Kapoor’s Optimal Stoppage Theorem
Arjun Kapoor · PRESS Journals · 2025
In this paper, we address a continuous variant of the Secretary Problem, where one must extract a fraction, 1/p, of a flowing liquid to optimize a characteristic of the extracted portion, such as purity. The characteristic of the immediately extractable liquid is assumed to be known, the extraction decision is irrevocable, and only a single extraction can be made. Initially, some amount must be rejected to gauge the random distribution of the characteristic, then an interval with the most desired trait is selected. The problem involves determining the optimal rejection strategy to maximize the likelihood of selecting the interval with the optimal characteristic. Excessive initial rejection risks bypassing the optimal interval, while insufficient rejection may result in settling for a sub-optimal interval. We derive the optimal rejection strategy, expressed as a function of p, that maximizes the probability of selecting the interval with the optimal characteristic. Specifically, we provide a precise formula for the rejection threshold and the associated maximum success probability. Our results generalize the classical discrete theory to a continuous domain, offering new insights into optimal stopping rules for real-time decision-making in various continuous contexts.