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

• 成果类型: Article
• 署名作者: Mahara, Ryoga
• 署名单位: University of Tokyo
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• 发表日期: 2024
• 页码: 1263-1277
• 关键词: matrix; rank

摘要

Envy freeness is one of the most widely studied notions in fair division. Because envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling notion is envy freeness up to any item (EFX). Informally speaking, EFX requires that no agent i envies another agent j after the removal of any item in j's bundle. The existence of EFX allocations is a major open problem. We study the existence of EFX allocations when agents have general valuations. For general valuations, it is known that an EFX allocation always exists (i) when n2 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 n1 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 n2 items unallocated. In addition to the positive results, we construct an instance with n3, in which an existing approach does not work.