Beating $1-\\frac{1}{e}$ for Ordered Prophets

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.