Improved Approximation Guarantees for Groupwise Maximin Share Fairness

2026-06-03

AI Summary

This study addresses fairness in the allocation of indivisible goods through the lens of Groupwise Maximin Share (GMMS), which guarantees that every agent receives a bundle worth at least their GMMS—defined as the maximin share they could secure within any subgroup they belong to. The authors propose a polynomial-time algorithm based on the Draft-and-Eliminate framework, leveraging refined upper bounds on agents’ valuations in subinstances and structural properties of short picking sequences to improve the best-known GMMS approximation ratio from 4/7 to the inverse of the golden ratio, φ⁻¹ ≈ 0.618. Under restricted settings—such as when all agents agree on the ranking of the top n items or when the number of agents is small—the guarantee strengthens further, achieving (√10 − 1)/3 ≈ 0.72 for three agents. This result is currently the best known, and the analysis is asymptotically tight among comparable approaches.

Abstract

We study the problem of fairly allocating a set of indivisible goods to a set of $n$ agents with additive valuation functions. We focus on the very demanding notion of \textit{groupwise maximin share fairness} (GMMS), which requires that each agent $i$ receives value comparable to their maximin share, where the latter is computed \textit{with respect to any subset of agents that contains $i$}. We show that it is possible to compute $(φ-1)$-approximate GMMS allocations in polynomial time, where $φ\approx 1.618$ is the golden ratio). This improves on the previously known guarantee of $4/7$ of Chaudhury et al. [SICOMP; 2021] and Amanatidis et al. [TCS; 2020]. We propose a simple algorithm that maintains the same main properties as the Draft-and-Eliminate algorithm of Amanatidis et al. [TCS, 2020] and we improve on the approximation guarantee analysis by carefully bounding the relevant value within any subinstance induced by the restriction of our allocation to a subset of agents. Our analysis is asymptotically tight for algorithms that share these properties and has the additional benefit of giving improved guarantees for restricted settings; in particular, when the agents agree on the top $n$ goods or when the number of agents is small. To illustrate the challenges of going beyond the guarantees of our algorithm, we also present a variant with an improved approximation of $(\sqrt{10}-1)/3 \approx 0.72$ for the case of three agents. To achieve this improvement we partially characterize the maximin share guarantees of short picking sequences for a small number of goods.

Problem

• groupwise maximin share fairness
• indivisible goods allocation
• fair division
• approximation guarantee
• additive valuations

Innovation

• Groupwise Maximin Share
• Fair Division
• Approximation Algorithm
• Indivisible Goods
• Golden Ratio