When the Correct Model Fails: The Optimality of Stackelberg Equilibria with Follower Intention Updates
This addresses a practical limitation in game theory for applications like collision avoidance, though it appears incremental as it extends classical Stackelberg equilibrium analysis.
The paper tackles the problem of a leader's optimal strategy in a dynamic Stackelberg game when the follower's best response is unknown and beliefs are updated, proving that assuming an incorrect follower's best response can sometimes yield lower leader costs than knowing the true one, with numerical examples in linear quadratic games showing non-trivial instances.
We study a two-player dynamic Stackelberg game where the follower's intention is unknown to the leader. Classical formulations of the Stackelberg equilibrium (SE) assume that the follower's best response (BR) function is known to the leader. However, this is not always true in practice. We study a setting in which the leader receives updated beliefs about the follower BR before the end of the game, such that the update prompts the leader and subsequently the follower to re-optimize their strategies. We characterize the optimality guarantees of the SE solutions under this belief update for both open loop and feedback information structures. Interestingly, we prove that in general, assuming an incorrect follower's BR may lead to a lower leader cost over the entire game than knowing the true follower's BR. We support these results with numerical examples in a linear quadratic (LQ) Stackelberg game, and use Monte Carlo simulations to show that the instances of incorrect BR achieving lower leader costs are non-trivial in collision avoidance LQ Stackelberg games.