An Improved Lower Bound for Maximin Share Allocations of Goods

TL;DR

This paper advances the understanding of fair division by establishing that maximin share allocations are guaranteed to exist when the number of goods is up to five more than the number of players, but not beyond that.

Contribution

It extends the known existence guarantee of MMS allocations from m ≤ n+4 to m = n+5, and demonstrates failure at m = n+6.

Findings

MMS allocations guaranteed for m ≤ n+5

Guarantee fails for m = n+6

Provides bounds for fair division scenarios

Abstract

The problem of fair division of indivisible goods has been receiving much attention recently. The prominent metric of envy-freeness can always be satisfied in the divisible goods setting (see for example \cite{BT95}), but often cannot be satisfied in the indivisible goods setting. This has led to many relaxations thereof being introduced. We study the existence of {\em maximin share (MMS)} allocations, which is one such relaxation. Previous work has shown that MMS allocations are guaranteed to exist for all instances with $n$ players and $m$ goods if $m \leq n+4$. We extend this guarantee to the case of $m = n+5$ and show that the same guarantee fails for $m = n+6$.