We propose a discrete time graphon game formulation on continuous state and action spaces using a representative player to study stochastic games with heterogeneous interaction among agents. This formulation admits both philosophical and mathematical advantages, compared to a widely adopted formulation using a continuum of players. We prove the existence and uniqueness of the graphon equilibrium with mild assumptions, and show that this equilibrium can be used to construct an approximate solution for finite player game on networks, which is challenging to analyze and solve due to curse of dimensionality. An online oracle-free learning algorithm is developed to solve the equilibrium numerically, and sample complexity analysis is provided for its convergence.
This paper studies stochastic games on large graphs and their graphon limits. We propose a new formulation of graphon games based on a single typical player's label-state distribution. In contrast, other recently proposed models of graphon games work directly with a continuum of players, which involves serious measure-theoretic technicalities. In fact, by viewing the label as a component of the state process, we show in our formulation that graphon games are a special case of mean field games, albeit with certain inevitable degeneracies and discontinuities that make most existing results on me…
We consider a game for a continuum of non-identical players evolving on a finite state space. Their heterogeneous interactions are represented by a graphon, which can be viewed as the limit of a dense random graph. The player's transition rates between the states depend on their own control and the interaction strengths with the other players. We develop a rigorous mathematical framework for this game and analyze Nash equilibria. We provide a sufficient condition for a Nash equilibrium and prove existence of solutions to a continuum of fully coupled forward-backward ordinary differential equat…
A Deep Learning Method for Optimal Investment Under Relative Performance Criteria Among Heterogeneous Agents
Shanghai Frontiers Science Center of Artificial Intelligence and Deep Learning（上海人工智能与深度学习前沿科学中心）；NYU-ECNU Institute of Mathematical Sciences（纽约大学-华东师范大学数学科学研究所）；NYU Shanghai（上海纽约大学）；Princeton University（普林斯顿大学）
In this paper, we study graphon mean field games using a system of forward-backward stochastic differential equations. We establish the existence and uniqueness of solutions under two different assumptions and prove the stability with respect to the interacting graphons which are necessary to show propagation of chaos results. As an application of propagation of chaos, we prove the convergence of n-player game Nash equilibrium for a general model, which is new in the theory of graphon mean field games.
Extended Graphon Mean-Field Games in Discrete Time
University of Florida（佛罗里达大学）；The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；NYU Shanghai（纽约大学上海分校）；INRIA Paris（巴黎国家信息与自动化研究所）
Optimal Coarse Correlated Equilibria in Mean Field Games: Linear Programming and No-Regret Learning
Department of Mathematics “Federigo Enriques”, University of Milan, Via Saldini 50, 20133, Milan, Italy.；Center for Mathematical Economics (IMW), Bielefeld University, Universit\"atsstrasse 25, 33615, Bielefeld, Germany.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
In this paper, we study a class of linear-quadratic (LQ) mean field games of controls with common noises and their corresponding $N$-player games. The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control. By the stochastic maximum principle, we first analyze the limiting behavior of the representative player and obtain his/her optimal control in a feedback form with the given distributional flow of the population and its control. The mean field equilibrium is determined by the Nash certainty equival…
In the paper, we use the equivalent formulation of a finite state mean field game as a control problem with mixed constraints to study the dependence of solutions to finite state mean field game on an initial distribution of players. We introduce the concept of value multifunction of the mean field game that is a mapping assigning to an initial time and an initial distribution a set of expected outcomes of the representative player corresponding to solutions of the mean field game. Using the control reformulation of the finite state mean field game, we give the sufficient condition on a given …
The planning problem for the mean field game implies the one tries to transfer the system of infinitely many identical rational agents from the given distribution to the final one using the choice of the terminal payoff. It can be formulated as the mean field game system with the boundary condition only on the measure variable. In the paper, we consider the continuous-time finite state mean field game assuming that the space of actions for each player is compact. It is shown that the planning problem in this case may not admit a solution even if the final distribution is reachable from the ini…
We introduce a mean field model for optimal holding of a representative agent of her peers as a natural expected scaling limit from the corresponding $N-$agent model. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations. We study the corresponding mean field game of mutual holding in the absence of common noise. Our first main result provides an explicit equilibrium of this mean field game, defined by a bang--bang control consisting in holding those competitors with positive drift coefficient of their dynami…
Operator Learning for Families of Finite-State Mean-Field Games
University of California, Berkeley（加州大学伯克利分校）；University of Michigan（密歇根大学）；NYU Shanghai Center for Data Science NYU-ECNU Institute of Mathematical Sciences, NYU Shanghai（纽约大学上海中心数据科学研究院 纽约大学-华东师范大学数学科学研究所）
他在法兰西学院的演讲中，P.L. Lions引入了Master equation的概念，参见[5] Mean Field Games。它是在一个启发式的方式，从偏微分方程组偏微分方程组，相关的纳什均衡的一个大的，但有限的，玩家的数量。该方法，也解释了在[2]，包括在一个正式的类比的条款。这个方程的有趣之处在于，它包含了一些有趣的特殊情况，可以直接研究，特别是当轨迹独立时，作为有限纳什均衡博弈的极限而得到的HJB-FP（Hamilton-Jacobi-Bellman，Fokker-Planck）方程组，见[4]。通常，在平均域论中，人们可以通过引入代表性代理人的概念来绕过大的纳什均衡，其行为受到类似代理人的分布的影响，并且直接获得感兴趣的HJB-FP方程组，例如参见[1]。显然，对于主方程没有这样的方法。我们在这里证明这是可能的。我们首先对平均场类型的控制问题进行了研究，我们完全解释了主方程。对于平均场博弈本身，我们解决了一个相关的问题，并再次获得主方程。
