Threshold Rules for the Classical Prophet Inequality

TL;DR

This paper analyzes threshold rules in the classical prophet inequality, providing a decomposition method to certify various deterministic and randomized thresholds.

Contribution

It introduces a common threshold/surplus decomposition applicable to multiple threshold strategies in prophet inequalities.

Findings

Certifies median, half-mean, and balanced-surplus thresholds.

Provides an averaged certificate for randomized thresholds.

Unifies analysis of different threshold rules.

Abstract

This note records a common threshold/surplus decomposition for single-threshold stopping rules in the classical prophet inequality. The same decomposition is used to certify several deterministic thresholds, including the median, half-mean, and balanced-surplus thresholds, and to give an averaged certificate for randomized thresholds distributed as the maximum.