The Properness of Weak Stability Notions

Abstract

Multiple notions of weak stability have been proposed for two-sided matching problems, each based on a different perspective to justify the existence of certain blocking pairs. We propose 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 less blocking pairs) is also weakly stable. We show that among existing notions of weak stability, (1) the von Neumann and Morgenstern (vNM) stability, reasonable stability, $\alpha$-equitability, partial stability (with respect to any set of acceptable priority violations), and the Tang-Zhang (TZ) weak stability are proper, while (2) the Klijn-Mosso (KM) weak stability, essential stability, and priority-neutrality are not proper.

Keywords: two-sided matching, school choice, weak stability, blocking pair, properness