Linear Quadratic Mean Field Games Research Articles

Linear‐quadratic mean‐field game for stochastic large‐population systems with jump diffusion

This study is concerned with linear-quadratic mean-field game problem for a class of stochastic large-population systems with Poisson processes. The control is allowed to enter the jump diffusion terms of the individuals' state equation. By virtue of the stochastic Hamiltonian system and Riccati equation for limiting control problem, the decentralised optimal strategies are represented in the open-loop and closed-loop forms, respectively. Different from the existing results, the limit representation of average term is proposed in closed-loop form via the separation technique. Meanwhile, the decentralised optimal strategies are verified to be the ϵ -Nash equilibrium of the original problem. Finally, two practical control problems in engineering and economics areas are presented to demonstrate the good performance of theoretical results.

ON THE EXISTENCE OF THE SOLUTION OF RICCATI EQUATIONS ARISING IN LINEAR QUADRATIC MEAN FIELD DYNAMIC GAMES

In this paper we obtain existence conditions for the solution of a class of generalized Riccati equations arising in finite horizon linear quadratic (LQ) mean-field games.

Linear quadratic mean field games with a major player: The multi-scale approach

This paper considers linear quadratic (LQ) mean field games with a major player and analyzes an asymptotic solvability problem. It starts with a large-scale system of coupled dynamic programming equations and applies a re-scaling technique introduced in Huang and Zhou (2018) and Huang and Zhou (2020) to derive a set of Riccati equations in lower dimensions, the solvability of which determines the necessary and sufficient condition for asymptotic solvability. We next derive the mean field limit of the strategies and the value functions. Finally, we show that the two decentralized strategies can be interpreted as the best responses of a major player and a representative minor player embedded in an infinite population, which have the property of consistent mean field approximations.

Linear Quadratic Mean Field Games: Asymptotic Solvability and Relation to the Fixed Point Approach

Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit. This is accomplished by developing a re-scaling method to derive a low-dimensional ordinary differential equation (ODE) system, where a non-symmetric Riccati ODE has a central role. We next compare with the fixed point approach which determines a two-point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem.

Selection of equilibria in a linear quadratic mean-field game

In this paper, we address an instance of uniquely solvable mean-field game with a common noise whose corresponding counterpart without common noise has several equilibria. We study the selection problem for this mean-field game without common noise via three approaches.A common approach is to select, amongst all the equilibria, those yielding the minimal cost for the representative player. Another one is to select equilibria that are included in the support of the zero noise limit of the mean-field game with common noise. A last one is to select equilibria supported by the limit of the mean-field component of the corresponding N-player game as the number of players goes to infinity. The contribution of this paper is to show that, for the class under study, the last two approaches select the same equilibria, but the first approach selects another one.

Relationship between backward and forward linear-quadratic mean-field-game with terminal constraint and optimal asset allocation for insurers and pension funds

Herein, motivated by problems faced by insurance firms, we consider the dynamic games of N weakly coupled linear forward stochastic systems with terminal constraints involving mean-field interactions. By penalisation method, the associated mean-field game (MFG) is formulated and its consistency condition is given by a fully coupled forward–backward stochastic differential equation (FBSDE). Moreover, the decentralised strategies are obtained, and the ε-Nash equilibrium is verified. In addition, we study the connection of backward linear quadratic (LQ) MFG and forward LQ MFG with terminal constraint. Furthermore, the decoupled optimal strategies of this MFG are solved explicitly by introducing some Riccati equations. As an illustration, some simulations of the optimal asset allocation for the firm and pension funds are further studied.

Price of anarchy for Mean Field Games

The price of anarchy, originally introduced to quantify the inefficiency of selfish behavior in routing games, is extended to mean field games. The price of anarchy is defined as the ratio of a worst case social cost computed for a mean field game equilibrium to the optimal social cost as computed by a central planner. We illustrate properties of such a price of anarchy on linear quadratic extended mean field games, for which explicit computations are possible. A sufficient and necessary condition to have no price of anarchy is presented. Various asymptotic behaviors of the price of anarchy are proved for limiting behaviors of the coefficients in the model and numerics are presented.

An integral control formulation of mean field game based large scale coordination of loads in smart grids

Pressure on ancillary reserves, i.e. frequency preserving, in power systems has significantly mounted due to the recent generalized increase of the fraction of (highly fluctuating) wind and solar energy sources in grid generation mixes. The energy storage associated with millions of individual customer electric thermal (heating–cooling) loads is considered as a tool for smoothing power demand/generation imbalances. The piecewise constant level tracking problem of their collective energy content is formulated as a linear quadratic mean field game problem with integral control in the cost coefficients. The introduction of integral control brings with it a robustness potential to mismodeling, but also the potential of cost coefficient unboundedness. A suitable Banach space is introduced to establish the existence of Nash equilibria for the corresponding infinite population game, and algorithms are proposed for reliably computing a class of desirable near Nash equilibria. Numerical simulations illustrate the flexibility and robustness of the approach.

Mean Field Game with Delay: A Toy Model

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

Linear–Quadratic Mean-Field Game for Stochastic Delayed Systems

This paper studies the linear–quadratic mean-field game (MFG) for a class of stochastic delayed systems. We consider a large-population system, where the dynamics of each agent is modeled by a stochastic differential delayed equation. The consistency condition is derived through an auxiliary system, which is an anticipated forward–backward stochastic differential equation with delay (AFBSDDE). The wellposedness of such an AFBSDDE system can be obtained using a continuation method. Thus, the MFG strategies can be defined on an arbitrary time horizon, not necessary on a small time horizon by a commonly used contraction mapping method. Moreover, the decentralized strategies are verified to satisfy the $\epsilon$ -Nash equilibrium property. For illustration, three special cases of delayed systems are further explored, for which the closed-loop and open-loop MFG strategies are derived, respectively.

Mean field stochastic linear quadratic games for continuum-parameterized multi-agent systems

In this paper, we consider a stochastic linear quadratic mean field game for the continuum-parameterized multi-agent systems with multiplicative noise. Based on the Nash certain equivalence principle, we obtain a series of decentralized control laws. Then, Dynkin’s formula and comparison principle are employed to prove the boundedness of the state of the closed loop system in the mean square sense. Finally, we show that the set of decentralized controls has an ϵ-Nash equilibrium property.

Linear quadratic mean field game with control input constraint

In this paper, we study a class of linear-quadratic (LQ) mean-field games in which the individual control process is constrained in a closed convex subsetΓof full space ℝm. The decentralized strategies and consistency condition are represented by a class of mean-field forward-backward stochastic differential equation (MF-FBSDE) with projection operators onΓ. The wellposedness of consistency condition system is obtained using the monotonicity condition method. The relatedϵ-Nash equilibrium property is also verified.

Linear-Quadratic Mean Field Stackelberg Games with State and Control Delays

In this article, we consider a linear-quadratic mean field game between a leader (dominating player) and a group of followers (agents) under the Stackelberg game setting as proposed in [A. Bensoussan, M. Chau, and S. Yam, Appl. Math. Optim., 74 (2016), pp. 91-128], so that the evolution of each individual follower is now also subjected to delay effects from both his/her state and control variables, as well as those of the leader. The overall Stackelberg game is solved by tackling three subproblems hierarchically. Their resolution corresponds to the establishment of the existence and uniqueness of the solutions of three different forward-backward stochastic functional differential equations, which we manage by applying the unified continuation method as first developed in, for example, [Y. Hu and S. Peng, Probab. Theory Related Fields, 103 (1995), pp. 273-283] and [X. Xu, Fully Coupled Forward-Backward Stochastic Functional Differential Equations and Applications to Quadratic Optimal Control, preprint, arX...

Mean-Field Games and Ambiguity Aversion

This paper looks at a general framework for mean-field games with ambiguity averse players based on the probabilistic framework described in Carmona (2013). A framework for mean-field games with ambiguity averse players is presented, using a version of the stochastic maximum principle to find the optimal controls of the players. The dynamics under the optimal control are characterized through a forwards-backwards stochastic differential equation and a relationship between the finite player game and the mean-field game is established. Explicit solutions are derived in the case of the linear-quadratic mean-field game.

Backward-forward linear-quadratic mean-field games with major and minor agents

This paper studies the backward-forward linear-quadratic-Gaussian (LQG) games with major and minor agents (players). The state of major agent follows a linear backward stochastic differential equation (BSDE) and the states of minor agents are governed by linear forward stochastic differential equations (SDEs). The major agent is dominating as its state enters those of minor agents. On the other hand, all minor agents are individually negligible but their state-average affects the cost functional of major agent. The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies. We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation (BFSDE) system. Next, we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method. Consequently, we obtain the decentralized strategies for major and minor agents which are proved to satisfy the e-Nash equilibrium property.

Linear Quadratic Mean Field Type Control and Mean Field Games with Common Noise, with Application to Production of an Exhaustible Resource

We study a general linear quadratic mean field type control problem and connect it to mean field games of a similar type. The solution is given both in terms of a forward/backward system of stochastic differential equations and by a pair of Riccati equations. In certain cases, the solution to the mean field type control is also the equilibrium strategy for a class of mean field games. We use this fact to study an economic model of production of exhaustible resources.

Uniqueness for Linear-Quadratic Mean Field Games with Common Noise

The purpose of this note is to show that a common noise may restore uniqueness in mean field games. To this end, we focus on a class of examples driven by linear dynamics and quadratic cost functions. Given these linear-quadratic mean field games, we prove existence and uniqueness of solutions in the presence of common noise and construct a counter-example in the absence of common noise. This illustrates the principle, already observed in dynamical systems like ODEs, that introducing an appropriate noise may restore uniqueness.

Linear-Quadratic Mean Field Games

We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward---backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.

Mean Field Stackelberg Games: Aggregation of Delayed Instructions

In this paper, we consider an $N$-player interacting strategic game in the presence of a (endogenous) dominating player, who gives direct influence on individual agents, through its impact on their control in the sense of Stackelberg game, and then on the whole community. Each individual agent is subject to a delay effect on collecting information, specifically at a delay time, from the dominating player. The size of his delay is completely known by the agent, while to others, including the dominating player, his delay plays as a hidden random variable coming from a common fixed distribution. By invoking a noncanonical fixed point property, we show that for a general class of finite $N$-player games, each of them converges to the mean field counterpart which may possess an optimal solution that can serve as an $\epsilon$-Nash equilibrium for the corresponding finite $N$-player game. Second, we provide, with explicit solutions, a comprehensive study on the corresponding linear quadratic mean field games of...

Linear–Quadratic Time-Inconsistent Mean Field Games

In this paper, we study a class of time-inconsistent analogs (in the sense of Hu et al. (Time-inconsistent stochastic linear–quadratic control. Preprint, 2012) which is originated from the mean-variance portfolio selection problem with state-dependent risk aversion in the context of financial economics) of the standard Linear–Quadratic Mean Field Games considered in Huang et al. (Commun. Inf. Syst. 6(3):221–252, 2006) and Bensoussan et al. (Linear–quadratic mean field games. http://www.sta.cuhk.edu.hk/scpy, submitted, 2012). For the one-dimensional case, we first establish the unique time-consistent optimal strategy under an arbitrary guiding path, with which we further obtain the unique time-consistent mean-field equilibrium strategy under a mild convexity condition. Second, for the dimension greater than one, by applying the adjoint equation approach, we formulate a sufficient condition under which the unique existence of both, time-consistent optimal strategy under a given guiding path and time-consistent equilibrium strategy, can be guaranteed.