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

Abstract

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 (2018a, 2018b) 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 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 [30,31], a necessary and sufficient condition for asymptotic solvability of N -player LQ mean field games is obtained, through analyzing a low-dimensional ODE system derived by applying a rescaling method to the sequence of high dimensional centralized solutions. Asymptotic solvability of LQ mean field games with a major player is studied in [36].…”

Linear quadratic mean field games: Decentralized $O(1/N)$-Nash equilibria

“…The proof is postponed near the end of this section. Following the rescaling method in [20,22,30], we define…”

“…It uses a rescaling method to derive a set of Riccati ordinary differential equations (ODEs), which characterizes a necessary and sufficient condition for asymptotic solvability [22]. This method can be extended to LQ mean field games with a major player [30]. Recently, Huang and Yang [20] extend this asymptotic solvability notion to mean field social optimization, where the agents cooperatively optimize a social cost.…”