Differential N-players game: Nash equilibria and Mather measures

TL;DR

This paper investigates Nash equilibria in deterministic N-players games, establishing their existence as Mather measures and connecting the mean field limit to ergodic PDE systems for a continuum of players.

Contribution

It introduces a framework linking Nash equilibria to Mather measures and demonstrates the mean field limit via ergodic PDEs.

Findings

Existence of Nash equilibria as Mather measures.

Connection between N-players game and ergodic PDEs.

Mean field limit described by continuum PDE system.

Abstract

We study Nash equilibria for the deterministic ergodic N-players game. We introduce pure strategies, mixed strategies and Nash equilibria associated with those. We show that a Nash equilibrium in mixed strategies exists and it is a Mather measure for the Lagrangian system defined by the cost functional. In conclusion, we show that the mean field limit of the N-players game is described by the ergodic PDE's system for a continuum of players.