LG

High-Probability Nash Regret for Decentralized Learning in Markov $α$ -Potential Games: Episodic and Fully Online Asynchronous Algorithms with Applications to Markov Congestion Games

We study decentralized learning of Nash equilibria (NE) in infinite-horizon discounted Markov games under bandit feedback, focusing on Markov

α

-potential games. We develop KL-projected natural policy gradient (NPG) algorithms in two settings: an episodic setting with frozen policies during sampling and a fully online setting in which players receive a single realized cost sample per time step and update their policies asynchronously along a continuing trajectory. We establish finite-time high-probability NE regret bounds of order

\widetilde O(T^{-1/4})

and

\widetilde O(T^{-2/15})

for the episodic and fully online settings, respectively, up to fixed approximation terms. Crucially, our bounds eliminate the distribution-mismatch coefficient, which can scale prohibitively with the size of the state space, while accommodating potential approximation, estimation-oracle bias, and transition sensitivity. We further identify a state-wise potential structure that yields sharper guarantees with additive dependence on the potential approximation error

α

. We specialize the framework to independent-resource Markov congestion games (IMCGs), establish their approximate-potential and transition-sensitivity properties, and construct decentralized estimation oracles from realized costs. As an application, we introduce strategic online job scheduling on stochastic machines and obtain a scalable decentralized algorithm for learning stable dispatching policies. Overall, our results provide the first finite-time high-probability NE regret guarantees for fully online asynchronous decentralized learning in Markov

α

-potential games, remove distribution-mismatch coefficients from the regret bounds, accommodate fixed estimation-oracle bias, and provide scalable decentralized learning with finite-time guarantees for IMCGs.

S. Rasoul Etesami

Department of Industrial and Systems Engineering Coordinated Science Laboratory University of Illinois Urbana-Champaign, Urbana, IL, USA

Mean Field Games Observable Stochastic Game Nash Equilibrium Decentralized Learning

Sep 1, 2026cs.MA

Differential Games for Compositional Handling of Competing Control Tasks

We introduce a novel Divide and Conquer control design methodology leveraging differential games in single-agent, multi-objective dynamical systems. The proposed framework associates each control objective with a virtual input and establishes a non-cooperative, finite or infinite horizon differential game among representative players. Each player optimizes a distinct virtual cost function tailored to its specific goal, the full system state, and the other virtual inputs, while accounting for the remaining players' optimal policies. By establishing a Nash Equilibrium for this game, we synthesize a composite controller that achieves a stable balance across competing objectives, providing control engineers with an intuitive and modular framework for parameter re-tuning throughout the design cycle. We provide formal mathematical derivations for both continuous-time and discrete-time dynamical systems, targeting large-scale single-agent applications where complex, dynamically conflicting control objectives make global weighting intractable. To demonstrate the methodology, we developed an open-source Python package implementing a novel numerical algorithm for solving Coupled Algebraic Riccati Equations arising in infinite-horizon differential games. We evaluate the approach on two benchmark case studies: an inverted pendulum on a cart and a non-linear hierarchically controlled quadrotor. The resulting closed-loop performance is compared against the classical Linear Quadratic Regulator (LQR) across various transient and steady-state control metrics, demonstrating superior trajectory tracking and robust multi-objective regulation.

Joshua Shay Kricheli

Ben-Gurion University of the Negev, The Faculty of Natural Sciences, The Department of Computer Science

Observable Stochastic Game Mean Field Games Multi-Objective Optimization

Sep 1, 2026cs.GT

Independent Reinforcement Learning in Discounted Markov Games

In this work, we study radically uncoupled learning in discounted general-sum Markov games. Assuming ``

\mathsf{ETH}

for

\mathsf{PPAD}

", we show that, for every fixed discount factor, there is no polynomial-time algorithm for computing inverse-polynomially accurate coarse correlated equilibria in discounted general-sum Markov games when players learn independently in decentralized settings. Complementing this hardness result, we provide what appears to be the first \emph{radically uncoupled} algorithm with sub-exponential convergence guarantees to coarse correlated equilibria in discounted general-sum Markov games without imposing any structural restrictions on the game. Our algorithm is a \emph{layered} variant of optimistic mirror descent with an increasing step-size schedule tailored to the multi-agent setting. Finally, we develop both full-feedback and partial feedback versions of the aforementioned algorithm and establish sub-exponential convergence guarantees for each case.

Asrin Efe Yorulmaz, Ugur Aydin, Tamer Basar

Department of Electrical and Computer Engineering, University of Illinois Urbana-Champaign, Urbana, IL, 61801, USA

Mean Field Games Multi-Agent Reinforcement Learning Mirror Descent Uncoupled Learning Offline Reinforcement Learning Convergence+1

Aug 3, 2026cs.RO

A Forward-Inverse Dynamic Game Framework for Enhanced Multi-Agent Trajectory Planning

This paper studies feedback Nash equilibrium (FBNE) seeking for multi-agent trajectory planning in nonlinear dynamical systems with unknown agents' objectives and state-dependent inter-agent coupling. While dynamic game theory provides a principled framework for such problems, existing approaches typically assume fully rational agents with known objectives or rely on fixed regularization, limiting their ability to capture bounded rationality and spatially varying interaction intensity in safety-critical settings. To this end, we propose a KL-regularized dynamic game with a state-dependent weight that adaptively balances optimality and behavioral priors. To infer unknown cost parameters from demonstrated behaviors, we develop a context-aware inverse game module based on maximum-entropy inverse reinforcement learning with physics-informed regularization, ensuring structural consistency with the forward game. We establish per-iteration well-posedness of the regularized local game and show that the adaptive weighting function remains Lipschitz continuous under bounded nominal-trajectory updates. Numerical simulations and multi-robot experiments on cooperative navigation and merging scenarios validate the effectiveness of the proposed framework.

Tianle Liu, Youcheng Niu, Jing Zeng +2

College of Control Science and Engineering, Zhejiang University, Hangzhou 310027, China

Mean Field Games Robust Trajectory Nonlinear Dynamics Planning

Jul 27, 2026cs.AI

PLATO: Pointer Learner for Agent and Task Openness

Open agent systems (OASYS) are increasingly prevalent in real-world domains where the sets of agents and tasks change unpredictably over time. Such openness, including agent openness (AO) and task openness (TO), poses a fundamental challenge to multi-agent reinforcement learning (MARL), which typically assumes fixed state and action spaces. Existing methods address openness only partially: padding and masking approaches introduce artificial bounds, while recent graph-based or hypergraph methods handle one dimension of openness but still depend on restrictive assumptions. In this paper, we introduce Pointer Learner for Agent and Task Openness (PLATO), a pointer-network-based actor combined with a centralized graph neural network (GNN) critic, trained with multi-agent proximal policy optimization under a centralized training and decentralized execution paradigm. Our pointer-based actor outputs distributions directly over the current task set. This directly supports changing action spaces without masking or retraining. Our GNN critic encodes agent-task interactions as a graph that changes shape with task and agent composition. Together, these components consider AO and TO without the boundedness of existing approaches. We formalize PLATO in a Task-and-Agent-Open Markov Game (TaAgO-MG), extending prior task-open formulations, and prove it is well-defined over the resulting unbounded state and action spaces. We evaluate PLATO with the Methods for Open Agent Systems Evaluation Initiative (MOASEI) wildfire suppression domain, an environment designed for open multi-agent system evaluation, and we demonstrate strong performance and more consistent zero-shot generalization than state-of-the-art baselines in OASYS.

Alireza Saleh Abadi, Leen-Kiat Soh, Daniel Alan Redder +2

School of Computing University of Nebraska-Lincoln · School of Computing University of Georgia · Oberlin College

Mean Field Games

Jul 25, 2026cs.LG

All in One: Generative Modeling as Mean-Field Game Design

Mean-field games (MFGs) offer a unifying lens on continuous-time generative modeling: a cost tuple recovering twelve prominent models---Continuous Normalizing Flows, OT-Flow, Score-based Models, Schrödinger Bridges, and more---as special cases of one variational problem. Yet two dimensions of this space remain entirely unexplored: the interaction term

\mathcal{I}

is set to zero in many existing models, and the rich family of MFG solvers has never been applied to generative modeling. We address both gaps with MFGLab an open-source PyTorch library whose primary API is the cost tuple: all twelve models are specified by four composable cost functions, and the training loop, log-Jacobian, and reverse-ODE sampler are shared automatically. We additionally propose DI-Flow, a novel cost design that uses a differentiable entropy functional to encourage mode coverage, and provide learning-based MFG solvers that substantially outperform neural training on stochastic-dynamics rows. Experiments on two 2-D benchmarks confirm that the unified API is lossless relative to hand-coded implementations.

Kun Zhao, Xu Chen

Generative Models Mean Field Games

Jun 22, 2026cs.GT

Neural Parameter Calibration for Finite-State Mean Field Games

Mean field games efficiently approximate a very large population of strategic agents. While these games can aid the understanding of complex systems, their deployment in real-world settings is challenged by the specification of their parameters: mean field games (MFGs) often involve hidden preferences, constraints, and interactions that can rarely be theoretically derived or directly observed. To address this gap, we present a neural network-based framework for learning parametric, finite-state MFGs from observed population dynamics. To do so, we formulate the parameter calibration as an inverse problem and use implicit differentiation to backpropagate through the games' equilibrium. The resulting approach is fully differentiable and enables us to estimate flexible trajectory-wise parameter paths, including state- and time-dependent specifications without requiring observations of the individual agents' actions or rewards. We provide a proof for the exactness of the gradient computation in a discrete-time formulation. We validate our framework through numerical experiments across four systems of increasing complexity, ranging from synthetic linear-quadratic benchmarks to real-world urban mobility datasets.

Anna C. M. Thöni, Grégoire Lambrecht, Gökçe Dayanıklı +3

Radboud University, Nijmegen, the Netherlands · New York University, New York, NY · University of Illinois Urbana-Champaign, Champaign, IL +2

Mean Field Games Mean-Field Limit Population Dynamics Strategic Agents

Jun 21, 2026cs.LG

Stationary Robust Mean-Field Games under Model Mismatches

Deploying multi-agent reinforcement learning (MARL) in the real world is often limited by model mismatches between the training simulators and the true environment, which could be further amplified through strategic interactions and result in severe performance degradation upon deployment. Distributional robustness offers a principled response by optimizing policies against worst-case transition models drawn from an uncertainty set, but standard robust MARL frameworks become increasingly intractable as the number of agents grows. This paper develops an infinite-horizon, stationary mean-field game framework that incorporates distributional model uncertainty directly into the population-coupled dynamics. We establish a robust dynamic programming principle with a contractive Bellman operator and prove the existence of a stationary robust mean-field equilibrium via a fixed-point argument. We further develop the first concrete algorithm with convergence guarantees. We then connect the mean-field solution to a finite-population robust game whose ambiguity sets depend on the empirical distribution, showing that the mean-field equilibrium policy induces approximate equilibrium behavior as the population size increases. Under a contractive robust-dynamics regime, we further obtain explicit non-asymptotic error bounds. Numerical experiments further illustrate the qualitative and quantitative impact of robustness under multiple uncertainty models, validating our theoretical findings.

Yue Wang

Department of Electrical and Computer Engineering University of Central Florida Orlando, Florida, USA

Mean Field Games Multi-Agent Reinforcement Learning Mean-Field Limit Mismatch

Jun 18, 2026math.OC

Optimal Coarse Correlated Equilibria in Mean Field Games: Linear Programming and No-Regret Learning

We introduce optimal coarse correlated equilibria for continuous-time mean field games. A coarse correlated equilibrium is a randomized recommendation scheme from which no player can gain by ignoring the recommendation and switching to an alternative strategy. The problem is as follows: a moderator selects, among all mean-field coarse correlated equilibria, one that optimizes a prescribed performance criterion, which may differ from the representative player's objective. After formulating the problem, we develop a linear programming (LP) formulation, prove the existence of optimal LP coarse correlated equilibria, and relate the LP characterization to the original probabilistic setting. Building on this characterization, we design a no-regret primal-dual algorithm, based on an equivalent Lagrangian formulation of the external-regret constraint, for learning such equilibria. We provide explicit convergence rates for the learning algorithm, and numerical examples illustrate the method.

Luciano Campi, Federico Cannerozzi, Ioannis Tzouanas

DEPARTMENT OF MATHEMATICS “FEDERIGO ENRIQUES”, UNIVERSITY OF MILAN, VIA SALDINI 50, 20133, MILAN, ITALY. · CENTER FOR MATHEMATICAL ECONOMICS (IMW), BIELEFELD UNIVERSITY, BIELEFELD, GERMANY.

Mean Field Games Equilibrium Mean-Field Limit Plan Regret Linear Programming Recommendation Quality

Jun 17, 2026cs.CV

Beyond the Current Observation: Evaluating Multimodal Large Language Models in Controllable Non-Markov Games

Deploying multimodal foundation models as closed-loop policies increasingly requires conditioning actions on observations that are no longer visible. However, existing benchmarks either expose the full state, conflate hidden-state reconstruction with other agent skills, or test recall only after an episode has ended. We introduce RNG-Bench (Reconstructive Non-Markov Games), a benchmark suite designed to isolate a base model's ability to reconstruct past observations and act on them during multi-step interaction. RNG-Bench includes two complementary games: Matching Pairs, where card identities briefly revealed at specific locations must later be recalled, and 3D Maze, where egocentric views must be integrated into a spatial map. Both games are evaluated under a unified harness with three controlled difficulty axes: grid size, visual pattern, and observation modality. The benchmark further introduces a head-to-head duel protocol to control for instance-level variance and a Memory Gap metric that disentangles forgetting from poor action selection. The hardest configurations require contexts of roughly 128K tokens and 350 image inputs per episode, and remain far from saturated by frontier MLLMs. Memory Gap analysis shows that most residual errors stem from forgetting earlier observations rather than from suboptimal decision making. Finally, fine-tuning Qwen3.5-9B on optimal-policy rollouts and filtered model demonstrations improves performance on RNG-Bench and transfers to existing benchmarks without degrading general multimodal capability.

Shengyuan Ding, Xilin Wei, Xinyu Fang +4

1Fudan University · 2Shanghai Innovation Institute · 3Shanghai Artificial Intelligence Laboratory +2

Multimodal Large Language Models Multimodal Foundation Model Mean Field Games

Jun 15, 2026cs.LG

Maximum Entropy Inverse Reinforcement Learning for Mean-Field Games with Average Reward

We study inverse reinforcement learning for discrete-time, infinite-horizon mean-field games (MFGs) under an average-reward criterion. Expert demonstrations are assumed to arise from a stationary mean-field equilibrium under an unknown reward, and the goal is to recover a policy explaining the observed behaviour via the maximum causal entropy principle. We formulate the inverse problem by enforcing consistency with the expert mean-field term and long-run feature expectations, treating two reward classes within a unified occupation-measure framework. For finite-dimensional linear rewards, we give a convex dual reformulation with an explicit log-partition objective, and prove smoothness and curvature properties justifying constant-step-size gradient descent. For infinite-dimensional RKHS rewards, we develop a Lagrangian relaxation whose inner-maximising policy is characterised by a soft Bellman equation. The main obstacle is the absence of a discount-factor contraction. We resolve this by introducing a minorisation-based sub-stochastic kernel that yields a strict contraction of the soft Bellman operator. We establish Fréchet differentiability and Lipschitz smoothness of the log-likelihood score, leading to a gradient ascent algorithm with convergence guarantees. Two numerical examples, a malware-spread MFG and an RKHS-based consumer-choice model, show that the recovered policies closely match expert behaviour.

Şevket Kaan Alkır, Naci Saldı, Berkay Anahtarcı +1

Department of Mathematics Bilkent University 06800 Ankara, Turkey · Department of Mathematical Engineering Özyeğin University 34794 Istanbul, Turkey

Mean Field Games Entropy Regularized Reinforcement Learning Offline Reinforcement Learning Bellman Equation Reproducing Kernel Hilbert Spaces Mean

Jun 1, 2026cs.LG

Minimax-Optimal Policy Regret in Partially Observable Markov Games

We study sequential decision-making in partially observable environments against strategic, adaptive opponents, modeled as partially observable Markov games (POMGs). The central challenge is to learn latent dynamics from partial observations while facing an adversary whose behavior depends on the learner's strategy, making standard regret notions inadequate. We prove that an epoch-based optimistic maximum-likelihood algorithm achieves

\tilde{O}(\sqrt{T})

policy regret for fixed problem parameters, with explicit dependence on the horizon, adversary memory, confidence radius, and the aggregate Eluder dimension of the observable-operator class. The algorithm selects one policy per geometrically growing epoch using confidence sets built cumulatively from past data, which keeps the cost of comparing adversary responses across policies logarithmic in

T

. We also prove a lower bound matching the

\sqrt{T}

and aggregate-Eluder-dimension dependence, up to problem-dependent and logarithmic factors. Finally, we extend the framework to horizon-adaptive guarantees and adversaries with geometric fading memory.

Raman Arora

Department of Computer Science, Johns Hopkins University, Baltimore, MD, USA.

Partially Observable Markov Decision Process Mean Field Games Expectation-Maximization

May 28, 2026math.OC

Kernel-based potential mean-field games with unbiased random Fourier $U$ -statistics

We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing-kernel maximum mean discrepancy (MMD) penalties, and develop a computational framework that exploits this kernel structure. Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size. The drift of the controlled diffusion is parametrized by a neural network and trained via stochastic gradient descent. For population near-minimizers we prove convergence to the terminal-constrained problem as the penalty diverges, and show that the same limit is recovered almost surely when the learned controls are evaluated on independent finite samples under explicit coupling conditions on the penalty, random-feature count and sample size. The framework includes the kernel-MMD-penalty Schr{ö}dinger bridge problem as the special case of a vanishing interaction cost. Numerical experiments illustrate the method on the Schr{ö}dinger bridge problem in dimensions up to one hundred, and on an electric vehicle charging coordination problem with per-vehicle physical heterogeneity, where an aggregate-demand congestion cost represents price-feedback competition at the population level and the terminal MMD penalty shapes the state-of-charge distribution at the deadline.

Yumiharu Nakano

Department of Mathematical and Computing Science, Institute of Science Tokyo

Mean Field Games Maximum Mean Discrepancy Mean-Field Limit Congestion Fourier

May 26, 2026cs.LG

Bilevel Optimization over Saddle Points of Zero-Sum Markov Games

Reinforcement learning (RL) often has a hierarchical structure, where an upper-level (UL) learner selects model parameters and a lower-level (LL) decision-making process responds, naturally leading to a bilevel optimization problem. Most existing bilevel RL methods assume a single-policy LL Markov decision process (MDP), and therefore fail to capture competitive structures arising in applications such as incentive design, where multiple policies interact. We study bilevel optimization problems in which the LL problem is a regularized min-max zero-sum Markov game and the UL objective is optimized through the saddle-point equilibrium induced by the LL game. In this work, we propose penalty-augmented Nikaido-Isoda descent-ascent (PANDA), a penalty-based first-order policy-gradient method based on the Nikaido-Isoda function. By exploiting the min-max game structure, PANDA avoids computing UL hypergradients and does not require second-order information. We prove that PANDA converges to stationary points without convexity assumptions on either the UL or LL objectives. Moreover, PANDA reaches an

ε

-stationary point in

\tilde{\mathcal{O}}(ε^{-1})

iterations with sample complexity

\tilde{\mathcal{O}}(ε^{-3})

, matching the best-known rates for bilevel RL with single-policy LL MDPs. Experiments demonstrate the superior performance of PANDA over closely related baselines.

Zihao Zheng, Irwin King, Songtao Lu

Department of Computer Science and Engineering, The Chinese University of Hong Kong · Shun Hing Institute of Advanced Engineering, The Chinese University of Hong Kong

Bilevel Optimization Mean Field Games Stationary Point

May 13, 2026cs.LG

Offline Two-Player Zero-Sum Markov Games with KL Regularization

We study the problem of learning Nash equilibria in offline two-player zero-sum Markov games. While existing approaches often rely on explicit pessimism to address distribution shift, we show that KL regularization alone suffices to stabilize learning and guarantee convergence. We first introduce Regularized Offline Sequential Equilibrium (ROSE), a theoretical framework that achieves a fast

\widetilde{\mathcal{O}}(1/n)

convergence rate under \textit{unilateral concentrability}, improving over the standard

\widetilde{\mathcal{O}}(1/\sqrt{n})

rates in unregularized settings. We then propose Sequential Offline Self-play Mirror Descent (SOS-MD), a practical model-free algorithm based on least-squares value estimation and iterative self-play updates. We prove that the last iterate of SOS-MD attains the same

\widetilde{\mathcal{O}}(1/n)

statistical rate up to a vanishing optimization error of order

\widetilde{\mathcal{O}}(1/\sqrt{T})

in the number of self-play iterations

T

.

Claire Chen, Yuheng Zhang, Xinyu Liu +3

1California Institute of Technology. · University of Illinois Urbana-Champaign. · University of Virginia. +2

Mean Field Games Observable Stochastic Game Nash Equilibrium Kullback-Leibler Regularization Self-Play

May 5, 2026cs.LG

Population-Aware Imitation Learning in Mean-field Games with Common Noise

Mean Field Games (MFGs) provide a powerful framework for modeling the collective behavior of large populations of interacting agents. In this paper, we address the problem of Imitation Learning (IL) in MFGs subject to common noise, where the population distribution evolves stochastically. This stochasticity compels agents to adopt population-aware policies to respond to aggregate shocks. We formulate two distinct learning objectives: recovering a Nash equilibrium and maximizing performance against an expert population. We investigate two imitation proxies: Behavioral Cloning (BC) and Adversarial (ADV) divergence. We then establish finite-sample error bounds showing that minimizing these proxies effectively controls both the policy's exploitability and its performance gap relative to the expert. Furthermore, we propose a numerical framework using generalized Fictitious Play and Deep Learning to compute expert population-aware policies. Through experiments on three environments we demonstrate that standard population-unaware policies fail to capture the equilibrium dynamics. Our results highlight that learning population-aware policies is crucial to avoid being misled by the randomness inherent in common noise.

Grégoire Lambrecht, Mathieu Laurière

Center for Data Science, New York University and NYU Shanghai, New York, United States of America · Shanghai Center for Data Science; NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai; NYU Shanghai, Shanghai, People’s Republic of China.

Mean Field Games Imitation Learning Mean-Field Limit Multi-Agent Reinforcement Learning Equilibrium

May 4, 2026cs.LG

Taming the Curses of Multiagency in Robust Markov Games with Large State Space through Linear Function Approximation

Multi-agent reinforcement learning (MARL) holds great potential but faces robustness challenges due to environmental uncertainty. To address this, distributionally robust Markov games (RMGs) optimize worst-case performance when the environment deviates from the nominal model within a uncertainty set. Beyond robustness, an equally urgent goal for MARL is data efficiency -- sampling from vast state and action spaces that grow exponentially with the number of agents potentially leads to the curse of multiagency. However, current provably data-efficient algorithms for RMGs are limited to tabular settings with finite state and action spaces, which are only computationally manageable for small-scale problems, leaving RMGs with large-scale (or infinite) state spaces largely unexplored. The only existing work beyond tabular settings focuses on linear function approximation (LFA) for a restrictive class of RMGs using vanish minimal value assumption and still suffers from sample complexity with the curse of multiagency. In this work, we focuses on general RMGs with LFA. For uncertainty sets defined by total variation distance, we develop provably data-efficient algorithms that break the curse of multiagency in both the generative model setting and a newly proposed online interactive setting. To our knowledge, our results are the first to break the curse of multiagency of sample complexity for RMGs with large (possibly infinite) state spaces, regardless of the uncertainty set construction.

Jingchu Gai, Laixi Shi

Machine Learning Department, Carnegie Mellon University. · Department of Electrical and Computer Engineering, Johns Hopkins University, MD 21286, USA.

Mean Field Games Multi-Agent Reinforcement Learning Markov Decision Processes State Space Action Space Sample Complexity

Apr 30, 2026math.OC

Continuous-time q-learning for mean-field control with common noise, part-II: q-learning algorithms

This paper is a continuation work of Ren et al. (2026) aiming to further devise q-learning algorithms for mean-field control (MFC) with controlled common noise. Based on the relaxed control formulation, we first establish the martingale condition of the value function and the Iq-function by evaluating along the conditional state distributions generated by all test policies. As the data in the relaxed control formulation are not observable in practice, we quantify the error incurred when they are replaced by the observable ones in the exploratory formulation under discretely sampled actions. This, together with a two-layer fixed point characterization of an optimal policy in Ren et al. (2026), allows us to propose several algorithms including the Actor-Critic q-learning algorithm, in which the policy is updated in the Actor-step based on the iteration rule induced by the improved Iq-function, and the value function and Iq-function are updated in the Critic-step based on the martingale orthogonality condition using the data from the exploratory formulation. We also establish the convergence of the inner iterations in the Actor-step in an infinite-horizon linear quadratic (LQ) framework. In two examples, within and beyond LQ framework, our q-learning algorithms are implemented with satisfactory performance.

Zhenjie Ren, Xiaoli Wei, Xiang Yu +1

LaMME, Universit´e ´Evry Paris-Saclay, ´Evry, France · Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong · Department of Industrial Engineering and Operations Research, Columbia University, New York, USA

Q-Learning Stochastic Optimal Control Mean Field Games Conformal Test Martingales

Apr 4, 2026cs.AI

Communication Gain and Delay Cost Under Cross-Timestep Delays in Cooperative Multi-Agent Reinforcement Learning

Communication is essential for coordination in \emph{cooperative} multi-agent reinforcement learning under partial observability, yet \emph{cross-timestep} delays cause messages to arrive multiple timesteps after generation, inducing temporal misalignment and making information stale when consumed. We formalize this setting as a delayed-communication partially observable Markov game (DeComm-POMG) and decompose a message's effect into \emph{communication gain} and \emph{delay cost}, yielding the Communication Gain and Delay Cost (CGDC) metric. We further establish a value-loss bound showing that the degradation induced by delayed messages is upper-bounded by a discounted accumulation of an information gap between the action distributions induced by timely versus delayed messages. Guided by CGDC, we propose \textbf{CDCMA}, an actor--critic framework that requests messages only when predicted CGDC is positive, predicts future observations to reduce misalignment at consumption, and fuses delayed messages via CGDC-guided attention. Experiments on no-teammate-vision variants of Cooperative Navigation and Predator Prey, and on SMAC maps across multiple delay levels show consistent improvements in performance, robustness, and generalization, with ablations validating each component.

Zihong Gao, Hongjian Liang, Lei Hao +1

The State Key Laboratory for Manufacturing Systems Engineering School of Automation Science and Engineering, Xi’an Jiaotong University

Multi-Agent Reinforcement Learning Mean Field Games

Feb 26, 2026cs.LG

Multi-agent imitation learning with function approximation: Linear Markov games and beyond

In this work, we present the first theoretical analysis of multi-agent imitation learning (MAIL) in linear Markov games where both the transition dynamics and each agent's reward function are linear in some given features. We demonstrate that by leveraging this structure, it is possible to replace the state-action level "all policy deviation concentrability coefficient" (Freihaut et al., arXiv:2510.09325) with a concentrability coefficient defined at the feature level which can be much smaller than the state-action analog when the features are informative about states' similarity. Furthermore, to circumvent the need for any concentrability coefficient, we turn to the interactive setting. We provide the first, computationally efficient, interactive MAIL algorithm for linear Markov games and show that its sample complexity depends only on the dimension of the feature map

d

. Building on these theoretical findings, we propose a deep MAIL interactive algorithm which clearly outperforms BC on games such as Tic-Tac-Toe and Connect4.

Luca Viano, Till Freihaut, Emanuele Nevali +3

1EPFL · University of Zurich · 3Earth Species Project

Mean Field Games Multi-Agent Reinforcement Learning Approximation

Aug 5, 2025cs.MA

Distributionally Robust Markov Games with Average Reward

We propose and study distributionally robust Markov games (DR-MGs) with the average-reward criterion as a crucial framework for multi-agent decision-making under model mismatches and over extended horizons. Under a standard irreducible assumption, we first derive a correspondence between the optimal policies and the solutions of the robust Bellman equation, based on which we further show the existence of a stationary Nash Equilibrium (NE) of the game. We further study DR-MGs under a more general weakly communicating setting. We construct a set-valued map based on the constant-gain optimal robust Bellman operator and show that its value is a subset of the best-response policies. We further prove that this map admits a fixed point, which implies the existence of NE. We then design two algorithms, Robust Nash-Iteration and robust TD Descent, with provably convergent guarantees. Finally, we show that the NE under average-reward can be approximated by the ones for the discounted DR-MGs as the discount factor approaches one. Our studies provide a comprehensive theoretical and algorithmic foundation for decision-making in complex, uncertain, and long-running multi-player environments.

Zachary Roch, Yue Wang

Department of Electrical and Computer Engineering, University of Central Florida, Orlando, Florida, USA. · Department of Computer Science, University of Central Florida, Orlando, Florida, USA.

Mean Field Games Markov Decision Processes Multi-Agent Reinforcement Learning Nash Equilibrium Distributional Reinforcement Learning Bellman Equation+1

May 1, 2025cs.LG

Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problems

In this paper, we propose a Gaussian Process (GP)-based policy iteration framework for addressing both forward and inverse problems in Hamilton--Jacobi--Bellman (HJB) equations and mean field games (MFGs). Policy iteration is formulated as an alternating procedure between evaluating the value function under a fixed control policy and improving the policy. In our approach, we model the unknown fields using GPs within a policy-iteration framework that converts the nonlinear system into a sequence of linear PDE subproblems. Then, leveraging the linear structure, the updates for the value function and, in the MFG setting, the population density admit explicit representer formulas under linear PDE collocation constraints. The policy is subsequently updated pointwise via a Legendre transform step, which involves a low-dimensional maximization over the control variable. This maximization is explicit for standard quadratic costs. For smooth, strictly convex costs, this pointwise maximization is solved through its first-order optimality condition, whereas in constrained or non-smooth cases, it becomes a low-dimensional constrained maximization problem. To improve convergence, we incorporate the additive Schwarz acceleration as a preconditioning step following each policy update. Numerical experiments demonstrate the effectiveness of the Schwarz acceleration in improving computational efficiency.

Xianjin Yang, Jingguo Zhang

Department of Computing and Mathematical Sciences, California Institute of Technology, CA, USA. · Department of Mathematics and Risk Management Institute, National University of Singapore, Singapore.

Mean Field Games Hamilton-Jacobi Reachability Boltzmann Policies Gaussian Process Stochastic Optimal Control Frictive Policy Optimization+2

Date pendingcs.GT

Independent Learning of Nash Equilibria in Partially Observable Markov Potential Games with Decoupled Dynamics

We study Nash equilibrium learning in partially observable Markov games (POMGs), a multi-agent reinforcement learning framework in which agents cannot fully observe the underlying state. Prior work in this setting relies on centralization or information sharing, and suffers from sample and computational complexity that scales exponentially in the number of players. We focus on a subclass of POMGs with independent state transitions, where agents remain coupled through their rewards, and assume that the underlying fully observed Markov game is a Markov potential game. For this class, we present an independent learning algorithm in which players, observing only their own actions and observations and without communication, jointly converge to an approximate Nash equilibrium. Due to partial observability, optimal policies may in general depend on the full action-observation history. Under a filter stability assumption, we show that policies based on finite history windows provide sufficient approximation guarantees. This enables us to approximate the POMG by a surrogate Markov game that is near-potential, leading to quasi-polynomial sample and computational complexity for independent Nash equilibrium learning in the underlying POMG.

Philip Jordan, Maryam Kamgarpour

SYCAMORE, EPFL

Mean Field Games Nash Equilibrium Partially Observable Markov Decision Process Multi-Agent Reinforcement Learning Markov