# EFX Exists for Three Types of Agents

arXiv: https://arxiv.org/abs/2410.13580 · HTML: https://mlsanity.com/paper/2410.13580/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Oct 2024.

## Built on by (most cited first, 3 of 3)

- [The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items](https://mlsanity.com/paper/2601.12849/) (2026)
- [Probing EFX via PMMS: (Non-)Existence Results in Discrete Fair Division](https://mlsanity.com/paper/2507.14957/) (2025)
- [EFX Allocation In (Multi)Hypergraphs](https://mlsanity.com/paper/2608.03171/) (2026)

## Abstract (arXiv)

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]).