Almost Envy-Freeness for Groups: Improved Bounds via Discrepancy Theory

TL;DR

This paper investigates fair division of indivisible goods among groups, establishing bounds on approximate envy-freeness using discrepancy theory, and highlighting computational complexity aspects of finding such allocations.

Contribution

It introduces tight bounds on envy-free up to $ heta(\sqrt{n})$ goods allocations for groups and demonstrates the computational hardness of achieving better approximations.

Findings

Existence of envy-free up to $ heta(\sqrt{n})$ goods allocations for constant groups

Efficient algorithms for finding these allocations

NP-hardness of computing allocations with better guarantees

Abstract

We study the allocation of indivisible goods among groups of agents using well-known fairness notions such as envy-freeness and proportionality. While these notions cannot always be satisfied, we provide several bounds on the optimal relaxations that can be guaranteed. For instance, our bounds imply that when the number of groups is constant and the $n$ agents are divided into groups arbitrarily, there exists an allocation that is envy-free up to $\Theta(\sqrt{n})$ goods, and this bound is tight. Moreover, we show that while such an allocation can be found efficiently, it is NP-hard to compute an allocation that is envy-free up to $o(\sqrt{n})$ goods even when a fully envy-free allocation exists. Our proofs make extensive use of tools from discrepancy theory.