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Linear Quadratic Mean Field Games: Asymptotic Solvability and Relation to the Fixed Point Approach.
- Source :
- IEEE Transactions on Automatic Control; Apr2020, Vol. 65 Issue 4, p1397-1412, 16p
- Publication Year :
- 2020
-
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 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00189286
- Volume :
- 65
- Issue :
- 4
- Database :
- Complementary Index
- Journal :
- IEEE Transactions on Automatic Control
- Publication Type :
- Periodical
- Accession number :
- 143316641
- Full Text :
- https://doi.org/10.1109/TAC.2019.2919111