The properness of weak stability notions

Abstract

Multiple weak stability notions have been proposed for two-sided matching problems, each based on a different perspective to justify the existence of certain blocking pairs. We introduce a criterion, called properness, to evaluate these weak stability notions. We say a notion of weak stability is proper if, whenever a matching is weakly stable, every matching that is more stable than it (admitting a subset of blocking pairs) is also weakly stable. We show that among existing notions of weak stability: the von Neumann and Morgenstern stable set, reasonable stability, $\alpha$-equitability, partial stability, and the Tang-Zhang weak stability are proper, whereas the Klijn-Massó weak stability, essential stability, and priority-neutrality are not proper.