Sharp inequalities for optimal stopping with rewards based on ranks (Q1198584)

Let \(X_ 1,X_ 2,\dots,X_ n\) be independent random variables and consider a problem of optimal stopping where the pay-off only depends on the observed ranks. The authors give a universal lower bound for the maximal expected reward. They show that this bound is sharp by finding distributions where the inequality becomes an equality for the following three cases: (i) maximizing the probability of choosing one of the \(k\) best; (ii) minimizing the expected rank; (iii) an exponential function of the rank.