Breaking the $3/4$ Barrier for Approximate Maximin Share

TL;DR

This paper proves the existence of approximate maximin share allocations with a factor slightly better than 3/4 for fair division of indivisible goods among agents, overcoming previous limitations.

Contribution

The authors introduce new reduction rules and analysis techniques to surpass the 3/4 barrier for approximate MMS allocations.

Findings

Existence of (/4 + 3/3836)-MMS allocations

New reduction rules for fair division problems

Improved theoretical bounds on MMS approximation

Abstract

We study the fundamental problem of fairly allocating a set of indivisible goods among $n$ agents with additive valuations using the desirable fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her MMS value. An allocation is called MMS if all agents receive at least their MMS value. Since MMS allocations need not exist when $n>2$, a series of works showed the existence of approximate MMS allocations with the current best factor of $\frac34 + O(\frac{1}{n})$. However, a simple example in [DFL82, BEF21, AGST23] showed the limitations of existing approaches and proved that they cannot improve this factor to $3/4 + \Omega(1)$. In this paper, we bypass these barriers to show the existence of $(\frac{3}{4} + \frac{3}{3836})$-MMS allocations by developing new reduction rules and analysis techniques.