EFX and PO Allocation Exists for Two Types of Goods

TL;DR

This paper proves that for two types of goods with positive valuations, there always exists an allocation that is both envy-free up to any good and Pareto optimal, and provides an efficient algorithm to find such allocations.

Contribution

It extends previous results by establishing the existence and computability of EFX+PO allocations for two types of goods with positive valuations.

Findings

EFX+PO allocations always exist for two types of goods with positive valuations.

A quasilinear time algorithm can compute EFX+PO allocations in this setting.

The result generalizes previous work by ensuring Pareto optimality alongside envy-freeness.

Abstract

We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuations. We focus on envy-freeness up to any good (EFX) -- an important fairness notion in fair division of indivisible goods. A central open question in this field is whether EFX allocations always exist for any number of agents. While prior work has established EFX existence for settings with at most three distinct valuations (Prakash HV et al. 2025) and for two types of goods (Gorantla, Marwaha, and Velusamy 2023), the general case remains unresolved. In this paper, we extend the existent knowledge by proving that EFX allocations satisfying Pareto optimality (PO) always exist and can be computed in quasiliniear time when there are two types of goods, given that the valuations are positive. This result strengthens the existing work of (Gorantla, Marwaha, and Velusamy 2023), which only guarantees the existence of EFX allocations without ensuring Pareto optimality. Our findings demonstrate a fairly simple and efficient algorithm constructing an EFX+PO allocation.