Lone wolf theorem for one-sided matching problems with outside option

In this paper, we study one-sided matching problems (so-called roommate problems) with the outside option. In the classical roommate problems, remaining single is conceived as the outside option. However, there are many real life applications where this is not the case. We study roommate problems in which the outside option is defined as having no room. In this general framework, we discuss the generalization of so-called "Lonely Wolf Theorem" which states that any agent who is single in one stable matching is single in all other stable matchings. In this study, we show that for the general model with outside option Lonely Wolf Theorem still holds.