Stopping on the last success with unknown odds

par M. Davy Paindaveine (Université Libre de Bruxelles)

jeudi 2 avr. 2026, 11:00 → 12:15 Europe/Paris

Auditorium 3 (Toulouse School of Economics)

Optimal stopping problems lie at the interface of probability, statistics, and sequential decision theory. They model situations in which decisions must be made online, using only the information revealed so far and without access to future outcomes. After briefly discussing a few motivating examples, we will focus on the classical last-success problem: one observes a sequence of Bernoulli trials and seeks to stop exactly on the final success. When the success probabilities are known, the problem admits an elegant and complete solution via Bruss’ sum-the-odds theorem (Bruss, 2000, Annals of Probability), which yields a simple threshold rule and an explicit formula for the optimal win probability. In most applications, however, these probabilities are unknown, so the oracle rule cannot be implemented. We therefore study the last-success problem under a minimal information structure, where the decision maker observes only the sequential Bernoulli outcomes. Our contribution provides a quantitative theory of oracle-freeness: we analyze the performance of the natural plug-in odds rule and, more importantly, characterize what is possible---and what is fundamentally impossible---when the success probabilities are unknown.