One-dimensional forward-forward mean-field games

TL;DR

This paper investigates one-dimensional forward-forward mean-field games, establishing existence, long-term behavior, and connections to conservation laws, including explicit solutions and shock formation.

Contribution

It introduces a new formulation of forward-forward MFGs as conservation law systems and proves existence and convergence results in one dimension.

Findings

Existence of global solutions for first-order logarithmic forward-forward MFGs

Identification of shocks in explicit solutions

Long-time convergence of parabolic forward-forward MFG solutions

Abstract

While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward-forward problem is still poorly understood - even in the one-dimensional setting. Here, we consider one-dimensional forward-forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward-forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward-forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward-forward MFGs and establish the long-time convergence of the solutions.