布尔估值下的公平分配：超越归一化

中文摘要 AI 辅助

我们研究当智能体具有任意两层偏好时不可分物品的公平分配问题：每个智能体对任意物品集合的价值是布尔型的，无需满足单调性或可加性。值得注意的是，我们未施加标准的归一化假设，即不同智能体对空集的估值可处于不同的布尔层级。由于偏好是非单调的，分件无嫉妒（EF1）和任意分件无嫉妒（EFX）各自存在多种变体，具体取决于移除哪些物品、以及从嫉妒智能体的束还是被嫉妒智能体的束中移除这些物品。本文研究了这些EF1和EFX变体的存在性，以及它们与经济效率、激励相容性、可行性约束、基于彩票的随机化相结合时的情况。我们的结果表明，存在性情况关键取决于归一化智能体的数量，归一化智能体指那些对空束估值处于较低布尔层级的智能体。作者在推导理论结果时获得了GPT-5.6-Sol的大量协助，验证了所有AI生成证明的正确性，并在GPT-5.6-Sol和Claude Opus 5的辅助下扩展了阐述内容并简化了论证。

英文摘要

We study fair division of indivisible items when agents have arbitrary two-level preferences: the value of each agent for any set of items is Boolean, which need not be monotone or additive. Notably, we do not impose the standard assumption of normalization, i.e., different agents may value the empty set at different Boolean levels. Since the preferences are nonmonotone, envy-freeness up to one item (EF1) and envy-freeness up to any item (EFX) each admit several variants, depending on which items are tested for removal and whether they are removed from the envious agent's bundle or the envied agent's bundle. This paper investigates the existence of these variants of EF1 and EFX, on their own and together with economic efficiency, incentive compatibility, feasibility constraints, and lottery-based randomization. Our results highlight that the existence landscape depends crucially on the number of normalized agents, who value the empty bundle at the lower Boolean level. The authors used significant assistance from GPT-5.6-Sol for deriving theoretical results, verified any AI-generated proofs for correctness, and expanded on the exposition and simplified arguments, with the aid of GPT-5.6-Sol and Claude Opus 5.

发表机构

• University of Toronto（多伦多大学）

机构由 AI 辅助整理，请以论文原文为准。