不可分割家务分配中加权PROPX与帕累托最优的不相容性
On the Incompatibility of Weighted PROPX and Pareto Optimality for Indivisible Chores
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中文总结 AI 辅助
研究不可分割家务分配中，加权PROPX与帕累托最优的不相容性，给出2主体4家务、n主体n+1家务的反例，且该不相容性在物品数不超主体数时不成立，与EF1的兼容性结果形成对比。
中文摘要 AI 辅助
比例性（PROP）是具有加性偏好的主体间物品分配的最简公平准则之一，但对于不可分割家务，PROP并非总能满足。我们研究任意物品的比例性（PROPX），其要求移除任一主体束中的任意家务后，该主体仍满足比例性。在严格正成本下，我们否定了加权兼容性问题：加权PROPX与帕累托最优（Pareto optimality）在2个主体和4项家务时已不相容。此外，对所有n≥3，我们给出n个主体、n+1项家务的反例，其份额可任意接近均等。这些反例是物品极小的：在严格正成本下，当家务数不超过主体数时，加权PROPX与帕累托最优始终相容；且2个主体最多3项家务时二者相容。我们的不可能性结果与Mahara（2026）关于加权无嫉妒性（EF1）与帕累托最优的兼容性定理形成对比。
英文摘要
Proportionality (PROP) is one of the simplest fairness criteria for allocating items among agents with additive preferences. With indivisible chores, however, PROP is not always satisfiable. We study proportionality up to any item (PROPX), which requires every agent to satisfy proportionality after any chore is removed from her bundle. Under strictly positive costs, we settle the weighted compatibility question negatively: weighted PROPX and Pareto optimality are incompatible already for two agents and four chores. Moreover, for every $n\geq3$, we give an $n$-agent, $(n+1)$-chore counterexample whose shares can be arbitrarily close to equal. These counterexamples are item-minimal: under strictly positive costs, weighted PROPX and Pareto optimality are always compatible when the number of chores is at most the number of agents, and they are compatible for two agents with at most three chores. Our impossibility result contrasts with the compatibility theorem of Mahara (2026) for weighted envy-freeness up to one item (EF1) and Pareto optimality .
