发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不可分割商品公平分配中的最大最小份额（MMS）问题，证明了一类推广少值域和弱字典序估值的估值域存在精确MMS分配，并给出时间复杂度为O(nm log²(2+m))的算法。
AI 中文摘要
我们研究在非负可加估值下，将不可分割商品公平分配给多个智能体的问题。智能体的最大最小份额（MMS）是指她通过将商品划分为与智能体数量相同的若干束并接受价值最低的一束所能保证获得的价值。我们证明，对于一个推广了若干少值域以及弱字典序估值的估值域，精确MMS分配是存在的。对于每个智能体，在正比例缩放后，较低价值的商品可能取值于$\{0,1,d,d+1\}$、$\{0,1,2,2e\}$或$\{0,1,2,3,4\}$，其中$d,e\ge 2$为整数。在这些商品之上，估值可包含任意多个价值类，但需满足每个类别中单个商品的价值至少等于所有严格较低价值商品的总价值。价值类、缩放因子和参数可因智能体而异。我们给出一种算法，在$O(nm\log^2(2+m))$时间内计算MMS分配，其中$n$为智能体数量，$m$为商品数量。
英文摘要
We study the fair division of indivisible goods among agents with nonnegative additive valuations. An agent's maximin share (MMS) is the value she can guarantee by partitioning the goods into as many bundles as there are agents and receiving a least-valued bundle. We prove the existence of exact MMS allocations for a valuation domain that generalizes several few-valued domains as well as weakly lexicographic valuations. For each agent, the lower-valued goods, after a positive rescaling, may take values in $\{0,1,d,d+1\}$, $\{0,1,2,2e\}$, or $\{0,1,2,3,4\}$, where $d,e\ge 2$ are integers. Above these goods, the valuation may contain arbitrarily many value classes, subject to the condition that a single good in each class is worth at least the total value of all strictly lower-valued goods. The value classes, scaling factors, and parameters may vary across agents. We give an algorithm that computes an MMS allocation in $O(nm\log^2(2+m))$ time, where $n$ is the number of agents and $m$ is the number of goods.
