Beyond the Federer paradox in tennis

arXiv:2611.03702v1 Announce Type: new 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.