Laplace principle for large population games with control interaction

TL;DR

This paper establishes a large deviation principle for Nash equilibria in large population stochastic differential games with control interactions, linking finite-player equilibria to mean field game solutions under certain regularity conditions.

Contribution

It provides the first large deviation results for equilibria in large population games with control interactions, connecting finite-player Nash equilibria to mean field game solutions.

Findings

Large deviation principle for Nash equilibria established

Conditions for existence of Lipschitz solutions to the master equation

Results extended to cooperative games and linear-quadratic cases

Abstract

This work investigates continuous time stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. The control processes are assumed to be open-loop. We give regularity conditions guaranteeing that if the finite-player game admits a Nash equilibrium, then both the sequence of equilibria and the corresponding states processes satisfy a Sanov-type large deviation principle. The result requires existence of a Lipschitz continuous solution of the master equation of the corresponding mean field game, and is based on concentration inequalities for Lipschitz FBSDEs. The result carries over to cooperative (i.e. central planner) games. We study the linear-quadratic case of such games in details.