A mathematical take on the competitive balance of a football league

TL;DR

This paper introduces a new mathematical framework and a goal-based index to measure and analyze competitive balance in football leagues, addressing limitations of existing measures and providing better insights into league dynamics and revenue variability.

Contribution

It presents a novel mathematical proof of perfect balance, develops a new competitive balance index, and demonstrates its effectiveness on European league data.

Findings

The new index better explains seasonal changes in league balance.

The proposed method effectively analyzes revenue variability.

The framework offers a more nuanced understanding of competitive balance.

Abstract

Competitive balance in a football league is extremely important from the perspective of economic growth of the industry. Many researchers have earlier proposed different measures of competitive balance, which are primarily adapted from the standard economic theory. However, these measures fail to capture the finer nuances of the game. In this work, we discuss a new framework which is more suitable for a football league. First, we present a mathematical proof of an ideal situation where a football league becomes perfectly balanced. Next, a goal based index for competitive balance is developed. We present relevant theoretical results and show how the proposed index can be used to formally test for the presence of imbalance. The methods are implemented on the data from top five European leagues, and it shows that the new approach can better explain the changes in the seasonal competitive balance of the leagues. Further, using appropriate panel data models, we show that the proposed index is more suitable to analyze the variability in total revenues of the football leagues.