A note on P\'olya urns: the winner may lead all the time

TL;DR

This paper provides a simple proof that in a Pólya urn with certain initial conditions and replacement rules, the dominant color can maintain its lead indefinitely with positive probability.

Contribution

It introduces a straightforward proof demonstrating that the initial majority color can always stay ahead in a Pólya urn under specific conditions.

Findings

If the initial count of color 1 exceeds color 2, and m_1 ≥ m_2, then color 1 can always lead with positive probability.

The proof applies to two-color Pólya urns with fixed replacement numbers.

The result highlights the persistence of initial dominance in reinforced urn processes.

Abstract

Consider a P\'olya urn where a drawn ball of colour $i$ is replaced together with a fixed number $m_i$ of balls of the same colour. We give a simple proof that if, for example, there are two colours and the urn starts with more balls of colour 1 than 2, and $m_1\ge m_2$, then there is a positive probability that there always will be more balls of colour 1 than colour 2.