EFX and PO Allocation Exists for Two Types of Goods

2026-01-06

arXiv.org

AI Summary

This study addresses a central open problem in fair division: the simultaneous achievement of envy-freeness up to any good (EFX) and Pareto optimality (PO) in the allocation of indivisible goods. Focusing on agents with additive positive valuations and instances restricted to only two distinct types of items, the paper establishes—for the first time—the existence of allocations that satisfy both EFX and PO. Moreover, it presents a quasi-linear time algorithm that efficiently constructs such allocations. This work not only strengthens prior theoretical results that guaranteed EFX existence without ensuring PO but also demonstrates, through combinatorial optimization techniques, the computational tractability and constructive feasibility of finding allocations meeting both fairness and efficiency criteria.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Combinatorial Optimization
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• EFX
• Pareto optimality
• indivisible goods
• fair division
• additive valuations

Innovation

• EFX
• Pareto optimality
• two types of goods
• indivisible goods allocation
• quasilinear-time algorithm