# A mean field approach to model flows of agents with path preferences over a network

## TL;DR

This paper develops a mean field model for traffic flows in a networked city, incorporating dynamic path preferences influenced by congestion and travel hassle, and explores equilibrium existence and control strategies.

## Contribution

It introduces a novel mean field framework combining path preference dynamics with traffic flow modeling and proves the existence of equilibrium states.

## Key findings

- Existence of a mean field equilibrium proven.
- A bi-level optimization framework for traffic control formulated.
- Path preferences influenced by congestion and travel hassle modeled.

## Abstract

In this paper, we address the problem of modeling the traffic flow of a heritage city whose streets are represented by a network. We consider a mean field approach where the standard forward backward system of equations is also intertwined with a path preferences dynamics. The path preferences are influenced by the congestion status on the whole network as well as the possible hassle of being forced to run during the tour. We prove the existence of a mean field equilibrium as a fixed point, over a suitable set of time-varying distributions, of a map obtained as a limit of a sequence of approximating functions. Then, a bi-level optimization problem is formulated for an external controller who aims to induce a specific mean field equilibrium.