Improved EFX Approximation Guarantees under Ordinal-based Assumptions

TL;DR

This paper improves approximation guarantees for fair division of indivisible items under ordinal assumptions, achieving better results than previous bounds in specific cases where agents agree on top items.

Contribution

It introduces a new algorithm with a 2/3-approximation for cases where agents agree on top items, surpassing the previous best of (φ-1) approximation.

Findings

Achieved a 2/3-approximation in specific ordinal settings.

Developed a general framework for approximation guarantees.

Improved bounds over the previous (φ-1) approximation.

Abstract

Our work studies the fair allocation of indivisible items to a set of agents, and falls within the scope of establishing improved approximation guarantees. It is well known by now that the classic solution concepts in fair division, such as envy-freeness and proportionality, fail to exist in the presence of indivisible items. Unfortunately, the lack of existence remains unresolved even for some relaxations of envy-freeness, and most notably for the notion of EFX, which has attracted significant attention in the relevant literature. This in turn has motivated the quest for approximation algorithms, resulting in the currently best known $(\phi-1)$-approximation guarantee by Amanatidis et al (2020), where $\phi$ equals the golden ratio. So far, it has been notoriously hard to obtain any further advancements beyond this factor. Our main contribution is that we achieve better approximations, for certain special cases, where the agents agree on their perception of some items in terms of their worth. In particular, we first provide an algorithm with a $2/3$-approximation, when the agents agree on what are the top $n$ items (but not necessarily on their exact ranking), with $n$ being the number of agents. To do so, we also study a general framework that can be of independent interest for obtaining further guarantees.