We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely proportionality and maximin share, are not universally feasible, nor are any constant approximations of them. We propose a novel share notion, where an agent assesses the fairness of a bundle by comparing it to her valuation in a random allocation. In this framework, a bundle is considered $q$-quantile fair, for $q\in[0,1]$, if it is at least as good as a bundle obtained in a uniformly random allocation with probability at least $q$. Our main question is whether there exists a constant value of $q$ for which the $q$-quantile share is universally feasible. Our main result establishes a strong connection between the feasibility of quantile shares and the classical Erd\H{o}s Matching Conjecture. Specifically, we show that if a version of this conjecture is true, then the $\frac{1}{2e}$-quantile share is universally feasible. Furthermore, we provide unconditional feasibility results for additive, unit-demand and matroid-rank valuations for constant values of $q$. Finally, we discuss the implications of our results for other share notions.

翻译：我们研究公平分配问题，即如何将一组不可分割的商品公平地分配给一组具有组合估值的智能体。为了量化公平性，我们采用份额概念，其中每个智能体基于某种公平准则有权获得一份公平份额，当每个智能体的估值（弱）超过其公平份额时，分配被认为是公平的。如果一个基于份额的概念对每个单调估值配置都允许公平分配，则称其具有普遍可行性。一个关键问题由此产生：是否存在一个非平凡的基于份额的概念具有普遍可行性？最著名的基于份额的概念——比例性和最大最小份额——不具有普遍可行性，它们的任何常数近似也不具备可行性。我们提出了一种新的份额概念，其中智能体通过将捆绑商品与其在随机分配中的估值进行比较来评估其公平性。在此框架下，对于$q\in[0,1]$，如果捆绑商品在均匀随机分配中至少与以概率$q$获得的捆绑商品一样好，则认为该捆绑商品是$q$-分位数公平的。我们的主要问题是：是否存在常数$q$使得$q$-分位数份额具有普遍可行性？我们的主要结果建立了分位数份额的可行性与经典Erdős匹配猜想之间的紧密联系。具体来说，我们证明如果该猜想的一个版本成立，那么$\frac{1}{2e}$-分位数份额具有普遍可行性。此外，我们针对常数$q$值，在可加性、单位需求和拟阵秩估值下提供了无条件的可行性结果。最后，我们讨论了这些结果对其他份额概念的影响。