# Variational time-fractional Mean Field Games

## TL;DR

This paper extends the variational framework of Mean Field Games to include time-fractional dynamics, modeling non-Markovian subdiffusive behavior of agents, and offers an Eulerian perspective on these systems.

## Contribution

It introduces a variational approach to time-fractional MFG systems, incorporating subdiffusive agent dynamics and providing a new Eulerian interpretation.

## Key findings

- Extended variational MFG theory to subdiffusive dynamics
- Formulated time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations
- Provided an Eulerian interpretation of fractional MFG systems

## Abstract

We consider the variational structure of a time-fractional second order Mean Field Games (MFG) system with local coupling. The MFG system consists of time-fractional Fokker-Planck and Hamilton-Jacobi-Bellman equations. In such a situation the individual agent follows a non-Markovian dynamics given by a subdiffusion process. Hence, the results of this paper extend the theory of variational MFG to the subdiffusive situation, providing an Eulerian interpretation of time-fractional MFG systems.