One-dimensional, forward-forward mean-field games with congestion

TL;DR

This paper analyzes one-dimensional forward-forward mean-field games with congestion, using conservation law techniques to establish bounds, prove global solutions, and construct special solutions, thereby advancing understanding of their qualitative behavior.

Contribution

It introduces a novel method to prove lower bounds and global existence for forward-forward MFGs with congestion, and constructs traveling-wave and time-periodic solutions.

Findings

Established lower bounds for density using Riemann invariants

Proved existence of global solutions for parabolic MFGs

Constructed traveling-wave and time-periodic solutions

Abstract

Here, we consider one-dimensional forward-forward mean-field games (MFGs) with congestion, which were introduced to approximate stationary MFGs. We use methods from the theory of conservation laws to examine the qualitative properties of these games. First, by computing Riemann invariants and corresponding invariant regions, we develop a method to prove lower bounds for the density. Next, by combining the lower bound with an entropy function, we prove the existence of global solutions for parabolic forward-forward MFGs. Finally, we construct traveling-wave solutions, which settles in a negative way the convergence problem for forward-forward MFGs. A similar technique gives the existence of time-periodic solutions for non-monotonic MFGs.