We study the fair allocation of indivisible goods among agents with additive valuations. The fair division literature has traditionally focused on two broad classes of fairness notions: envy-based notions and share-based notions. Within the share-based framework, most attention has been devoted to the maximin share (MMS) guarantee and its relaxations, while envy-based fairness has primarily centered on EFX and its relaxations. Recent work has shown the existence of allocations that simultaneously satisfy multiplicative approximate MMS and envy-based guarantees such as EF1 or EFX. Motivated by this line of research, we study for the first time the compatibility between ordinal approximations of MMS and envy-based fairness notions. In particular, we establish the existence of allocations satisfying the following combined guarantees: (i) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EFX for ordered instances; (ii) simultaneous $1$-out-of-$\lceil 3n/2 \rceil$ MMS and EF1 for top-$n$ instances; and (iii) simultaneous $1$-out-of-$4\lceil n/3 \rceil$ MMS and EF1 for ordered instances.

翻译：我们研究在代理人具有可加性估值的情况下，不可分割物品的公平分配问题。公平分配文献传统上关注两类广泛的公平概念：基于嫉妒的概念和基于份额的概念。在基于份额的框架内，大多数注意力集中在最大最小份额（MMS）保证及其松弛上，而基于嫉妒的公平则主要围绕EFX及其松弛展开。最近的研究表明，存在同时满足乘性近似MMS和基于嫉妒的保证（如EF1或EFX）的分配。受此研究方向的启发，我们首次研究了MMS的序数近似与基于嫉妒的公平概念之间的兼容性。具体而言，我们证明了满足以下组合保证的分配的存在：（i）对于有序实例，同时满足$1$-out-of-$\lceil 3n/2 \rceil$ MMS和EFX；（ii）对于顶部-$n$实例，同时满足$1$-out-of-$\lceil 3n/2 \rceil$ MMS和EF1；以及（iii）对于有序实例，同时满足$1$-out-of-$4\lceil n/3 \rceil$ MMS和EF1。