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

arXiv: https://arxiv.org/abs/1401.1421 · HTML: https://mlsanity.com/paper/1401.1421/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2014.

## Built on by (most cited first, 5 of 5)

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- [Graphon Particle Systems, Part II: Dynamics of Distributed Stochastic Continuum Optimization](https://mlsanity.com/paper/2407.02765/) (2024)
- [Decentralized Strategies for Finite Population LQG Social Control: A Reinforcement Learning Approach](https://mlsanity.com/paper/2608.30244/) (2026)
- [Entropy-Regularized Mean-Field Games with Average-Reward Criterion: Span-Seminorm Contractions and Model-Free Learning](https://mlsanity.com/paper/2610.15871/) (2026)

## Abstract (arXiv)

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.