Prophet inequalities consist of many beautiful statements that establish tight performance ratios between online and offline allocation algorithms. Typically, tightness is established by constructing an algorithmic guarantee and a worst-case instance separately, whose bounds match as a result of some "ingenuity". In this paper, we instead formulate the construction of the worst-case instance as an optimization problem, which directly finds the tight ratio without needing to construct two bounds separately. Our analysis of this complex optimization problem involves identifying the structure in a new "Type Coverage" dual problem. It can be seen as akin to the celebrated Magician and OCRS problems, except more general in that it can also provide tight ratios relative to the optimal offline allocation, whereas the earlier problems only concerns the ex-ante relaxation of the offline problem. Through this analysis, our paper provides a unified framework that derives new prophet inequalities and recovers existing ones, including two important new results. First, we show that the "oblivious" method of setting a static threshold due to Chawla et al. (2020), surprisingly, is best-possible among all static threshold algorithms, under any number $k$ of units. We emphasize that this result is derived without needing to explicitly find any counterexample instances. This implies the tightness of the asymptotic convergence rate of $1-O(\sqrt{\log k/k})$ for static threshold algorithms from Hajiaghayi et al. (2007), is tight; this confirms for the first time a separation with the convergence rate of adaptive algorithms, which is $1-\Theta(\sqrt{1/k})$ due to Alaei (2014). Second, turning to the IID setting, our framework allows us to numerically illustrate the tight guarantee (of adaptive algorithms) under any number $k$ of starting units. Our guarantees for $k>1$ exceed the state-of-the-art.

翻译：先知不等式包含一系列优美的结论，这些结论建立了在线与离线分配算法之间的紧致性能比。通常，紧致性是通过分别构造算法保证和最坏情况实例来建立的，而这两者的匹配依赖于某种“独创性”。在本文中，我们转而将最坏情况实例的构造表述为一个优化问题，无需分别构造两个边界即可直接找到紧致比。我们对该复杂优化问题的分析涉及在新的“类型覆盖”对偶问题中识别结构。该问题可被视为类似于著名的魔术师问题和OCRS问题，但更普遍的是，它还能提供相对于最优离线分配的紧致比，而此前的问题仅涉及离线问题的事前松弛。通过这一分析，本文提供了一个统一框架，能够推导出新的先知不等式并恢复已有的结论，其中包括两项重要的新成果。首先，我们证明，Chawla等人（2020）提出的“无偏”静态阈值方法，在任意单位数量$k$下，出人意料地是所有静态阈值算法中最佳可能的。我们强调，这一结果无需显式构造任何反例实例即可推导得出。这意味着Hajiaghayi等人（2007）提出的静态阈值算法渐近收敛速率$1-O(\sqrt{\log k/k})$的紧致性成立；这首次证实了其与自适应算法收敛速率（Alaei（2014）提出的$1-\Theta(\sqrt{1/k})$）的分离。其次，转向IID设置，我们的框架能够数值化地展示在任意初始单位数量$k$下（自适应算法的）紧致保证。对于$k>1$，我们的保证超越了现有最优结果。