In the classical prophet inequality settings, a gambler is given a sequence of $n$ random variables $X_1, \dots, X_n$, taken from known distributions, observes their values in this (potentially adversarial) order, and select one of them, immediately after it is being observed, so that its value is as high as possible. The classical \emph{prophet inequality} shows a strategy that guarantees a value at least half of that an omniscience prophet that picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle $\mathcal{O}$. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with $m$ oracle calls is equivalent to the \textsc{Top-$1$-of-$(m+1)$} model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the \textsc{Top-$1$-of-$(m+1)$} model. We resolve the oracle model for any $m$, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the \textsc{Top-$1$-of-$m$} model.

翻译：在经典预言不等式设定中，赌徒面对$n$个来自已知分布的随机变量$X_1, \dots, X_n$，需按（可能对抗性的）顺序观察其取值，并在观察到后立即选择其中一个，以使所选值尽可能大。经典预言不等式表明存在一种策略能保证至少达到全知先知（能选取最大值）的半数收益，且该比率是最优的。本文对预言不等式进行推广，允许赌徒获取未来部分信息（这些信息原本仅为先知所掌握）。具体而言，在过程中的任意时刻，赌徒可向预言机$\mathcal{O}$发起一次查询。该预言机以单比特回应：若当前实现值大于剩余实现值则返回是，否则返回否。我们证明，当优化目标为最大化选中最大值的概率时，具有$m$次预言机调用的预言模型等价于\textsc{Top-$1$-of-$(m+1)$}模型。当优化目标为最大化竞争比时，该等价性不再成立，但我们仍证明预言模型的任意算法可推导出\textsc{Top-$1$-of-$(m+1)$}模型的等价竞争比。我们完整解决了任意$m$下的预言模型，给出了与全能对手相比的最优竞争比的紧致上下界。由此，我们为\textsc{Top-$1$-of-$m$}模型提供了新结果并对已知结果进行了改进。