Calculus of Variations and Geometric Measure Theory

M. Bardi - F. S. Priuli

Linear-Quadratic N-person and Mean-Field Games with Ergodic Cost

created by bardi on 08 Jan 2014
modified on 07 Jan 2015


Published Paper

Inserted: 8 jan 2014
Last Updated: 7 jan 2015

Journal: SIAM J. Control Optim.
Volume: 52
Number: 5
Pages: 3022-3052
Year: 2014
Doi: 10.1137/140951795


We consider stochastic differential games with $N$ players, linear-Gaussian dynamics in arbitrary state-space dimension, and long-time-average cost with quadratic running cost. Admissible controls are feedbacks for which the system is ergodic. We first study the existence of affine Nash equilibria by means of an associated system of $N$ Hamilton-Jacobi-Bellman and $N$ Kolmogorov-Fokker-Planck partial differential equations. We give necessary and sufficient conditions for the existence and uniqueness of quadratic-Gaussian solutions in terms of the solvability of suitable algebraic Riccati and Sylvester equations. Under a symmetry condition on the running costs and for nearly identical players we study the large population limit, $N$ tending to infinity, and find a unique quadratic-Gaussian solution of the pair of Mean Field Game HJB-KFP equations. Examples of explicit solutions are given, in particular for consensus problems.