We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the desirable 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 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 $d=\lfloor 3n/2 \rfloor$ [HSSH21]. We show that $1$-out-of-$\lceil 4n/3\rceil$ MMS allocations always exist. 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. A series of works in the last decade led to the state-of-the-art factor of $\alpha = \frac{3}{4} + \frac{3}{3836}$ [AG23]. We introduce a general framework of $(\alpha, \beta, \gamma)$-MMS that guarantees $\alpha$ fraction of agents $\beta$ times their MMS values and the remaining $(1-\alpha)$ fraction of agents $\gamma$ times their MMS values. The $(\alpha, \beta, \gamma)$-MMS captures both ordinal and multiplicative approximations as its special cases. We show that $(2(1 -\beta)/\beta, \beta, 3/4)$-MMS allocations always exist. Furthermore, since we can choose the $2(1-\beta)/\beta$ fraction of agents arbitrarily in our algorithm, this implies (using $\beta=\sqrt{3}/2$) the existence of a randomized allocation that gives each agent at least 3/4 times her MMS value (ex-post) and at least $(17\sqrt{3} - 24)/4\sqrt{3} > 0.785$ times her MMS value in expectation (ex-ante).

翻译：我们考虑在$n$个具有可加性估值的智能体之间分配一组不可分割物品的公平性问题，采用理想公平概念——最大化份额（MMS）。MMS是最流行的基于份额的公平概念，在该概念中，若智能体获得的物品价值至少达到其MMS值，则认为分配对该智能体是公平的。当所有智能体都获得其MMS值时，该分配被称为MMS分配。然而，由于MMS分配并非始终存在，研究重点转向了其序数近似和乘法近似。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$d>n$）。一系列工作达到了当前最优因子$d=\lfloor 3n/2 \rfloor$ [HSSH21]。我们证明了$1$-out-of-$\lceil 4n/3\rceil$ MMS分配始终存在。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$\alpha < 1$），该分配保证每个智能体至少获得其MMS值的$\alpha$倍。过去十年的一系列工作达到了当前最优因子$\alpha = \frac{3}{4} + \frac{3}{3836}$ [AG23]。我们引入了$(\alpha, \beta, \gamma)$-MMS的一般框架，该框架保证$\alpha$比例的智能体获得其MMS值的$\beta$倍，其余$(1-\alpha)$比例的智能体获得其MMS值的$\gamma$倍。$(\alpha, \beta, \gamma)$-MMS将序数近似和乘法近似作为特例统一起来。我们证明了$(2(1 -\beta)/\beta, \beta, 3/4)$-MMS分配始终存在。此外，由于在我们的算法中可以任意选择这$2(1-\beta)/\beta$比例的智能体，这（利用$\beta=\sqrt{3}/2$）意味着存在一种随机分配，使得每个智能体在事后（ex-post）至少获得其MMS值的3/4倍，并在期望意义上（ex-ante）至少获得其MMS值的$(17\sqrt{3} - 24)/4\sqrt{3} > 0.785$倍。