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Hill and Kertz studied the prophet inequality on iid distributions [TheAnnals of Probability 1982]. They proved a theoretical bound of $1-\frac{1}{e}$on the approximation factor of their algorithm. They conjectured that the bestapproximation factor for arbitrarily large n is $\frac{1}{1+1/e} \approx0.731$. This conjecture remained open prior to this paper for over 30 years. Inthis paper we present a threshold-based algorithm for the prophet inequalitywith n iid distributions. Using a nontrivial and novel approach we show thatour 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, wegeneralize our results to non-iid distributions and discuss its applications inmechanism design.

Beating 1-1/e for Ordered Prophets