  Towards characterizing the optimization landscape of games, this paper
analyzes the stability of gradient-based dynamics near fixed points of
two-player continuous games. We introduce the quadratic numerical range as a
method to characterize the spectrum of game dynamics and prove the robustness
of equilibria to variations in learning rates. By decomposing the game Jacobian
into symmetric and skew-symmetric components, we assess the contribution of a
vector field's potential and rotational components to the stability of
differential Nash equilibria. Our results show that in zero-sum games, all Nash
are stable and robust; in potential games, all stable points are Nash. For
general-sum games, we provide a sufficient condition for instability. We
conclude with a numerical example in which learning with timescale separation
results in faster convergence.
