Prophet Inequalities for I.I.D. Random Variables from an Unknown Distribution

Abstract

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables X 1 , . . . , X n drawn independently from a distribution F, the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we haveWhat 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 quite some time the best known bound for the problem was α ≥ 1 − 1/e ≈ 0.632 [Hill and Kertz, 1982]. Only recently this bound was improved by Abolhassani et al. [2017], and a tight bound of α ≈ 0.745 was obtained by Correa et al. [2017].The case where F is unknown, such that the decision whether τ = t may depend only on the values of the first t random variables but not on F, is equally well motivated (e.g., [Azar et al., 2014]) but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem. Our main result is that this bound is tight. Motivated by this impossibility result we investigate the case where the stopping time may additionally depend on a limited number of samples from F. An extension of our main result shows that even with o(n) samples α ≤ 1/e, so that the interesting case is the one with Ω(n) samples. Here we show that n samples allow for a significant improvement over the secretary problem, while 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.

The Two-Sided Game of Googol and Sample-Based Prophet Inequalities

“…Interestingly, our bound surpasses that of Azar et al [3] which until very recently was the best known bound for (the full information) prophet secretary. Furthermore, our bound also beats the bound of 1 − 1/e obtained by Correa et al [5] for the i.i.d. single-sample prophet inequality.…”

Competitive Analysis with a Sample and the Secretary Problem

“…Our algorithm for the secretary problem in the ROS model improves upon recent results by Correa et al [5] for the prophet inequality in the i.i.d. model in which the online player gets access to a limited number of training samples from the (unknown) distribution.…”

“…We next prove an upper bound on the competitive-ratio for the problem. A similar asymptotic result was established in [5], we provide here a proof that applies for any n ∈ N. Proof. Fix n, h ∈ N. We consider a classical settings of the secretary problem in which the input to the online player at each online round is only the rank of the arriving candidate among the subsequence of already observed candidates, and the goal is to maximize the probability of accepting the best candidate overall.…”

The Secretary Problem with Independent Sampling

“…Indeed, for the prophet inequality with i.i.d. values from an unknown distribution (a model that arguable gives more information than ours) Correa et al [9] proved that with O(n 2 ) samples one can achieve the best possible performance guarantee of the case with known distribution, and only very recently Rubinstein et al [23] improved this to O(n) samples. This is in line with our result here since for p close to, but strictly less than 1, the size of the sample set is linear in the size of V .…”