The Last-Success Stopping Problem with Random Observation Times

TL;DR

This paper investigates an optimal stopping problem for a sequence of Bernoulli trials observed at random times, identifying conditions under which a myopic strategy is optimal and deriving formulas for success probabilities.

Contribution

It introduces a novel last-success stopping problem with random observation times, characterizes the optimal strategy using hypergeometric functions, and provides formulas for success probabilities.

Findings

Myopic strategy is optimal if and only if the shape parameter f6 of the negative binomial prior satisfies f6 b7 f6.

Derived explicit formulas for the probability of successfully stopping at the last success.

Discussed asymptotic behavior and limit forms for large N.

Abstract

Suppose $N$ independent Bernoulli trials are observed sequentially at random times of a mixed binomial process. The task is to maximise, by using a nonanticipating stopping strategy, the probability of stopping at the last success. We focus on the version of the problem where the $k^\text{th}$ trial is a success with probability $p_k=\theta/(\theta+k-1)$ and the prior distribution of $N$ is negative binomial with shape parameter $\nu$. Exploring properties of the Gaussian hypergeometric function, we find that the myopic stopping strategy is optimal if and only if $\nu\geq\theta$. We derive formulas to assess the winning probability and discuss limit forms of the problem for large $N$.