NDLI: Linear-Quadratic Mean Field Games

Content Provider	Springer Nature Link
Author	Bensoussan, A. Sung, K. C. J. Yam, S. C. P. Yung, S. P.
Copyright Year	2015
Abstract	We provide a comprehensive study of a general class of linear-quadratic mean field games. We adopt the adjoint equation approach to investigate the unique existence of their equilibrium strategies. Due to the linearity of the adjoint equations, the optimal mean field term satisfies a forward–backward ordinary differential equation. For the one-dimensional case, we establish the unique existence of the equilibrium strategy. For a dimension greater than one, by applying the Banach fixed point theorem under a suitable norm, a sufficient condition for the unique existence of the equilibrium strategy is provided, which is independent of the coefficients of controls in the underlying dynamics and is always satisfied whenever the coefficients of the mean field term are vanished, and hence, our theories include the classical linear-quadratic stochastic control problems as special cases. As a by-product, we also establish a neat and instructive sufficient condition, which is apparently absent in the literature and only depends on coefficients, for the unique existence of the solution for a class of nonsymmetric Riccati equations. Numerical examples of nonexistence of the equilibrium strategy will also be illustrated. Finally, a similar approach has been adopted to study the linear-quadratic mean field type stochastic control problems and their comparisons with mean field games.
Starting Page	496
Ending Page	529
Page Count	34
File Format	PDF
ISSN	00223239
Journal	Journal of Optimization Theory and Applications
Volume Number	169
Issue Number	2
e-ISSN	15732878
Language	English
Publisher	Springer US
Publisher Date	2015-10-26
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mean field games Mean field type stochastic control problems Adjoint equations Linear quadratic Calculus of Variations and Optimal Control; Optimization Optimization Theory of Computation Applications of Mathematics Engineering Operation Research/Decision Theory
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Control and Optimization Management Science and Operations Research

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