PMMS Allocations Need Not Exist for 3 Agents with Additive Valuations

2026-09-08

AI Summary

本文通过一个实例证明了在3个参与者和9个不可分割商品的情况下，无法实现PMMS公平分配，并指出PMMS的近似比不能超过0.987。

Abstract

This note gives an instance demonstrating that the pairwise maximin share (PMMS) property cannot be satisfied by any allocation in certain fair division problems with indivisible goods and additive valuations. The instance requires only $n=3$ agents and $m=9$ goods, and is accompanied by a proof that no PMMS allocation exists. A separate instance shows (certified by exhaustive enumeration with a computer) that PMMS cannot be approximated within a ratio above $78/79 \approx 0.987$.

Problem

• PMMS
• fair division
• indivisible goods
• additive valuations

Innovation

• PMMS
• Additive Valuations
• Indivisible Goods