The Last Success Problem with Samples

TL;DR

The paper tackles the last success problem under unknown distributions by introducing sampling-based variants. It shows a stark separation from the complete-information benchmark: with no samples, no positive win guarantee exists; with a single sample, a deterministic ASLS policy achieves at least $1/4$ when the odds-sum $R$ satisfies $R\ge(\sqrt{3}-1)/2$, while a randomized approach reaches $1/4$ only for $R\ge1/2$. For any fixed $\varepsilon>0$, a constant number of samples per distribution suffices to guarantee $1/e-\varepsilon$, and further, with more samples one can approach the complete-information bound when $R\ge1$, though no finite-sample policy reaches exactly $1/e$. The results establish both tight upper-lower bounds and practical threshold-based policies, advancing understanding of online stopping with limited information.

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 Vondrák~\cite{NV23}. Furthermore, for any positive constant $ε$, we show that a constant number of samples from each distribution is sufficient to guarantee a winning probability of $1/e-ε$.

Figures (2)

• Figure 1: Classifications of the relative positions of $i_1,i_2,j_1,j_2$. Red lines indicate locations without success.
• Figure 2: The winning probability of the ASLS policy (red line) and an upper bound of any policy for the $(1,R)$-last success problem (black dashed line).

Theorems & Definitions (25)

• Theorem 1: Nuti and Vondrák NV23
• Lemma 1
• proof
• Proposition 1: Bruss B00
• Proposition 2
• proof
• Theorem 2
• Theorem 3
• proof
• Theorem 4