发表机构
Tel Aviv University（特拉维夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对最大化份额分配，改进物品与杂务的不可行性界，将缺口从多项式加强为逆多对数，并优化小规模实例的常数界。
AI 中文摘要
最大化份额（MMS）是分配不可分割物品的核心公平基准，但即使在可加偏好下，它也不一定能够同时实现。尽管大量工作已发展出近似保证，但定量不可行性界受到的关注相对较少。我们为物品和杂务建立了改进的渐近和常数不可行性界。对于每一个足够大的代理人数$n$，我们构造可加物品实例，其中每个分配都使某个代理获得至多其MMS的$1-\Omega((\log n)^{-2})$比例。这将Feige、Sapir和Tauber（2021）的$1/n^4$缺口加强为逆多对数缺口，在$n$的对数尺度上是指数级改进。对于杂务，我们构造实例，其中每个分配都使某个代理的成本至少为其MMS的$1+\Omega((\log n)^{-2})$倍。因此，对于每个固定的$\varepsilon>0$，物品的$1-O(n^{-\varepsilon})$保证和杂务的$1+O(n^{-\varepsilon})$保证都是不可能的。我们还给出了四个代理、十一件物品的实例，将物品的普遍不可行性界从$39/40$改进到$20/21$，将杂务的从$44/43$改进到$31/30$。
英文摘要
The maximin share (MMS) is a central fairness benchmark for allocating indivisible items, but it need not be simultaneously attainable even under additive preferences. While extensive work has developed approximation guarantees, quantitative impossibility bounds have received comparatively little attention. We establish improved asymptotic and constant impossibility bounds for both goods and chores. For every sufficiently large number $n$ of agents, we construct additive goods instances in which every allocation gives some agent at most a $1-Ω((\log n)^{-2})$ fraction of her MMS. This strengthens the $1/n^4$ shortfall of Feige, Sapir, and Tauber (2021) to an inverse-polylogarithmic shortfall, an exponential improvement on the logarithmic scale of $n$. For chores, we construct instances in which every allocation gives some agent cost at least a $1+Ω((\log n)^{-2})$ factor of her MMS. Consequently, for every fixed $\varepsilon>0$, guarantees of $1-O(n^{-\varepsilon})$ for goods and $1+O(n^{-\varepsilon})$ for chores are impossible. We also give four-agent, eleven-item instances that improve the universal impossibility bounds from $39/40$ to $20/21$ for goods and from $44/43$ to $31/30$ for chores.
