# Linear Mean-Field Games with Discounted Cost

arXiv: https://arxiv.org/abs/2301.06074 · HTML: https://mlsanity.com/paper/2301.06074/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jan 2023.

## Built on by (most cited first, 1 of 1)

- [Maximum Causal Entropy IRL in Mean-Field Games and GNEP Framework for Forward RL](https://mlsanity.com/paper/2401.06566/) (2024)

## Abstract (arXiv)

In this paper, we introduce discrete-time linear mean-field games subject to an infinite-horizon discounted-cost optimality criterion. The state space of a generic agent is a compact Borel space. At every time, each agent is randomly coupled with another agent via their dynamics and one-stage cost function, where this randomization is generated via the empirical distribution of their states (i.e., the mean-field term). Therefore, the transition probability and the one-stage cost function of each agent depend linearly on the mean-field term, which is the key distinction between classical mean-field games and linear mean-field games. Under mild assumptions, we show that the policy obtained from infinite population equilibrium is varepsilon(N)-Nash when the number of agents N is sufficiently large, where varepsilon(N) is an explicit function of N. Then, using the linear programming formulation of MDPs and the linearity of the transition probability in mean-field term, we formulate the game in the infinite population limit as a generalized Nash equilibrium problem (GNEP) and establish an algorithm for computing equilibrium with a convergence guarantee.