Simple ratio prophet inequalities for a mortal with multiple choices

Abstract

Let X i ≥ 0 be independent, i = 1,…, n, with known distributions and let X n *= max(X 1,…,X n ). The classical ‘ratio prophet inequality’ compares the return to a prophet, which is EX n *, to that of a mortal, who observes the X i s sequentially, and must resort to a stopping rule t. The mortal's return is V(X 1,…,X n ) = max EX t , where the maximum is over all stopping rules. The classical inequality states that EX n * < 2V(X 1,…,X n ). In the present paper the mortal is given k ≥ 1 chances to choose. If he uses stopping rules t 1,…,t k his return is E(max(X t 1,…,X t k )). Let t(b) be the ‘simple threshold stopping rule’ defined to be the smallest i for which X i ≥ b, or n if there is no such i. We show that there always exists a proper choice of k thresholds, such that EX n * ≤ ((k+1)/k)Emax(X t 1,…,X t k )), where t i is of the form t(b i ) with some added randomization. Actually the thresholds can be taken to be thej/(k+1) percentile points of the distribution of X n *, j = 1,…,k, and hence only knowledge of the distribution of X n * is needed.