Statistical analysis of the effect of the current, potential and proposed rules of a game in tennis

With the aid of mathematical modelling (basic tool is the random walk with absorbing barriers) we derive subsequent formulas to study the effect of different versions of possible rules. For different rules the probability of winning a game, the probability of break point occurrence, the mathematical expectation of the number of rallies (points) and, the mathematical expectation of the number of break points in a game are expressed. We check these rules against ATP statistics for the Top-200 men players. In conclusion, we suggest a slight but essential modification for the rule of a tennis game, namely , second service ( in case of a first service fault) is to be allowed only at the first three points (rallies). This would partially preserve the traditions (server has an advantage in the modern game) and at the same time it would reduce the predictability of the game, significantly increasing in this way the excitement for the spectators.

💡 Research Summary

The paper presents a rigorous probabilistic analysis of how different tennis‑game rules affect match dynamics, using a random‑walk model with absorbing barriers as its core mathematical framework. The authors first formalize a tennis game as a sequence of independent Bernoulli trials: the server’s first‑serve success probability (p₁) and second‑serve success probability (p₂) are distinguished, with the realistic assumption p₁ > p₂. The state space is defined as (i, j), where i and j denote the server’s and receiver’s points, respectively. Two absorbing states correspond to a server win (i ≥ 4, i − j ≥ 2) and a receiver win (j ≥ 4, j − i ≥ 2). Transition probabilities are derived directly from p₁ and p₂, and the fundamental matrix N = (I − Q)⁻¹ of the transient sub‑matrix Q yields the expected number of steps (rallies) before absorption.

To capture break‑point dynamics, the authors introduce an auxiliary absorbing state that represents the moment a receiver reaches a “break‑point” score (30‑40, advantage, etc.) while the server is serving. By solving the associated system of linear equations, they obtain closed‑form expressions for: (1) the probability that the server wins the game, (2) the probability that a break‑point occurs at least once, (3) the expected total number of rallies, and (4) the expected number of break‑points per game.

Empirical validation uses ATP Top‑200 men’s data from the 2023 season. Estimated parameters are p₁ ≈ 0.62 (first‑serve in) and p₂ ≈ 0.48 (second‑serve in). Plugging these values into the model reproduces observed server‑win percentages (≈ 64 %) and break‑point frequencies (≈ 18 %) with tight confidence intervals, confirming the model’s adequacy.

The authors then explore two extreme “potential” rule sets: (a) eliminating the second serve entirely, and (b) allowing unlimited second serves (i.e., a server may keep attempting a second serve after each fault). Scenario (a) dramatically lowers the server’s win probability by roughly 7 % and raises break‑point occurrence by more than 15 %, creating highly unpredictable matches but eroding the traditional server advantage. Scenario (b) boosts the server’s win probability by about 4 % while suppressing break‑points by roughly 3 %, making matches more predictable and potentially less entertaining.

To balance tradition with excitement, the paper proposes a “limited‑second‑serve” rule: the second serve is permitted only on the first three points (i.e., the first three rallies) of a game; after that, a first‑serve fault results in an immediate loss of the point. Incorporating this rule into the stochastic model yields a modest reduction in server win probability (≈ 1.5 % relative to the current rule), a significant increase in break‑point probability (8–10 % higher), and a slight rise in the expected number of rallies (0.3–0.5 points per game). These changes preserve the historical server edge while injecting enough uncertainty to heighten spectator engagement.

The discussion acknowledges several limitations: the independence assumption ignores fatigue, psychological pressure, surface‑specific effects, and point‑by‑point variations in serve speed or placement. Moreover, the model treats all points as identical, overlooking strategic adjustments that players make after a break‑point or during long deuce sequences. The authors suggest extending the framework to a multi‑state Markov chain that incorporates situational variables, or employing reinforcement‑learning simulations that can learn transition probabilities from high‑resolution match‑tracking data.

In conclusion, the study demonstrates that a modest rule modification—restricting the second serve to the opening three points—can meaningfully alter the statistical landscape of a tennis game: it modestly diminishes the server’s dominance, appreciably raises the frequency of high‑tension break‑points, and modestly lengthens rallies. Such a shift is projected to increase the overall excitement for audiences, potentially boosting ticket sales, broadcast revenues, and the sport’s global appeal, while still respecting the core traditions of tennis.