There are two common classes of fairness notions that are considered when allocating $m$ indivisible items to $n$ agents of equal entitlements. One is that of share-based fairness notions, with the maximin share (MMS) and its relaxations to $ρ$-MMS being prominent representatives of this class. The other is that of comparison-based fairness notions, with envy-freeness (EF) and its relaxations such as EF1 being prominent representatives of this class. In general, no class offers good guarantees for the other class. In this work, we design allocations that simultaneously satisfy notions from both classes, and specifically, are $ρ$-MMS for constant $ρ$ and EF1 (in fact, also EFL). Such results were previously known when agents have additive valuations, and we prove such results for the more general class of submodular valuations.

翻译：在将$m$个不可分割物品公平分配给$n$个权益相等的智能体时，通常考虑两类公平性概念。一类是基于份额的公平性概念，其典型代表是最大最小份额（MMS）及其松弛形式$\rho$-MMS；另一类是基于比较的公平性概念，其典型代表是无嫉妒性（EF）及其松弛形式如EF1。一般而言，这两类概念之间不存在良好的相互保证。在本工作中，我们设计了能同时满足两类概念的分配方案，具体而言，该方案对常数$\rho$满足$\rho$-MMS和EF1（事实上还满足EFL）。此类结果此前仅适用于可加估值智能体，而我们在更一般的次模估值类别中证明了该结论。