Mean-field games for harvesting problems: Uniqueness, long-time behaviour and weak KAM theory

TL;DR

This paper analyzes a complex mean-field game model for resource harvesting, establishing existence, uniqueness, long-term behavior, and explicit solutions within a non-local reaction-diffusion framework.

Contribution

It introduces a novel non-local MFG system for harvesting, providing analytical results and explicit solutions, advancing understanding of long-term dynamics and solution properties.

Findings

Proved existence and uniqueness of solutions

Analyzed convergence to ergodic system

Derived explicit solutions for the ergodic system

Abstract

The goal of this paper is to study a Mean Field Game (MFG) system stemming from the harvesting of resources. Modelling the latter through a reaction-diffusion equation and the harvesters as competing rational agents, we are led to a non-local (in time and space) MFG system that consists of three equations, the study of which is quite delicate. The main focus of this paper is on the derivation of analytical results (e.g existence, uniqueness) and of long time behaviour (here, convergence to the ergodic system). We provide some explicit solutions to this ergodic system.