Statistical Properties of Rankings in Sports and Games

TL;DR

This paper analyzes the statistical properties and dynamics of rankings across various sports and games, introducing new measures and a simple model to understand how rankings evolve over time.

Contribution

It introduces a new measure called 'system closure' and demonstrates that simple random walk models can replicate observed rank dynamics across diverse datasets.

Findings

Rank distributions vary across sports but rank dynamics show similar patterns.

The new 'system closure' measure captures elements entering or leaving rankings.

A simple random walk model can reproduce observed rank dynamics.

Abstract

Any collection can be ranked. Sports and games are common examples of ranked systems: players and teams are constantly ranked using different methods. The statistical properties of rankings have been studied for almost a century in a variety of fields. More recently, data availability has allowed us to study rank dynamics: how elements of a ranking change in time. Here, we study the rank distributions and rank dynamics of twelve datasets from different sports and games. To study rank dynamics, we consider measures we have defined previously: rank diversity, change probability, rank entropy, and rank complexity. We also introduce a new measure that we call ``system closure'' that reflects how many elements enter or leave the rankings in time. We use a random walk model to reproduce the observed rank dynamics, showing that a simple mechanism can generate similar statistical properties as the ones observed in the datasets. Our results show that, while rank distributions vary considerably for different rankings, rank dynamics have similar behaviors, independently of the nature and competitiveness of the sport or game and its ranking method. Our results also suggest that our measures of rank dynamics are general and applicable for complex systems of different natures.