Non-uniformly Stable Matchings
Naoyuki Kamiyama
TL;DR
Stable matching with ties is analyzed through non-uniform stability, a unifying generalization of super-stability and strong stability. The authors provide a polynomial-time decision algorithm and a polyhedral description, proving P = S for the associated polyhedra and demonstrating that the non-uniformly stable matchings form a distributive lattice. The approach unifies and extends prior algorithmic, polyhedral, and lattice results for the special cases E1 = E and E2 = E, enabling efficient computation and structural insights in markets with ties. This work advances theoretical understanding of stable matchings with ties and informs practical computation and analysis in matching markets.
Abstract
Super-stability and strong stability are properties of a matching in the stable matching problem with ties. In this paper, we introduce a common generalization of super-stability and strong stability, which we call non-uniform stability. First, we prove that we can determine the existence of a non-uniformly stable matching in polynomial time. Next, we give a polyhedral characterization of the set of non-uniformly stable matchings. Finally, we prove that the set of non-uniformly stable matchings forms a distributive lattice.