Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences

Abstract

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.

A Survey on Fair Allocation of Chores

“…Note that, for the general example of indivisible chores, this problem is NP-hard. Hosseini et al ( 2022) [43] proved that, even for lexicographic chore-only instances, determining the existence of EF allocation is NP-complete.…”

“…Some studies have explored the relationship between EFX and PO. Given a chore instance, Hosseini et al (2022) [43] proved that EFX and PO allocation can be obtained in polynomial time. In the generalized binary disutility scenario of chores, the algorithm of [50] proved that EFX+fPO allocation always exists for three agents with bivalue disutility.…”