The Winning Probability of a Game and the Importance of Points in Tennis Matches

ABSTRACT

Purpose: This study builds a stochastic model of a discrete-time Markov chain (DTMC) that fits well with a dataset of professional playing records. Methods: The point-by-point dataset of Men’s single matches played in the Association of Tennis Professionals (ATP) tour from 2011 to 2015 is analyzed. A long-debated assumption on the iid-ness in the point winning probability of the server is statistically tested. A DTMC model is then developed to analyze the dataset further. Results: The statistical test results indicate that the identicality of point winning probabilities is not a valid assumption. For example, the server’s point winning probability from scores 40:0, 30:15, 15:30, and 0:40 are significantly different. On the other hand, the independence is a generally valid assumption except for 40:15 where who won the previous point influences the point winning probability. Game winning probabilities and the importance of each point in winning a game are analyzed using the DTMC model by court surfaces and player groups of the different levels of serve effectiveness. Conclusion: Extensive empirical validation concludes unsealed debates over the stochastic models for tennis. The presented results reveal interesting properties in professional tennis matches.

KEYWORDS:

• Sports analytics
• probability model
• big data
• ATP tour

Acknowledgments

We appreciate Jeff Sackmann for his initiatives in sharing the database used in this study.

Notes

¹ Under the terms of the license, the database can be freely used for noncommercial purposes.

² 30:15 and 30:30 on clay courts are exceptions such that the server plays better when the last point was won by the receiver with some statistical significance. This observation is for clay courts only, and we were unable to find appropriate conjecture except that receivers may have become accustomed to this in these neck-and-neck scores on relatively receiver-friendly clay courts.

³ This study assumes the ace rate as a representative measure for the effectiveness of service, for simplicity and data availability. We note that the ace rate may not capture, in particular, the effectiveness of second service. For example, Federer is known for, among his many other strengths, his effective second services that put him in dominating position easily, but his accurate second service skill may not be measured into the ace rate because second services tend to be a lot slower for acing, thus scarcely aimed for scoring aces.

Additional information

Funding

This work was supported by the National Research Foundation of Korea [NRF-2014S1A3A2036037] and this paper contains no IRB approval.