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arXiv · 2605.06451

Counterexamples to EFX for Submodular and Subadditive Valuations

Simon Mackenzie·Mashbat Suzuki
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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.

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Simon Mackenzie, Mashbat Suzuki. 2026-05-07. Counterexamples to EFX for Submodular and Subadditive Valuations. https://arxiv.org/abs/2605.06451

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