Well-posedness of deterministic bilevel games through a Bayesian approach

In 1996, Mallozzi and Morgan [37] proposed a new model for bilevel games with one leader (which they called Intermediate), who has only partial information over how followers select their response among possibly multiple optimal ones. This partial information is modeled as a decision-dependent distribution, the so-called belief ([14, p 85]) of the leader. In this work, we study the well-posedness of the model, for general bilevel games, in terms of two fundamental questions: if it admits solutions and how we can compute a solution. Our existence results are based on a new property that we call Rectangular Continuity, which is verified by linear lower-level problems. We provide new existence results for a large family of bilevel problems and a large family of beliefs. We also provide an enumerative algorithm for linear bilevel programming problems with the belief induced by uniform distributions, which we call the Neutral belief. The proposed algorithm is based on a full description of the centroid mapping of the fibers of polytopes as a piecewise affine function. Due to the similarity of the approach to Bayesian games, we propose to rename it as the Bayesian approach for bilevel games.