  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.
