  The framework of Mean-field Games (MFGs) is used for modelling the collective
dynamics of large populations of non-cooperative decision-making agents. We
formulate and analyze a kinetic MFG model for an interacting system of
non-cooperative motile agents with inertial dynamics and finite-range
interactions, where each agent is minimizing a biologically inspired cost
function. By analyzing the associated coupled forward-backward in time system
of nonlinear Fokker-Planck and Hamilton-Jacobi-Bellman equations, we obtain
conditions for closed-loop linear stability of the spatially homogeneous MFG
equilibrium that corresponds to an ordered state with non-zero mean speed.
Using a combination of analysis and numerical simulations, we show that when
energetic cost of control is reduced below a critical value, this equilibrium
loses stability, and the system transitions to a travelling wave solution. Our
work provides a game-theoretic perspective to the problem of collective motion
in non-equilibrium biological and bio-inspired systems.
