Stability of Gradient Learning Dynamics in Continuous Games: Scalar Action Spaces

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.