The classic fair division problems assume the resources to be allocated are either divisible or indivisible, or contain a mixture of both, but the agents always have a predetermined and uncontroversial agreement on the (in)divisibility of the resources. In this paper, we propose and study a new model for fair division in which agents have their own subjective divisibility over the goods to be allocated. That is, some agents may find a good to be indivisible and get utilities only if they receive the whole good, while others may consider the same good to be divisible and thus can extract utilities according to the fraction of the good they receive. We investigate fairness properties that can be achieved when agents have subjective divisibility. First, we consider the maximin share (MMS) guarantee and show that the worst-case MMS approximation guarantee is at most $2/3$ for $n \geq 2$ agents and this ratio is tight in the two- and three-agent cases. This is in contrast to the classic fair division settings involving two or three agents. We also give an algorithm that produces a $1/2$-MMS allocation for an arbitrary number of agents. Second, we study a hierarchy of envy-freeness relaxations, including EF1M, EFM and EFXM, ordered by increasing strength. While EF1M is compatible with non-wastefulness (an economic efficiency notion), this is not the case for EFM, even for two agents. Nevertheless, an EFXM and non-wasteful allocation always exists for two agents if at most one good is discarded.

翻译：经典的公平分配问题假设待分配资源要么是可分的，要么是不可分的，或者两者兼有，但参与者对资源的（不）可分性总是存在预先确定且无争议的共识。本文提出并研究了一种新的公平分配模型，在该模型中，参与者对待分配物品具有各自的主观可分性。也就是说，某些参与者可能认为某物品是不可分的，仅当获得整个物品时才能获得效用；而其他参与者可能认为同一物品是可分的，从而可以根据所获物品的比例提取效用。我们研究了当参与者具有主观可分性时可实现的公平性性质。首先，我们考虑最大最小份额（MMS）保证，并证明对于 \(n \geq 2\) 个参与者，最坏情况下的 MMS 近似保证至多为 \(2/3\)，且该比例在参与者为两到三人的情况下是紧的。这与涉及两到三名参与者的经典公平分配设置形成对比。我们还给出了一种可为任意数量参与者生成 \(1/2\)-MMS 分配的算法。其次，我们研究了一系列按强度递增排序的嫉妒消除松弛性质，包括 EF1M、EFM 和 EFXM。虽然 EF1M 与非浪费性（一种经济效率概念）是相容的，但 EFM 则不然，即使对于两名参与者也是如此。然而，对于两名参与者，如果最多丢弃一件物品，则始终存在 EFXM 且非浪费的分配。