Allocating items in a fair and economically efficient manner is a central problem in fair division. We study this problem for agents with additive preferences, when items are all goods or all chores, divisible or indivisible. The celebrated notion of Nash welfare is known to produce fair and efficient allocations for both divisible and indivisible goods; there is no known analogue for dividing chores. The Nash welfare objective belongs to a large, parameterized family of objectives called the p-mean welfare functions, which includes other notable members, like social welfare and egalitarian welfare. However, among the members of this family, only the Nash welfare produces fair allocations for goods. Incidentally, Nash welfare is also the only member that satisfies the axiom of scale invariance, which is crucially associated with its fairness properties. We define the class of "normalized p-mean" objectives, which imparts the missing key axiom of scale invariance to the p-mean family. Our results show that optimizing the normalized p-mean objectives produces fair and efficient allocations when the items are goods or chores, divisible or indivisible. For instance, the normalized p-means gives us an infinite class of objectives that produce (i) proportional and Pareto efficient allocations for divisible goods, (ii) approximately proportional and Pareto efficient allocations for divisible chores, (iii) EF1 and Pareto efficient allocations for indivisible goods for two agents, and (iv) EF1 and Pareto efficient allocations for indivisible chores for two agents.

翻译：在公平分配中，以公平且经济高效的方式分配物品是一个核心问题。我们针对具有可加性偏好的智能体，研究当物品均为物品或均为劳务、可分割或不可分割时的分配问题。著名的纳什福利概念已知能为可分割与不可分割物品产生公平高效的分配方案；然而对于劳务分配，尚无类似已知方法。纳什福利目标属于一个大型参数化目标族——p均值福利函数族，该族包含其他重要成员，如社会福利与平等主义福利。然而，在该族成员中，仅纳什福利能为物品产生公平分配。值得注意的是，纳什福利也是唯一满足尺度不变性公理的成员，该公理与其公平性特质密切相关。我们定义了“归一化p均值”目标类，为p均值族补全了缺失的尺度不变性关键公理。研究结果表明，优化归一化p均值目标能够在物品或劳务、可分割或不可分割的情况下产生公平高效的分配方案。例如，归一化p均值为我们提供了无限类目标函数，可产生：（i）可分割物品的比例性与帕累托有效分配，（ii）可分割劳务的近似比例性与帕累托有效分配，（iii）两智能体间不可分割物品的EF1与帕累托有效分配，以及（iv）两智能体间不可分割劳务的EF1与帕累托有效分配。