Improved Impossibility Bounds for Maximin Share Allocations

2026-09-14

AI Summary

本文针对不可分割物品分配问题，通过构建实例证明了在最大化最小份额(MMS)标准下的改进的不可能性边界，为商品和任务提供了更严格的界限。

Abstract

The maximin share (MMS) is a central fairness benchmark for allocating indivisible items, but it need not be simultaneously attainable even under additive preferences. While extensive work has developed approximation guarantees, quantitative impossibility bounds have received comparatively little attention. We establish improved asymptotic and constant impossibility bounds for both goods and chores. For every sufficiently large number $n$ of agents, we construct additive goods instances in which every allocation gives some agent at most a $1-Ω((\log n)^{-2})$ fraction of her MMS. This strengthens the $1/n^4$ shortfall of Feige, Sapir, and Tauber (2021) to an inverse-polylogarithmic shortfall, an exponential improvement on the logarithmic scale of $n$. For chores, we construct instances in which every allocation gives some agent cost at least a $1+Ω((\log n)^{-2})$ factor of her MMS. Consequently, for every fixed $\varepsilon>0$, guarantees of $1-O(n^{-\varepsilon})$ for goods and $1+O(n^{-\varepsilon})$ for chores are impossible. We also give four-agent, eleven-item instances that improve the universal impossibility bounds from $39/40$ to $20/21$ for goods and from $44/43$ to $31/30$ for chores.

Problem

• maximin share
• indivisible items
• fairness
• impossibility bounds

Innovation

• impossibility bounds
• maximin share (MMS)
• additive preferences
• inverse-polylogarithmic shortfall
• asymptotic improvement