Envy-free up to one good (EF1) and envy-free up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is $(\phi -1) \simeq 0.61$ by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time $0.73$-approximation allocation algorithm. For our algorithm, we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness. We believe Nash Social Welfare Matching will find its applications in future work.

中文翻译：

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几乎无嫉妒、嫉妒等级和纳什社会福利匹配

Envy-free up to one good (EF1) 和 envy-free up to any good (EFX) 是不可分割项目的两个众所周知的无嫉妒扩展。结果表明，对于具有次可加性估值的代理，EF1 总是可以保证的。与此形成鲜明对比的是，即使对于四个代理和附加估值，EFX 分配是否始终存在也是未知的。此外，EFX 的最佳近似保证是 Amanitidis 等人的 $(\phi -1) \simeq 0.61$。为了找到一个中间立场来弥合这一差距，在本文中，我们提出了另一个公平标准，即嫉妒- 自由度达到随机商品或 EFR，它比 EFX 弱，但比 EF1 强。对于这个概念，我们提供了一个多项式时间 $0.73$-近似分配算法。对于我们的算法，我们介绍了纳什社会福利匹配，它在纳什社会福利和嫉妒自由之间建立了联系。我们相信 Nash Social Welfare Matching 会在未来的工作中找到它的应用。