Bad News for Couples: Tight Lower Bounds for Fair Division of Indivisible Items

TL;DR

This paper proves tight lower bounds on the fairness of dividing indivisible items among couples, showing that envy-freeness up to a certain number of items cannot always be achieved, even in small groups.

Contribution

It establishes tight lower bounds for envy-freeness in fair division of indivisible items among couples, matching previous upper bounds and challenging prior conjectures.

Findings

Existence of instances where envy-freeness up to (rac{}{}) items cannot be guaranteed.

The lower bounds match the known upper bounds, closing the theoretical gap.

Surprising result that small groups do not necessarily allow better fairness guarantees.

Abstract

We consider the problem of fairly allocating indivisible goods to couples, where each couple consists of two agents with distinct additive valuations. We show that there exist instances of allocating indivisible items to $n$ couples for which envy-freeness up to $\Omega(\sqrt{n})$ items cannot be guaranteed. This closes the gap by matching the upper bound of Manurangsi and Suksompong, which applies to arbitrary instances with $n$ agents in total. This result is somewhat surprising, as that upper bound was conjectured not to be tight for instances consisting only of small groups.