  This work explores bilevel problems in the context of cardinality
constraints. More specifically Single-Leader-Multi-Follower games (SLMFG)
involving cardinality constraints are considered in two different
configurations: one with the cardinality constraint at the leader's level and a
mixed structure in which the cardinality constraint is split between leader and
followers problem. We prove existence results in both cases and provided
equivalent reformulations allowing the numerical treatment of these complex
problems. The obtained results are illustrated thanks to an application to a
facility location problem.
