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The Last Success Problem with Samples

Authors Toru Yoshinaga , Yasushi Kawase

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  • Theory of computation → Online algorithms
Keywords
  • The Last Success Problem
  • Secretary Problem
  • Sample Information Model
  • Optimal Stopping
  • Online Algorithms

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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 [Bruss, 2000] 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 [Nuti and Vondr{á}k, 2023]. 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.

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Toru Yoshinaga and Yasushi Kawase. The Last Success Problem with Samples. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 105:1-105:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.105

Author Details

Toru Yoshinaga
  • The University of Tokyo, Japan
Yasushi Kawase
  • The University of Tokyo, Japan

Funding

This work was supported by JST SPRING Grant Number JPMJSP2108, JST ERATO Grant Number JPMJER2301, JST PRESTO Grant Number JPMJPR2122, JSPS KAKENHI Grant Number JP20K19739, and Value Exchange Engineering, a joint research project between Mercari, Inc. and the RIISE.

Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable and insightful comments.

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