We study the problem of fairly allocating indivisible goods among agents which are equipped with {\em leveled} valuation functions. Such preferences, that have been studied before in economics and fair division literature, capture a simple and intuitive economic behavior; larger bundles are always preferred to smaller ones. We provide a fine-grained analysis for various subclasses of leveled valuations focusing on two extensively studied notions of fairness, (approximate) MMS and EFX. In particular, we present a general positive result, showing the existence of $2/3$-MMS allocations under valuations that are both leveled and submodular. We also show how some of our ideas can be used beyond the class of leveled valuations; for the case of two submodular (not necessarily leveled) agents we show that there always exists a $2/3$-MMS allocation, complementing a recent impossibility result. Then, we switch to the case of subadditive and fractionally subadditive leveled agents, where we are able to show tight (lower and upper) bounds of $1/2$ on the approximation factor of MMS. Moreover, we show the existence of exact EFX allocations under general leveled valuations via a simple protocol that in addition satisfies several natural economic properties. Finally, we take a mechanism design approach and we propose protocols that are both truthful and approximately fair under leveled valuations.

翻译：我们研究了在具有{\em层级}估值函数的智能体之间公平分配不可分割物品的问题。这类偏好已在经济学和公平分配文献中被研究过，捕捉了一种简单直观的经济行为：较大的物品束总是比较小的更受偏好。我们对层级估值的各个子类进行了细粒度分析，重点关注两种被广泛研究的公平性概念——(近似)最大最小份额(MMS)和无嫉妒性除一(EFX)。具体而言，我们提出了一个普遍的正向结果，证明了在同时满足层级性和子模性的估值下，存在$2/3$-MMS分配。我们还展示了如何将部分思路推广到层级估值类之外：针对两个子模（未必层级）智能体的情形，我们证明了$2/3$-MMS分配始终存在，这补充了近期的一项不可能性结果。随后，我们转向次可加与分数次可加的层级智能体情形，给出了MMS近似比$1/2$的紧确（上下界）界限。此外，我们通过一个简单协议证明了在一般层级估值下精确EFX分配的存在性，该协议还满足若干自然的经济属性。最后，我们采用机制设计方法，提出了在层级估值下同时满足真实性与近似公平性的协议。