Fair Division Under Boolean Valuations: Beyond Normalization

2026-08-29

AI Summary

本文研究了在布尔估值下，非可分物品的公平分配问题，不假设标准化偏好，并探讨了EF1和EFX变体的存在性及其与经济效率等条件的关系。

Abstract

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.

Problem

• Fair Division
• Boolean Valuations
• EF1
• EFX
• Nonmonotone Preferences

Innovation

• Boolean Valuations
• Non-normalized Preferences
• Envy-freeness Variants
• Economic Efficiency