We study the problem of fairly allocating indivisible goods between groups of agents using the recently introduced relaxations of envy-freeness. We consider the existence of fair allocations under different assumptions on the valuations of the agents. In particular, our results cover cases of arbitrary monotonic, responsive, and additive valuations, while for the case of binary valuations we fully characterize the cardinalities of two groups of agents for which a fair allocation can be guaranteed with respect to both envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX). Moreover, we introduce a new model where the agents are not partitioned into groups in advance, but instead the partition can be chosen in conjunction with the allocation of the goods. In this model, we show that for agents with arbitrary monotonic valuations, there is always a partition of the agents into two groups of any given sizes along with an EF1 allocation of the goods. We also provide an extension of this result to any number of groups.

我们研究了使用最近引入的嫉妒自由放松在代理人群体之间公平分配不可分割商品的问题。我们认为在对代理人估值的不同假设下存在公平分配。尤其是，我们的结果涵盖了任意单调，响应式和加性估值的情况，而对于二元估值的情况，我们充分描述了两组代理人的基数，可以保证在这两个代理人之间实现对羡慕自由的公平分配。达到一种商品（EF1）的程度，而羡慕至极的程度达到任何一种商品（EFX）。此外，我们引入了一种新的模型，在该模型中，不将代理预先划分为多个组，而是可以结合商品分配来选择划分。在此模型中，我们表明对于具有任意单调估值的代理商，总会有代理商分成任意给定大小的两组以及商品的EF1分配。我们还将此结果扩展到任意数量的组。