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. Our result implies that the asymptotic convergence rate of $1-O(\sqrt{\log k/k})$ for static threshold algorithms, first established in 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：先知不等式包含许多优美的结论，建立了在线分配算法与离线分配算法之间的紧致性能比率。通常，紧致性是通过分别构造算法保证和最坏情况实例来确立的，其边界因某种“巧妙构思”而相互匹配。在本文中，我们将最坏情况实例的构造直接表述为一个优化问题，无需分别构建两个边界即可直接求解紧致比率。我们对这一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。该问题可类比于著名的魔术师问题和在线选择与竞争问题，但更一般，因为它还能提供相对于最优离线分配的紧致比率，而先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，推导出新的先知不等式并复现了现有结果，包括两项重要的新成果。首先，我们证明，Chawla等人（2020）提出的“无意识”静态阈值方法在任意数量$k$个单位下，出人意料地在所有静态阈值算法中是最优的。需要强调的是，这一结果无需显式构造任何反例实例即可推导得出。我们的结果意味着静态阈值算法由Hajiaghayi等人（2007）首先确立的$1-O(\sqrt{\log k/k})$渐近收敛率是紧致的；这首次证实了其与自适应算法收敛率（Alaei（2014）得到的$1-\Theta(\sqrt{1/k})$）之间的分离。其次，转向独立同分布设定下，我们的框架能够数值化地展示任意初始单位数量$k$下自适应算法的紧致保证。对于$k>1$的情况，我们的保证超越了现有最优水平。