精确Hill份额是同时保证

中文摘要 AI 辅助

不可分割坏品的公平分配旨在寻找一种分配方案，使得每个智能体获得的捆绑成本不超过一个有意义的公平基准。经典的极小极大份额已被广泛使用；遗憾的是，它并非一种同时保证。Hill（《概率年鉴》，1987）开创了一种补充方法，其中份额仅取决于智能体数量和最大单项价值的可能取值。Li等人（《ACM经济计算汇刊》，2024）给出了精确的Hill公式，并证明了其单调闭包是一种同时保证。该闭包仅将最大项目成本视为上界，且可能严格大于基于实际最大项目条件化的份额。他们提出疑问：对于三个或更多智能体，这个更小的精确份额本身是否也是一种同时保证。我们肯定地回答了这一开放问题。对于任意数量的智能体以及任意异质的最大项目成本，存在一种分配方案满足每个智能体的精确Hill份额。我们进一步提供了一个多项式时间算法来计算这种分配。我们的算法结合了有序移动刀与尾部支配不变量，以及一个用于双智能体端点的单侧修剪子集和例程，而无需计算精确的极小极大划分。

英文摘要

Fair division of indivisible bads seeks allocations that guarantee every agent a bundle whose cost is no larger than a meaningful fairness benchmark. The canonical minimax share has widely been used; unfortunately, it is not a simultaneous guarantee. Hill (Ann. Probab., 1987) initiated a complementary approach in which the share depends only on the number of agents and the largest possible single-item value. Li et al. (ACM Trans. Econ. Comput., 2024) gave the exact Hill formula and proved that its monotone closure is a simultaneous guarantee. The closure treats the largest-item cost only as an upper bound and can be strictly larger than the share conditioned on the \emph{actual} largest item. They asked whether this smaller exact share is itself simultaneously guaranteed for three or more agents. We resolve this open question affirmatively. For any number of agents and arbitrary heterogeneous largest-item costs, there is one allocation that satisfies every agent's exact Hill's share. We further provide a polynomial-time algorithm computes such an allocation. Our algorithm combines an ordered moving knife with a tail-domination invariant, and a one-sided trimmed subset-sum routine for the two-agent endpoint without computing an exact minimax partition.

发表机构

• The Hong Kong Polytechnic University（香港理工大学）
• Peking University（北京大学）
• Chinese Academy of Sciences（中国科学院）

机构由 AI 辅助整理，请以论文原文为准。