# Universal behavior in non stationary Mean Field Games

## TL;DR

This paper investigates the behavior of non-stationary Mean Field Games, revealing a universal scaling solution analogous to an ergodic state, derived through a mapping to electrostatics, enhancing understanding of complex dynamic systems.

## Contribution

It introduces a universal scaling solution for non-stationary Mean Field Games lacking an ergodic state, expanding theoretical understanding of their long-term dynamics.

## Key findings

- Existence of a universal scaling solution in non-stationary MFGs
- Mapping of MFG behavior to electrostatic problems
- Insights into long-term dynamics without ergodic states

## Abstract

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled objects in interaction. Though these models are much simpler than the underlying differential games they describe in some limit, their behavior is still far from being fully understood. When the system is confined, a notion of "ergodic state" has been introduced that characterizes most of the dynamics for long optimization times. Here we consider a class of models without such an ergodic state, and show the existence of a scaling solution that plays similar role. Its universality and scaling behavior can be inferred from a mapping to an electrostatic problem.