Reforming an Envy-Free Matching

TL;DR

This paper studies how to efficiently reform an envy-free matching through exchanges, proving uniqueness, polynomial-time algorithms under certain conditions, and computational hardness in others.

Contribution

It introduces the concept of reformist envy-free matching, proves its uniqueness, and provides algorithms and hardness results for finding shortest reform sequences.

Findings

Reformist envy-free matching is uniquely determined up to initial conditions.

Polynomial-time algorithms exist when each agent accepts at most three items or each item is accepted by at most two agents.

Finding shortest reform sequences is computationally hard in general.

Abstract

We consider the problem of reforming an envy-free matching when each agent is assigned a single item. Given an envy-free matching, we consider an operation to exchange the item of an agent with an unassigned item preferred by the agent that results in another envy-free matching. We repeat this operation as long as we can. We prove that the resulting envy-free matching is uniquely determined up to the choice of an initial envy-free matching, and can be found in polynomial time. We call the resulting matching a reformist envy-free matching, and then we study a shortest sequence to obtain the reformist envy-free matching from an initial envy-free matching. We prove that a shortest sequence is computationally hard to obtain even when each agent accepts at most four items and each item is accepted by at most three agents. On the other hand, we give polynomial-time algorithms when each agent accepts at most three items or each item is accepted by at most two agents. Inapproximability and fixed-parameter (in)tractability are also discussed.