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 form...

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Bibliographic Details
Published in:IEEE transactions on automatic control 2020-04, Vol.65 (4), p.1397-1412
Main Authors: Huang, Minyi, Zhou, Mengjie
Format: Article
Language:English
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
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Summary: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.
ISSN:0018-9286
1558-2523
DOI:10.1109/TAC.2019.2919111