Tightness without Counterexamples: A New Approach and New Results for Prophet Inequalities
Jiashuo Jiang, Will Ma, Jiawei Zhang
TL;DR
This work develops a unified optimization-based framework for tight prophet inequalities in multi-unit settings by formulating the worst-case instance as a two-stage LP problem and exposing a Type Coverage dual that mirrors Magician/OCRS ideas. The approach yields two main contributions: (i) the oblivious static-threshold policy is best-possible among static-threshold rules for any number of slots $k$ in the non-IID setting, with the known asymptotic rate $1-O( oot{ ext{log}k}{k})$; and (ii) in the IID setting, the framework characterizes the tight adaptive guarantee $ u_{k,n}$ for all $k$ and $n$ via a semi-infinite LP, with a numerical procedure and a closed-form derivation recovering the classical $0.745$ bound for $k=1$. By reducing adversarial design to Type Coverage constraints, the paper unifies existing prophet-inequality results and recovers classical benchmarks relative to both the prophet and the ex-ante relaxation, while enabling efficient computation of tight guarantees even for non-identical distributions. The IID analysis further shows equivalences among DP/ExAnte and OST-based guarantees, and provides a practical route to compute $ u_{k,n}$ numerically. Overall, the framework offers a scalable, theory-grounded method to derive and verify tight prophet inequalities across settings, with potential extensions to continuous distributions and broader matroid constraints.
Abstract
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 $n$.