# A mean-field game model for homogeneous flocking

## TL;DR

This paper develops a mean-field game model that captures the collective flocking behavior of agents with gradient self-propulsion, providing a non-cooperative control framework aligned with empirical flocking models.

## Contribution

It introduces a mean-field game approach to model homogeneous flocking, linking empirical models with a Nash equilibrium-based control framework for large agent populations.

## Key findings

- Model reproduces bifurcations similar to empirical flocking models
- Linear stability analysis confirms the model's dynamic behavior
- Low-rank perturbation approach simplifies the analysis

## Abstract

Empirically derived continuum models of collective behavior among large populations of dynamic agents are a subject of intense study in several fields, including biology, engineering and finance. We formulate and study a mean-field game model whose behavior mimics an empirically derived non-local homogeneous flocking model for agents with gradient self-propulsion dynamics. The mean-field game framework provides a non-cooperative optimal control description of the behavior of a population of agents in a distributed setting. In this description, each agent's state is driven by optimally controlled dynamics that result in a Nash equilibrium between itself and the population. The optimal control is computed by minimizing a cost that depends only on its own state, and a mean-field term. The agent distribution in phase space evolves under the optimal feedback control policy. We exploit the low-rank perturbative nature of the non-local term in the forward-backward system of equations governing the state and control distributions, and provide a linear stability analysis demonstrating that our model exhibits bifurcations similar to those found in the empirical model. The present work is a step towards developing a set of tools for systematic analysis, and eventually design, of collective behavior of non-cooperative dynamic agents via an inverse modeling approach.