EFX与PMMS的精确MMS保证
The Exact MMS Guarantees of EFX and PMMS
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- University of Chinese Academy of Sciences（中国科学院大学）
- Chinese Academy of Sciences（中国科学院）
- Huawei Technologies Co., Ltd.（华为技术有限公司）
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中文总结 AI 辅助
该研究确定非负加性估值下EFX与PMMS的精确MMS保证为10/17，证明其紧性，建立了公平分配保证与调度均衡的对应关系。
中文摘要 AI 辅助
任何商品的无嫉妒（EFX）和成对最大最小份额（PMMS）是不可分割商品的标准局部公平准则，而最大最小份额（MMS）是全局基准。我们在非负加性估值下确定了这些局部公平准则与全局MMS保证之间的精确定量关系。我们表明，这两个准则的最优通用因子为ρ^EFX→MMS = ρ^PMMS→MMS = 10/17。我们通过组合收费论证证明了下界。在初始归约后，EFX和PMMS都对焦点代理视角下的每个外来束隐含相同的局部条件：删除其最不值钱的商品后，价值最多为焦点束的价值。一个三段凹权重函数将此局部条件转化为全局10/17保证。然后我们构造了一个显式的完全分配族，该族同时是PMMS和EFX₀，其MMS比率收敛到10/17，表明即使存在零价值商品，这两个常数也是紧的。该论证还给出α-EFX ⇒ (10α/17)-MMS。最后，我们建立了这些公平分配保证与调度均衡之间的精确对应关系：对于每个固定的代理数量n，EFX到MMS的极值比率等于自私相同机器覆盖的纯价格无政府状态的倒数；类似地，PMMS到MMS的比率等于基于精确成对机器重新划分的局部差距的倒数。这些对应关系解释了常数10/17为何同时支配这两个问题。
英文摘要
Envy-freeness up to any good (EFX) and pairwise maximin share (PMMS) are standard local fairness criteria for indivisible goods, whereas maximin share (MMS) is a global benchmark. We determine the exact quantitative relationship between these local fairness notions and the global MMS guarantee under nonnegative additive valuations. We show that the optimal universal factor for both notions is $ρ^{\mathrm{EFX}\to\mathrm{MMS}}=ρ^{\mathrm{PMMS}\to\mathrm{MMS}}=\frac{10}{17}$. We prove the lower bound by a combinatorial charging argument. After an initial reduction, both EFX and PMMS imply the same local condition on every foreign bundle from the perspective of a focal agent: deleting its least valuable good leaves value at most the focal bundle. A three-piece concave weight function translates this local condition into the global $10/17$ guarantee. We then construct an explicit family of complete allocations that are simultaneously PMMS and EFX$_0$, whose MMS ratios converge to $10/17$, showing that both constants are tight even in the presence of zero-valued goods. The argument also gives $α$-EFX $\Rightarrow (10α/17)$-MMS. Finally, we establish an exact correspondence between these fair-division guarantees and scheduling equilibria. For every fixed number of agents $n$, the EFX-to-MMS extremal ratio equals the reciprocal of the pure price of anarchy for selfish identical-machine covering. Similarly, the PMMS-to-MMS ratio equals the reciprocal of a locality gap based on exact pairwise machine repartition. These correspondences explain why the constant $10/17$ governs both problems.
