Mean Field Games With Parametrized Followers

• Published in: IEEE transactions on automatic control Vol. 65; no. 1; pp. 12 - 27
• Main Authors: Bensoussan, Alain, Cass, Thomas, Chau, Man Ho Michael, Yam, Sheung Chi Phillip
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Delay; Delay effects; Delay systems; Delays; Differential equations; Games; mean field games (MFGs); nonlinear system; Partial differential equations; Sociology; Statistics; stocashtic optimal control; Stochastic processes; stochastic systems
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2019.2910945

We consider mean field games between a dominant leader and many followers, such that each follower is subject to a heterogeneous delay effect from the leader's action, who in turn can exercise governance on the population through this influence. The delay effects are assumed to be discretely distributed among the followers. Given regular enough coefficients, we describe a necessary condition for the existence of a solution for the equilibrium by a system of coupled forward-backward stochastic differential equations and stochastic partial differential equations. We provide a thorough study for the particular linear quadratic case. By adopting a functional approach, we obtain the time-independent sufficient condition, which warrants the unique existence of the solution of the whole mean field game problem. Several numerical illustrations with different time horizons and populations are demonstrated.