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August 2002 Ratio prophet inequalities when the mortal has several choices
David Assaf, Larry Goldstein, Ester Samuel-Cahn
Ann. Appl. Probab. 12(3): 972-984 (August 2002). DOI: 10.1214/aoap/1031863177

Abstract

Let Xi be nonnegative, independent random variables with finite expectation, and Xn=max{X1,,Xn}. The value EXn is what can be obtained by a "prophet." A "mortal" on the other hand, may use k1 stopping rules t1,,tk, yielding a return of E[maxi=1,,kXti]. For nk the optimal return is Vkn(X1,,Xn)=supE[maxi=1,,kXti] where the supremum is over all stopping rules t1,,tk such that P(tin)=1. We show that for a sequence of constants gk which can be evaluated recursively, the inequality EXn<gkVkn(X1,,Xn) holds for all such X1,,Xn and all nk; \hbox{g1=2}, g2=1+e1=1.3678,g3=1+e1e=1.1793,\breakg4=1.0979 and g5=1.0567. Similar results hold for infinite sequences X1,X2,.

Citation

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David Assaf. Larry Goldstein. Ester Samuel-Cahn. "Ratio prophet inequalities when the mortal has several choices." Ann. Appl. Probab. 12 (3) 972 - 984, August 2002. https://doi.org/10.1214/aoap/1031863177

Information

Published: August 2002
First available in Project Euclid: 12 September 2002

zbMATH: 1012.60045
MathSciNet: MR1925448
Digital Object Identifier: 10.1214/aoap/1031863177

Subjects:
Primary: 60G40
Secondary: 60E15

Keywords: best choice multiple stopping options , Optimal stopping rules , ratio prophet inequality

Rights: Copyright © 2002 Institute of Mathematical Statistics

Vol.12 • No. 3 • August 2002
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