Existence of solutions for deterministic bilevel games under a general Bayesian approach

In 1996, Mallozzi and Morgan [33] proposed a new model for Stackelberg games which we refer here to as the Bayesian approach. The leader has only partial information about how followers select their reaction among possibly multiple optimal ones. This partial information is modeled as a decision-dependent distribution, the so-called belief of the leader. In this work, we formalize the setting of this approach for bilevel games admitting multiple leaders and we provide new results of existence of solutions. We pay particular attention to the fundamental case of linear bilevel problems, which has not been studied before, and which main difficulty is given by possible variations in the dimension of the reaction set of the follower. Our main technique to address this difficulty is based on a stronger notion of continuity for set-valued maps that we call rectangular continuity, and which is verified by the solution set of parametric linear problems. Finally, we provide some numerical experiments to address linear bilevel problems under the Bayesian approach.