Introduction

Tennis is a game often played in a singles format, with one opponent facing off against another. In its standard format, a player must win 6 games to win a set, and 2 or 3 sets to win a match depending on the level of game-play. Throughout each set, players alternate between serving the ball to their opponent (service game) and returning their opponent’s serves. As a general rule-of-thumb, it is often considered to be difficult to win on your opponent’s service (break); said differently, it is basic tennis strategy to win service games since the tone is set by the one serving. Therefore, following this approach, basic strategy to win a match is to win all service games and break the opponent at least once in each set.

This paper explores the relationship of a player’s serve and return ratings to their winning percentage over the course of a season testing the efficiency of the proposed strategy.

Data Summary

2021 season data was retrieved from “Ultimate Tennis Statistics” and was limited to the Men’s tour. The final dataset included 103 different players from 36 different countries. Since players can use their own discretion as to when to play, it was unsurprising to see the total matches played per player ranged from 20 to 77; the winning percentage on these matches ranged from 21% to 89%. Additional summary statistics are offered in the table below. Of note, the AVERAGE_SERVE and AVERAGE_RETURN statistics are the averages of serve and return ratings, respectively. These metrics are proprietary to the ATP and are intended to highlight the strength of a players serve and their return of serve, the core of the basic strategy laid out in this paper’s introduction. AVERAGE_PROF is a sum of a player’s serve and return ratings attempting to highlight overall strength on the start of every point.

As the summary table shows, players tend to have stronger serve ratings than return ratings; this is generally unsurprising given the focus on winning service games. Across the 103 players studied, serve rating tended to be 2.3 times stronger than return rating (AVERAGE_SERVE_RETURN).

STATISTIC
PLAYER_COUNT 103.000
AVERAGE_WINS 22.100
AVERAGE_MATCHES 41.000
AVERAGE_WIN_PCT 0.539
AVERAGE_SERVE 326.400
AVERAGE_RETURN 139.900
AVERAGE_PROF 466.300
AVERAGE_SERVE_RETURN 2.300

It was of interest to model a player’s winning percentage against their serve and return ratings; linear regression techniques were deployed to accomplish this. With this as the defined approach, it is important to highlight that it would not typically be recommended to model percents with a line because it introduces unrealistic outcomes, such as a winning percentage over 100%. However, given the confined range of winning percentages in the dataset, it served as an easy starting point without introducing too much risk to the model’s overall interpretability. If further research were to be conducted, the model could be strengthened with a logit/probit modeling approach.

Prior to establishing a model, the data was tested for use in a linear regression approach; overall, it was tested for normality and some degree of correlation between dependent (winning percentage) and independent (serve and return ratings) variables. Below is a correlation matrix exploring these two items:

Correlation Matrix of 2021 Tennis Data

The correlation matrix displays strong normality and Pearson correlation in the variables of interest:

Model Construction

Two models were constructed to explore how service and return of serve affect a player’s chance of increasing their winning percentage.

Model 1 Results

The first model took the following shape

\(win_{pct} = \beta_0 + \beta_1 * Prof_{total}\)

with the following null and alternative hypotheses

  • \(H_0: \beta_1 = 0\)
  • \(H_A: \beta_1 \neq 0\)

The results of this model are shown below.

## 
## Call:
## lm(formula = win_pct ~ total_prof, data = atp_2021_data_model)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.186160 -0.032754  0.001819  0.045253  0.131349 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.2119092  0.1368054  -16.17   <2e-16 ***
## total_prof   0.0058363  0.0002931   19.91   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0589 on 101 degrees of freedom
## Multiple R-squared:  0.7969, Adjusted R-squared:  0.7949 
## F-statistic: 396.4 on 1 and 101 DF,  p-value: < 2.2e-16

The model suggests a significantly significant positive correlation between a player’s combined serve and return ratings and their winning percentage. Of note, \(\beta_0 = -2.2119092\) suggests that a player must have a combined serve and return rating of \(378.99\) to be at a realistic point of consideration for this model (e.g., a winning percentage that is not less than 0). Further, with a \(\beta_1 = 0.0058363\), any rating above \(550.33\) pushes a player beyond what is positively realistic (e.g., a winning percentage greater than 100%). In the dataset used, the range of players’ combined rating was \(420.5\) to \(528.50\) falling within the limitations presented by the model.

The model does highlight positive statistical significance between a player’s combined serve and return ratings and their winning percentage. With a \(\beta_1 = 0.0058363\), an improvement in a player’s combined serve and return rating of \(10\) leads to an additive 5% increase to their winning percentage. This accounts for 79% of variance in winning percentage according to the \(R^{2}_{adj} = 0.7949\).

Model 2 Results

It was of interest to determine whether a model could more specifically target a player’s serve and return rating, giving players an opportunity to choose which of these areas to focus on in relation to their entire game.

A second model was constructed with form

\(win_{pct} = \beta_0 + \beta_1 * Prof_{serve} + \beta_2 * Prof_{return}\)

The results of this second model are below.

## 
## Call:
## lm(formula = win_pct ~ serve_prof + return_prof, data = atp_2021_data_model)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.16045 -0.02988  0.00031  0.03534  0.15293 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2.2145196  0.1335504  -16.58   <2e-16 ***
## serve_prof   0.0054771  0.0003216   17.03   <2e-16 ***
## return_prof  0.0066930  0.0004521   14.80   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0575 on 100 degrees of freedom
## Multiple R-squared:  0.8084, Adjusted R-squared:  0.8046 
## F-statistic:   211 on 2 and 100 DF,  p-value: < 2.2e-16
##     Variables Tolerance     VIF
## 1  serve_prof   0.95595 1.04608
## 2 return_prof   0.95595 1.04608

Once again, the model has limitations as it relates to the ranges a player must fall in in order to stay within the realm of reality (e.g., winning percentage between 0 and 100%). On an weighted basis that considers a serve rating to be \(2.3\) times greater than return rating, a player’s combined serve and return rating must fall within the range \(378.8382\) and \(549.9084\). This is similar to the range found in model one and, once again, the data used in the model falls within this tolerable range.

The second model’s results highlight statistically significant relationships between winning percentage and serve and return ratings. Based on the results, an increase of \(10\) in serve or return rating increases a player’s winning percentage by an additive 5.5% or 6.7%, respectively. An \(R^{2}_{adj} = 0.8046\) suggests that 80.1% of the variance in a player’s winning percentage is covered by the model.

An additional area of exploration for model 2 included determining the extent of any multicollinearity present. It was theorized that a player with a higher than average service rating may also have a higher than average return rating as some aspects of the ratings may be derived from a certain amount of base skill; this would potentially suggest that using the variables separately could introduce false success in the model. With \(VIF = 1.046084\), the existence of multicollinearity is unlikely as the VIF factors are well below the multicollinearity threshold of \(10\).

Choosing a Model

Both models used to determine the effect of serve and return ratings on winning percentage yielded statistically significant results that were similar to each other. However, as it relates to choosing a model for final use, the second model adds an additional layer of segmentation allowing for better opportunity cost assessment. By separating serve and return ratings into their own independent variables, a player can more easily see that heightening their return rating has a larger impact on winning percentage (+1.2%) than focusing on their serve.