We revisit the setting of fairly allocating indivisible items when agents have different weights representing their entitlements. First, we propose a parameterized family of relaxations for weighted envy-freeness and the same for weighted proportionality; the parameters indicate whether smaller-weight or larger-weight agents should be given a higher priority. We show that each notion in these families can always be satisfied, but any two cannot necessarily be fulfilled simultaneously. We then introduce an intuitive weighted generalization of maximin share fairness and establish the optimal approximation of it that can be guaranteed. Furthermore, we characterize the implication relations between the various weighted fairness notions introduced in this and prior work, and relate them to the lower and upper quota axioms from apportionment.

翻译：本文重新审视了当代理具有代表其权益的不同权重时，公平分配不可分割物品的问题。首先，我们提出了加权无嫉妒性的参数化松弛族，以及加权比例性的相应松弛族；参数指示了较小权重或较大权重的代理是否应被赋予更高优先级。我们证明了这些族中的每个概念总是可满足的，但任意两个概念未必能同时实现。接着，我们引入了最大化最小份额公平性的一种直观加权推广，并建立了可保证的最优近似度。此外，我们刻画了本文及先前工作中引入的各种加权公平性概念之间的蕴含关系，并将其与分配中的下配额和上配额公理联系起来。