Research Paper

Total Games Played in Tennis using Serve Data

Tae Jong Kang December 2025

Index Terms: Tennis Modeling, Monte Carlo, Serve Data, Machine Learning.

Abstract

This paper studies how serve performance influences the total number of games played in professional tennis matches. Using ATP-level serve statistics, we model the relationship between service hold probability and expected set length. Since ATP matches are played in best-of-three format, and best of five in Grand Slams, the total number of games in a match is bounded but varies depending on each player's serving ability. We introduce a composite serve-strength metric, \(W_{sp}\), and utilize a dynamic Monte Carlo simulation to estimate the distribution of total games. Furthermore, we construct a synthetic dataset to train a machine-learning regression model and implement an integrated prediction pipeline. This framework provides a quantitative foundation for evaluating match duration in betting and broadcasting contexts.

I. Introduction

Professional tennis is a serve-dominated sport where small margins in serving quality often dictate the length of a match. While bookmakers routinely publish over/under lines for the total number of games, public models typically focus on predicting the match winner rather than the full distribution of games played.

We propose a methodology that:

Converts serve metrics into a single Serve Win-Potential indicator, \(W_{sp}\).
Simulates matches using a dynamic state model that accounts for fatigue and momentum.
Uses synthetic data generation to train a tree-based regression model for fast inference.

II. Methodology: The Serve-Based Model

The framework begins by transforming ATP serve statistics into a compact matchup representation. The serve model estimates each player's service strength, converts that strength into match-level probabilities, and then uses simulation and regression to estimate total games and market-relevant over/under outcomes.

This structure separates the problem into two layers. The first layer defines interpretable serve features, while the second layer uses those features inside a dynamic Monte Carlo simulator and a machine-learning model trained on synthetic match outcomes.

Interactive Serve Matchup Visualization

The visualization below compares how stronger and weaker service hold profiles change the expected total-games distribution. Use the scenarios or manual sliders to see how serve gap, matchup balance, and market line assumptions move the over/under probabilities.

III. Serve Win-Potential Metric

To quantify serving strength, we define the serve win-potential metric, \(W_{sp}\). This scalar aggregates box-score statistics into a single value used to estimate the probability of holding serve. Let the normalized player statistics be denoted as:

\begin{aligned} A &= \text{Ace percentage} / 100, \\ DF &= \text{Double Fault percentage} / 100, \\ F_I &= \text{First Serve In percentage} / 100, \\ F_W &= \text{First Serve Win percentage} / 100, \\ S_W &= \text{Second Serve Win percentage} / 100, \\ DR &= \text{Dominance Ratio = (Rate of return points won) / (Rate of service points lost)}. \end{aligned}

We define the components of serve utility as:

W_{\text{1st}} = F_I(1 + F_W)

W_{\text{2nd}} = (1 - F_I - A - DF)(1 + S_W)

Here, \(W_{\text{1st}}\) rewards high first-serve accuracy combined with high win rates. \(W_{\text{2nd}}\) captures performance on second-serve points, explicitly accounting for the frequency of second serves by subtracting first serves in, aces, and double faults from the total.

The final metric is scaled by the Dominance Ratio to account for overall rally strength:

W_{sp} = DR \times (W_{\text{1st}} + W_{\text{2nd}})

IV. Dynamic Monte Carlo Simulation

Using \(W_{sp}\), we build a point-free Monte Carlo simulator. Rather than simulating every point, the model generates outcomes game by game. The probability of holding serve is derived from the difference in \(W_{sp}\) between opponents, passed through a logistic function.

A key feature of this model is the dynamic player state. Serve performance is not constant; it evolves based on fatigue and momentum.

Fatigue: A cumulative decay factor degrades \(F_I\) and \(S_W\) while increasing \(DF\) as the match count increases.
Momentum: A temporary boost to serve effectiveness follows consecutive games won. Momentum decays immediately upon losing a game.

The simulator enforces standard ATP scoring rules, including tiebreaks at 6--6 and best-of-three or best-of-five set formats.

V. Machine Learning Framework

To bypass the computational cost of running Monte Carlo simulations for every prediction, we train a Random Forest regressor to approximate the simulator's output. The model learns from generated match examples and estimates expected total games for a given matchup.

The machine-learning layer is designed for fast inference, while the Monte Carlo simulator remains useful as a validation and uncertainty-estimation mechanism.

VI. Feature Engineering

We transform the raw statistics of Player 1, \(P_1\), and Player 2, \(P_2\), into a 15-dimensional feature vector \(\mathbf{x}\). These features capture directional advantages, absolute magnitudes, and proportional differences.

A. Signed Differences

These features indicate which player is superior.

\begin{aligned} x_1 &= DR_1 - DR_2 & x_4 &= F_{I1} - F_{I2} \\ x_2 &= A_1 - A_2 & x_5 &= F_{W1} - F_{W2} \\ x_3 &= DF_1 - DF_2 & x_6 &= S_{W1} - S_{W2} \end{aligned}

B. Absolute Differences

These features measure the tightness of the matchup. Small absolute differences imply a closer contest, increasing the likelihood of tiebreaks and extra sets.

\begin{aligned} x_7 &= |DR_1 - DR_2| & x_{10} &= |F_{I1} - F_{I2}| \\ x_8 &= |A_1 - A_2| & x_{11} &= |F_{W1} - F_{W2}| \\ x_9 &= |DF_1 - DF_2| & x_{12} &= |S_{W1} - S_{W2}| \end{aligned}

C. Stabilized Ratios

To capture proportional dominance, such as the impact of a high serve win percentage relative to a low one, we use ratios stabilized by \(\varepsilon = 10^{-3}\):

x_{13} = \frac{DR_1}{DR_2 + \varepsilon}, \quad x_{14} = \frac{F_{W1}}{F_{W2} + \varepsilon}, \quad x_{15} = \frac{S_{W1}}{S_{W2} + \varepsilon}

VII. Synthetic Training Data Generation

Real-world tennis data is sparse and noisy. To train a robust regressor, we generate synthetic datasets using the Monte Carlo simulator described in the serve-based model.

Sampling: We sample pairs of players with statistics drawn from realistic ATP distributions, such as \(\text{Ace \%} \sim \mathcal{N}(10, 4)\).
Simulation: Each pair plays a simulated best-of-three match with fatigue and momentum enabled.
Targeting: For the total games model, the target is the integer count of games played. For the win probability model, the target is binary: 1 if Player A wins and 0 otherwise.

This approach allows us to generate \(10^5\) or \(10^6\) training samples, enabling the Random Forest to learn the non-linear interactions between serve disparities and match duration.

VIII. Integrated Prediction Pipeline

To combine machine-learning inference with the Monte Carlo simulator, we implement an integrated prediction pipeline. This structure enables fast evaluation of any matchup by combining analytic models, ML predictions, and Monte Carlo reliability checks.

Training: A Random Forest regressor is trained on synthetic data generated by the match simulator to learn the relationship between the feature vector \(\mathbf{x}\) and the total games count.
Prediction: For a specific matchup, the model predicts expected total games using the trained regressor. Simultaneously, an analytic win probability is calculated using the logistic transformation of the \(W_{sp}\) difference, \(CSA = W_{spA} - W_{spB}\).
Validation: A high-iteration Monte Carlo simulation, such as \(N = 10{,}000\), is run in parallel to generate confidence intervals and volatility metrics.

Pipeline Outputs

Primary Predictions: The ML-predicted total games and the analytic win percentage.
Validation Metrics: The Monte Carlo average game count and win rate, used to verify the ML model's stability.
Risk Metrics: Standard deviation of games, margin-of-victory spread, and confidence levels.
Market Analysis: Probabilities of total games landing over or under specific benchmark lines.
Visual Diagnostics: Auto-generated plots for game-count distribution, scoreline frequency, and margin of victory.

IX. Conclusion

This study presents a serve-based framework for estimating total games played in professional tennis matches. By combining the Serve Win-Potential metric, dynamic Monte Carlo simulation, and a machine-learning regression pipeline trained on synthetic data, the model provides a structured way to analyze match length, win probability, and over/under outcomes. The approach demonstrates how serve statistics such as dominance ratio, ace percentage, double-fault percentage, first-serve rate, and serve win percentages can be converted into practical prediction features.

However, the current system has important limitations. The machine-learning model is trained primarily on synthetic data, so its accuracy still depends heavily on the assumptions used inside the simulator. The fatigue and momentum adjustments are useful for modeling match dynamics, but they are simplified and require further calibration against real ATP match data. Because of this, the model should be treated as an experimental prediction framework rather than a fully validated betting model.

Future work should focus on validating the predictions against real historical match outcomes, tuning the fatigue and momentum parameters, and comparing the model's total-games projections with actual sportsbook or prediction-market lines. Additional features such as surface type, player injury history, recent schedule density, and head-to-head performance could also improve the realism of the simulation. Overall, this project establishes a foundation for tennis total-games modeling while clearly showing the need for stronger validation and continued model refinement.