The Annals of Applied Probability

Ratio prophet inequalities when the mortal has several choices

David Assaf, Larry Goldstein, and Ester Samuel-Cahn

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Let $X_i$ be nonnegative, independent random variables with finite expectation, and $X^*_n = \max \{X_1, \ldots, X_n\}$. The value $EX^*_n$ is what can be obtained by a "prophet." A "mortal" on the other hand, may use $k \ge 1$ stopping rules $t_1, \ldots, t_k$, yielding a return of $E[\max_{i=1, \ldots, k} X_{t_i}]$. For $n \ge k$ the optimal return is $V^n_k (X_1, \ldots, X_n) = \sup E [\max_{i = 1, \ldots, k} X_{t_i}]$ where the supremum is over all stopping rules $t_1, \ldots, t_k$ such that $P(t_i \le n) = 1$. We show that for a sequence of constants $g_k$ which can be evaluated recursively, the inequality $EX^*_n < g_k V^n_k (X_1, \ldots, X_n)$ holds for all such $X_1, \ldots, X_n$ and all $n \ge k$; \hbox{$g_1 = 2$}, $ g_2 = 1 + e^{-1} = 1.3678\ldots,\; g_3 = 1+ e^{1-e}= 1.1793\ldots,\break g_4 = 1.0979 \ldots$ and $g_5 = 1.0567 \ldots\,$. Similar results hold for infinite sequences $X_1, X_2, \ldots\,$.

Article information

Ann. Appl. Probab., Volume 12, Number 3 (2002), 972-984.

First available in Project Euclid: 12 September 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60E15: Inequalities; stochastic orderings

Optimal stopping rules ratio prophet inequality best choice multiple stopping options


Assaf, David; Goldstein, Larry; Samuel-Cahn, Ester. Ratio prophet inequalities when the mortal has several choices. Ann. Appl. Probab. 12 (2002), no. 3, 972--984. doi:10.1214/aoap/1031863177.

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