Safe Leads and Lead Changes in Competitive Team Sports

TL;DR

This paper models lead changes in team sports using random walk theory, revealing statistical laws governing lead dynamics and validating predictions with extensive real-world game data.

Contribution

It introduces a novel application of random walk theory to analyze lead changes and safety in team sports, providing more accurate predictive models than existing heuristics.

Findings

Lead changes follow a Gaussian distribution.

The last lead change and largest lead size are governed by the arcsine law.

Predictions match data from over 1.25 million scoring events.

Abstract

We investigate the time evolution of lead changes within individual games of competitive team sports. Exploiting ideas from the theory of random walks, the number of lead changes within a single game follows a Gaussian distribution. We show that the probability that the last lead change and the time of the largest lead size are governed by the same arcsine law, a bimodal distribution that diverges at the start and at the end of the game. We also determine the probability that a given lead is "safe" as a function of its size $L$ and game time $t$. Our predictions generally agree with comprehensive data on more than 1.25 million scoring events in roughly 40,000 games across four professional or semi-professional team sports, and are more accurate than popular heuristics currently used in sports analytics.