Fair Division with Two-Sided Preferences

TL;DR

This paper investigates fair division with two-sided preferences, establishing existence and computation of EF1, stability, and Pareto optimality under various conditions, highlighting both possibilities and computational challenges.

Contribution

It introduces algorithms and proofs for EF1, stability, and Pareto optimality in two-sided preferences, including polynomial-time solutions and complexity results.

Findings

EF1, swap stability, and individual stability always exist and are computable in polynomial time.

EF1 and Pareto optimal allocations exist for nonnegative valuations, with binary valuations allowing polynomial-time computation.

Existence of justified envy-free allocations is not guaranteed, and their decision problem is computationally hard.

Abstract

We study a fair division setting in which participants are to be fairly distributed among teams, where not only do the teams have preferences over the participants as in the canonical fair division setting, but the participants also have preferences over the teams. We focus on guaranteeing envy-freeness up to one participant (EF1) for the teams together with a stability condition for both sides. We show that an allocation satisfying EF1, swap stability, and individual stability always exists and can be computed in polynomial time, even when teams may have positive or negative values for participants. When teams have nonnegative values for participants, we prove that an EF1 and Pareto optimal allocation exists and, if the valuations are binary, can be found in polynomial time. We also show that an EF1 and justified envy-free allocation does not necessarily exist, and deciding whether such an allocation exists is computationally difficult.