We study the fundamental problem of fairly dividing a set of indivisible items among agents with (general) monotone valuations. The notion of envy-freeness up to any item (EFX) is considered to be one of the most fascinating fairness concepts in this line of work. Unfortunately, despite significant efforts, existence of EFX allocations is a major open problem in fair division, thereby making the study of approximations and relaxations of EFX a natural line of research. Recently, Caragiannis et al. introduced a promising relaxation of EFX, called epistemic EFX (EEFX). We say an allocation to be EEFX if, for every agent, it is possible to shuffle the items in the remaining bundles so that she becomes "EFX-satisfied". Caragiannis et al. prove existence and polynomial-time computability of EEFX allocations for additive valuations. A natural question asks what happens when we consider valuations more general than additive? We address this important open question and answer it affirmatively by establishing the existence of EEFX allocations for an arbitrary number of agents with general monotone valuations. To the best of our knowledge, EEFX is the only known relaxation of EFX to have such strong existential guarantees. Furthermore, we complement our existential result by proving computational and information-theoretic lower bounds. We prove that even for an arbitrary number of (more than one) agents with identical submodular valuations, it is PLS-hard to compute EEFX allocations and it requires exponentially-many value queries to do so.

翻译：我们研究在具有（一般）单调估值的智能体之间公平分配不可分物品的基本问题。在这一研究领域中，无嫉妒至任意物品（EFX）的概念被认为是最引人入胜的公平性概念之一。遗憾的是，尽管付出了巨大努力，EFX分配的存在性仍是公平分配领域的一个主要开放问题，这使得对EFX的近似与松弛形式的研究成为一条自然的路径。最近，Caragiannis等人引入了一种有前景的EFX松弛形式，称为认知EFX（EEFX）。若对每个智能体，都可以通过重新排列其他智能体所获物品包中的物品，使得该智能体变得“EFX满足”，则我们称该分配为EEFX分配。Caragiannis等人证明了在加性估值下EEFX分配的存在性及其多项式时间可计算性。一个自然的问题是：当我们考虑比加性更一般的估值时会发生什么？我们针对这一重要的开放问题给出了肯定回答，证明了对于具有一般单调估值的任意数量智能体，EEFX分配总是存在。据我们所知，EEFX是目前已知的唯一具有如此强存在性保证的EFX松弛形式。此外，我们通过证明计算与信息论下界来补充我们的存在性结果。我们证明，即使对于任意数量（多于一个）具有相同子模估值的智能体，计算EEFX分配也是PLS难的，并且需要指数级次数的价值查询才能实现。