We consider the fair allocation problem of indivisible items. Most previous work focuses on fairness and/or efficiency among agents given agents' preferences. However, besides the agents, the allocator as the resource owner may also be involved in many real-world scenarios, e.g., heritage division. The allocator has the inclination to obtain a fair or efficient allocation based on her own preference over the items and to whom each item is allocated. In this paper, we propose a new model and focus on the following two problems: 1) Is it possible to find an allocation that is fair for both the agents and the allocator? 2) What is the complexity of maximizing the allocator's social welfare while satisfying the agents' fairness? We consider the two fundamental fairness criteria: envy-freeness and proportionality. For the first problem, we study the existence of an allocation that is envy-free up to $c$ goods (EF-$c$) or proportional up to $c$ goods (PROP-$c$) from both the agents' and the allocator's perspectives, in which such an allocation is called doubly EF-$c$ or doubly PROP-$c$ respectively. When the allocator's utility depends exclusively on the items (but not to whom an item is allocated), we prove that a doubly EF-$1$ allocation always exists. For the general setting where the allocator has a preference over the items and to whom each item is allocated, we prove that a doubly EF-$1$ allocation always exists for two agents, a doubly PROP-$2$ allocation always exists for binary valuations, and a doubly PROP-$O(\log n)$ allocation always exists in general. For the second problem, we provide various (in)approximability results in which the gaps between approximation and inapproximation ratios are asymptotically closed under most settings. Most results are based on novel technical tools including the chromatic numbers of the Kneser graphs and linear programming-based analysis.

翻译：我们研究了不可分割物品的公平分配问题。以往研究主要关注基于代理人偏好的公平性和/或效率。然而，在许多现实场景（如遗产分割）中，除代理人外，作为资源所有者的分配者也可能参与其中。分配者会根据自身对物品分配对象及物品分配的偏好，倾向于获得公平或高效的分配结果。本文提出新模型，聚焦以下两个问题：1）能否找到同时满足代理人和分配者公平性的分配方案？2）在满足代理人公平性的前提下，最大化分配者社会福利的计算复杂性如何？我们考虑了嫉妒性和比例性这两个基本公平准则。针对第一个问题，我们从代理人和分配者双重视角研究是否存在至多c个物品的嫉妒无差异（EF-c）或至多c个物品的比例公平（PROP-c）分配方案，分别称为双重EF-c和双重PROP-c。当分配者效用仅取决于物品（而非物品分配给谁）时，我们证明双重EF-1分配总是存在。在一般设定下（分配者对物品及物品分配对象均有偏好），我们证明：当存在两方代理人时双重EF-1分配总是存在；在二元估值场景下双重PROP-2分配总是存在；一般情况下双重PROP-O(log n)分配总是存在。针对第二个问题，我们给出了多种（不可）近似性结果，其中多数设定下近似比与非近似比之间的间隙渐近闭合。这些结果主要基于新颖的技术工具，包括Kneser图染色数和线性规划分析。