Bounds for the Success Probability in the Odds Theorem

TL;DR

This paper derives sharp upper and lower bounds for the success probability in Bruss's odds theorem, providing explicit examples where these bounds are achieved, thus enhancing understanding of optimal stopping rules for independent events.

Contribution

The paper introduces explicit, sharp bounds for success probabilities in Bruss's odds theorem, with examples demonstrating equality cases, advancing the theoretical understanding of optimal stopping.

Findings

Derived explicit bounds for success probability

Provided examples where bounds are attained with equality

Enhanced understanding of optimal stopping in independent events

Abstract

Bruss's odds theorem \cite{Bruss1} addresses the problem of determining the optimal stopping time for sequences of independent indicator functions. In this note, we derive upper and lower bounds for the success probability under the optimal stopping rule. These bounds depend on the number of independent events under consideration and on a deterministic index specifying the stopping time. Moreover, the bounds are sharp: we provide explicit examples in which the corresponding inequalities are attained with equality.