We explore a novel variant of the classical prophet inequality problem, where the values of a sequence of items are drawn i.i.d. from some distribution, and an online decision maker must select one item irrevocably. We establish that the competitive ratio between the expected optimal performance of the online decision maker compared to that of a prophet, who uses the average of the top $\ell$ items, must be greater than $\ell/c_{\ell}$, with $c_{\ell}$ the solution to an integral equation. We prove that this lower bound is larger than $1-1/(\exp(\ell)-1)$. This implies that the bound converges exponentially fast to $1$ as $\ell$ grows. In particular, the bound for $\ell=2$ is $2/c_{2} \approx 0.966$ which is much closer to $1$ than the classical bound of $0.745$ for $\ell=1$. Additionally, the proposed algorithm can be extended to a more general scenario, where the decision maker is permitted to select $k$ items. This subsumes the $k$ multi-unit i.i.d. prophet problem and provides the current best asymptotic guarantees, as well as enables broader understanding in the more general framework. Finally, we prove a nearly tight competitive ratio when only static threshold policies are allowed.

翻译：我们探索了经典先知不等式问题的一个新颖变体，其中一系列项目的价值独立同分布于某个分布，在线决策者必须不可撤销地选择一个项目。我们建立了在线决策者的期望最优性能与先知（使用前 $\ell$ 个项目的平均值）相比的竞争比必须大于 $\ell/c_{\ell}$，其中 $c_{\ell}$ 是一个积分方程的解。我们证明该下界大于 $1-1/(\exp(\ell)-1)$。这意味着随着 $\ell$ 的增长，该界以指数速度收敛到 $1$。特别地，对于 $\ell=2$，该界为 $2/c_{2} \approx 0.966$，这比 $\ell=1$ 时的经典界 $0.745$ 更接近 $1$。此外，所提出的算法可以扩展到更一般的场景，其中允许决策者选择 $k$ 个项目。这涵盖了 $k$ 个多单元独立同分布先知问题，并提供了当前最佳的渐近保证，同时使得在更一般的框架下获得更广泛的理解成为可能。最后，我们证明了在仅允许静态阈值策略时，竞争比是近乎紧的。