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

Abstract : 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.
Type de document :
Pré-publication, Document de travail
31 pages. 2014
Domaine :

https://hal.inria.fr/hal-01068307
Contributeur : Estelle Bouzat <>
Soumis le : jeudi 25 septembre 2014 - 13:56:40
Dernière modification le : jeudi 14 juin 2018 - 10:54:02

### Identifiants

• HAL Id : hal-01068307, version 1
• ARXIV : 1401.1421

### Citation

Martino Bardi, Fabio S. Priuli. Linear-Quadratic $N$-person and Mean-Field Games with Ergodic Cost. 31 pages. 2014. 〈hal-01068307〉

### Métriques

Consultations de la notice