A tight negative example for MMS fair allocations

TL;DR

This paper constructs specific examples demonstrating the limitations of maximin share fairness in indivisible item allocations, showing that agents can receive significantly less than their fair share, with bounds proven for various scenarios.

Contribution

The paper presents larger gap examples for MMS fairness, establishes minimal item-agent differences, and extends results to chores with new quantitative bounds.

Findings

For three agents and nine items, an instance where an agent gets at most 39/40 of her MMS.

No negative example exists with fewer than six items more than agents, and the 1/40 gap is optimal for nine items.

Extensions to chores show similar gaps, with a 1/43 MMS gap for three agents and nine chores.

Abstract

We consider the problem of allocating indivisible goods to agents with additive valuation functions. Kurokawa, Procaccia and Wang {[JACM, 2018]} present instances for which every allocation gives some agent less than her maximin share. We present such examples with larger gaps. For three agents and nine items, we design an instance in which at least one agent does not get more than a $\frac{39}{40}$ fraction of her maximin share. {Moreover, we show that there is no negative example in which the difference between the number of items and the number of agents is smaller than six, and that the gap (of $\frac{1}{40}$) of our example is worst possible among all instances with nine items.} For $n \ge 4$ agents, we show examples in which at least one agent does not get more than a $1 - \frac{1}{n^4}$ fraction of her maximin share. {In the instances designed by Kurokawa, Procaccia and Wang, the gap is exponentially small in $n$.} Our proof techniques extend to allocation of chores (items of negative value), though the quantitative bounds for chores are different from those for goods. For three agents and nine chores, we design an instance in which the MMS gap is $\frac{1}{43}$.