Almost Envy-Freeness under Weakly Lexicographic Preferences

TL;DR

We study fair allocation of indivisible items under weakly lexicographic preferences, focusing on $EF$, $EF1$, $EFX$, $MMS$, and $PO$. The paper develops a customizable algorithm for goods that can guarantee $PO$ together with $EF1$, $EFX$, $MMS$, or their combinations, and introduces the concepts of preference graphs and potential envy to handle ties. It proves NP-hardness of deciding the existence of an $EF$ allocation in goods with ties even when each agent has at most two indifference classes, and shows a constructive $EF1$+ $PO$ allocation for chores. The results reveal key algorithmic and axiomatic differences between goods and chores under weakly lexicographic preferences and introduce new techniques that may be of independent interest for handling indifferences in fair division.

Abstract

In fair division of indivisible items, domain restriction has played a key role in escaping from negative results and providing structural insights into the computational and axiomatic boundaries of fairness. One notable subdomain of additive preferences, the lexicographic domain, has yielded several positive results in dealing with goods, chores, and mixtures thereof. However, the majority of work within this domain primarily consider strict linear orders over items, which do not allow the modeling of more expressive preferences that contain indifferences (ties). We investigate the most prominent fairness notions of envy-freeness up to any (EFX) or some (EF1) item under weakly lexicographic preferences. For the goods-only setting, we develop an algorithm that can be customized to guarantee EF1, EFX, maximin share (MMS), or a combination thereof, along the efficiency notion of Pareto optimality (PO). From the conceptual perspective, we propose techniques such as preference graphs and potential envy that are independently of interest when dealing with ties. Finally, we demonstrate challenges in dealing with chores and highlight key algorithmic and axiomatic differences of finding EFX solutions with the goods-only setting. Nevertheless, we show that there is an algorithm that always returns an EF1 and PO allocation for the chores-only instances.