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.

翻译：EFX分配的存在性是公平分配中的一个基本问题。本文构造了一个包含三个智能体、八种商品的实例，其估价为单调次可加函数，使得对于任意 $α> \frac{1}{\sqrt[6]{2}} \approx 0.89$，不存在满足 $α$-EFX 的分配。我们还提供了一个密切相关的三智能体、八商品实例，其估价为子模函数（实际上是加权覆盖函数），该实例中不存在EFX分配。我们构造的一个关键特征是其对称性：智能体的估价在商品重标号下完全相同。因此，即便智能体仅因商品标号方式不同而相异，EFX也可能失效。这种对称性使得反例简洁且易于人工验证，从而揭示了EFX存在性的简单组合障碍。