AI 中文总结
研究离散公平分配中EF1与PO对于次模估值是否兼容的问题，通过构建反例表明二者不兼容，确定不可能边界，找到恢复兼容性的条件，量化坚持EF1时的效率损失。
AI 中文摘要
离散公平分配中的核心问题之一是公平与效率能否同时实现。对于不可分割商品，无嫉妒到一个商品（EF1）是无嫉妒的一种典型松弛，而标准效率基准是帕累托最优（PO）。Caragiannis等人表明，对于加法估值，EF1和PO总是兼容的，并询问这种兼容性是否扩展到次模估值。本文给出否定答案，构建了一个有两个主体和八个商品的实例，其中两个主体都有次模估值，使得没有EF1分配是弱帕累托最优的。接着确定了不可能的边界，表明即使对于加权拟阵秩估值，EF1和分数帕累托最优（fPO）也不兼容。另一方面，识别出一个恢复兼容性的共同包络条件，在此条件下，任意数量主体都存在EF1+PO分配。最后量化了坚持EF1时不可避免的效率损失。
英文摘要
One of the central questions in discrete fair division is whether fairness and efficiency can be achieved simultaneously. For indivisible goods, a canonical relaxation of envy-freeness is envy-freeness up to one good (EF1), while the standard efficiency benchmark is Pareto optimality (PO). In their seminal work, Caragiannis et al. showed that, for additive valuations, EF1 and PO are always compatible, and asked whether this compatibility extends to submodular valuations. This question has since become an important open problem in the study of fair division. In this paper, we settle the question in the negative. We construct an instance with two agents and eight goods, where both agents have submodular valuations, such that no EF1 allocation is even weakly Pareto optimal. Thus, the celebrated compatibility between EF1 and PO for additive valuations breaks down already for two agents under submodular valuations. We then map the boundary of this impossibility. On the negative side, we show that even for weighted matroid rank valuations, EF1 and fractional Pareto optimality (fPO) are incompatible. This rules out, in general, broad classes of weighted-welfare and Fisher-market-based approaches. On the positive side, we identify a common-envelope condition that restores compatibility. Under this condition, EF1+PO allocations exist for any number of agents. This yields new positive results showing that common-weight matroid-rank valuations always admit EF1+PO allocations. Finally, we quantify the efficiency loss that is unavoidable when insisting on EF1. Our submodular counterexample implies that there is a constant $α<1$ such that no EF1 allocation is $α$-\PO. For the broader class of subadditive valuations, we prove a tight two-agent bound: for any $\varepsilon>0$, there exists an instance in which no EF1 allocation is $\left(\frac{1}{\sqrt{2}}+\varepsilon\right)$-PO.
