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 structure in a new "Type Coverage" dual problem. It can be seen as akin to the celebrated Magician and OCRS (Online Contention Resolution Scheme) problems, except more general in that it can also provide tight ratios relative to the optimal offline allocation, whereas the earlier problems only establish tight ratios relative to 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, with our principal results being two-fold. 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 starting 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, which dates back to from Hajiaghayi et al. (2007). Turning to the IID setting, our second principal result is to use our framework to characterize the tight guarantee (of adaptive algorithms) under any number $k$ of selection slots and any fixed number of agents $n$.

翻译：先知不等式包含一系列优美的论断，用于建立在线分配算法与离线分配算法之间的紧性能比。传统上，紧性证明通常需要分别构造算法保证界与最坏情况实例，并通过某种“巧妙设计”使两者界值匹配。本文提出了一种新方法：将最坏情况实例的构造形式化为一个优化问题，从而直接得到紧比值，无需分别构造两个界值。我们通过对这一复杂优化问题的分析，识别出一种新型“类型覆盖”对偶问题的结构。该问题可视为著名的魔术师问题与在线竞争解析方案（OCRS）的推广，其更具一般性：它不仅能获得相对于离线问题外生松弛的紧比值（如早期问题），还能获得相对于最优离线分配的紧比值。通过此分析，本文建立了一个统一框架，既能推导新的先知不等式，又能复现已有结果。我们的主要成果体现在两个方面：首先，我们证明Chawla等人（2020）提出的“遗忘型”静态阈值设定方法——令人惊讶地——在所有静态阈值算法中达到最优（适用于任意初始单位数k）。需要强调的是，该结论的推导无需显式构造任何反例实例。这同时意味着Hajiaghayi等人（2007）提出的静态阈值算法渐近收敛速率$1-O(\sqrt{\log k/k})$具有紧性。针对独立同分布（IID）场景，我们的第二个主要成果是利用该框架刻画了在任意选择槽数量k与任意固定智能体数量n条件下（自适应算法的）紧保证界。