New Prophet Inequalities via Poissonization and Sharding

This work introduces \emph{sharding} and \emph{Poissonization} as a unified framework for analyzing prophet inequalities. Sharding involves splitting a random variable into several independent random variables, shards, that collectively mimic the original variable's behavior. We combine this with Poissonization, where these shards are modeled using a Poisson distribution. Despite the simplicity of our framework, we improve the competitive ratio analysis of a dozen well studied prophet inequalities in the literature, some of which have been studied for decades. This includes the \textsc{Top-$1$-of-$k$} prophet inequality, prophet secretary inequality, and semi-online prophet inequality, among others. This approach not only refines the constants but also offers a more intuitive and streamlined analysis for many prophet inequalities in the literature. Furthermore, it simplifies proofs of several known results and may be of independent interest for other variants of the prophet inequality, such as order-selection.