Optimal single threshold stopping rules and sharp prophet inequalities

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables. The objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a "prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of "prophet inequalities." In this paper we characterize sharp prophet inequalities for single threshold stopping rules as solutions to infinite two person zero sum games on the unit square with special payoff kernels. The proposed game theoretic characterization allows one to derive sharp non-asymptotic prophet inequalities for different classes of distributions. This, in turn, gives rise to a simple and computationally tractable algorithmic paradigm for deriving optimal single threshold stopping rules. Our results also indicate that several classical observations in the literature are either incorrect or incomplete in treating this problem.