Risk-averse mean field games: exploitability and non-asymptotic analysis

TL;DR

This paper investigates how mean field games can approximate large N-player games with risk-averse agents, providing non-asymptotic bounds and methods to construct approximate equilibria without requiring equilibrium uniqueness.

Contribution

It introduces a non-asymptotic analysis framework for risk-averse mean field games, establishing approximation bounds and methods to derive N-player game equilibria from mean field solutions.

Findings

Large N-player games can be approximated by mean field games with small exploitabilities.

$oldsymbol{oldsymbol{ ext{delta}}}$-mean field equilibria can generate $oldsymbol{oldsymbol{ ext{epsilon}}}$-equilibria in N-player games.

Existence of mean field equilibria is proven under general conditions.

Abstract

In this paper, we use mean field games (MFGs) to investigate approximations of $N$-player games with uniformly symmetrically continuous heterogeneous closed-loop actions. To incorporate agents' risk aversion (beyond the classical expected utility of total costs), we use an abstract evaluation functional for their performance criteria. Centered around the notion of exploitability, we conduct non-asymptotic analysis on the approximation capability of MFGs from the perspective of state-action distributions without requiring the uniqueness of equilibria. Under suitable assumptions, we first show that scenarios in the $N$-player games with large $N$ and small average exploitabilities can be well approximated by approximate solutions of MFGs with relatively small exploitabilities. We then show that $\delta$-mean field equilibria can be used to construct $\varepsilon$-equilibria in $N$-player games. Furthermore, in this general setting, we prove the existence of mean field equilibria. This proof reveals a possible avenue for incorporating penalization for randomized action into MFGs.