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：我们研究了在具有可加性估值函数的智能体之间分配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大类公平性概念：基于嫉妒的概念和基于份额的概念。在基于份额的框架内，大部分研究集中于最大最小份额（MMS）保证及其松弛形式，而基于嫉妒的公平性则主要围绕EFX及其松弛形式展开。近期研究表明，存在同时满足近似MMS（乘性近似）与基于嫉妒的保证（如EF1或EFX）的分配方案。受此研究脉络启发，我们首次探讨了MMS的序数近似与基于嫉妒的公平性概念之间的兼容性。具体而言，我们证明了满足以下组合保证的分配方案的存在性：（i）对于有序实例，同时满足$1$-out-of-$\lceil 3n/2 \rceil$ MMS与EFX；（ii）对于top-$n$实例，同时满足$1$-out-of-$\lceil 3n/2 \rceil$ MMS与EF1；（iii）对于有序实例，同时满足$1$-out-of-$4\lceil n/3 \rceil$ MMS与EF1。