A sharp bound for winning within a proportion of the maximum of a sequence

TL;DR

This paper investigates the probability of stopping within a proportion of the maximum in a sequence of i.i.d. random variables, establishing a sharp lower bound for this probability relative to the classic maximum-winning probability.

Contribution

It introduces a new bound for the probability of stopping within a proportion of the maximum, extending the classical secretary problem to a broader stopping criterion.

Findings

The lower bound for stopping within a proportion of the maximum is sharp.

For any continuous distribution, the probability of stopping within a proportion is at least as large as the maximum winning probability.

The result generalizes the classical secretary problem to proportional stopping criteria.

Abstract

This note considers a variation of the full-information secretary problem where the random variables to be observed are independent and identically distributed. Consider $X_1,\dots,X_n$ to be an independent sequence of random variables, let $M_n:=\max\{X_1,\dots,X_n\}$, and the objective is to select the maximum of the sequence. What is the maximum probability of "stopping at the maximum"? That is, what is the stopping time $\tau$ adapted to $X_1,...,X_n$ that maximizes $P(X_{\tau}=M_n)$? This problem was examined by Gilbert and Mosteller \cite{GilMost} when in addition the common distribution is continuous. The optimal win probability in this case is denoted by $v_{n,max}^*$. What if it is desired to "stop within a proportion of the maximum"? That is, for $0<\alpha<1$, what is the stopping rule $\tau$ that maximizes $P(X_{\tau} \geq \alpha M_n)$? In this note both problems are treated as games, it is proven that for any continuous random variable $X$, if $\tau^*$ is the optimal stopping rule then $P(X_{\tau^*} \geq \alpha M_n)\geq v_{n,max}^*$, and this lower bound is sharp. Some examples and another interesting result are presented.