On Approximate Nash Equilibria in Mean Field Games

TL;DR

This paper studies approximate Nash equilibria in large symmetric games using mean field game solutions, showing they are nearly optimal for individual players uniformly across initial conditions.

Contribution

It establishes an $ ext{L}^ ext{infinity}$ notion of approximate Nash equilibrium in both static and dynamic mean field games, including cases with conditional laws.

Findings

Strategies form an $ ext{L}^ ext{infinity}$ approximate Nash equilibrium.

Results apply to static and continuous-time dynamic models.

Includes cases where agents' payoffs depend on conditional distributions.

Abstract

In the context of large population symmetric games, approximate Nash equilibria are introduced through equilibrium solutions of the corresponding mean field game in the sense that the individual gain from optimal unilateral deviation under such strategies converges to zero in the large population size asymptotic. We show that these strategies satisfy an $\L^\infty$ notion of approximate Nash equilibrium which guarantees that the individual gain from optimal unilateral deviation is small uniformly among players and uniformly on their initial characteristics. We establish these results in the context of static models and in the dynamic continuous time setting, and we cover situations where the agents' criteria depend on the conditional law of the controlled state process.