An interpretable and probabilistic competitive balance metric for sports leagues

Abstract We propose a new probabilistic and interpretable approach to quantify competitive balance in sports leagues based on the precise moment when a tournament can no longer be perceived as perfectly balanced: the longer it takes, the more balanced it is. We analyzed 1,539 seasons from 175 sports leagues in basketball, soccer, handball, and volleyball and observed that only 5 % of the seasons could be seen as perfectly balanced throughout their entire duration. For the others, there is a turning point round after which the points distribution permanently diverges from a range of behaviors likely to occur in perfectly balanced tournaments. Our initial results agree with the general literature since soccer has a considerably more random behavior, i.e., its perceived balance is higher overall. Given the explicit temporal dependence, we also proposed a modification to remove the bias that the order of matches could impose, thereby enabling fair comparisons across different leagues and sports. Our modified coefficient highlights the substantial imbalance in heavily debated leagues, such as the Premier League and the NBA. Furthermore, combining both metrics enables the discovery of anomalous tournament schedules where even simple random changes to the match order could have improved the perceived competitive balance.