Calculus of Variations and Geometric Measure Theory

M. Bardi

Explicit solutions of some Linear-Quadratic Mean Field Games

created by bardi on 19 May 2011
modified on 27 Aug 2012


Published Paper

Inserted: 19 may 2011
Last Updated: 27 aug 2012

Journal: Networks and Heterogeneous Media
Volume: 7
Number: 2
Pages: 243-261
Year: 2012


We consider N-person differential games involving linear systems affected by white noise, running cost quadratic in the control and in the displacement of the state from a reference position, and with long-time-average integral cost functional. We solve an associated system of Hamilton-Jacobi-Bellman and Kolmogorov-Fokker-Plank equations and find explicit Nash equilibria in the form of linear feedbacks. Next we compute the limit as the number N of players goes to infinity, assuming they are almost identical and with suitable scalings of the parameters. This provides a quadratic-Gaussian solution to a system of two differential equations of the kind introduced by Lasry and Lions in the theory of Mean Field Games. Under a natural normalization the uniqueness of this solution depends on the sign of a single parameter. We also discuss some singular limits, such as vanishing noise, cheap control, vanishing discount. Finally, we compare the L-Q model with other Mean Field models of population distribution.