The Last Success Problem with Samples

TL;DR

This paper studies a variant of the last success problem where only samples from distributions are available, proposing policies that guarantee high success probabilities with limited information and samples.

Contribution

It introduces a sampling-based approach to the last success problem, achieving near-optimal success probabilities with minimal samples per distribution.

Findings

Single-sample policy guarantees 1/4 success probability.

Multiple samples per distribution can approach 1/e success probability.

Optimal success probability bounds are established with limited samples.

Abstract

The last success problem is an optimal stopping problem that aims to maximize the probability of stopping on the last success in a sequence of independent $n$ Bernoulli trials. In the classical setting where complete information about the distributions is available, Bruss~\cite{B00} provided an optimal stopping policy that ensures a winning probability of $1/e$. However, assuming complete knowledge of the distributions is unrealistic in many practical applications. This paper investigates a variant of the last success problem where samples from each distribution are available instead of complete knowledge of them. When a single sample from each distribution is allowed, we provide a deterministic policy that guarantees a winning probability of $1/4$. This is best possible by the upper bound provided by Nuti and Vondr\'{a}k~\cite{NV23}. Furthermore, for any positive constant $\epsilon$, we show that a constant number of samples from each distribution is sufficient to guarantee a winning probability of $1/e-\epsilon$.