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 results and recovers many existing ones. First, we show that the "oblivious" method of setting a static threshold due to Chawla et al. (2020) is, surprisingly, best-possible among all static threshold algorithms, for any number $k$ of starting units. We emphasize that this result is derived without needing to explicitly find any counterexample instances. We establish similar "no separation" results for static thresholds in the IID setting, which although previously known, required the construction of complicated counterexamples. Finally, our framework and in particular our Type Coverage problem yields a simplified derivation of the tight 0.745 ratio when $k=1$ in the IID setting.