A full characterization of the set of optimal affine solutions to the reverse Stackelberg game

The class of reverse Stackelberg games can be used as a structure for hierarchical decision making and can be adopted in multi-level optimization approaches for large-scale control problems like road tolling. In this game, a leader player acts first by presenting a leader function that maps the follower decision space into the leader decision space. Subsequently, the follower acts by presenting his optimal decision variables. In order to solve the - in general complex - reverse Stackelberg game, a specific structure of the leader function is considered in this paper, given a desired equilibrium that the leader strives to achieve. We present conditions for the existence of such an optimal affine leader function in the static reverse Stackelberg game and delineate the set of all possible solutions of the affine leader function structure. The parametrized characterization of such a set facilitates further optimization, e.g., when considering the sensitivity to deviations from the optimal follower response, as is illustrated by a simple example. Moreover, it can be used to verify the existence of an optimal affine leader function in a constrained decision space.