A sharp lower bound for choosing the maximum of an independent sequence

Abstract

This paper considers a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if $X_1,\dots ,X_n$ are independent random variables with known continuous distributions and $V_n(X_1,\dots ,X_n):={\sup }_{\tau }P(X_{\tau }=M_n)$, where $M_n:=\max \{X_1,\dots ,X_n\}$ and the supremum is over all stopping times adapted to $X_1,\dots ,X_n$, then $$V_n(X_1,\dots ,X_n)\ge {\left(1-\frac{1}{n}\right)}^{n-1},$$ and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of two-valued random variables, and then applying Bruss' sum-the-odds theorem (2000). In order to obtain a sharp bound for each $n$, we improve Bruss' lower bound (2003) for the sum-the-odds problem.

arXiv comment13 pages