Counterexamples to EFX for Submodular and Subadditive Valuations

2026-05-07

AI Summary

This study investigates the existence of envy-free up to any good (EFX) allocations under submodular and subadditive valuations. By constructing a highly symmetric counterexample with three agents and eight goods—where agent valuations differ only by item labels and are derived from weighted coverage functions (submodular) within a subadditive framework—the authors establish, for the first time, that EFX allocations may not exist under such general valuation classes. The counterexample is simple and human-verifiable, and it further demonstrates that α-EFX allocations do not exist for any α > 1/⁶√2 ≈ 0.89. This result reveals a fundamental limitation of EFX fairness in non-quasi-linear settings.

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 $α$-EFX for any $α> \frac{1}{\sqrt[6]{2}} \approx 0.89$. 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.

Problem

• EFX
• fair division
• submodular valuations
• subadditive valuations
• counterexamples

Innovation

• EFX
• submodular valuations
• subadditive valuations
• fair division
• counterexample