AI 中文总结
研究当代理有单调次模估值时，不可分割商品分配能否同时满足 EF1 和 PO 的问题，通过给出反例及证明 NP 难来否定解决，将不存在性边界拓展到未加权覆盖估值，还证明对杂事也不存在 EF1 + PO。
AI 中文摘要
当代理具有单调次模估值时，不可分割商品的分配是否能同时公平（至多一件物品无嫉妒（EF1））且有效（帕累托最优（PO））一直是长期未解决的开放问题。我们给出一个两个代理的例子，表明不存在同时为 EF1 和 PO 的分配，从而否定地解决了该问题。我们还表明，对于单调次模估值，确定此类分配的存在性是 NP 难的。我们的例子使用（未加权的）覆盖估值，这是单调次模估值的一个严格子类。由于已知通过纳什社会福利最大化，加法估值和拟阵秩估值总是存在 EF1 + PO 分配，此前仅知单调次可加估值不存在 EF1 + PO 分配，我们的工作将不存在性边界扩展到未加权覆盖估值。我们还表明，通过将估值解释为负效用，我们为商品设计的例子也证明了对于具有未加权覆盖成本的杂事，一般不存在 EF1 + PO。
英文摘要
The existence of allocations of indivisible goods that are simultaneously fair (envy-free up to one item (EF1)) and efficient (Pareto optimal (PO)) when agents have monotone submodular valuations has been a longstanding open problem. We settle this question negatively by giving an example with two agents where no allocation is simultaneously EF1 and PO. We also show that determining the existence of such allocations is NP-hard for monotone submodular valuations. Our example uses (unweighted) coverage valuations, which is a strict subclass of monotone submodular valuations. Since EF1+PO allocations are known to always exist for additive valuations via the maximization of Nash Social Welfare (Caragiannis et al. (ACM TEAC 2019)), and for matroid-rank valuations (Benabbou et al. (ACM TEAC 2021)), nonexistence was known only for monotone subadditive valuations (Caragiannis et al. (ACM TEAC 2019)). Our work moves the nonexistence frontier to unweighted coverage valuations. We also show that the example we designed for goods also proves nonexistence of EF1+PO in general, for chores with unweighted coverage costs, by interpreting the valuations as disutilities.
Comments11 pages, which includes a short appendix
