We consider allocations of a set of $m$ indivisible goods to $n$ agents of equal entitlements that have valuations from the class XOS. A previous sequence of works showed allocations that obtain an $α$-approximation for the maximin share (MMS), for values of $α$ that gradually approach $\frac{1}{4}$ from below (the currently known ratio is $\frac{4}{17}$). In this work we attempt to obtain ratios better than $\frac{1}{4}$, and manage to do so for sufficiently large $n$. Our methodology is to first investigate the gap between the anyprice share (APS) and the MMS when all agents have the same XOS valuations, for which we design an allocation algorithm and prove that each agent receives at least $α> \frac{11}{40}$ times the APS. Then, we derive inspiration from this algorithm, and modify it so that it applies also when agents have different XOS valuations. Using this modified version, we show that for some sufficiently large $n_0$, there is an $α$-MMS allocation (in fact, an $α$-APS allocation) for every $n \geq n_0$.

翻译：我们考虑将一组 $m$ 个不可分物品分配给 $n$ 个具有平等权利且估值属于 XOS 类别的智能体。已有的一系列工作展示了对于最大化份额 (MMS) 实现 $\alpha$ 近似分配的方案，其中 $\alpha$ 的值从下方逐渐趋近于 $\frac{1}{4}$（当前已知比率为 $\frac{4}{17}$）。本研究尝试获得优于 $\frac{1}{4}$ 的比率，并成功地在 $n$ 足够大时实现了这一目标。我们的方法首先探究当所有智能体具有相同 XOS 估值时任意价格份额 (APS) 与 MMS 之间的差距，为此我们设计了一种分配算法，并证明每个智能体至少获得 $\alpha > \frac{11}{40}$ 倍的 APS。随后，我们从该算法中汲取灵感，对其进行修改，使其同样适用于智能体具有不同 XOS 估值的情形。利用这一修改版本，我们证明对于某些足够大的 $n_0$，当 $n \geq n_0$ 时存在 $\alpha$-MMS 分配（实际上为 $\alpha$-APS 分配）。