  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.
