Bounds for the Success Probability in the Odds Theorem

TL;DR

This note analyzes Bruss's odds theorem for independent indicator sequences by deriving sharp upper and lower bounds on the maximal win probability $V_n$ under the optimal stopping rule. The bounds are expressed in terms of the total odds sum $R_s$ from the stopping index $s$ and are shown to be attainable by explicit constructions of the success probabilities $p_j$. The authors provide a unified framework that recovers known one-dimensional bounds and characterizes equality cases across three regimes of $R_s$, using calculus-based optimization and AM-GM-type arguments. The results have a precise, sharp nature and illuminate how the stopping threshold $s$ and the distribution of odds across indices influence the optimal success probability.

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.