Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities

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 $n$.