The Last Success Problem with a Single Sample
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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 n Bernoulli trials. In a typical setting where complete information about the distributions is available, Bruss provided an optimal stopping policy ensuring a winning probability of 1/e. However, assuming complete knowledge of the distributions is unrealistic in many practical applications. In this paper, we investigate a variant of the last success problem where we have single-sample access from each distribution instead of having comprehensive knowledge of the distributions. Nuti and Vondrák demonstrated that a winning probability exceeding 1/4 is unachievable for this setting, but it remains unknown whether a stopping policy that meets this bound exists. We reveal that Bruss's policy, when applied with the estimated success probabilities, cannot ensure a winning probability greater than (1-e^-4)/4≈ 0.2454 (< 1/4), irrespective of the estimations from the given samples. Nevertheless, we demonstrate that by setting the threshold the second-to-last success in samples and stopping on the first success observed after this threshold, a winning probability of 1/4 can be guaranteed.
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