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：以公平且经济高效的方式分配物品是公平分配领域的核心问题。我们研究了当智能体具有可加性偏好，且物品全部为货物或全部为家务、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够为可分割和不可分割货物产生公平且高效的分配，但在家务分配中尚未发现类似方法。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，该族还包括社会福利和平等福利等其他重要成员。然而在该族成员中，仅有纳什福利能对货物分配产生公平结果。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，该公理与其公平性属性密切相关。我们定义了"归一化p均值"目标类，为p均值族赋予了缺失的尺度不变性关键公理。研究结果表明，当物品为货物或家务、可分割或不可分割时，优化归一化p均值目标能产生公平且高效的分配。例如，归一化p均值为我们提供了无限类目标，可产生：(i) 可分割货物中的比例分配与帕累托有效分配，(ii) 可分割家务中的近似比例分配与帕累托有效分配，(iii) 两个智能体不可分割货物中的EF1分配与帕累托有效分配，(iv) 两个智能体不可分割家务中的EF1分配与帕累托有效分配。