Potentially Invalid: Stable Matching with Outside Option

Matching is based on the order of selection given by the participants. We found that in real life, there are indeed other ideas when matching. We want to reject the matching when there is no suitable object in the matching. We abstract all non-opposite choices as “outside option”. This means that each applicant and the hospital have a total order of opposites and an “outside option”. In this paper, we use virtual objects as “outside option” to implement extended deferred acceptance algorithm. The extended deferred acceptance algorithm does not destroy the original structure, and has proven to retain the same properties. In addition, we proved that when there are “outside option”, potentially stable matches may have invalid matches.