Prophet Inequalities for Independent and Identically Distributed Random Variables from an Unknown Distribution

A central object of study in optimal stopping theory is the single-choice prophet inequality for independent and identically distributed random variables: given a sequence of random variables X 1 , … , X n drawn independently from the same distribution, the goal is to choose a stopping time τ such that for the maximum value of α and for all distributions, E [ X τ ] ≥ α · E [ max t X t ] . What makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X 1 , … , X t and on the distribution F . For a long time, the best known bound for the problem had been α ≥ 1 − 1 / e ≈ 0.632 , but recently a tight bound of α ≈ 0.745 was obtained. The case where F is unknown, such that the decision whether τ = t may depend only on the values of the random variables X 1 , … , X t , is equally well motivated but has received much less attention. A straightforward guarantee for this case of α ≥ 1 / e ≈ 0.368 can be derived from the well-known optimal solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F , and we show that, even with o ( n ) samples, α ≤ 1 / e . On the other hand, n samples allow for a significant improvement, whereas O ( n 2 ) samples are equivalent to knowledge of the distribution: specifically, with n samples, α ≥ 1 − 1 / e ≈ 0.632 and α ≤ ln ( 2 ) ≈ 0.693 , and with O ( n 2 ) samples, α ≥ 0.745 − ε for any ε > 0 .