Simple ratio prophet inequalities for a mortal with multiple choices

Abstract

Let Xi ≥ 0 be independent, i = 1,…, n, with known distributions and let Xn*= max(X1,…,Xn). The classical ‘ratio prophet inequality’ compares the return to a prophet, which is EXn*, to that of a mortal, who observes the Xis sequentially, and must resort to a stopping rule t. The mortal's return is V(X1,…,Xn) = max EXt, where the maximum is over all stopping rules. The classical inequality states that EXn* < 2V(X1,…,Xn). In the present paper the mortal is given k ≥ 1 chances to choose. If he uses stopping rules t1,…,tk his return is E(max(Xt1,…,Xtk)). Let t(b) be the ‘simple threshold stopping rule’ defined to be the smallest i for which Xi ≥ b, or n if there is no such i. We show that there always exists a proper choice of k thresholds, such that EXn* ≤ ((k+1)/k)Emax(Xt1,…,Xtk)), where ti is of the form t(bi) with some added randomization. Actually the thresholds can be taken to be thej/(k+1) percentile points of the distribution of Xn*, j = 1,…,k, and hence only knowledge of the distribution of Xn* is needed.