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

TL;DR

This paper establishes a precise lower bound for the maximum probability of selecting the overall maximum in a sequence of independent, non-identically distributed variables, extending secretary problem results.

Contribution

It provides a sharp lower bound for the optimal win probability in a generalized secretary problem with independent variables, improving existing bounds for the sum-the-odds problem.

Findings

Lower bound matches the known maximum value (1-1/n)^{n-1}

Bound is tight and attained in specific cases

Method reduces problem to two-valued variables and applies Bruss' theorem

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 $$V_n(X_1,\dots,X_n)\geq \left(1-\frac{1}{n}\right)^{n-1},$$ 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.