In fair division of indivisible items, domain restriction has played a key role in escaping from negative results and providing structural insights into the computational and axiomatic boundaries of fairness. One notable subdomain of additive preferences, the lexicographic domain, has yielded several positive results in dealing with goods, chores, and mixtures thereof. However, the majority of work within this domain primarily consider strict linear orders over items, which do not allow the modeling of more expressive preferences that contain indifferences (ties). We investigate the most prominent fairness notions of envy-freeness up to any (EFX) or some (EF1) item under weakly lexicographic preferences. For the goods-only setting, we develop an algorithm that can be customized to guarantee EF1, EFX, maximin share (MMS), or a combination thereof, along the efficiency notion of Pareto optimality (PO). From the conceptual perspective, we propose techniques such as preference graphs and potential envy that are independently of interest when dealing with ties. Finally, we demonstrate challenges in dealing with chores and highlight key algorithmic and axiomatic differences of finding EFX solutions with the goods-only setting. Nevertheless, we show that there is an algorithm that always returns an EF1 and PO allocation for the chores-only instances.

翻译：在不可分物品的公平分配中，领域限制在摆脱负面结果、揭示公平性计算与公理边界的结构洞见方面发挥了关键作用。作为可加偏好的一个重要子领域，字典序领域在处理物品、杂务及其混合情形时已取得多项积极成果。然而，该领域的大多数研究主要关注物品上的严格线性序，这种序结构无法建模包含无差异（平局）的更富表达力的偏好。我们研究了弱字典序偏好下最突出的公平性概念——任意物品无嫉妒（EFX）与某物品无嫉妒（EF1）。针对纯物品场景，我们提出了一种可定制的算法，能够保证EF1、EFX、最大最小份额（MMS）或其组合，同时满足帕累托最优（PO）的效率概念。从概念层面，我们提出了偏好图与潜在嫉妒等独立适用于平局处理的技术。最后，我们展示了处理杂务时面临的挑战，并强调了与纯物品场景相比，寻求EFX解在算法与公理上的关键差异。尽管如此，我们证明存在一种算法能够始终为纯杂务实例返回EF1且PO的分配方案。