The distance to competitive balance: a cardinal measure

Competitive balance is a key variable in the analysis of sports competitions. There are several indexes that measure competitive balance but, either they are not a mathematical metric or they do not have the unit interval as a range. Therefore, these indexes do not indicate the magnitude of the differences, and the measurements cannot be interpreted as percentages. We characterize the space of all admissible configurations of the results of a competition. Then, we construct a new index, based on the concept of distance, whose range is the unit interval, and define it as a function in a metric space. So, it solves issues linked to the limited cardinality of most existing indexes, and we can answer this question: What is the meaning of the differences between the levels of competitive balance? We applied this index to the major European soccer leagues over the last twenty seasons.