  Learning processes in games explain how players grapple with one another in
seeking an equilibrium. We study a natural model of learning based on
individual gradients in two-player continuous games. In such games, the
arguably natural notion of a local equilibrium is a differential Nash
equilibrium. However, the set of locally exponentially stable equilibria of the
learning dynamics do not necessarily coincide with the set of differential Nash
equilibria of the corresponding game. To characterize this gap, we provide
formal guarantees for the stability or instability of such fixed points by
leveraging the spectrum of the linearized game dynamics. We provide a
comprehensive understanding of scalar games and find that equilibria that are
both stable and Nash are robust to variations in learning rates.
