Counterexamples to EFX for Submodular and Subadditive Valuations

TL;DR

This paper constructs specific examples demonstrating that EFX fairness allocations may not exist even with symmetric valuations among agents, highlighting fundamental limitations in fair division.

Contribution

It provides the first explicit counterexamples for EFX existence with submodular and subadditive valuations, emphasizing symmetry as a key factor.

Findings

No $rac{1}{ oot6 of{2}}$-EFX allocation exists for certain subadditive valuations.

No EFX allocation exists for certain submodular valuations.

Counterexamples are symmetric and human-verifiable.

Abstract

The existence of EFX allocations is a fundamental question in fair division. In this paper, we construct a three-agent, eight-good instance with monotone subadditive valuations such that no allocation satisfies $\alpha$-EFX for any $\alpha > \frac{1}{\sqrt[6]{2}} \approx 0.89$. We also provide a closely related three-agent, eight-good instance with submodular (in fact weighted coverage) valuations for which no EFX allocation exists. A key feature of our construction is its symmetry: the agents' valuations are identical up to a relabeling of the goods. Thus, EFX can fail even when agents differ only in how the goods are labeled. This symmetry makes the counterexamples compact and human-verifiable, yielding simple combinatorial obstructions to the existence of EFX.