EFX Exists for Three Types of Agents

TL;DR

This paper proves that envy-free up to any good (EFX) allocations always exist when there are any number of agents with at most three distinct additive valuations, advancing the understanding of fair division.

Contribution

The paper establishes the existence of EFX allocations for any number of agents with up to three distinct additive valuations, generalizing previous results.

Findings

EFX allocations exist for any number of agents with up to three distinct valuations.

The result generalizes the three-agent case and the two-type valuation case.

This settles an open question in the field of fair division.

Abstract

We study the problem of finding an envy-free allocation of indivisible goods among agents with additive valuations. We focus on the fairness notion of envy-freeness up to any good (EFX). A central open question in fair division is whether EFX allocations always exist for any number of agents. While EFX has been established for three agents [CGM24] and for any number of agents with at most two distinct valuations [Mah23], its existence in more general settings remains open. In this paper, we make significant progress by proving that EFX allocations exist for any number of agents when there are at most three distinct additive valuations. This result simultaneously generalizes both the three-agent case and the two-type case, settling an open question in the field (see [Mah23]).