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 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.

Dynamics in matching and coalition formation games with structural constraints

“…However, deciding existence and computing a stable matching, if they exist, can be done in polynomial time [24,40]. Several works have studied the computation of (variants of) stable matchings using iterative entry dynamics [14,16,17,22], or matching problems in scenarios with payments or profit sharing [4,13].…”

Locally Stable Marriage with Strict Preferences

“…Ergodic sets of the underlying Markov chain have been studied [25] and related to random dynamics [29]. Alternatively, several works have studied the computation of (variants of) stable matchings using iterative entry dynamics [8,10,11,13].…”

Maintaining Near-Popular Matchings

“…In two-sided instances, Ma [25] proposed the random-order mechanism, where agents arrive in uniform random order and blocking pairs are resolved in a best-response manner. This mechanism is known to arrive at stable matchings that are neither man-nor woman-optimal and has interesting structural and computational properties [13,14,25]. In terms of fully dynamic populations, Blum et al [13] study resolution chains of blocking pairs when in each round an arbitrary agent arrives or leaves.…”