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

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 μ is ROM-reachable if ROM can output μ. In this paper, we investigate computational questions related to ROM-reachability. First, we show that there is an efficient algorithm for determining if ROM can reach a non-trivial stable matching i.e., one different from the man-optimal and woman-optimal stable matchings. However, if we wish to determine if ROM can reach a particular stable matching then the problem becomes NPcomplete. 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 computational complexity of determining if they are ROM-reachable depends on the number of unstable partners of the agents.