Ratio prophet inequalities when the mortal has several choices

Abstract

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 $X_1, X_2, \ldots\,$.