Mathieu Laurière

Kevin Wang, Anna Thöni, Benjamin Kempinski et al.

Large language models (LLMs) are increasingly deployed as interactive agents, yet their capacity for social and strategic reasoning over extended interaction remains poorly understood. Existing evaluations rely on static vignettes or single-game benchmarks that cannot capture the sustained, multi-faceted reasoning that real-world multi-agent settings demand. We introduce Mindgames, a multi-game arena and evaluation platform for LLM agents that operationalizes complementary reasoning demands relevant to ``theory of mind'': belief attribution under hidden information, opponent modeling through repeated strategic interaction, cooperative inference under knowledge asymmetries, and sustained deception in social deduction. Built on TextArena, Mindgames provides a unified interaction interface, TrueSkill-based rating, and full trajectory logging across four game environments. We instantiate Mindgames through a 2025 competition cycle hosted at a major AI conference, which assessed 944 submitted agents from 76 teams across four games: Colonel Blotto, Iterated Prisoner's Dilemma, Codenames, and Secret Mafia. Our analysis surfaces both agent-level and evaluation-level limitations: brittle rule adherence remains a major bottleneck, top-performing systems repeatedly rely on explicit structural scaffolding, and leaderboard validity differs sharply across environments. In particular, failure-heavy environments can reward robustness to opponent errors as much as strategic ability, with Secret Mafia exhibiting a pronounced error-survival confound in this cycle. We release a dataset of 29,571 multi-agent games with turn-level observations, actions, and rewards, together with MG-Ref, a deterministic offline tournament protocol that scores new agents against a frozen reference pool of top-ranked, low-error Stage~II submissions under the same error-attribution lens used in this analysis.

Daisuke Inoue, Mathieu Laurière, Dante Kalise

The Schrödinger Bridge Problem constructs a stochastic process that connects an initial distribution to a terminal distribution with minimum energy. This work considers its mean-field extension, the Mean-Field Schrödinger Bridge, for interacting particle systems. With nonlocal interactions, evaluating the resulting particle-dependent distributional terms can scale quadratically with the population size, which makes large-scale problems intractable. We address this bottleneck by approximating the nonlocal interactions with neural network surrogates. The resulting four-stage alternating algorithm reduces the per-step cost from quadratic to linear in the population size at inference. We also derive Grönwall-type stability bounds that show how surrogate errors propagate to the generated trajectories. In numerical experiments on navigation and opinion-dynamics tasks, the proposed method reproduces trajectories obtained with analytical evaluation and reduces training time.

Lorenzo Magnino, Jiacheng Shen, Matthieu Geist et al.

The intersection of Mean Field Games (MFGs) and Reinforcement Learning (RL) has fostered a growing family of algorithms designed to solve large-scale multi-agent systems. However, the field currently lacks a standardized evaluation protocol, forcing researchers to rely on bespoke, isolated, and often simplistic environments. This fragmentation makes it difficult to assess the robustness, generalization, and failure modes of emerging methods. To address this gap, we propose a comprehensive benchmark suite for MFGs (Bench-MFG), focusing on the discrete-time, discrete-space, stationary setting for the sake of clarity. We introduce a taxonomy of problem classes, ranging from no-interaction and monotone games to potential and dynamics-coupled games, and provide prototypical environments for each. Furthermore, we propose MF-Garnets, a method for generating random MFG instances to facilitate rigorous statistical testing. We benchmark a variety of learning algorithms across these environments, including a novel black-box approach (MF-PSO) for exploitability minimization. Based on our extensive empirical results, we propose guidelines to standardize future experimental comparisons. Code available at \href{https://github.com/lorenzomagnino/Bench-MFG}{https://github.com/lorenzomagnino/Bench-MFG}.

Mathieu Laurière, Sarah Perrin, Julien Pérolat et al.

Non-cooperative and cooperative games with a very large number of players have many applications but remain generally intractable when the number of players increases. Introduced by Lasry and Lions, and Huang, Caines and Malhamé, Mean Field Games (MFGs) rely on a mean-field approximation to allow the number of players to grow to infinity. Traditional methods for solving these games generally rely on solving partial or stochastic differential equations with a full knowledge of the model. Recently, Reinforcement Learning (RL) has appeared promising to solve complex problems at scale. The combination of RL and MFGs is promising to solve games at a very large scale both in terms of population size and environment complexity. In this survey, we review the quickly growing recent literature on RL methods to learn equilibria and social optima in MFGs. We first identify the most common settings (static, stationary, and evolutive) of MFGs. We then present a general framework for classical iterative methods (based on best-response computation or policy evaluation) to solve MFGs in an exact way. Building on these algorithms and the connection with Markov Decision Processes, we explain how RL can be used to learn MFG solutions in a model-free way. Last, we present numerical illustrations on a benchmark problem, and conclude with some perspectives.

Mathieu Laurière, Sarah Perrin, Sertan Girgin et al.

Mean Field Games (MFGs) have been introduced to efficiently approximate games with very large populations of strategic agents. Recently, the question of learning equilibria in MFGs has gained momentum, particularly using model-free reinforcement learning (RL) methods. One limiting factor to further scale up using RL is that existing algorithms to solve MFGs require the mixing of approximated quantities such as strategies or $q$-values. This is far from being trivial in the case of non-linear function approximation that enjoy good generalization properties, e.g. neural networks. We propose two methods to address this shortcoming. The first one learns a mixed strategy from distillation of historical data into a neural network and is applied to the Fictitious Play algorithm. The second one is an online mixing method based on regularization that does not require memorizing historical data or previous estimates. It is used to extend Online Mirror Descent. We demonstrate numerically that these methods efficiently enable the use of Deep RL algorithms to solve various MFGs. In addition, we show that these methods outperform SotA baselines from the literature.

Luis Briceño-Arias, Dante Kalise, Ziad Kobeissi et al.

We study a numerical approximation of a time-dependent Mean Field Game (MFG) system with local couplings. The discretization we consider stems from a variational approach described in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] for the stationary problem and leads to the finite difference scheme introduced by Achdou and Capuzzo-Dolcetta in [SIAM J. Numer. Anal., 48(3):1136-1162, 2010]. In order to solve the finite dimensional variational problems, in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] the authors implement the primal-dual algorithm introduced by Chambolle and Pock in [J. Math. Imaging Vision, 40(1):120-145, 2011], whose core consists in iteratively solving linear systems and applying a proximity operator. We apply that method to time-dependent MFG and, for large viscosity parameters, we improve the linear system solution by replacing the direct approach used in [Briceno-Arias, Kalise, and Silva, SIAM J. Control Optim., 2017] by suitable preconditioned iterative algorithms.

Ruimeng Hu, Mathieu Laurière

Stochastic optimal control and games have a wide range of applications, from finance and economics to social sciences, robotics, and energy management. Many real-world applications involve complex models that have driven the development of sophisticated numerical methods. Recently, computational methods based on machine learning have been developed for solving stochastic control problems and games. In this review, we focus on deep learning methods that have unlocked the possibility of solving such problems, even in high dimensions or when the structure is very complex, beyond what traditional numerical methods can achieve. We consider mostly the continuous time and continuous space setting. Many of the new approaches build on recent neural-network-based methods for solving high-dimensional partial differential equations or backward stochastic differential equations, or on model-free reinforcement learning for Markov decision processes that have led to breakthrough results. This paper provides an introduction to these methods and summarizes the state-of-the-art works at the crossroad of machine learning and stochastic control and games.

Giorgia Ramponi, Pavel Kolev, Olivier Pietquin et al.

We explore the problem of imitation learning (IL) in the context of mean-field games (MFGs), where the goal is to imitate the behavior of a population of agents following a Nash equilibrium policy according to some unknown payoff function. IL in MFGs presents new challenges compared to single-agent IL, particularly when both the reward function and the transition kernel depend on the population distribution. In this paper, departing from the existing literature on IL for MFGs, we introduce a new solution concept called the Nash imitation gap. Then we show that when only the reward depends on the population distribution, IL in MFGs can be reduced to single-agent IL with similar guarantees. However, when the dynamics is population-dependent, we provide a novel upper-bound that suggests IL is harder in this setting. To address this issue, we propose a new adversarial formulation where the reinforcement learning problem is replaced by a mean-field control (MFC) problem, suggesting progress in IL within MFGs may have to build upon MFC.

Sebastian Baudelet, Brieuc Frénais, Mathieu Laurière et al.

Mean field control (MFC) problems have been introduced to study social optima in very large populations of strategic agents. The main idea is to consider an infinite population and to simplify the analysis by using a mean field approximation. These problems can also be viewed as optimal control problems for McKean-Vlasov dynamics. They have found applications in a wide range of fields, from economics and finance to social sciences and engineering. Usually, the goal for the agents is to minimize a total cost which consists in the integral of a running cost plus a terminal cost. In this work, we consider MFC problems in which there is no terminal cost but, instead, the terminal distribution is prescribed. We call such problems mean field optimal transport problems since they can be viewed as a generalization of classical optimal transport problems when mean field interactions occur in the dynamics or the running cost function. We propose three numerical methods based on neural networks. The first one is based on directly learning an optimal control. The second one amounts to solve a forward-backward PDE system characterizing the solution. The third one relies on a primal-dual approach. We illustrate these methods with numerical experiments conducted on two families of examples.

Jean-Pierre Fouque, Mathieu Laurière, Mengrui Zhang

We establish the convergence of the deep actor-critic reinforcement learning algorithm presented in [Angiuli et al., 2023a] in the setting of continuous state and action spaces with an infinite discrete-time horizon. This algorithm provides solutions to Mean Field Game (MFG) or Mean Field Control (MFC) problems depending on the ratio between two learning rates: one for the value function and the other for the mean field term. In the MFC case, to rigorously identify the limit, we introduce a discretization of the state and action spaces, following the approach used in the finite-space case in [Angiuli et al., 2023b]. The convergence proofs rely on a generalization of the two-timescale framework introduced in [Borkar, 1997]. We further extend our convergence results to Mean Field Control Games, which involve locally cooperative and globally competitive populations. Finally, we present numerical experiments for linear-quadratic problems in one and two dimensions, for which explicit solutions are available.

Lev Fedorov, Michaël E. Sander, Romuald Elie et al.

Transformers have revolutionized deep learning across various domains but understanding the precise token dynamics remains a theoretical challenge. Existing theories of deep Transformers with layer normalization typically predict that tokens cluster to a single point; however, these results rely on deterministic weight assumptions, which fail to capture the standard initialization scheme in Transformers. In this work, we show that accounting for the intrinsic stochasticity of random initialization alters this picture. More precisely, we analyze deep Transformers where noise arises from the random initialization of value matrices. Under diffusion scaling and token-wise RMS normalization, we prove that, as the number of Transformer layers goes to infinity, the discrete token dynamics converge to an interacting-particle system on the sphere where tokens are driven by a \emph{common} matrix-valued Brownian noise. In this limit, we show that initialization noise prevents the collapse to a single cluster predicted by deterministic models. For two tokens, we prove a phase transition governed by the interaction strength and the token dimension: unlike deterministic attention flows, antipodal configurations become attracting with positive probability. Numerical experiments confirm the predicted transition, reveal that antipodal formations persist for more than two tokens, and demonstrate that suppressing the intrinsic noise degrades accuracy.

Jiacheng Shen, Masato Hagiwara, Milad Alizadeh et al.

Large language models have shown strong performance on broad-domain knowledge and reasoning benchmarks, but it remains unclear how well language models handle specialized animal-related knowledge under a unified closed-book evaluation protocol. We introduce BAGEL, a benchmark for evaluating animal knowledge expertise in language models. BAGEL is constructed from diverse scientific and reference sources, including bioRxiv, Global Biotic Interactions, Xeno-canto, and Wikipedia, using a combination of curated examples and automatically generated closed-book question-answer pairs. The benchmark covers multiple aspects of animal knowledge, including taxonomy, morphology, habitat, behavior, vocalization, geographic distribution, and species interactions. By focusing on closed-book evaluation, BAGEL measures animal-related knowledge of models without external retrieval at inference time. BAGEL further supports fine-grained analysis across source domains, taxonomic groups, and knowledge categories, enabling a more precise characterization of model strengths and systematic failure modes. Our benchmark provides a new testbed for studying domain-specific knowledge generalization in language models and for improving their reliability in biodiversity-related applications.

Kai Shao, Jiacheng Shen, Mathieu Laurière

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 finite number of other coalitions. Although the theory has been extensively developed, we are still lacking efficient and scalable computational methods. Here, we develop reinforcement learning methods for such games in a finite space setting with general dynamics and reward functions. We start by proving that the MFTG solution yields approximate Nash equilibria in finite-size coalition games. We then propose two algorithms. The first is based on the quantization of mean-field spaces and Nash Q-learning. We provide convergence and stability analysis. We then propose a deep reinforcement learning algorithm, which can scale to larger spaces. Numerical experiments in 4 environments with mean-field distributions of dimension up to $200$ show the scalability and efficiency of the proposed method.

Lin Chen, Samuel Drapeau, Fanghao Shao et al.

Generative Flow Network (GFlowNet) objectives implicitly fix an equal mixing of forward and backward policies, potentially constraining the exploration-exploitation trade-off during training. By further exploring the link between GFlowNets and Markov chains, we establish an equivalence between GFlowNet objectives and Markov chain reversibility, thereby revealing the origin of such constraints, and provide a framework for adapting Markov chain properties to GFlowNets. Building on these theoretical findings, we propose $α$-GFNs, which generalize the mixing via a tunable parameter $α$. This generalization enables direct control over exploration-exploitation dynamics to enhance mode discovery capabilities, while ensuring convergence to unique flows. Across various benchmarks, including Set, Bit Sequence, and Molecule Generation, $α$-GFN objectives consistently outperform previous GFlowNet objectives, achieving up to a $10 \times$ increase in the number of discovered modes.

Grégoire Lambrecht, Mathieu Laurière

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.

Muhammad Aneeq uz Zaman, Alec Koppel, Mathieu Laurière et al.

We address in this paper Reinforcement Learning (RL) among agents that are grouped into teams such that there is cooperation within each team but general-sum (non-zero sum) competition across different teams. To develop an RL method that provably achieves a Nash equilibrium, we focus on a linear-quadratic structure. Moreover, to tackle the non-stationarity induced by multi-agent interactions in the finite population setting, we consider the case where the number of agents within each team is infinite, i.e., the mean-field setting. This results in a General-Sum LQ Mean-Field Type Game (GS-MFTG). We characterize the Nash equilibrium (NE) of the GS-MFTG, under a standard invertibility condition. This MFTG NE is then shown to be $O(1/M)$-NE for the finite population game where $M$ is a lower bound on the number of agents in each team. These structural results motivate an algorithm called Multi-player Receding-horizon Natural Policy Gradient (MRNPG), where each team minimizes its cumulative cost \emph{independently} in a receding-horizon manner. Despite the non-convexity of the problem, we establish that the resulting algorithm converges to a global NE through a novel problem decomposition into sub-problems using backward recursive discrete-time Hamilton-Jacobi-Isaacs (HJI) equations, in which \emph{independent natural policy gradient} is shown to exhibit linear convergence under time-independent diagonal dominance. Numerical studies included corroborate the theoretical results.

Kai Cui, Gökçe Dayanıklı, Mathieu Laurière et al.

Recent techniques based on Mean Field Games (MFGs) allow the scalable analysis of multi-player games with many similar, rational agents. However, standard MFGs remain limited to homogeneous players that weakly influence each other, and cannot model major players that strongly influence other players, severely limiting the class of problems that can be handled. We propose a novel discrete time version of major-minor MFGs (M3FGs), along with a learning algorithm based on fictitious play and partitioning the probability simplex. Importantly, M3FGs generalize MFGs with common noise and can handle not only random exogeneous environment states but also major players. A key challenge is that the mean field is stochastic and not deterministic as in standard MFGs. Our theoretical investigation verifies both the M3FG model and its algorithmic solution, showing firstly the well-posedness of the M3FG model starting from a finite game of interest, and secondly convergence and approximation guarantees of the fictitious play algorithm. Then, we empirically verify the obtained theoretical results, ablating some of the theoretical assumptions made, and show successful equilibrium learning in three example problems. Overall, we establish a learning framework for a novel and broad class of tractable games.

Mathieu Laurière, Ludovic Tangpi, Xuchen Zhou

Graphon games have been introduced to study games with many players who interact through a weighted graph of interaction. By passing to the limit, a game with a continuum of players is obtained, in which the interactions are through a graphon. In this paper, we focus on a graphon game for optimal investment under relative performance criteria, and we propose a deep learning method. The method builds upon two key ingredients: first, a characterization of Nash equilibria by forward-backward stochastic differential equations and, second, recent advances of machine learning algorithms for stochastic differential games. We provide numerical experiments on two different financial models. In each model, we compare the effect of several graphons, which correspond to different structures of interactions.

Lorenzo Magnino, Kai Shao, Zida Wu et al.

Mean field games (MFGs) have emerged as a powerful framework for modeling interactions in large-scale multi-agent systems. Despite recent advancements in reinforcement learning (RL) for MFGs, existing methods are typically limited to finite spaces or stationary models, hindering their applicability to real-world problems. This paper introduces a novel deep reinforcement learning (DRL) algorithm specifically designed for non-stationary continuous MFGs. The proposed approach builds upon a Fictitious Play (FP) methodology, leveraging DRL for best-response computation and supervised learning for average policy representation. Furthermore, it learns a representation of the time-dependent population distribution using a Conditional Normalizing Flow. To validate the effectiveness of our method, we evaluate it on three different examples of increasing complexity. By addressing critical limitations in scalability and density approximation, this work represents a significant advancement in applying DRL techniques to complex MFG problems, bringing the field closer to real-world multi-agent systems.

Antonio Ocello, Daniil Tiapkin, Lorenzo Mancini et al.

We introduce Mean-Field Trust Region Policy Optimization (MF-TRPO), a novel algorithm designed to compute approximate Nash equilibria for ergodic Mean-Field Games (MFG) in finite state-action spaces. Building on the well-established performance of TRPO in the reinforcement learning (RL) setting, we extend its methodology to the MFG framework, leveraging its stability and robustness in policy optimization. Under standard assumptions in the MFG literature, we provide a rigorous analysis of MF-TRPO, establishing theoretical guarantees on its convergence. Our results cover both the exact formulation of the algorithm and its sample-based counterpart, where we derive high-probability guarantees and finite sample complexity. This work advances MFG optimization by bridging RL techniques with mean-field decision-making, offering a theoretically grounded approach to solving complex multi-agent problems.

Lorenzo Magnino, Yuchen Zhu, Mathieu Laurière

Optimal stopping is a fundamental problem in optimization with applications in risk management, finance, robotics, and machine learning. We extend the standard framework to a multi-agent setting, named multi-agent optimal stopping (MAOS), where agents cooperate to make optimal stopping decisions in a finite-space, discrete-time environment. Since solving MAOS becomes computationally prohibitive as the number of agents is very large, we study the mean-field optimal stopping (MFOS) problem, obtained as the number of agents tends to infinity. We establish that MFOS provides a good approximation to MAOS and prove a dynamic programming principle (DPP) based on mean-field control theory. We then propose two deep learning approaches: one that learns optimal stopping decisions by simulating full trajectories and another that leverages the DPP to compute the value function and to learn the optimal stopping rule using backward induction. Both methods train neural networks to approximate optimal stopping policies. We demonstrate the effectiveness and the scalability of our work through numerical experiments on 6 different problems in spatial dimension up to 300. To the best of our knowledge, this is the first work to formalize and computationally solve MFOS in discrete time and finite space, opening new directions for scalable MAOS methods.

Mathieu Laurière, Mehdi Talbi

This paper presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex recursive dependencies that remain challenging. We address this by combining the Dynamic Programming Principle with neural network approximation of the value function. Unlike policy-search methods, our algorithm explicitly learns the value surface. We first consider the discrete-time problem and analyze neural network training error. We then turn to continuous problems and analyze the additional error due to the discretization of the underlying stochastic processes. Numerical experiments on high-dimensional American basket options and nonlinear utility maximization demonstrate that our method provides an efficient and scalable method for the multiple optimal stopping problem.

Zhouzhou Gu, Mathieu Laurière, Sebastian Merkel et al.

We propose and compare new global solution algorithms for continuous time heterogeneous agent economies with aggregate shocks. First, we approximate the agent distribution so that equilibrium in the economy can be characterized by a high, but finite, dimensional non-linear partial differential equation. We consider different approximations: discretizing the number of agents, discretizing the agent state variables, and projecting the distribution onto a finite set of basis functions. Second, we represent the value function using a neural network and train it to solve the differential equation using deep learning tools. We refer to the solution as an Economic Model Informed Neural Network (EMINN). The main advantage of this technique is that it allows us to find global solutions to high dimensional, non-linear problems. We demonstrate our algorithm by solving important models in the macroeconomics and spatial literatures (e.g. Krusell and Smith (1998), Khan and Thomas (2007), Bilal (2023)).

Sarah Perrin, Mathieu Laurière, Julien Pérolat et al.

Mean Field Games (MFGs) can potentially scale multi-agent systems to extremely large populations of agents. Yet, most of the literature assumes a single initial distribution for the agents, which limits the practical applications of MFGs. Machine Learning has the potential to solve a wider diversity of MFG problems thanks to generalizations capacities. We study how to leverage these generalization properties to learn policies enabling a typical agent to behave optimally against any population distribution. In reference to the Master equation in MFGs, we coin the term ``Master policies'' to describe them and we prove that a single Master policy provides a Nash equilibrium, whatever the initial distribution. We propose a method to learn such Master policies. Our approach relies on three ingredients: adding the current population distribution as part of the observation, approximating Master policies with neural networks, and training via Reinforcement Learning and Fictitious Play. We illustrate on numerical examples not only the efficiency of the learned Master policy but also its generalization capabilities beyond the distributions used for training.

Mathieu Laurière, Gilles Pagès, Olivier Pironneau

Fishing quotas are unpleasant but efficient to control the productivity of a fishing site. A popular model has a stochastic differential equation for the biomass on which a stochastic dynamic programming or a Hamilton-Jacobi-Bellman algorithm can be used to find the stochastic control -- the fishing quota. We compare the solutions obtained by dynamic programming against those obtained with a neural network which preserves the Markov property of the solution. The method is extended to a similar multi species model to check its robustness in high dimension.

René Carmona, Mathieu Laurière

Financial markets and more generally macro-economic models involve a large number of individuals interacting through variables such as prices resulting from the aggregate behavior of all the agents. Mean field games have been introduced to study Nash equilibria for such problems in the limit when the number of players is infinite. The theory has been extensively developed in the past decade, using both analytical and probabilistic tools, and a wide range of applications have been discovered, from economics to crowd motion. More recently the interaction with machine learning has attracted a growing interest. This aspect is particularly relevant to solve very large games with complex structures, in high dimension or with common sources of randomness. In this chapter, we review the literature on the interplay between mean field games and deep learning, with a focus on three families of methods. A special emphasis is given to financial applications.

Matthieu Geist, Julien Pérolat, Mathieu Laurière et al.

Concave Utility Reinforcement Learning (CURL) extends RL from linear to concave utilities in the occupancy measure induced by the agent's policy. This encompasses not only RL but also imitation learning and exploration, among others. Yet, this more general paradigm invalidates the classical Bellman equations, and calls for new algorithms. Mean-field Games (MFGs) are a continuous approximation of many-agent RL. They consider the limit case of a continuous distribution of identical agents, anonymous with symmetric interests, and reduce the problem to the study of a single representative agent in interaction with the full population. Our core contribution consists in showing that CURL is a subclass of MFGs. We think this important to bridge together both communities. It also allows to shed light on aspects of both fields: we show the equivalence between concavity in CURL and monotonicity in the associated MFG, between optimality conditions in CURL and Nash equilibrium in MFG, or that Fictitious Play (FP) for this class of MFGs is simply Frank-Wolfe, bringing the first convergence rate for discrete-time FP for MFGs. We also experimentally demonstrate that, using algorithms recently introduced for solving MFGs, we can address the CURL problem more efficiently.

Sarah Perrin, Mathieu Laurière, Julien Pérolat et al.

We present a method enabling a large number of agents to learn how to flock, which is a natural behavior observed in large populations of animals. This problem has drawn a lot of interest but requires many structural assumptions and is tractable only in small dimensions. We phrase this problem as a Mean Field Game (MFG), where each individual chooses its acceleration depending on the population behavior. Combining Deep Reinforcement Learning (RL) and Normalizing Flows (NF), we obtain a tractable solution requiring only very weak assumptions. Our algorithm finds a Nash Equilibrium and the agents adapt their velocity to match the neighboring flock's average one. We use Fictitious Play and alternate: (1) computing an approximate best response with Deep RL, and (2) estimating the next population distribution with NF. We show numerically that our algorithm learn multi-group or high-dimensional flocking with obstacles.

Julien Perolat, Sarah Perrin, Romuald Elie et al.

We address scaling up equilibrium computation in Mean Field Games (MFGs) using Online Mirror Descent (OMD). We show that continuous-time OMD provably converges to a Nash equilibrium under a natural and well-motivated set of monotonicity assumptions. This theoretical result nicely extends to multi-population games and to settings involving common noise. A thorough experimental investigation on various single and multi-population MFGs shows that OMD outperforms traditional algorithms such as Fictitious Play (FP). We empirically show that OMD scales up and converges significantly faster than FP by solving, for the first time to our knowledge, examples of MFGs with hundreds of billions states. This study establishes the state-of-the-art for learning in large-scale multi-agent and multi-population games.

René Carmona, Kenza Hamidouche, Mathieu Laurière et al.

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic utility are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The game is analyzed and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the first case, the gradients are computed exactly using the model whereas they are estimated using Monte-Carlo simulations in the second case. Numerical experiments show the convergence of the two players' controls as well as the utility function when the two algorithms are used in different scenarios.

René Carmona, Kenza Hamidouche, Mathieu Laurière et al.

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic cost are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of indistinguishable agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The optimality conditions of the game are analysed for both open-loop and closed-loop controls, and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the model-based case, the gradients are computed exactly using the model, whereas they are estimated using Monte-Carlo simulations in the sample-based case. Numerical experiments are conducted to show the convergence of the utility function as well as the two players' controls.

Sarah Perrin, Julien Perolat, Mathieu Laurière et al.

In this paper, we deepen the analysis of continuous time Fictitious Play learning algorithm to the consideration of various finite state Mean Field Game settings (finite horizon, $γ$-discounted), allowing in particular for the introduction of an additional common noise. We first present a theoretical convergence analysis of the continuous time Fictitious Play process and prove that the induced exploitability decreases at a rate $O(\frac{1}{t})$. Such analysis emphasizes the use of exploitability as a relevant metric for evaluating the convergence towards a Nash equilibrium in the context of Mean Field Games. These theoretical contributions are supported by numerical experiments provided in either model-based or model-free settings. We provide hereby for the first time converging learning dynamics for Mean Field Games in the presence of common noise.

Andrea Angiuli, Jean-Pierre Fouque, Mathieu Laurière

We present a Reinforcement Learning (RL) algorithm to solve infinite horizon asymptotic Mean Field Game (MFG) and Mean Field Control (MFC) problems. Our approach can be described as a unified two-timescale Mean Field Q-learning: The \emph{same} algorithm can learn either the MFG or the MFC solution by simply tuning the ratio of two learning parameters. The algorithm is in discrete time and space where the agent not only provides an action to the environment but also a distribution of the state in order to take into account the mean field feature of the problem. Importantly, we assume that the agent can not observe the population's distribution and needs to estimate it in a model-free manner. The asymptotic MFG and MFC problems are also presented in continuous time and space, and compared with classical (non-asymptotic or stationary) MFG and MFC problems. They lead to explicit solutions in the linear-quadratic (LQ) case that are used as benchmarks for the results of our algorithm.

Laura Leal, Mathieu Laurière, Charles-Albert Lehalle

We use a deep neural network to generate controllers for optimal trading on high frequency data. For the first time, a neural network learns the mapping between the preferences of the trader, i.e. risk aversion parameters, and the optimal controls. An important challenge in learning this mapping is that in intraday trading, trader's actions influence price dynamics in closed loop via the market impact. The exploration--exploitation tradeoff generated by the efficient execution is addressed by tuning the trader's preferences to ensure long enough trajectories are produced during the learning phase. The issue of scarcity of financial data is solved by transfer learning: the neural network is first trained on trajectories generated thanks to a Monte-Carlo scheme, leading to a good initialization before training on historical trajectories. Moreover, to answer to genuine requests of financial regulators on the explainability of machine learning generated controls, we project the obtained "blackbox controls" on the space usually spanned by the closed-form solution of the stylized optimal trading problem, leading to a transparent structure. For more realistic loss functions that have no closed-form solution, we show that the average distance between the generated controls and their explainable version remains small. This opens the door to the acceptance of ML-generated controls by financial regulators.

Haoyang Cao, Xin Guo, Mathieu Laurière

Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing, and have recently attracted growing interests in financial modelings. This paper analyzes GANs from the perspectives of mean-field games (MFGs) and optimal transport. More specifically, from the game theoretical perspective, GANs are interpreted as MFGs under Pareto Optimality criterion or mean-field controls; from the optimal transport perspective, GANs are to minimize the optimal transport cost indexed by the generator from the known latent distribution to the unknown true distribution of data. The MFGs perspective of GANs leads to a GAN-based computational method (MFGANs) to solve MFGs: one neural network for the backward Hamilton-Jacobi-Bellman equation and one neural network for the forward Fokker-Planck equation, with the two neural networks trained in an adversarial way. Numerical experiments demonstrate superior performance of this proposed algorithm, especially in the higher dimensional case, when compared with existing neural network approaches.

René Carmona, Mathieu Laurière, Zongjun Tan

We study infinite horizon discounted Mean Field Control (MFC) problems with common noise through the lens of Mean Field Markov Decision Processes (MFMDP). We allow the agents to use actions that are randomized not only at the individual level but also at the level of the population. This common randomization allows us to establish connections between both closed-loop and open-loop policies for MFC and Markov policies for the MFMDP. In particular, we show that there exists an optimal closed-loop policy for the original MFC. Building on this framework and the notion of state-action value function, we then propose reinforcement learning (RL) methods for such problems, by adapting existing tabular and deep RL methods to the mean-field setting. The main difficulty is the treatment of the population state, which is an input of the policy and the value function. We provide convergence guarantees for tabular algorithms based on discretizations of the simplex. Neural network based algorithms are more suitable for continuous spaces and allow us to avoid discretizing the mean field state space. Numerical examples are provided.

René Carmona, Mathieu Laurière, Zongjun Tan

We investigate reinforcement learning in the setting of Markov decision processes for a large number of exchangeable agents interacting in a mean field manner. Applications include, for example, the control of a large number of robots communicating through a central unit dispatching the optimal policy computed by maximizing an aggregate reward. An approximate solution is obtained by learning the optimal policy of a generic agent interacting with the statistical distribution of the states and actions of the other agents. We first provide a full analysis this discrete-time mean field control problem. We then rigorously prove the convergence of exact and model-free policy gradient methods in a mean-field linear-quadratic setting and establish bounds on the rates of convergence. We also provide graphical evidence of the convergence based on implementations of our algorithms.

René Carmona, Mathieu Laurière

We propose two numerical methods for the optimal control of McKean-Vlasov dynamics in finite time horizon. Both methods are based on the introduction of a suitable loss function defined over the parameters of a neural network. This allows the use of machine learning tools, and efficient implementations of stochastic gradient descent in order to perform the optimization. In the first method, the loss function stems directly from the optimal control problem. The second method tackles a generic forward-backward stochastic differential equation system (FBSDE) of McKean-Vlasov type, and relies on suitable reformulation as a mean field control problem. To provide a guarantee on how our numerical schemes approximate the solution of the original mean field control problem, we introduce a new optimization problem, directly amenable to numerical computation, and for which we rigorously provide an error rate. Several numerical examples are provided. Both methods can easily be applied to certain problems with common noise, which is not the case with the existing technology. Furthermore, although the first approach is designed for mean field control problems, the second is more general and can also be applied to the FBSDE arising in the theory of mean field games.

René Carmona, Mathieu Laurière

We propose two algorithms for the solution of the optimal control of ergodic McKean-Vlasov dynamics. Both algorithms are based on approximations of the theoretical solutions by neural networks, the latter being characterized by their architecture and a set of parameters. This allows the use of modern machine learning tools, and efficient implementations of stochastic gradient descent.The first algorithm is based on the idiosyncrasies of the ergodic optimal control problem. We provide a mathematical proof of the convergence of the approximation scheme, and we analyze rigorously the approximation by controlling the different sources of error. The second method is an adaptation of the deep Galerkin method to the system of partial differential equations issued from the optimality condition. We demonstrate the efficiency of these algorithms on several numerical examples, some of them being chosen to show that our algorithms succeed where existing ones failed. We also argue that both methods can easily be applied to problems in dimensions larger than what can be found in the existing literature. Finally, we illustrate the fact that, although the first algorithm is specifically designed for mean field control problems, the second one is more general and can also be applied to the partial differential equation systems arising in the theory of mean field games.

Romuald Elie, Julien Pérolat, Mathieu Laurière et al.

Learning by experience in Multi-Agent Systems (MAS) is a difficult and exciting task, due to the lack of stationarity of the environment, whose dynamics evolves as the population learns. In order to design scalable algorithms for systems with a large population of interacting agents (e.g. swarms), this paper focuses on Mean Field MAS, where the number of agents is asymptotically infinite. Recently, a very active burgeoning field studies the effects of diverse reinforcement learning algorithms for agents with no prior information on a stationary Mean Field Game (MFG) and learn their policy through repeated experience. We adopt a high perspective on this problem and analyze in full generality the convergence of a fictitious iterative scheme using any single agent learning algorithm at each step. We quantify the quality of the computed approximate Nash equilibrium, in terms of the accumulated errors arising at each learning iteration step. Notably, we show for the first time convergence of model free learning algorithms towards non-stationary MFG equilibria, relying only on classical assumptions on the MFG dynamics. We illustrate our theoretical results with a numerical experiment in a continuous action-space environment, where the approximate best response of the iterative fictitious play scheme is computed with a deep RL algorithm.