New Prophet Inequalities via Poissonization and Sharding

TL;DR

This paper presents a unified framework using sharding and Poissonization to analyze and improve prophet inequalities, simplifying proofs and refining constants for several classical inequalities.

Contribution

Introduces sharding and Poissonization as a novel, unified approach to analyze and improve prophet inequalities, simplifying existing proofs and constants.

Findings

Improved competitive ratio bounds for multiple prophet inequalities

Simplified and more intuitive analysis framework

Refined constants in classical prophet inequality results

Abstract

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.