Large Deviations Principle for Discrete-time Mean-field Games

TL;DR

This paper proves a large deviations principle for discrete-time mean-field games, providing a mathematical framework to analyze rare events in large populations under equilibrium policies.

Contribution

It extends large deviations analysis to discrete-time mean-field games under weak Feller continuity, adapting methods from continuous-time models and prior uncontrolled particle systems.

Findings

Established LDP for discrete-time mean-field games.

Connected LDP for initial states and noise to the game model.

Compared results with prior uncontrolled particle system studies.

Abstract

In this paper, we establish a large deviations principle (LDP) for interacting particle systems that arise from state and action dynamics of discrete-time mean-field games under the equilibrium policy of the infinite-population limit. The LDP is proved under weak Feller continuity of state and action dynamics. The proof is based on transferring LDP for empirical measures of initial states and noise variables under setwise topology to the original game model via contraction principle, which was first suggested by Delarue, Lacker, and Ramanan to establish LDP for continuous-time mean-field games under common noise. We also compare our work with LDP results established in prior literature for interacting particle systems, which are in a sense uncontrolled versions of mean-field games.