发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了不可分割混合甘露的加权公平分配问题，证明了WEF1分配的存在性、WMMS分配的存在性与可计算性，分析了WEF1的功利主义价格及相关公平性保证。
AI 中文摘要
我们研究加法估值下不可分割混合甘露的加权公平分配问题。首先，我们解决了加权至多一件物品的无嫉妒性（WEF1）的一般性存在性开放问题，证明了任意正权益的所有实例都存在可在多项式时间内计算的完整WEF1分配。随后我们表明，存在性不蕴含任何社会福利保证，即WEF1的功利主义价格是无穷大的，即使对于两个无权重的代理人，他们具有归一化估值、共同物品符号，且在固定四值集中具有单例值；在该构造中，社会福利最大化的WEF1分配是分数帕累托最优的。其次，假设每个代理人i有一个数a_i>0，其对任何物品的估值为-a_i、0或a_i。那么，对于任意权益，加权最大最小份额（WMMS）分配总是存在，可在多项式时间内计算，且可被选择为分数帕累托最优。每个WMMS值的精确公式导出了多项式时间流算法。在该类别中，每个WEF1分配满足最佳可能的加性WMMS保证，其损失取决于代理人权益相对于最大权益的比例。因此，最大权益代理人获得精确WMMS，而在权益相等时，每个WEF1分配也是MMS公平的。允许第二个正量级会违反精确WMMS，而不受限制的权益比例排除了任何与WEF1兼容的固定乘性WMMS保证（适用于家务事）。
英文摘要
We study weighted fair division of indivisible mixed manna under additive valuations. First, we resolve the general existence open question for weighted envy-freeness up to one item (WEF1), and show that every instance with arbitrary positive entitlements admits a complete WEF1 allocation computable in polynomial time. We then show that existence does not imply any welfare guarantee, i.e., the utilitarian price of WEF1 is infinite, even for two unweighted agents with normalized valuations, common item signs, and singleton values in a fixed four-value set; a welfare-maximizing WEF1 allocation in the construction is fractionally Pareto optimal. Second, suppose each agent $i$ has a number $a_i>0$ such that their valuation for any item is $-a_i$, $0$, or $a_i$. Then, for arbitrary entitlements, a weighted maximin share (WMMS) allocation always exists, is computable in polynomial time, and can be chosen to be fractionally Pareto optimal. An exact formula for each WMMS value leads to a polynomial-time flow algorithm. In this class, every WEF1 allocation satisfies a best possible additive WMMS guarantee whose loss depends on the agent's entitlement relative to the largest entitlement. Thus maximum entitlement agents receive exact WMMS and, under equal entitlements, every WEF1 allocation is also MMS-fair. Allowing a second positive magnitude can violate exact WMMS, while unrestricted entitlement ratios rule out any fixed multiplicative WMMS guarantee compatible with WEF1 for chores.
