Beyond the Federer paradox in tennis
Abstract
Roger Federer won about 80% of his professional matches with only 54% of the points. Hierarchical scoring amplifies small advantages on points into large advantages on matches, but also allows a player to win with fewer points than the opponent. Using Monte Carlo simulations and exact Markov chain calculations, we show that in tennis, badminton, volleyball and table tennis this amplification is set almost entirely by the number of points per match. We then let a player win ordinary points with probability p1 and crucial points, those closing a game, set or match, with probability p2. In every sport a narrow region of the (p1, p2) plane exists where a player wins most matches with fewer than half of the points, because crucial points carry more influence than their share of points. Of ATP matches played in 1991-2025, 4.7% were won with fewer points, 16% more than independent points predict (5% including the serve), a constant excess. Pressure performance measured in other seasons predicts match outcomes beyond the points won, and Fabio Fognini won 51% of his matches with 49% of the points, in what we call the Fognini paradox.
Related
- Magnon Klein paradox
- Paradox, Probability, and Quantum Reality: Explained with Physics and with Philosophy — and How Perception Works Within Them
- Jevons' Paradox and Fast Generative Simulation for HEP: Why Realistic Benchmarking is Essential
- Precise Solution to Zeno’s Paradox of the Arrow by Discrete Motion Model
- The friendship paradox: Causal evidence of its behavioral consequences
BibTeX
@misc{martin2026beyond,
title = {{Beyond the Federer paradox in tennis}},
author = {Andrés Martín and Johann H. Martínez and Javier M. Buldú},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03702},
eprint = {2611.03702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03702},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beyond the Federer paradox in tennis
AU - Andrés Martín
AU - Johann H. Martínez
AU - Javier M. Buldú
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.03702
UR - https://arxiv.org/abs/2611.03702
AB - Roger Federer won about 80% of his professional matches with only 54% of the points. Hierarchical scoring amplifies small advantages on points into large advantages on matches, but also allows a player to win with fewer points than the opponent. Using Monte Carlo simulations and exact Markov chain calculations, we show that in tennis, badminton, volleyball and table tennis this amplification is set almost entirely by the number of points per match. We then let a player win ordinary points with probability p1 and crucial points, those closing a game, set or match, with probability p2. In every sport a narrow region of the (p1, p2) plane exists where a player wins most matches with fewer than half of the points, because crucial points carry more influence than their share of points. Of ATP matches played in 1991-2025, 4.7% were won with fewer points, 16% more than independent points predict (5% including the serve), a constant excess. Pressure performance measured in other seasons predicts match outcomes beyond the points won, and Fabio Fognini won 51% of his matches with 49% of the points, in what we call the Fognini paradox.
ER -