Extension of Additive Valuations to General Valuations on the Existence of EFX

Abstract

Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when n = 2 or (ii) when all agents have identical valuations, where n is the number of agents. it is also known that an EFX allocation always exists when one can leave at most n − 1 items unallocated.We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n + 3. We also show that an EFX allocation always exists when one can leave at most n − 2 items unallocated. In addition to the positive results, we construct an instance with n = 3 in which an existing approach does not work as it is.

Fairly Dividing Mixtures of Goods and Chores under Lexicographic Preferences

“…It is also known that when agents belong to one of two given "types", an EFX allocation is guaranteed to exist for goods-only instances under monotone valuations [Mahara 2021]. Our result in Theorem 2, which also has two types of agents, demonstrates a barrier to extending this result in the non-monotone setting, even under lexicographic preferences.…”

“…The counterexample in the proof of Theorem 2 (given below) uses only four agents and seven items. Interestingly, for the said number of agents and items, an EFX allocation is guaranteed to exist for goods-only instances even under monotone valuations [Mahara 2021], which is significantly more general than additive (or lexicographic) preferences.…”

“…For goods-only problems with additive valuations, the existence and computation of an EFX allocation-except for a few special cases [Plaut and Roughgarden 2020;Mahara 2021;Chaudhury et al 2020]-remains open. Moreover, EFX in known to be incompatible with wellstudied notions of economic efficiency such as Pareto optimality (PO).…”