We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her ($1$-out-of-$n$) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3\rceil$ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of $\alpha$-MMS allocations (for the largest possible $\alpha < 1$), which guarantees each agent at least $\alpha$ times her MMS value. We introduce a general framework of "approximate MMS with agent priority ranking". An allocation is said to be $T$-MMS, for a non-increasing sequence $T = (\tau_1, \ldots, \tau_n)$ of numbers, if the agent at rank $i$ in the order gets a bundle of value at least $\tau_i$ times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of $T$-MMS allocations where $\tau_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$ for all $i$. Furthermore, we can get allocations that are $(\frac{3}{4} + \frac{1}{12n})$-MMS ex-post and $(0.8253 + \frac{1}{36n})$-MMS ex-ante. We also prove that our algorithm does not give better than $(0.8631 + \frac{1}{2n})$-MMS ex-ante.

翻译：我们考虑使用最大最小份额（MMS）这一公平性概念，将一组不可分割商品公平分配给$n$个具有可加估值的智能体。MMS 是最流行的基于份额的概念，其中一个智能体认为分配对她公平，如果她获得的商品价值至少等于她的（$1$-out-of-$n$）MMS 值。如果所有智能体都获得其 MMS 值，则称该分配为 MMS。然而，由于 MMS 分配并不总是存在，研究重点转向了对其序数近似和乘法近似的探索。在序数近似中，目标是证明$1$-out-of-$d$ MMS 分配的存在性（对于尽可能小的$d>n$）。一系列工作将最优因子推进到$d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]。我们证明了$1$-out-of-$4\lceil n/3\rceil$ MMS 分配始终存在，从而改进了序数近似的最优结果。在乘法近似中，目标是证明$\alpha$-MMS 分配的存在性（对于尽可能大的$\alpha < 1$），这保证每个智能体至少获得其 MMS 值的$\alpha$倍。我们引入了一种通用的“带有智能体优先级排序的近似 MMS”框架。对于非递增序列$T = (\tau_1, \ldots, \tau_n)$，如果排序中第$i$位的智能体获得价值至少为其 MMS 值$\tau_i$倍的商品束，则称该分配为$T$-MMS。该框架将序数近似和乘法近似作为特例进行统一。我们证明了$T$-MMS 分配的存在性，其中对所有$i$有$\tau_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$。此外，我们能够获得事后$(\frac{3}{4} + \frac{1}{12n})$-MMS 且事前$(0.8253 + \frac{1}{36n})$-MMS 的分配。我们还证明了我们的算法不会给出优于事前$(0.8631 + \frac{1}{2n})$-MMS 的结果。