An interpretable and probabilistic competitive balance metric for sports leagues
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Estéfano Bortolini Vassoler
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Pedro O. S. Vaz-de-Melo
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.
Funding source: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Funding source: Fundação de Amparo à Pesquisa do Estado de Minas Gerais
Funding source: Conselho Nacional de Desenvolvimento Científico e Tecnológico
Acknowledgments
This research was supported by CAPES, CNPq and Fapemig.
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Research ethics: Not applicable.
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Informed consent: Not applicable.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission. CRediT authorship E.B.V: Conceptualization, Methodology, Software, Validation, Investigation, Data Curation, Writing - Original Draft. P.O.S.V.M: Conceptualization, Methodology, Writing - Review \& Editing, Supervision.
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Use of Large Language Models, AI and Machine Learning Tools: Only used to improve language throughout the paper.
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Conflict of interest: The authors state no conflict of interest.
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Research funding: This research was supported by Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and Fundação de Amparo à Pesquisa do Estado de Minas Gerais (FAPEMIG).
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Data availability: The dataset is available at: https://github.com/Skill-and-Luck-Coefficients/scrape_tournament_matches/tree/main/data.
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Software availability: Source code is available at: https://github.com/Skill-and-Luck-Coefficients/turning_point.
Appendix A other point systems
Considering that our coefficient directly depends on the variance of the standings, how many points teams make after matches is a relevant factor. For soccer and handball, our 3, 1, and 0 points for, respectively, a win, draw, and loss is standard for most competitions. For basketball and volleyball, the impossibility of drawing makes our point system equivalent to counting the number of wins each team achieved. However, given that volleyball matches are based on the set result, rather than the points the teams accrued in the match, there is another common point system. Specifically, 3-0 and 3-1 victories still provide 3 points for the winner and 0 for the loser; whereas a 3-2 victory indicates that both teams are similarly matched, so it is worth 2 points for the winner and 1 for the loser.
Figure 6 compares both point systems for all volleyball tournaments in our dataset. While the difference is not large, a slight discrepancy can be observed. The perceived competitive balance in this new system tends to be smaller than in the previous one, suggesting that it makes tournaments more predictable than they could be. We believe that an explanation for this difference is that it mostly affects the teams in the middle of the standings, grouping them closer and reducing the average points slightly. Strong (weak) teams will most likely win (lose) matches by 3-0 or 3-1, so they will not be influenced very much by this change. Thus, the real setting has a bias for these teams: the more (less) points teams have, the more likely they are to get a 3-0 or 3-1 victory (defeat). This is not accounted for in the simulations, so the teams at the top (bottom) of the standings do not separate themselves from the rest as much as they do in real life, accelerating the τ detachment.
Comparison between two different points systems: a 3-2 victory gives 2 (3) points to the winner and 1 (0) to the loser. On the left, we show the one-to-one comparison for all volleyball seasons, and on the right, we illustrate them as a boxplot.
Appendix B most imbalanced tournament
Given their broad application in other fields, the Gini index and the HHI coefficients already have normalized versions. However, they rely on upper bounds which are impossible to obtain in conventional round-robin tournaments, since a single team cannot win all matches. Therefore, these coefficients should be normalized in sports leagues using the upper bound from the most unbalanced tournament possible (Fort and Quirk 1997; Utt and Fort 2002).
In such a tournament, the first-place team wins all matches they play, the second-place team wins all the matches except those against the first-place team, and so on. From this definition, it is possible to estimate the upper bound threshold – for example, the HHI upper bound for a balanced round-robin tournaments has a closed form given by
Unfortunately, two factors prevent us from using analytical expressions directly: (i) our database is not composed only of competitions with perfect round-robin schedules; (ii) calculating the imbalance progression in subsection 8.2 requires applying these coefficients to portions of real tournaments that might not be perfect either. With that in mind, we choose to build the most unbalanced tournament and calculate the upper limit directly from it. To achieve this, we slightly generalize the definition of the most unbalanced tournament to work with schedules where different teams play a different number of games. In this new version, the team that wins all its games is the one (or one of) that played the most amount of games; the second-place team is the one who played the second-largest number of games; and so forth. In particular, if all teams play the same number of matches, we have a situation identical to the old definition. Additionally, in the unrealistic limit where a team plays in every tournament match, this team would win every game, and the upper limit would be the same as the one used in the other fields.
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