关于次可加估值下的两全其美分配

中文摘要 AI 辅助

我们考虑将不可分割物品分配给具有平等权利和次可加估值的代理人的问题。作为事后公平概念，我们考虑最大最小份额（MMS）；作为事前公平概念，我们考虑最大期望份额（MES），它总是至少与MMS一样大，有时甚至大得多。我们提出一个简单的变换，对于每个$0 < \rho \le 1$，给定任何产生$\rho$-MMS分配的算法，将其转换为一个随机分配算法，该算法同时提供事后$\rho$-MMS和事前$\eta$-MES。我们证明了MES的几个新性质，并用它们表明$\eta \ge \min[\frac{\rho}{2 + \rho}, \frac{1}{4}]$。我们还展示了变换导致$\eta$值更高的情形。将我们的变换应用于当前已知的分配算法，对于次可加估值，证明了存在同时满足事前$\Omega(\frac{1}{\log\log n})$-MES和事后$\Omega(\frac{1}{\log\log n})$-MMS的随机分配；对于XOS估值，证明了存在同时满足事前$\frac{4}{27}$-MES和事后$\frac{4}{17}$-MMS的随机分配。

英文摘要

We consider allocation of indivisible goods to agents with equal entitlements and subadditive valuations. As an ex-post fairness notion we consider the maximin share (MMS), and as an ex-ante fairness notion we consider the maximum expectation share (MES), which is always at least as large as the MMS, and sometimes much larger. We present a simple transformation that for every $0 < ρ\le 1$, given any algorithm that produces $ρ$-MMS allocations, transforms it into a randomized allocation algorithm that offers $ρ$-MMS ex-post simultaneously with $η$-MES ex-ante. We prove several new properties of MES, and use them to show that $η\ge \min[\fracρ{2 + ρ}, \frac{1}{4}]$. We also present cases in which the transformation results in a higher value of $η$. Applying our transformation to currently known allocation algorithms shows for subadditive valuations the existence of randomized allocations that are simultaneously $Ω(\frac{1}{\log\log n})$-MES ex-ante and $Ω(\frac{1}{\log\log n})$-MMS ex-post, and for XOS valuations the existence of randomized allocations that are simultaneously $\frac{4}{27}$-MES ex-ante and $\frac{4}{17}$-MMS ex-post.

发表机构

• Weizmann Institute（魏茨曼科学研究所）

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