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

Abstract

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 rescaling 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.Index Terms-Asymptotic solvability, direct approach, fixed point approach, linear quadratic, mean field game, re-scaling, Riccati differential equation.where Ψ is any matrix from A, Q i , Q i f

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

“…Subsequently, we view the ODE system (B.1)-(B.9) as a slightly perturbed form of (31). The remaining proof is similar to that of (Huang and Zhou, 2018b, Theorem 5) and we only give its sketch.…”

Linear quadratic mean field social optimization: Asymptotic solvability

“…Accordingly, (3.18) is highly nonlinear. This feature distinguishes our model from [30,31,36]. Lemma 3.5.…”

“…A key feature of our system is that the state and control weight matrices may be indefinite. Due to the highly nonlinear Riccati ODEs resulting from controlled diffusion terms, the development of the rescaling technique is more challenging than in [30,31,36]. We further obtain a tight upper bound of the optimality loss of the obtained decentralized strategies, and quantify the efficiency gain with respect to mean field game solutions.…”

“…In contrast, the PbP optimality based approach is much harder to apply due to the common noise. Second, the large Riccati equation is particularly suitable for applying the rescaling technique as in [31]. The LQ social optimization model in [14,26] involves positive semi-definite state weight and positive definite control weight, and the players have only independent noises.…”

Indefinite Linear Quadratic Mean Field Social Control with Multiplicative Noise

“…In this paper, we solve the problem by directly decoupling high-dimensional forward-backward stochastic differential equations (FBSDEs) instead of fixed-point analysis. This procedure shares a similar philosophy with the direct method [24], [25]. In recent years, some progress for optimal LQ control has been made by tackling FBSDEs [48], [11], [51], [6], [31], [39], [33].…”