Counterexamples to EFX for Submodular and Subadditive Valuations
Simon Mackenzie, Mashbat Suzuki
TLDR
This paper presents counterexamples showing that EFX allocations do not always exist for submodular and subadditive valuations, even with symmetric agents.
Key contributions
- Constructs a 3-agent, 8-good instance with subadditive valuations where no α-EFX exists for α > 0.89.
- Provides a related 3-agent, 8-good instance with submodular (weighted coverage) valuations lacking EFX.
- Demonstrates EFX failure even when agents have symmetric valuations, differing only by good labels.
- The symmetric construction yields compact, human-verifiable counterexamples to EFX existence.
Why it matters
This work resolves a fundamental question in fair division by proving the non-existence of EFX allocations for important valuation classes. It highlights limitations of EFX even under highly symmetric conditions, guiding future research in fair allocation algorithms.
Original 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.
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