The Prophet Inequality Can Be Solved Optimally with a Single Set of Samples

The setting of the classic prophet inequality is as follows: a gambler is shown the probability distributions of $n$ independent, non-negative random variables with finite expectations. In their indexed order, a value is drawn from each distribution, and after every draw the gambler may choose to accept the value and end the game, or discard the value permanently and continue the game. What is the best performance that the gambler can achieve in comparison to a prophet who can always choose the highest value? Krengel, Sucheston, and Garling solved this problem in 1978, showing that there exists a strategy for which the gambler can achieve half as much reward as the prophet in expectation. Furthermore, this result is tight. In this work, we consider a setting in which the gambler is allowed much less information. Suppose that the gambler can only take one sample from each of the distributions before playing the game, instead of knowing the full distributions. We provide a simple and intuitive algorithm that recovers the original approximation of $\frac{1}{2}$. Our algorithm works against even an almighty adversary who always chooses a worst-case ordering, rather than the standard offline adversary. The result also has implications for mechanism design -- there is much interest in designing competitive auctions with a finite number of samples from value distributions rather than full distributional knowledge.