Guaranteeing Maximin Shares: Some Agents Left Behind

TL;DR

This paper explores fairness guarantees in allocating indivisible goods, showing limitations of optimal algorithms, establishing a 2/3 MMS guarantee for most agents, and providing polynomial algorithms and empirical insights.

Contribution

It introduces a new approximation notion for MMS, proves bounds on the number of agents satisfied, and offers algorithms achieving improved fairness guarantees.

Findings

No optimal algorithm can satisfy more than a constant number of agents.

A 2/3 MMS guarantee can be achieved for up to nine agents.

Allocations can guarantee MMS for approximately 1.5 times the number of agents, surpassing previous bounds.

Abstract

The maximin share (MMS) guarantee is a desirable fairness notion for allocating indivisible goods. While MMS allocations do not always exist, several approximation techniques have been developed to ensure that all agents receive a fraction of their maximin share. We focus on an alternative approximation notion, based on the population of agents, that seeks to guarantee MMS for a fraction of agents. We show that no optimal approximation algorithm can satisfy more than a constant number of agents, and discuss the existence and computation of MMS for all but one agent and its relation to approximate MMS guarantees. We then prove the existence of allocations that guarantee MMS for $\frac{2}{3}$ of agents, and devise a polynomial time algorithm that achieves this bound for up to nine agents. A key implication of our result is the existence of allocations that guarantee $\text{MMS}^{\lceil{3n/2}\rceil}$, i.e., the value that agents receive by partitioning the goods into $\lceil{\frac{3}{2}n}\rceil$ bundles, improving the best known guarantee of $\text{MMS}^{2n-2}$. Finally, we provide empirical experiments using synthetic data.