When the Correct Model Fails: The Optimality of Stackelberg Equilibria with Follower Intention Updates

Abstract: We study a two-player dynamic Stackelberg game between a leader and a follower. Classical formulations of the Stackelberg equilibrium (SE) assume that the follower's best response (BR) mapping is known to the leader. However, this is not always true in practice. In those cases the leader needs to simultaneously infer this BR function while fulfilling an internal objective. We study a setting in which the leader selects a control strategy that optimizes an objective given an initial belief about the follower's best response. This belief is updated during the finite decision horizon, prompting the leader to reoptimize its control. We characterize the optimality guarantees of the SE solutions under this belief update for both open loop (OL) and feedback (FB) information structures. In particular, we show that it is possible that assuming an incorrect follower BR map obtains a lower cost over the game horizon than knowing the true BR. We support these claims with numerical examples in a linear quadratic (LQ) Stackelberg game.