Convergence of Learning Dynamics in Stackelberg Games

This paper investigates the convergence of learning dynamics in Stackelberg games. In the class of games we consider, there is a hierarchical game being played between a leader and a follower with continuous action spaces. We show that in zero-sum games, the only stable attractors of the Stackelberg gradient dynamics are Stackelberg equilibria. This insight allows us to develop a gradient-based update for the leader that converges to Stackelberg equilibria in zero-sum games and the set of stable attractors in general-sum games. We then consider a follower employing a gradient-play update rule instead of a best response strategy and propose a two-timescale algorithm with similar asymptotic convergence results. For this algorithm, we also provide finite-time high probability bounds for local convergence to a neighborhood of a stable Stackelberg equilibrium in general-sum games.