Fairly Allocating Goods and (Terrible) Chores

TL;DR

This paper investigates fair allocation of indivisible goods and chores under lexicographic preferences, establishing complexity results for EFX and proposing efficient algorithms for certain subclasses and weaker fairness notions.

Contribution

It identifies NP-completeness of EFX existence with terrible chores and provides efficient algorithms for EFX, PO, EF1, and MMS allocations in specific lexicographic preference subclasses.

Findings

EFX existence is NP-complete with terrible chores.

Efficient algorithms for EFX and PO in certain lexicographic subclasses.

Efficient computation of EF1 and MMS allocations under lexicographic preferences.

Abstract

We study the fair allocation of mixtures of indivisible goods and chores under lexicographic preferences$\unicode{x2014}$a subdomain of additive preferences. A prominent fairness notion for allocating indivisible items is envy-freeness up to any item (EFX). Yet, its existence and computation has remained a notable open problem. By identifying a class of instances with "terrible chores", we show that determining the existence of an EFX allocation is NP-complete. This result immediately implies the intractability of EFX under additive preferences. Nonetheless, we propose a natural subclass of lexicographic preferences for which an EFX and Pareto optimal (PO) allocation is guaranteed to exist and can be computed efficiently for any mixed instance. Focusing on two weaker fairness notions, we investigate finding EF1 and PO allocations for special instances with terrible chores, and show that MMS and PO allocations can be computed efficiently for any mixed instance with lexicographic preferences.