# Odds-Theorem and Monotonicity

## TL;DR

This paper explores how the optimal stopping strategy in the Odds-theorem relates to the monotonicity of event probabilities, providing complete answers and applications to deepen understanding of the theorem's properties.

## Contribution

It offers a detailed analysis of the relationship between optimal win probabilities and monotonicity in the Odds-theorem, extending the original results with new insights and applications.

## Key findings

- Monotonicity properties influence optimal stopping strategies.
- Complete characterization of how probabilities affect win likelihood.
- Applications demonstrate practical relevance of theoretical results.

## Abstract

Given a finite sequence of events and a well-defined notion of events being interesting, the Odds-theorem (Bruss (2000)) gives an online strategy to stop on the last interesting event. It is optimal for independent events. Here we study questions in how far optimal win probabilities mirror monotonicity properties of the underlying sequence of probabilities of events. We make these questions precise, motivate them, and then give complete answers. This note, concentrating on the original Odds-theorem, is elementary, and the answers are hoped to be of interest. We include several applications.