Sharp Inequalities for Optimal Stopping with Rewards Based on Ranks

A universal bound for the maximal expected reward is obtained for stopping a sequence of independent random variables where the reward is a nonincreasing function of the rank of the variable selected. This bound is shown to be sharp in three classical cases: (i) when maximizing the probability of choosing one of the $k$ best; (ii) when minimizing the expected rank; and (iii) for an exponential function of the rank.