We study fair allocation of indivisible goods and chores among agents with lexicographic preferences---a subclass of additive valuations. In sharp contrast to the goods-only setting, we show that an allocation satisfying envy-freeness up to any item (EFX) could fail to exist for a mixture of objective goods and chores. To our knowledge, this negative result provides the first counterexample for EFX over (any subdomain of) additive valuations. To complement this non-existence result, we identify a class of instances with (possibly subjective) mixed items where an EFX and Pareto optimal allocation always exists and can be efficiently computed. When the fairness requirement is relaxed to maximin share (MMS), we show positive existence and computation for any mixed instance. More broadly, our work examines the existence and computation of fair and efficient allocations both for mixed items as well as chores-only instances, and highlights the additional difficulty of these problems vis-à-vis their goods-only counterparts.
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Fairly Allocating Goods and (Terrible) Chores
We study the fair allocation of mixture of indivisible goods and chores under lexicographic preferences---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 Pareto optimal 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.
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- PAR ID:
- 10443108
- Date Published:
- Journal Name:
- Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence
- Page Range / eLocation ID:
- 2738 to 2746
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.24963/ijcai.2023/305
