公平先知

中文摘要 AI 辅助

我们开创了对$\alpha$-公平先知不等式的研究。这在内插了功利主义福利$(\alpha=0)$、纳什福利$(\alpha=1)$和罗尔斯最大最小公平$(\alpha\to\infty)$之间。鉴于目标的非线性，期望的应用时机很重要。例如，对于罗尔斯目标，目标是最大化$\min \mathbb{E}[u_i]$还是$\mathbb{E}[\min u_i]$很重要。我们将前者称为事前模型，后者称为事后模型。对于事前公平，完全分布知识为每个$\alpha\ge 0$产生恰好$1/2$的紧竞争比。在样本访问下，当$\alpha\in(0,1]$时，每个分布的$O(n\log n)$个样本足以获得常数竞争比。相反，对于每个$\alpha>1$，任何有限数量的样本都无法改善平凡的$1/n$保证。因此，与功利主义设置不同，全信息和样本访问的先知不等式变得根本分离。对于事后公平，在全信息下，我们为所有$\alpha\in(0,1)$获得统一的常数比，而对于每个$\alpha>1$，竞争比降至$1/n$。在样本访问模型中，对于每个固定的$\alpha<1$，每个分布的一个样本就足够了，但仅依赖于$n$的样本预算无法在$\alpha\to 1$时产生统一的常数保证。除了这些$\alpha$-公平性的相变之外，我们的结果为非线性福利目标的先知不等式的更广泛理论打开了大门。

英文摘要

We initiate the study of $α$-fair prophet inequalities. This interpolates between utilitarian welfare $(α=0)$, Nash welfare $(α=1)$, and Rawlsian max-min fairness $(α\to\infty)$. Given the non-linearity of the objective, it matters when the expectation is applied. For instance, for the Rawlsian objective, it matters whether we aim to maximize $\min \mathbb{E}[u_i]$ or $\mathbb{E}[\min u_i]$. We refer to the former as the ex-ante model, and the latter as the ex-post model. For ex-ante fairness, full distributional knowledge yields a tight competitive ratio of exactly $1/2$ for every $α\ge 0$. Under sample access, $O(n\log n)$ samples per distribution suffice for a constant competitive ratio when $α\in(0,1]$. In contrast, for every $α>1$, no finite number of samples improves upon the trivial $1/n$ guarantee. Thus, unlike in the utilitarian setting, full-information and sample-access prophet inequalities become fundamentally separated. For ex-post fairness, under full information, we obtain a uniform constant ratio for all $α\in(0,1)$, while for every $α>1$ the competitive ratio collapses to $1/n$. In the sample-access model, one sample per distribution suffices for each fixed $α<1$, but no sample budget depending only on $n$ yields a uniform constant guarantee as $α\to 1$. Beyond these phase transitions for $α$-fairness, our results open the door to a broader theory of prophet inequalities for non-linear welfare objectives.

发表机构

• Google Research, Zürich, Switzerland（谷歌研究院）
• Tel Aviv University, Tel Aviv, Israel（特拉维夫大学）

机构由 AI 辅助整理，请以论文原文为准。