Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 \u00e2\u0088\u0092 1\/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1\/1+1\/e\u00e2\u0089\u0083 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 1\/1+1\/e, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-uniform distributions and discuss its applications in mechanism design.<\/p>\n","protected":false},"excerpt":{"rendered":"
Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of 1 \u00e2\u0088\u0092 1\/e on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is 1\/1+1\/e\u00e2\u0089\u0083 0.731. This conjecture remained open prior to this paper for over 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