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arXiv · 2203.07279

Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences

Hadi Hosseini·Sujoy Sikdar·Rohit Vaish·Lirong Xia
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Abstract

We study fair allocation of indivisible goods and chores among agents with \emph{lexicographic} preferences -- a subclass of additive valuations. In sharp contrast to the goods-only setting, we show that an allocation satisfying \emph{envy-freeness up to any item} (EFX) could fail to exist for a mixture of \emph{objective} goods and chores. To our knowledge, this negative result provides the \emph{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 \emph{maximin share} (MMS), we show positive existence and computation for \emph{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-{\`a}-vis their goods-only counterparts.

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Hadi Hosseini, Sujoy Sikdar, Rohit Vaish, Lirong Xia. 2022-03-14. Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences. https://arxiv.org/abs/2203.07279

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