On the Stable Matchings That Can Be Reached When the Agents Go Marching in One By One

Abstract

The random order mechanism (ROM) can be thought of as a sequential version of Gale and Shapley's deferred-acceptance (DA) algorithm, where agents are arriving one at a time, and each newly arrived agent has an opportunity to propose. Like the DA algorithm, ROM can be implemented in polynomial time. Unlike the DA algorithm, it is possible for ROM to output a stable matching that is different from the man-optimal and woman-optimal stable matchings. We say that a stable matching $\mu$ is ROM-reachable if ROM can output $\mu$. In this paper, we investigate computational questions related to ROM-reachability. First, we prove that determining if a particular stable matching is ROM-reachable is NP-complete. However, we show that there is an efficient algorithm for determining if ROM can reach a nontrivial stable matching in the case when every agent has at least two stable partners. We then study two restricted versions of this problem. In the first version, we consider stable matchings that can be reached by ROM in a “direct” manner. We show that they are computationally easy to recognize. In the second version, we restrict the class of stable matchings to what we call extreme stable matchings and prove that the computational complexity of determining if they are ROM-reachable depends on the number of unstable partners of the agents.