One-dimensional stationary mean-field games with local coupling

TL;DR

This paper constructs explicit solutions for one-dimensional mean-field games without the usual monotonicity assumption, revealing phenomena like non-uniqueness and regions devoid of agents, which are absent in standard models.

Contribution

It introduces a method to explicitly solve 1D MFGs without monotonic coupling, showing new phenomena such as non-uniqueness and agent-free regions.

Findings

Explicit solutions demonstrate non-uniqueness.

Regions with no agents can form.

Standard monotonicity assumptions are not necessary for solutions.

Abstract

A standard assumption in mean-field game (MFG) theory is that the coupling between the Hamilton-Jacobi equation and the transport equation is monotonically non-decreasing in the density of the population. In many cases, this assumption implies the existence and uniqueness of solutions. Here, we drop that assumption and construct explicit solutions for one-dimensional MFGs. These solutions exhibit phenomena not present in monotonically increasing MFGs: low-regularity, non-uniqueness, and the formation of regions with no agents.