A refined and asymptotic analysis of optimal stopping problems of Bruss and Weber

TL;DR

This paper refines the analysis and explores the asymptotic behavior of Bruss-Weber optimal stopping problems, focusing on sequences of independent variables with three possible outcomes, and considers various information scenarios.

Contribution

It provides a detailed refined analysis and asymptotic study of Bruss-Weber problems, including incomplete information cases and simulations of selection algorithms.

Findings

Asymptotic behavior characterized for various problem settings.

Optimal stopping strategies derived for different information scenarios.

Simulation results support theoretical findings.

Abstract

The classical secretary problem has been generalized over the years into several directions. In this paper we confine our interest to those generalizations which have to do with the more general problem of stopping on a last observation of a specific kind. We follow Dendievel, (where a bibliography can be found) who studies several types of such problems, mainly initiated by Bruss and Weber. Whether in discrete time or continuous time, whether all parameters are known or must be sequentially estimated, we shall call such problems simply "Bruss-Weber problems". Our contribution in the present paper is a refined analysis of several problems in this class and a study of the asymptotic behaviour of solutions. The problems we consider center around the following model. Let $X_1,X_2,\ldots,X_n$ be a sequence of independent random variables which can take three values: $\{+1,-1,0\}.$ Let $p:=\P(X_i=1), p':=\P(X_i=-1), \qt:=\P(X_i=0), p\geq p'$, where $p+p'+\qt=1$. The goal is to maximize the probability of stopping on a value $+1$ or $-1$ appearing for the last time in the sequence. Following a suggestion by Bruss, we have also analyzed an x-strategy with incomplete information: the cases $p$ known, $n$ unknown, then $n$ known, $p$ unknown and finally $n,p$ unknown are considered. We also present simulations of the corresponding complete selection algorithm.