Prophet inequalities for i.i.d. random variables from an Unknown distribution

Correa, Jose, Dütting, Paul, Fischer, Felix and Schewior, Kevin (2019) Prophet inequalities for i.i.d. random variables from an Unknown distribution. In: 20th ACM conference on Economics and Computation, 2019-06-24 - 2019-06-28, Phoenix, United States.

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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 X1, . . . , Xn 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 have E[Xτ ] ≥ α · E[maxt Xt ]. What makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X1, . . . , Xt 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 quite 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 X1, . . . , Xt , 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 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 show that even with o(n) samples α ≤ 1/e. On the other hand, n samples allow for a significant improvement, while O(n2) 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(n2) samples α ≥ 0.745 − ε for any ε > 0.

Item Type:	Conference or Workshop Item (Paper)
Official URL:	https://sigecom.org/ec19/
Additional Information:	© 2019 The Authors
Divisions:	Mathematics
Subjects:	Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Date Deposited:	29 May 2019 16:00
Last Modified:	16 Apr 2024 07:06
URI:	http://eprints.lse.ac.uk/id/eprint/100760

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