Lattice operations for the pairwise stable set in many-to-many markets via re-equilibration dynamics

TL;DR

This paper develops a method to compute lattice operations for the stable set in many-to-many markets using re-equilibration dynamics, based on Tarski operators and quasi-stable matchings.

Contribution

It introduces a novel approach to compute lattice operations for stable matchings via Tarski operators and dynamics, extending the understanding of many-to-many market stability.

Findings

Sets of firm-quasi-stable and worker-quasi-stable matchings form lattices.

Tarski operators' fixed points coincide with the stable set.

Iterating these operators yields lattice operations in the stable set.

Abstract

We compute the lattice operations for the (pairwise) stable set in many-to-many matching markets when only path-independence on agents' choice functions is imposed. To do this, we first show that the sets of firm-quasi-stable and worker-quasi-stable many-to-many matchings form lattices. Then, we construct Tarski operators on these lattices whose fixed points coincide with the set of stable matchings, and show that iterating these operators from suitable quasi-stable matchings yields the lattice operations in the stable set. These operators resemble lay-off and vacancy chain dynamics, respectively.