Differential N-players game: Nash equilibria and Mather measures.

This article focuses on the analysis of Nash equilibria in deterministic ergodic N-player differential games, introducing both pure and mixed strategy frameworks. It establishes that while Nash equilibria in pure strategies exist under the condition of smooth solutions to an associated system of Hamilton–Jacobi equations, such smoothness is generally not guaranteed. To address this, the authors define mixed strategies as invariant probability measures (Mather measures) for the Euler flow of the underlying Lagrangian system and prove the existence of Nash equilibria in mixed strategies under broad assumptions. Furthermore, for symmetric games, the paper demonstrates that as the number of players tends to infinity, the Nash equilibria converge to solutions of an ergodic mean field game (MFG) system described by a coupled PDE system, thereby linking finite-player games with continuum-player MFG theory.