A "trembling hand perfect" equilibrium for a certain class of mean field games

TL;DR

This paper explores a specific class of mean field games linked to scalar transport equations, demonstrating non-uniqueness of equilibria and proposing entropy-based criteria for selecting rational solutions, with explicit error estimates in the vanishing noise limit.

Contribution

It introduces explicit examples of non-uniqueness in mean field game equilibria and applies entropy solution theory to select and analyze solutions with error bounds.

Findings

Non-uniqueness of Nash equilibria in the studied class.

Explicit error estimates in the vanishing noise limit.

Application of entropy solutions to select rational equilibria.

Abstract

We study a particular class of mean field games whose solutions can be formally connected to a scalar transport equation on the Wasserstein space of measures. For this class, we construct some interesting explicit examples of non-uniqueness of Nash equilibria. We then address the selection problem of finding rational criteria by which to choose one equilibrium over others. We show that when the theory of entropy solutions is used, we can obtain explicit error estimates for the ``vanishing noise limit,'' where the error is measured in a certain norm that measures the distance between two functions over the set of empirical measures.