Mean field game (MFG) systems consisting of a major agent and a large number of minor agents were introduced in (Huang, 2010) in an LQG setup. The Nash certainty equivalence was used to obtain a Markovian closed-loop Nash equilibrium for the limiting system when the number of minor agents tends to infinity. In the past years several approaches to major--minor mean field game problems have been developed, principally (i) the Nash certainty equivalence and analytic approach, (ii) master equations, (iii) asymptotic solvability, and (iv) the probabilistic approach. For the LQG case, the recent work (Huang, 2021) establishes the equivalency of the Markovian closed-loop Nash equilibrium obtained via (i) with those obtained via (ii) and (iii). In this work, we demonstrate that the Markovian closed-loop Nash equilibrium of (i) is equivalent to that of (iv) for the LQG case. These two studies answer the long-standing questions about the consistency of the solutions to major-minor LQG MFG systems derived using different approaches.
Mean field games with a major player were introduced in (Huang, 2010) within a linear-quadratic (LQ) modeling framework. Due to the rich structure of major-minor player models, the past ten years have seen significant research efforts for different solution notions and analytical techniques. For LQ models, we address the relation between three solution frameworks: the Nash certainty equivalence (NCE) approach in (Huang, 2010), master equations, and asymptotic solvability, which have been developed starting with different ideas. We establish their equivalence relationships.
This paper studies approximate solutions to large-scale linear quadratic stochastic games with homogeneous nodal dynamics parameters and heterogeneous network couplings within the graphon mean field game framework in [2]-[4]. A graphon time-varying dynamical system model is first formulated to study the finite and then limit problems of linear quadratic Gaussian graphon mean field games (LQG-GMFG). The Nash equilibrium of the limit problem is then characterized by two coupled graphon time-varying dynamical systems. Sufficient conditions are established for the existence of a unique solution to…
We study a general class of entropy-regularized multi-variate LQG mean field games (MFGs) in continuous time with $K$ distinct sub-population of agents. We extend the notion of actions to action distributions (exploratory actions), and explicitly derive the optimal action distributions for individual agents in the limiting MFG. We demonstrate that the optimal set of action distributions yields an $ε$-Nash equilibrium for the finite-population entropy-regularized MFG. Furthermore, we compare the resulting solutions with those of classical LQG MFGs and establish the equivalence of their existenc…
This paper investigates the linear-quadratic-Gaussian (LQG) mean-field game (MFG) for a class of stochastic delay systems. We consider a large population system in which the dynamics of each player satisfies some forward stochastic differential delay equation (SDDE). The consistency condition or Nash certainty equivalence (NCE) principle is established through an auxiliary mean-field system of anticipated forward-backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such consistency condition system can be further established by some continuation method instead t…
This paper is concerned with a backward-forward stochastic differential equation (BFSDE) system, in which a large number of negligible agents are coupled in their dynamics via state average. Here some BSDE is introduced as the dynamics of major player, while dynamics of minor players are described by SDEs. Some auxiliary mean-field SDEs (MFSDEs) and a $3\times2$ mixed forward-backward stochastic differential equation (FBSDE) system are considered and analyzed instead of involving the fixed-point analysis as in other mean-field games. We also derive the decentralized strategies which are shown …
We study the performance of the certainty equivalent controller on Linear Quadratic (LQ) control problems with unknown transition dynamics. We show that for both the fully and partially observed settings, the sub-optimality gap between the cost incurred by playing the certainty equivalent controller on the true system and the cost incurred by using the optimal LQ controller enjoys a fast statistical rate, scaling as the square of the parameter error. To the best of our knowledge, our result is the first sub-optimality guarantee in the partially observed Linear Quadratic Gaussian (LQG) setting.…
Inverse Problems for Costs and Controls in LQG MFGs via Mean Field Trajectories
Center for Data Science, New York University（纽约大学数据科学中心）；NYU Shanghai（纽约大学上海分校）；Shanghai Center for Data Science（上海数据科学中心）；NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai（纽约大学上海分校纽约-华东师范大学数学科学研究所）
This paper studies a new class of dynamic optimization problems of large-population (LP) system which consists of a large number of negligible and coupled agents. The most significant feature in our setup is the dynamics of individual agents follow the forward-backward stochastic differential equations (FBSDEs) in which the forward and backward states are coupled at the terminal time. This current paper is hence different to most existing large-population literature where the individual states are typically modeled by the SDEs including the forward state only. The associated mean-field linear-…
This paper investigates a class of mixed stochastic linear-quadratic-Gaussian (LQG) social optimization and Nash game in the context of a large scale system. Two types of interactive agents are involved: a major agent and a large number of weakly-coupled minor agents. All minor agents are cooperative to minimize the social cost as the sum of their individual costs, whereas such social cost are conflictive to that of major agent. Thus, major agent and all minor agents are further competitive to reach some non-zero Nash equilibrium. The control processes enter both diffusion and drift terms of a…
The large-population system consists of considerable small agents whose individual behavior and mass effect are interrelated via their state-average. The mean-field game provides an efficient way to get the decentralized strategies of large-population system when studying its dynamic optimizations. Unlike other large-population literature, this current paper possesses the following distinctive features. First, our setting includes the partial information structure of large-population system which is practical from real application standpoint. Specially, two cases of partial information structu…
Northwestern University（西北大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We study discrete-time mean-field Markov games with infinite numbers of agents where each agent aims to minimize its ergodic cost. We consider the setting where the agents have identical linear state transitions and quadratic cost functions, while the aggregated effect of the agents is captured by the population mean of their states, namely, the mean-field state. For such a game, based on the Nash certainty equivalence principle, we provide sufficient conditions for the existence and uniqueness of its Nash equilibrium. Moreover, to find the Nash equilibrium, we propose a mean-field actor-criti…
This paper considers a class of mean field linear-quadratic-Gaussian (LQG) games with model uncertainty. The drift term in the dynamics of the agents contains a common unknown function. We take a robust optimization approach where a representative agent in the limiting model views the drift uncertainty as an adversarial player. By including the mean field dynamics in an augmented state space, we solve two optimal control problems sequentially, which combined with consistent mean field approximations provides a solution to the robust game. A set of decentralized control strategies is derived by…
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…
