On the convergence of closed-loop Nash equilibria to the mean field game limit

TL;DR

This paper investigates how Nash equilibria of finite-player stochastic games converge to mean field game equilibria as the number of players grows, extending convergence results to cases with non-unique equilibria.

Contribution

It adapts compactness arguments to the closed-loop setting, proving convergence to weak MFG equilibria even when the equilibrium is non-unique.

Findings

Convergence of closed-loop Nash equilibria to weak MFG equilibria is established.

The results hold even when the MFG equilibrium is non-unique.

Partial insights into which weak MFG equilibria can be limits of finite-player equilibria.

Abstract

This paper continues the study of the mean field game (MFG) convergence problem: In what sense do the Nash equilibria of $n$-player stochastic differential games converge to the mean field game as $n\rightarrow\infty$? Previous work on this problem took two forms. First, when the $n$-player equilibria are open-loop, compactness arguments permit a characterization of all limit points of $n$-player equilibria as weak MFG equilibria, which contain additional randomness compared to the standard (strong) equilibrium concept. On the other hand, when the $n$-player equilibria are closed-loop, the convergence to the MFG equilibrium is known only when the MFG equilibrium is unique and the associated "master equation" is solvable and sufficiently smooth. This paper adapts the compactness arguments to the closed-loop case, proving a convergence theorem that holds even when the MFG equilibrium is non-unique. Every limit point of $n$-player equilibria is shown to be the same kind of weak MFG equilibrium as in the open-loop case. Some partial results and examples are discussed for the converse question, regarding which of the weak MFG equilibria can arise as the limit of $n$-player (approximate) equilibria.