On monotonicity conditions for Mean Field Games

TL;DR

This paper introduces two novel monotonicity conditions that ensure the unconditional uniqueness of Nash equilibria in mean field games, independent of time horizon or initial distribution regularity.

Contribution

The paper proposes two new monotonicity conditions that are distinct from existing conditions and guarantee unconditional uniqueness in mean field games.

Findings

New conditions are not equivalent to existing monotonicity conditions.

Conditions guarantee uniqueness regardless of time horizon or initial distribution.

Illustrated through simple examples showing their applicability.

Abstract

In this paper we propose two new monotonicity conditions that could serve as sufficient conditions for uniqueness of Nash equilibria in mean field games. In this study we aim for $unconditional\ uniqueness$ that is independent of the length of the time horizon, the regularity of the starting distribution of the agents, or the regularization effect of a non-degenerate idiosyncratic noise. Through a rich class of simple examples we show that these new conditions are not only in dichotomy with each other, but also with the two widely studied monotonicity conditions in the literature, the Lasry-Lions monotonicity and displacement monotonicity conditions.