# Beating 1-1/e for Ordered Prophets

## TL;DR

This paper introduces a new threshold-based algorithm for the prophet inequality with iid distributions, achieving a 0.738 approximation factor that surpasses the long-standing conjectured bound of approximately 0.731, and extends results to non-iid cases.

## Contribution

It presents a novel algorithm that beats the 1-1/e bound for iid distributions and refutes the Hill-Kertz conjecture, also generalizing to non-iid distributions and applications.

## Key findings

- Achieves a 0.738-approximation for iid prophet inequality
- Refutes the Hill-Kertz conjecture of 1/(1+1/e)
- Extends results to non-iid distributions and discusses mechanism design applications

## Abstract

Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of $1-\frac{1}{e}$ on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is $\frac{1}{1+1/e} \approx 0.731$. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of $\frac{1}{1+1/e}$, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-iid distributions and discuss its applications in mechanism design.