Beyond the Federer paradox in tennis

Introduction

In his commencement address at Dartmouth College in June 2024, Roger Federer told the graduates that he had won almost 80% of the 1,526 singles matches of his professional career, but only 54% of the points[1]. His official tour record, 1,251 wins[2], corresponds to about 82% of those matches. Federer used the figure to make a point about resilience. Even the most successful players lose almost every second point. The same figure illustrates a basic property of hierarchical scoring systems. Because a tennis match is won by winning sets, sets by winning games and games by winning points, a modest advantage on points is magnified at each level of the hierarchy, a fact that has recently been revisited in connection with Federer’s remark[3, 4].

The mapping from the probability of winning a point to the probability of winning a game, a set and a match is classical. Assuming that points are independent and identically distributed (i.i.d.), closed form expressions were derived for tennis games, tiebreaks, sets and matches[5, 6, 7, 8, 9], later extended to separate serve and return probabilities, compared with data and used for prediction[10, 11, 12, 13]. The efficiency of scoring systems in rewarding the better player has been analysed in general[14, 15] and, with Markov chain models, for table tennis[16, 17], badminton[18] and volleyball[19]. From the perspective of statistical physics, the predictability of competitions and the dynamics of scoring have been characterised across sports and leagues[20, 21].

The i.i.d. assumption is a convenient baseline, but it is not exact. Points in professional tennis are not independent and identically distributed, although the deviations are small[22], and the probability of winning a point depends on the score[23, 24]. Morris defined the importance of a point as the difference between the probabilities of winning the match after winning and after losing it, and showed that a player who redistributes his performance towards important points increases his chance of winning[25]. The idea of raising one’s game on important points has been explored since[26, 27]. Whether players actually perform differently on important points has been investigated with data on individual points[28, 29, 30, 31, 32, 33], with evidence that performance changes under pressure and that better players perform relatively better when it matters most[28], but also that performance on break points shows little persistence from year to year[34, 35]. At the level of outcomes, a tennis match is sometimes won by the player who wins fewer points or games than his opponent[36, 37, 38, 39], a quasi-Simpson’s paradox of best-of-$N$ contests[36].

Federer himself is a striking example. Between 1991 and 2025, he won more points than his opponent in 1,197 ATP matches with point statistics. Of these, he won 1,161 (97.0%) and lost 񟿿 36 (3.0%). No other player has lost so many matches after winning more points than his opponent. Federer was the loser in 36 of the 45 paradoxical matches he played (80.0%). This record was first reported by Wright et al.[36], and Lisi et al. later discussed it as the Federer paradox[38]. However, the picture changes when the data are read the other way round. Of the 232 matches in which Federer won fewer points than his opponent, he won 9 (3.9%), a higher share than the 3.0% of matches he lost after winning more points. The imbalance arises because Federer won more points than his opponent far more often than the reverse, so he had many more opportunities to lose a paradoxical match than to win one. Wright et al. already read his record as further evidence of his greatness[36], and Lisi et al. showed with simulations that it is expected for a dominant player[38]. The Federer paradox is therefore less simple than it seems, and we show below that it is partly a counting effect and partly a small but genuine deficit.

What has not been established is how a difference in performance between ordinary and crucial points maps onto the probability of winning a match, and when it can make a player who wins fewer than half of the points the favourite. Here we address this question with a minimal model in which a player wins ordinary points with probability $p_1$ and crucial points with probability $p_2$. A crucial point is one whose winner, whichever of the two players, closes a game, a set or the match, so that $p_2$ governs both how often a player wins his own game, set or match points and how often he saves those of the opponent. We compute the probability of winning a match exactly in the $(p_1, p_2)$ plane for tennis, badminton, volleyball and table tennis, characterise analytically the region where a player wins most matches while winning fewer than half of the points, and show that its existence is a direct consequence of point importance. Going beyond the Federer paradox, we then confront the model with three decades of men’s professional tennis, extending previous analyses of points and wins[4] and of performance under pressure[28, 34] with point by point data, out of sample tests and the history of the paradox.

Results

Amplification of a point advantage

We first consider the classical case in which player A wins every point with the same probability $p$ (Methods). Figure 1 shows the probability $P(p)$ of winning a match for tennis over three and five sets, badminton, volleyball and table tennis over five and seven games. All curves pass through $P(1/2) = 1/2$ and become steeper as matches get longer. The slope at $p = 1/2$, which measures how strongly a small advantage is amplified, ranges from 7.3 in table tennis over five games to 13.4 in tennis over five sets (Supplementary Table S1). A player who wins 52.5% of the points wins 80% of his matches in tennis over five sets, whereas in table tennis over five games he needs 54.6% of the points to reach the same win rate.

Remarkably, the slope is well approximated by

where $N$ is the mean number of points per match at $p = 1/2$, with deviations of at most 5% for all six formats (Figure 1b). Equation (1) is the slope of the simplest contest, in which $N$ points are played and the player who wins more of them wins. A point decides the outcome only when the other points are split evenly, and the probability of this falls as $N^{-1/2}$ (Methods). The amplification of a point advantage is therefore set almost entirely by the length of the match, and the hierarchy of games and sets modifies it only slightly, increasing it in tennis over three sets (+5%) and reducing it in volleyball (−3%).

For $p = 0.54$, the i.i.d. model predicts that a player wins 86% of matches over three sets and 91% of matches over five sets, more than Federer’s 82%. The difference is expected. Federer’s 54% is an average over opponents of very different strength, and because $P(p)$ is concave above $1/2$, the average of $P$ over a distribution of $p$ is smaller than $P$ at the average $p$ (Jensen’s inequality[40]).

Crucial points and the paradox region

We now let player A win ordinary points with probability $p_{1}$ and crucial points with probability $p_{2}$, with crucial points defined at the level of the game, the set or the match (Methods). Figure 2 shows the probability of winning a tennis match in the $(p_{1}, p_{2})$ plane. Increasing $p_{2}$ always increases the probability of winning, but the effect depends on the level. Game points are frequent, about 29% of all points at $p_{1} = p_{2} = 1/2$, so $p_{2}$ at the game level controls a large part of the match. Set and match points are rare, about 3% and 1% of the points, and $p_{2}$ at those levels matters only when it differs strongly from $p_{1}$.

Let $F(p_1, p_2)$ be the expected share of points won by A over many matches, the analogue of Federer’s 54% (Methods). We call paradox region the set of $(p_1, p_2)$ for which $P > 1/2$ and $F < 1/2$, where player A wins most of his matches while winning fewer than half of the points.

The paradox region exists for every format we simulated and every level of crucial point, and its shape follows from a simple argument. When $p_1 = p_2 = 1/2$ both players are equivalent, so $P = F = 1/2$ and both boundaries, $P = 1/2$ and $F = 1/2$, pass through $(1/2, 1/2)$. Linearising around this point,

where $\delta_i = p_i - 1/2$ and $\{P_i, F_i\}$ are the partial derivatives at $(1/2, 1/2)$. A point with $\delta_1 < 0 < \delta_2$ belongs to the paradox region if

The paradox region is therefore a wedge with its vertex at $(1/2, 1/2)$, and it exists if and only if $P_2/P_1 > F_2/F_1$. The ratio $F_2/F_1$ is essentially the ratio between the numbers of crucial and ordinary points, whereas $P_2/P_1$ is the ratio of their total influence on the outcome. The condition states that crucial points must carry a larger share of the influence on the match than their share of the points, which is precisely Morris’ notion of point importance[25]. Points that close a game, a set or the match are more important than average, so the condition holds for every sport and level (Supplementary Table S3).

Although it always exists, the paradox region is narrow, with opening angles from $0.3^\circ$ to $1.6^\circ$ (Figure 3b and Supplementary Table S3). In tennis at the game level it is a thin sliver. For $p_1 = 0.49$ a player wins most matches with fewer than half of the points only if $p_2 \approx0.524$, and for $p_1 = 0.48$ only if $p_2 \approx0.546$. Game points are so frequent that any advantage on them also raises the overall share of points. At the set and match levels the opposite happens. Crucial points are rare, so they barely change the share of points, but an advantage of $p_2 \gtrsim0.7$ is needed to tip the match. Table tennis, with short games and therefore many set points, has the widest wedge (Figure 3).

Points and wins in professional tennis

Across the 328 players with at least 200 ATP matches between 1991 and 2025, the shares of points and of matches won are almost perfectly correlated (Pearson $r = 0.97$, 95% confidence interval (CI) 0.965–0.977, Figure 4a), as recently reported for a similar data set[4]. Federer won 54.1% of the points and 81.7% of the 1,437 matches with statistics, in agreement with his speech. As anticipated, the i.i.d. model evaluated at each player’s career share of points overestimates the win rate of the best players, by 5.6 percentage points on average for $p > 0.52$, a consequence of averaging over opponents rather than a deficit of the best players. We therefore measure each player’s match conversion, the difference between his actual wins and those expected from the share of points he won in each match (Methods), which is uncorrelated with the share of points ($r = -0.04$, 95% CI −0.15 to 0.07, Figure 4b).

Winning a match with fewer than half of the points is not rare. It happened in 4,530 of the 96,826 ATP matches in which the two players did not win the same number of points (4.68%), in line with the 4.5% reported for 1991–2017[36, 38]. Given the share of points won in each match, an i.i.d. model predicts 3,892 such matches (4.02%), so the observed number exceeds the expectation by 16% ($z = 11.7$, $P < 10^{-31}$, Figure 4c–Figure 4f). The excess is larger in matches over five sets (23%, $z = 6.8$) than over three sets (15%, $z = 9.8$), larger in Grand Slams (22%) than in other main tour tournaments (14% to 17%), and larger on grass (21%) and hard courts (19%) than on clay (13%) (Supplementary Table S7 and Supplementary Fig. S5). Points therefore do not translate into wins as they would if all points were equivalent. This does not contradict the conclusion that their overall frequency is compatible with i.i.d. points[38]. That conclusion depends on the assumed differences in level between opponents (resimulating every match with seasonal shares of points predicts 5.2%), and conditioning on the points won in each match removes this dependence. Part of the excess, however, reflects the serve, which groups points into service games. With a benchmark that includes the serve and conditions also on the serve dominance of each match (Supplementary Section S11), the excess falls to 5% ($z = 3.9$), and to 14% in matches over five sets ($z = 4.5$). The comparisons between players below are essentially unchanged.

The Grand Slam point by point data (4,062 matches, Methods) show where the excess comes from. In the 166 matches won with fewer than half of the points, the winners won 49.1% of the ordinary points but 51.1% of the set points and 61.9% of the match points, whereas their opponents show the mirror image. The mechanism is that of the model, $p_1 < 1/2 < p_2$, operating at the level of sets, while at the game level the difference between crucial and ordinary points is small (+0.5 percentage points), consistent with the narrowness of the paradox band at the game level. Within a match these figures are partly mechanical, because the winner of a set necessarily wins the point that closes it. The tests in the next section avoid this bias by measuring pressure performance in other matches.

Who benefits from crucial points?

The pressure index adjusted for the serve on break points (Methods) varies little across players. After subtracting the binomial noise, its standard deviation across players is about 0.5 percentage points, and its correlation between odd and even seasons is only 0.13 (0.26 for players with at least 500 matches). Its effect on match outcomes is nevertheless clear when it is measured out of sample. In a logistic regression of the outcome of 90,229 matches on the i.i.d. probability of winning given the points won in that match and on the difference between the two players’ pressure indices measured in all other seasons, the pressure difference has a coefficient of 0.076 per percentage point (standard error 0.011, $z = 7.2$, $P < 10^{-12}$), and the effect is the same in close matches with $0.48 < f < 0.52$ (0.075, $z = 6.5$). A player whose pressure index exceeds that of his opponent by 1 percentage point therefore multiplies his odds of winning by 1.08 at a given share of points. Consistently, the index measured in odd seasons is correlated with the match conversion in even seasons ($r = 0.17$, $P = 0.002$, Figure 5c), and match conversion itself is a stable trait ($r = 0.35$ between odd and even seasons). The low persistence from year to year reported previously[34, 35] reflects the low reliability of the index. Pooling seasons and testing outcomes reveals its small persistent component.

Figure 5a ranks the players with the largest and smallest difference between their performance on crucial and on ordinary points. The difference adjusted for the serve must be used. The raw difference $p_2-p_1$ is dominated by playing style (its correlation with the share of serve points won is $-0.40$), because strong servers play most of their break points on return, and it would place Federer, Alcaraz, Roddick, Raonic and Djokovic at the bottom (Supplementary Table S6). With the adjustment, Nick Kyrgios, Marcos Ondruska, Martin Klizan, Alexander Volkov and Cameron Norrie lead the ranking, and most of the top and bottom ten differ from zero individually, although none remains significant after correcting for multiple comparisons.

Across players, the pressure index and match conversion are correlated ($r = 0.33$, Figure 5b).

Match conversion, unlike the pressure index, also distinguishes individual players. After correcting for multiple comparisons (false discovery rate 5%), 20 of the 328 players have a match conversion significantly different from zero (Supplementary Table S12). Match conversion is correlated with the overperformance in tiebreaks ($r = 0.45$, 95% CI 0.36–0.53), which is unrelated to the pressure index on break points ($r = 0.09$, $P = 0.12$). Break points and tiebreaks are two distinct channels through which crucial points are converted, and Isner’s advantage, for instance, lies entirely in tiebreaks (Supplementary Fig. S6).

Beyond the Federer paradox

The Federer paradox suggests a direct way of measuring how well a player converts points into wins. For each player we count the paradoxical matches he won, that is, the matches won with fewer than half of the points, and those he lost, the matches lost with more than half of the points. We then compare both numbers with their expectation under the i.i.d. model, given the share of points won in each match (Methods). The difference between the observed and the expected balance, which we call the close match balance, is the number of close matches a player has gained, or lost, with respect to independent points. Across the 328 players its standard deviation is 1.4 matches per 100 matches played, whereas binomial noise alone would produce a standard deviation of 0.9, so the genuine differences between players amount to about 1.1 matches per 100. For 23 players the balance differs significantly from zero after correcting for multiple comparisons, 16 of them positively (Figure 6a and Table 1).

Table 1 | Winning more, or fewer, matches than the points imply. The ten ATP players (1991–2025, at least 200 matches) with the largest and the ten with the smallest close match balance. Points: share of points won (%). Matches: share of matches won (%), with the expectation under the i.i.d. model given the share of points won in each match in parentheses). Won with fewer points / lost with more points: paradoxical matches won and lost, with their i.i.d. expectations in parentheses. Balance: close match balance in matches, with its z-score. *Significant after Benjamini–Hochberg correction (false discovery rate 5%). Close sets: share of sets decided by two games or fewer that the player won (%). Asymmetry: mean margin of the sets lost minus that of the sets won (games). Pressure: pressure index on break points (percentage points). Tiebreak: tiebreak overperformance (percentage points).

Player	Matches played	Points (%)	Matches (%)	Won with fewer points	Lost with more points	Balance (z)	Close sets (%)	Asymmetry	Pressure	Tiebreak
Largest balance
Fabio Fognini	774	49.4	51.2 (47.5)	32 (14.8)	10 (18.8)	+26.0* (+5.2)	56.6	+0.31	+0.5	+6.5
John Isner	775	50.6	61.7 (58.4)	49 (21.5)	32 (28.1)	+23.6* (+3.9)	57.5	+0.02	−0.1	+9.0
Carlos Moyà	835	51.1	63.6 (60.9)	32 (19.0)	14 (21.9)	+20.9* (+3.8)	60.8	−0.03	+0.8	+5.8
Tommy Robredo	849	50.8	60.0 (57.7)	34 (19.6)	13 (16.6)	+18.0* (+3.5)	57.2	+0.03	+0.2	+3.7
Goran Ivanišević	710	51.5	65.4 (62.9)	25 (10.2)	18 (20.8)	+17.6* (+3.6)	63.0	+0.03	+1.8	+6.2
Feliciano López	948	49.6	50.6 (48.6)	38 (22.9)	23 (25.0)	+17.1 (+2.9)	53.0	+0.20	+1.0	+4.1
Kei Nishikori	623	51.4	66.9 (63.9)	20 (10.2)	8 (14.8)	+16.6* (+3.9)	61.0	−0.01	−0.6	+5.9
Yevgeny Kafelnikov	826	51.5	66.5 (64.5)	22 (14.2)	13 (20.7)	+15.4* (+3.0)	59.1	−0.11	+0.6	+3.3
Fernando González	518	51.1	64.3 (61.7)	22 (10.7)	8 (9.8)	+13.1* (+3.3)	63.2	+0.02	+1.3	+7.3
Jürgen Melzer	615	49.7	51.2 (48.7)	18 (13.0)	8 (15.8)	+12.7 (+2.7)	51.1	+0.06	+0.3	−0.3
Smallest balance
Roger Federer	1,437	54.1	81.7 (82.9)	9 (15.6)	36 (25.6)	−17.0* (−3.0)	69.2	−0.54	+0.2	+7.7
Fernando Verdasco	962	50.9	55.0 (56.7)	17 (18.6)	33 (18.9)	−15.7* (−2.9)	51.5	−0.25	+0.2	−2.0
Todd Martin	591	51.9	63.6 (66.0)	11 (13.3)	26 (14.7)	−13.6* (−3.0)	57.4	−0.18	+0.5	+4.0
Nikolay Davydenko	759	51.8	61.0 (62.9)	7 (14.8)	16 (10.6)	−13.3* (−3.0)	52.6	−0.34	+0.3	−0.7
Dominik Hrbatý	611	50.5	51.6 (53.9)	7 (12.8)	20 (13.3)	−12.5 (−2.8)	50.2	−0.14	0.0	−0.1
Marin Čilić	898	51.7	62.4 (64.1)	10 (15.1)	28 (20.9)	−12.2 (−2.3)	54.0	−0.25	−0.4	−0.9
Max Mirnyi	423	50.0	47.5 (50.2)	8 (12.1)	21 (13.3)	−11.7 (−2.7)	49.0	−0.03	−0.2	+2.4

Roger Federer is the clearest negative case. His balance of paradoxical matches, 9 won and 36 lost, is the most negative in the data set (the Federer paradox[38]). Counting paradoxical matches, however, is misleading. A dominant player wins more points than his opponent far more often than the reverse, so he has many more chances to lose such a match than to win one[38]. The fair comparison is with the expectation. In the 1,197 matches in which Federer won more points than his opponent, the i.i.d. model predicts 25.6 defeats, but he lost 36 ($z = 2.4$). In the 232 matches in which he won fewer points, the model predicts 15.6 victories, but he won only 9 ($z = -1.9$). Overall, Federer won 17 fewer close matches than his points imply ($z = -3.0$), the same deficit measured by his match conversion, −1.2 wins per 100 matches over 1,437 matches. The Federer paradox is therefore partly a counting effect, as argued previously[38], and partly a small but genuine deficit in close matches. The deficit is at the limit of significance after correcting for multiple comparisons ($q = 0.046$) and is not significant when the serve is taken into account ($z = -2.4$, $q = 0.12$), so it is an indication rather than definitive evidence.

None of the other champions shows it. Pete Sampras (+11.2 matches, $z = 2.4$), Andre Agassi (+8.4, $z = 2.0$), Andy Murray (+7.8, $z = 1.6$) and Rafael Nadal (+7.0, $z = 1.5$) won more close matches than their points imply, and Novak Djokovic is close to the expectation (Supplementary Table S9).

At the opposite end of the ranking stands Fabio Fognini. In his 774 matches he won 61,624 of 124,623 points, 49.4%, significantly fewer than half ($z = -3.9$). Yet he won 396 of those matches, 51.2%. Given the points he won in each match, the i.i.d. model predicts only 47.5% of wins. Fognini won 32 matches with fewer points than his opponent, against 14.8 expected, and lost only 10 with more points, against 18.8 expected. His close match balance is +26.0 matches ($z = 5.2$), the largest of the data set, significant after correcting for multiple comparisons and robust to the inclusion of the serve. Fognini thus won most of his matches with fewer than half of the points. He is a real realisation of the paradox region of the model, which we call the Fognini paradox. He is not the only one. Twenty-three of the 328 players won more than half of their matches with less than half of the points (Figure 6b and Supplementary Table S11), among them Feliciano López, Alexander Bublik and Jürgen Melzer, although for most of them the margins are small.

How does Fognini do it? Not on break points, where his pressure index is close to zero (+0.5 percentage points), but through the way his points are distributed across sets. He won 50.5% of all his sets, but 56.6% of the close ones, those decided by two games or fewer. His lost sets were more lopsided than his won sets, with mean margins of 3.0 and 2.7 games, respectively, and 18% of the sets he lost ended 6–0 or 6–1, against 5% for Federer (Supplementary Fig. S7).

Points lost in lopsided sets do not cost matches, a possibility already suggested by Wright et al., who noted that Federer rarely loses lopsided sets and quoted Isner on saving energy in return games once a break up[36]. Fognini spent many points where they did not matter and won the ones that did, which is precisely the mechanism of the model. The same pattern holds across players. The set margin asymmetry, that is, the mean margin of lost sets minus that of won sets, is correlated with the close match balance ($r = 0.33$, 95% CI 0.23–0.42, Figure 6c).

Three channels therefore turn points into wins, namely performance on break points, performance in tiebreaks and the distribution of points across sets. In a multiple regression across players, the three contribute almost equally to the close match balance, with standardised coefficients of 0.34 for the pressure index, 0.35 for the tiebreak overperformance and 0.34 for the set margin asymmetry (all $P < 10^{-11}$), and together they explain 37% of its variance across players. The players who handle pressure best are therefore not only those who win the big points, but also those whose points are concentrated in the sets that decide the match.

A history of the paradox

Figure 7 shows how the paradox has evolved. Between 1991 and 2025 the share of ATP matches won with fewer than half of the points followed a U shape (quadratic term of a logistic regression on the season, $P = 0.0003$), with a minimum around 2007 and no overall linear trend ($P = 0.46$). Previous analyses of the periods 1991–1997, 1998–2004, 2005–2011 and 2012–2017 found rates of 4.78%, 4.53%, 4.18% and 4.41%, with the lowest value in the same period, but no significant differences between periods[36, 38]. The longer series and the regression at the level of matches used here make the U shape detectable. This evolution is not a change in how players convert points into wins. The excess of paradoxes over the i.i.d. expectation, 0.66 percentage points on average, has remained constant (trend −0.04 percentage points per decade, $P = 0.51$, Figure 7b). It reflects instead the closeness of matches. The share of matches decided by a narrow margin of points fell during the 2000s and rose afterwards (Figure 7c), and accounting for it reduces the curvature of the trend ($P = 0.03$). The record of games, whose paradox was first analysed by Lisi et al.38`` (2.15% of men’s matches), is available from the scores since the beginning of the Open Era and shows a slow increase of matches won with fewer games than the opponent, from 1.64% in 1968–1990 to 1.93% in 1991–2007 and 2.00% in 2008–2025 (logistic trend 0.059 per decade, $P < 10^{-7}$, Figure 7d), and the two paradoxes are correlated across seasons ($r = 0.58$, $P = 0.0003$).

matches, in which the share of points of the two players differs from 1/2 by less than 0.01. d, Share of matches won by the player who won fewer games, 1968–2025, from the scores (symbols show seasons and the line a moving average over five seasons). The dotted line marks the start of the point statistics in 1991.

Discussion

Federer’s remark captures the strength of hierarchical scoring. A small but consistent advantage on points becomes a large advantage on matches. We have shown that, to a very good approximation, the size of this amplification depends only on the number of points played, as for a random walk, so that tennis over five sets rewards the better player most and table tennis over five games least, and the hierarchy of games and sets changes the amplification by only a few per cent.

The same hierarchy makes the outcome sensitive to when points are won. In our model a player who is slightly worse on ordinary points but better on crucial points can win most of his matches while winning fewer than half of the points. The paradox region exists in all four sports and at every level of their scoring hierarchy, and its existence is equivalent to a condition on the importance of crucial points (Supplementary Section S3). It is, however, narrow. Frequent crucial points, such as game points in tennis, cannot be won more often without also raising the share of points won, while infrequent ones, such as set and match points, require a large advantage to change the outcome. This explains why matches won with fewer points occur regularly while players who systematically win more matches than their points imply are rare.

Men’s professional tennis agrees with this picture. Matches won with fewer points are 16% more frequent than an i.i.d. model predicts at the same share of points (5% once the serve is included), their winners perform better on set points than on ordinary points, and a player’s pressure performance measured in other seasons predicts the outcome of a match beyond the points he won in it. These results agree with previous evidence on performance under pressure[28, 29, 30, 31, 32] and on matches won with fewer points or games[36, 37, 38, 39]. Our contribution is to show why such outcomes arise from a difference in performance between ordinary and crucial points, to give the condition under which they can be expected on average, and to quantify how this depends on the scoring system. History adds a nuance. The frequency of the paradox has changed since 1991, but the excess due to crucial points has not. Its variations follow the competitive balance of the tour, which determines how many matches are close enough for crucial points to decide them. The same mechanism separates players. Some, with Fabio Fognini as the clearest example, won most of their matches while winning fewer than half of their points, by winning close sets and conceding lopsided ones. Federer’s paradoxical record, in contrast, is partly a counting effect and partly a small deficit in close matches.

Our model is deliberately minimal. It ignores the serve, although including it changes the amplification by at most 8% and preserves the paradox region in all four sports (Supplementary Section S12). It assumes that $p_1$ and $p_2$ are constant during a match, whereas form, fatigue and momentum may vary. Finally, it treats all crucial points of a given level as equivalent, whereas their importance depends on the score[25, 27]. The data analysis is limited to men’s tennis, to break points in the ATP match statistics and to the Grand Slam tournaments recorded point by point. A continuous measure of point importance, women’s tennis and other sports with point by point records are natural extensions. In Federer’s words, it is only a point. But not all points are equal.

Methods

Scoring systems

In tennis, a game is won by the first player to win four points with a margin of two, a set by the first player to win six games with a margin of two, with a tiebreak to seven points at six games all, and the match by the first player to win two (best of three) or three (best of five) sets. Badminton (BWF rules, games to 21 points with a margin of two, capped at 30, best of three games), volleyball (FIVB, sets to 25 points with a margin of two, deciding fifth set to 15, best of five) and table tennis (ITTF, games to 11 points with a margin of two, best of five or seven) games) use rally scoring. We use the term set for the scoring unit immediately above the point in these sports (Supplementary Table S2). The serve is not modelled, and every point is won by player A with a probability that depends only on whether the point is crucial. This isolates the effect of crucial points. In Supplementary Section S12 we add the serve with the rules of each sport, as in earlier models of tennis in which the probability of a point depends on the server[10]. The serve changes the amplification by at most 8% and the paradox region exists in all four sports, so the conclusions of the model do not change.

Ordinary and crucial points

Player A wins each point independently, with probability $p_1$ if the point is ordinary and $p_2$ if it is crucial. A point is crucial at level $L \in\{\text{game}, \text{set}, \text{match}\}$ if the player who wins it, whichever of the two, closes a scoring unit of level $L$ or higher. A crucial point at the game level is therefore any game point, including break points and points at advantage. At deuce no player can close the game and the point is ordinary. Crucial points at the set and match levels are the subsets of game points that would also close a set or the match, so that the levels are nested. In the rally scoring sports the game and set levels coincide. With probability $p_2$ player A converts his own game, set and match points and saves those of the opponent. We also considered a variant in which player A plays with probability $p_2$ only the points he can close by winning them, his game, set and match points whoever serves, while the game, set and match points of the opponent are played with probability $p_1$, as ordinary points. The paradox region still exists in all four sports and at every level, although its opening angle is only 40% to 50% of that obtained when the crucial points of both players are considered (Supplementary Table S4 and Supplementary Fig. S1). For $p_1 = p_2$ the model reduces to the classical i.i.d. model.

Simulations and exact calculation

Notation. A match is played between player A and player B. Player A wins an ordinary point with probability $p_1$ and a crucial point with probability $p_2$, and B wins them with probabilities $1-p_1$ and $1-p_2$. When $p_1=p_2=p$, every point is won by A with the same probability $p$. $N_A$ and $N_B$ are the numbers of points won by A and by B in a match. They are random, and so is the length of the match in number of points, $N_A+N_B$. $P(p_1,p_2)$ is the probability that A wins the match. $F(p_1,p_2)=E[N_A]/(E[N_A]+E[N_B])$, where $E[\cdot]$ denotes the expected value, is the share of points won by A over many matches. It is a ratio of totals, like a share of points over a career, and not the average of the share of each match. In Eq. (1), $N=E[N_A+N_B]$ at $p=1/2$ is the mean number of points per match between two equal players.

Monte Carlo simulations. Matches are simulated point by point. The algorithm keeps the score of each match (points in the current game or tiebreak, games and sets) and, at each step, every unfinished match plays one point. The point is first classified as crucial or ordinary from the score (see Ordinary and crucial points), and A wins it if a uniform random number in $[0,1)$ is smaller than $p_2$ or $p_1$, respectively. The score is then updated with the rules of the sport. Many matches, each with its own $(p_1,p_2)$, are played simultaneously in vectorised form. For each match we record the winner, $N_A$, $N_B$ and the number of crucial points. Curves for a single $p$ were simulated with 20,000 matches per value of $p$, and the $(p_1,p_2)$ plane with 2,000 matches per point of a grid of step 0.02.

Exact calculation. Because the outcome of each point depends only on the current score, the score is a Markov chain and $P$ and $F$ can also be computed exactly. Let $s$ be a state of the score, $s_A$ and $s_B$ the states reached when A or B wins the next point, and $\rho(s)$ the probability that A wins that point, equal to $p_2$ if the point is crucial and to $p_1$ otherwise. The probability $V(s)$ that A wins the match from state $s$ obeys

with $V=1$ in the states in which A has won the match and $V=0$ in those in which B has won it. In the same way, the expected numbers of points that A and B will still win from state $s$, $a(s)$ and $b(s)$, obey

with $a = b = 0$ at the end of the match. Then $P = V(s_0)$, $E[N_A] = a(s_0)$ and $E[N_B] = b(s_0)$, where $s_0$ is the score at the start of the match. Equations (4) and (5) are solved backwards from the end of the match (dynamic programming). Two details make the calculation simple. First, the match is a sequence of scoring units (games and tiebreaks in tennis, sets in the rally scoring sports), and during a unit the score at the higher levels does not change. Whether a point is crucial therefore depends only on the score of the unit and on which levels each player would close by winning the unit. For example, a game point in tennis is also a set point if winning that game would win the set. Each unit is solved once and the units are then combined. Second, a unit without a cap can last indefinitely after deuce. Its last states (tie, A one point ahead, B one point ahead) form a loop, and Eqs. (4) and (5) for these three states are a system of three linear equations, solved exactly. Badminton games, capped at 30 points, have no loop. The exact calculation reproduces the classical probability of winning a tennis game[6], $p^4(1 + 4q + 10q^2) + 20p^5q^3/(1 - 2pq)$, with $q = 1 - p$, and the Monte Carlo estimates agree with the exact values within sampling error for every sport, level and definition of crucial point (Supplementary Section S4).

Grids and derivatives. $P$ and $F$ were computed exactly on a grid of step 0.01 in $[0.01, 0.99]^2$ and, around $(1/2, 1/2)$, on finer grids of step 0.002 (Figure 3) and of step 0.0005 to 0.001 (Figure 2d–Figure 2f). The partial derivatives of Eq. (2) were obtained by centred finite differences with $h = 10^{-4}$, for example $P_1 \approx[P(1/2 + h, 1/2) - P(1/2 - h, 1/2)]/(2h)$. The slope of Eq. (1) was obtained from the exact $P(p)$ with a centred difference of step 0.005 around $p = 1/2$.

Derivation of (1)

Consider the simplest contest between two players, in which $N$ points are played, with $N$ odd, and the player who wins more of them wins. If A wins each point with probability $p$, the number of points he wins follows a binomial distribution, and the probability that he wins the contest is

where $q=1-p$, $m=(N-1)/2$ and $C(N,k)$ is the binomial coefficient. The derivative of $P_N$ has a simple meaning. Raising the probability of one particular point changes the outcome only if that point is decisive, that is, if the other $N-1$ points are split evenly, $m$ for each player. This happens with probability $C(N-1,m)(pq)^m$. All $N$ points play the same role, so

The same result is obtained by differentiating Eq. (6) term by term, since all the terms but one cancel. Equation (7) is a particular case of a general property. In any scoring system, $\mathrm{d}P/\mathrm{d}p$ is the expected sum, over the points played, of the probability that each point is decisive, that is, the expected total importance of the points of a match in the sense of Morris[25]. At $p=1/2$, $pq=1/4$ and Stirling’s approximation[41], $C(2m,m)\approx4^m/(\pi m)^{1/2}$, gives

where the last step uses $m\approx N/2$. The approximation is very accurate. For $N=79$ the exact slope is 7.11 and Eq. (8) gives 7.09. The slope therefore grows as the square root of the number of points, because the probability that a given point is decisive decreases as $N^{-1/2}$ while the number of points grows as $N$.

The same result follows from the central limit theorem. The difference $D=N_A-N_B$ between the points won by the two players is a random walk of $N$ steps of $\pm1$, with mean $N(2p-1)$ and variance $4Npq$. A wins if $D>0$, so for large $N$, $P_N\approx\Phi\left(N(2p-1)/(4Npq)^{1/2}\right)$, where $\Phi$ is the standard normal cumulative distribution function. Near $p=1/2$, $4pq\approx1$ and $P_N\approx\Phi\left(2(p-1/2)N^{1/2}\right)$. Since $\Phi'(0)=(2\pi)^{-1/2}$, the slope at $p=1/2$ is $2N^{1/2}(2\pi)^{-1/2}=(2N/\pi)^{1/2}$.

Real scoring systems differ from this contest in two respects. First, the number of points is not fixed, and we take its mean at $p = 1/2$ (Supplementary Table S1). Second, the winner is not always the player who wins more points, and the hierarchy of games and sets redistributes importance among the points. Game, set and match points are much more important than the average point, and many other points are almost irrelevant. Equation (1) therefore assumes that the total importance of the points of a match is the same as in a contest of the same length decided by the total number of points. Figure 1b shows that this holds well. The exact slope is up to 5% higher than Eq. (1) in tennis, badminton and table tennis, and 3% lower in volleyball.

Data

We used two databases compiled by J. Sackmann[42], released under a CC BY-NC-SA 4.0 licence. As the original repositories were withdrawn in September 2026, we used an archival copy made in June 2026[43]. The two databases differ in content and size. The ATP match database gives, for each match of the main tour, aggregate statistics, namely the serve points played and won by each player and the break points faced and saved, but not the sequence or the score of the points. Point statistics are available from 1991. After removing retirements, walkovers and matches without statistics, it contains 97,950 matches of the seasons 1991–2025 and 2,559 players. All the analyses of players and seasons use this data set. The Grand Slam point by point database records the winner, the server and the score after every point, so that every crucial point can be identified, but it is much smaller. It covers 49 Grand Slam tournaments between 2011 and 2024 (the Australian Open and Roland Garros are not available after 2021), and in many tournaments only part of the matches were recorded, 5,336 men’s singles matches in total. We reconstructed the score before each point and classified every point as in the model, taking into account the rules of the final set of each tournament and year (advantage sets, tiebreaks at 6–6 or 12–12, and tiebreaks to ten points). As a validation, every point that ended a game or a set must be crucial at that level for its winner, which holds for 99.92% of games and 99.44% of sets. We excluded the Australian Open and Roland Garros files of 2018–2021 (1,012 matches), in which the order of points within matches is scrambled, and 262 matches that were not completed or had inconsistent scores (Supplementary Section S7). The final point by point data set contains 4,062 men’s singles matches (2011–2024). It is used only where the score of every point is needed, namely for the comparison of ordinary and crucial points in matches won with fewer points and for the pressure index with the full definition of crucial points.

Measures of performance

We use three measures of how well a player turns points into wins. Each one answers a different question.

Match conversion. The first question is whether a player wins more matches than the points he wins imply. His share of matches won does not answer it, because it is mostly determined by his share of points (Figure 4a). The i.i.d. model evaluated at his career share of points is not a good benchmark either, because that share is an average over opponents of very different strength (see Results). We therefore compare every match with its own benchmark. The match conversion of a player is

where the sum runs over his $n$ matches, $w_m = 1$ if he won match $m$ and $0$ otherwise, and $f_m$ is the share of points he won in that match. $\pi(f)$ is the probability of winning a match with a share of points $f$ when all points are i.i.d. It was estimated from $4 \times10^6$ simulated matches of each format, over three and over five sets, with $p$ uniformly distributed in $[0.3, 0.7]$. Matches in which both players won the same number of points were kept apart ($\pi= 1/2$), and the curve was made symmetric, $\pi(1 - f) = 1 - \pi(f)$. Each term $w_m - \pi(f_m)$ measures the surprise of one match. It is positive when the player won a match that his points made unlikely, and negative when he lost a match that his points made likely. The strength of the opponent enters only through $f_m$, so $C$ compares players who faced very different opponents. $C$ is the number of wins per match above those implied by the points. In the figures and tables it is given per 100 matches. A player who wins crucial points more often than ordinary ones has $C > 0$.

Pressure index. The second question is whether a player wins crucial points more often than ordinary ones, that is, whether $p_2 > p_1$ in the language of the model. The ATP match statistics record only one kind of crucial point, the break point, which is a game point for the receiver and is crucial for both players. The other crucial points of the model, such as the game points of the server (for example at 40–30 on his serve), set points and match points, are not recorded, so in the ATP data the index is restricted to break points. The raw difference between the shares of break points and of other points won is misleading, because break points are not a random sample of points. A player defends break points on his own serve, where he wins most points, and plays for them on his opponent’s serve, where he loses most points. A strong server plays most of his break points on return, and a weak server on serve, so their raw differences reflect this mix rather than their performance under pressure. The pressure index extends the break points over expectation of Sackmann[34], which are computed from the points won on return, to the break points defended on serve. Unlike the pressure indices that rate the importance of each game or point[30, 44], it is a property of the player and not of the situation. It compares every break point with the level of the player on the same kind of point:

where $B$ is the number of break points he won, $b_s$ and $b_r$ are the break points he played on serve and on return, and $r_s$ and $r_r$ are the shares of the other points that he won on serve and on return. The term in parentheses is the number of break points he would have won at his usual level. $I > 0$ means that he plays break points better than his other points, and $I$ is given in percentage points. In the Grand Slam data the same index is computed with the full definition of crucial points of the model (game, set and match points). Stability is assessed by correlating each index between odd and even seasons, and the standard deviation across players is corrected by subtracting the mean binomial variance. Sliding windows span ten consecutive seasons, with players with at least 50 matches in the window.

Close match balance. The third question is who benefits from the Federer paradox. Counting the matches that a player won with fewer than half of the points, $W$, and lost with more than half of the points, $L$, does not answer it, because both counts depend on how often he wins more points than his opponent. Federer won more points than his opponent in most of his matches, so he had many chances to lose such a match and few chances to win one. The close match balance subtracts what independent points would give:

where $E[W]$ is the sum of $\pi(f_m)$ over the matches with $f_m < 1/2$ and $E[L]$ is the sum of $1 - \pi(f_m)$ over the matches with $f_m > 1/2$. Its variance is that of the corresponding Poisson-binomial sums, $\sum\pi(1 - \pi)$. $\Delta$ is measured in matches, so that $\Delta= +10$ means that the player turned ten more close matches into wins than independent points predict. The balance is closely related to match conversion. Summing $w_m - \pi(f_m)$ over the matches with $f_m < 1/2$ gives $W - E[W]$, and over the matches with $f_m > 1/2$ it gives $-(L - E[L])$. The balance is therefore $nC$ computed without the matches in which both players won the same number of points. It expresses match conversion as a number of matches, in the terms of the Federer paradox, and it separates its two sources, the matches won with fewer points and those lost with more points.

Set measures. To see where the balance comes from, we used the scores of each match. A player can win most of his matches with fewer than half of the points if he wins his close sets and loses his other sets by large margins. For every player we computed the share of close sets won (sets decided by two games or fewer), the mean margin of the sets won and of the sets lost, their difference (the set margin asymmetry, positive when the sets he loses are more lopsided than those he wins) and the share of lost sets that ended 6–0 or 6–1.

Statistical analysis

The expected number of matches won with fewer than half of the points is the sum, over matches, of the i.i.d. probability $\pi(f_{\mathrm{low}})$ that the player with the smaller share of points $f_{\mathrm{low}}$ wins. Its variance is that of a Poisson-binomial distribution, $\sum\pi(1-\pi)$, from which the z-scores and the confidence intervals of the yearly rates are obtained. Temporal trends were tested with logistic regressions of the indicator for each match on the season (linear and quadratic terms, in decades), with and without the logit of $\pi(f_{\mathrm{low}})$ as a covariate to control for the closeness of matches, and with weighted least squares on the yearly excess. For the out of sample test, the pressure index of each player in each season was computed from all his other seasons (players with at least 200 break points in them), each match was oriented towards a randomly chosen player, and the outcome was regressed on the logit of $\pi(f)$ and on the difference between the two players’ indices. Confidence intervals of match conversion and of the pressure index of each player were obtained from 2,000 bootstrap resamples of his matches, and $P$-values were corrected for multiple comparisons with the Benjamini–Hochberg procedure. Correlation coefficients are reported with Fisher-z confidence intervals. The tiebreak overperformance of a player is the difference between the tiebreaks he won and those expected from the i.i.d. probability of winning a tiebreak to seven points at the share of points he won in each match, divided by the number of tiebreaks. The games paradox was computed from the scores of all completed matches of the main tour between 1968 and 2025. The contributions of the pressure index, the tiebreak overperformance and the set margin asymmetry to the close match balance were estimated with an ordinary least squares regression on standardised variables across the 328 players. To test the robustness of the benchmark to the serve, $\pi$ was also estimated from matches simulated with the serve, in which each player wins his service points with his own probability, conditioning on both the share of points and the serve dominance of the match, the sum of the shares of service points won by the two players (Supplementary Section S11).

Data availability

The ATP and Grand Slam data analysed in this study were compiled by J. Sackmann and are available under a CC BY-NC-SA 4.0 licence from an archival copy of the original repositories[43]. The processed data supporting the findings will be made available in a public repository upon publication.

Code availability

The simulation and analysis code (Python) is available at https://github.com/jmbuldu/crucial_point and will be made public upon publication.

Acknowledgements

J.M.B. and J.H.M. acknowledge support from the Ministerio de Ciencia e Innovación, Spain, under grant PID2023-147827NB-I00. We thank Jeff Sackmann for compiling and releasing the tennis data used in this work.

Author contributions

A.M., J.H.M. and J.M.B. conceived the study. A.M. performed the numerical simulations and the analysis of the data. J.H.M. developed the analytical treatment. J.M.B. supervised the work.

A.M. wrote the first draft of the manuscript, and all authors reviewed and edited it.

Competing interests

The authors declare no competing interests.

Table

Player	Matches played	Points (%)	Matches (%)	Won with fewer points	Lost with more points	Balance (z)	Close sets (%)	Asymmetry	Pressure	Tiebreak
Sam Querrey	676	50.6	53.6 (55.1)	14 (15.6)	28 (17.9)	−11.7 (−2.3)	50.3	−0.11	−0.7	−2.9
Marc Kevin Goellner	330	49.8	44.2 (47.9)	4 (8.1)	15 (7.5)	−11.6* (−3.4)	45.1	−0.12	+0.1	−9.9
Jonathan Stark	276	50.4	47.5 (51.6)	4 (6.0)	14 (4.5)	−11.5* (−4.1)	46.3	−0.17	−1.2	+0.5