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.

中文翻译：

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基于 EFX 的存在将附加估值扩展到一般估值

无嫉妒是公平划分中研究最广泛的概念之一。由于当项目不可分割时，并不总是存在无嫉妒分配，因此已经考虑了几种放松方式。其中，可能最引人注目的概念是对任何项目的嫉妒自由（EFX）。我们研究了用于一般估值的 EFX 分配的存在。EFX 分配的存在是一个主要的开放性问题。对于一般估值，已知 EFX 分配始终存在 (i) 当 $n=2$ 或 (ii) 当所有代理具有相同估值时，其中 $n$ 是代理数量。众所周知，当一个人最多可以保留 $n-1$ 项未分配时，EFX 分配总是存在的。我们开发新技术并将附加估值的一些结果扩展到 EFX 分配存在的一般估值。我们表明 EFX 分配总是存在 (i) 当所有代理都具有两个一般估值之一或 (ii) 当项目数量最多为 $n+3$ 时。我们还表明，当最多可以保留 $n-2$ 项未分配时，EFX 分配始终存在。除了积极的结果之外，我们还构建了一个 $n=3$ 的实例，其中现有方法无法正常工作。