  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.
