On a connection between total positivity and Bernoulli stopping problems

TL;DR

This paper explores the relationship between total positivity and Bernoulli stopping problems, demonstrating how these properties influence optimal stopping strategies and their effectiveness in various success-based decision scenarios.

Contribution

It establishes a theoretical connection between total positivity and the optimality of myopic strategies in Bernoulli stopping problems, with new insights into reward sequence properties.

Findings

Total positivity ensures quasi-unimodality preservation in the chain.

Quasi-unimodality of rewards suffices for myopic strategy optimality.

Examples illustrate applications in last-success settings.

Abstract

Consider a discrete-time optimal selection problem where one observes a sequence of independent Bernoulli trials and receives a nonnegative reward upon stopping on a success. The aim is to find a single-choice strategy that maximises the expected payoff. These Bernoulli stopping problems are characterised by two key properties: (i) a recurrence relation connecting the reward sequence to the continuation payoff sequence, and (ii) the total positivity of the Markov chain embedded in success epochs of the trials. The recurrence is fundamental in proving the optimality of the myopic strategy under unimodal continuation payoff sequence, while the total positivity ensures that the expectation of a quasi-unimodal function of the chain remains quasi-unimodal with respect to the initial state. In particular, if the number of successes is finite almost surely, the quasi-unimodality of the reward sequence is sufficient for the myopic rule to be optimal. Illustrative examples are given in various last-success settings.