A New Approach to the Pareto Stable Matching Problem

In two-sided matching markets, the concept of stability proposed by Gale and Shapley (1962) is one of the most important solution concepts. In this paper, we consider a problem related to the stability of a matching in a two-sided matching market with indifferences (i.e., ties). The introduction of ties into preference lists dramatically changes the properties of stable matchings. For example, stable matchings need not have the same size. Furthermore, it is known that stability do not guarantee Pareto efficiency that is also one of the most important solution concepts in two-sided matching markets. This fact naturally leads to the concept of Pareto stability, i.e., both stable and Pareto efficient. Erdil and Ergin (2006, 2008) proved that there always exists a Pareto stable matching in a one-to-one/many-to-one matching market with indifferences and gave a polynomial-time algorithm for finding it. Furthermore, Chen (2012) proved that there always exists a Pareto stable matching in a many-to-many matching market with indifferences and gave a polynomial-time algorithm for finding it. In this paper, we propose a new approach to the problem of finding a Pareto stable matching in a many-to-many matching market with indifferences. Our algorithm is an alternative proof of the existence of a Pareto stable matching in a many-to-many matching market with indifferences.