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

• Published in: IEEE transactions on automatic control Vol. 65; no. 4; pp. 1397 - 1412
• Main Authors: Huang, Minyi, Zhou, Mengjie
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.04.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic properties; Asymptotic solvability; Boundary value problems; Differential equations; direct approach; Dynamic programming; fixed point approach; Fixed points (mathematics); Game theory; Games; linear quadratic; Mathematical model; mean field game; Optimal control; Ordinary differential equations; re-scaling; Riccati differential equation; Sociology; Statistics
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2919111

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.