In this paper, we study $k$-unit single sample prophet inequalities. A seller has $k$ identical, indivisible items to sell. A sequence of buyers arrive one-by-one, with each buyer's private value for the item, $X_i$, revealed to the seller when they arrive. While the seller is unaware of the distribution from which $X_i$ is drawn, they have access to a single sample, $Y_i$ drawn from the same distribution as $X_i$. What strategies can the seller adopt for selling items so as to maximize social welfare? Previous work has demonstrated that when $k = 1$, if the seller sets a price equal to the maximum of the samples, they can achieve a competitive ratio of $\frac{1}{2}$ of the social welfare, and recently Pashkovich and Sayutina established an analogous result for $k = 2$. In this paper, we prove that for $k \geq 3$, setting a (static) price equal to the $k^{\text{th}}$ largest sample also obtains a competitive ratio of $\frac{1}{2}$, resolving a conjecture Pashkovich and Sayutina pose. We also consider the situation where $k$ is large. We demonstrate that setting a price equal to the $(k-\sqrt{2k\log k})^{\text{th}}$ largest sample obtains a competitive ratio of $1 - \sqrt{\frac{2\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$, and that this is the optimal possible ratio achievable with a static pricing scheme with access to a single sample. This should be compared against a competitive ratio $1 - \sqrt{\frac{\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$, which is the optimal possible ratio achievable with a static pricing scheme with knowledge of the distributions of the values.

翻译：本文研究$k$单元单样本先知不等式问题。卖家持有$k$件相同的不可分割商品。买家序列依次到达，每位买家对商品的私有估值$X_i$在到达时向卖家披露。虽然卖家不知道$X_i$的分布，但可获得与$X_i$同分布的一个独立样本$Y_i$。卖家应采用何种售卖策略以最大化社会福利？先前研究表明，当$k=1$时，若卖家将价格设为所有样本的最大值，可获得社会福利的$\frac{1}{2}$竞争比；近期Pashkovich与Sayutina证明了$k=2$时的类似结论。本文证明，对于$k\geq 3$，将（静态）价格设为第$k$大的样本值同样可达到$\frac{1}{2}$竞争比，从而解决了Pashkovich与Sayutina提出的猜想。我们还考虑$k$较大时的情形，证明将价格设为第$(k-\sqrt{2k\log k})$大的样本值时，可获得$1 - \sqrt{\frac{2\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$的竞争比，且这是单样本静态定价方案所能达到的最优竞争比。该结果需与已知分布情形下静态定价方案所能达到的最优竞争比$1 - \sqrt{\frac{\log k}{k}} - o\left(\sqrt{\frac{\log k}{k}}\right)$进行比较。