Tennis has a weird scoring system. Point, game, set, match. This leads to lots of excitement, as mini-dramas unfold near the end of many games and sets. But it can also lead to unfair results. The better player sometimes loses. If tennis had a more straightforward scoring system, like basketball or cricket, just totting up the points scored, the results would be fairer. But just how unfair is the present system?
It turns out that the present tennis scoring system can vastly improve the chances of an underdog winning a match, including at Wimbledon. The effect is larger for men than for women, even though women play shorter matches at Grand Slam events, because men win more points on serve.
The two tables below show the percent of matches won with the present scoring system (actual), a points-based scoring system (points) and the gain for the underdog (gain), for a player who is either moderate, good or strong on serve (shown in the rows) against a player who is good, strong or exceptional on serve (shown in the columns).
Men | good on serve | strong on serve | exceptional on serve | |||||||
Win (%) | actual | points | gain | actual | points | gain | actual | points | gain | |
moderate on serve | 21.3 | 18.2 | 3.1 | 6.3 | 3.2 | 3.1 | 1.3 | 0.2 | 1 | |
good on serve | 22.6 | 17.3 | 5.3 | 7.1 | 2.5 | 4.6 | ||||
strong on serve | 23.3 | 15.7 | 7.7 | |||||||
Women | good on serve | strong on serve | exceptional on serve | |||||||
Win (%) | actual | points | gain | actual | points | gain | actual | points | gain | |
moderate on serve | 25.5 | 24.9 | 0.6 | 9.5 | 8.6 | 1 | 2.8 | 1.9 | 0.9 | |
good on serve | 25.6 | 24.3 | 1.3 | 10.3 | 8 | 2.3 | ||||
strong on serve | 26.7 | 23.5 | 3.2 |
For both men and women, the tables show that the chances of the underdog are boosted most in absolute terms if a player who is strong on serve plays one who is exceptional.
For men, a player who is strong on serve against one who is exceptional will win 23.3% of the time with the actual scoring system against only 15.7% of the time if the scoring were points-based. So his chances are boosted by 7.7 percentage points. For women, a player who is strong on serve against one who is exceptional will win 26.7% of the time with the actual scoring system against 23.5% of the time if the scoring were points-based. So her chances are boosted by 3.2 percentage points.
The chances of the underdog are boosted most in relative terms if player who is moderate on serve plays one who is exceptional, although the effect is small in absolute terms, and the player who is moderate on serve still only wins the match very rarely, even with the actual scoring system.
For men, a player who is moderate on serve against one who is exceptional will win 1.3% of the time with the actual scoring system against only 0.2% of the time if the scoring were points-based. So his chances increase six-fold. For women, a player who is moderate on serve against one who is exceptional will win 2.8% of the time with the actual scoring system against 1.9% of the time if the scoring were points-based. So her chances are increased by half.
How might this apply to Britain’s two top-25 ranked players at Wimbledon?
Kyle Edmund won 67.5% of his points on serve on grass in 2017 which puts him firmly in the ‘good on serve’ category. In the early rounds, when he would be the favourite to win his matches, the scoring system will work against him, making him 3 percentage points less likely to win. But if he gets through to the later rounds, when he is more likely to meet a player who is strong or exceptional on serve, the scoring system will give a bigger boost of 4-5 percentage points in his favour.
Johanna Konta won 63% of her points on serve in 2017, and will probably win a higher percentage on the grass at Wimbledon, which puts her in the ‘strong on serve’ category. In the early rounds, when she would be the favourite to win her matches, the scoring system will work slightly against her, making her 1-1.5 percentage points less likely to win. But if she gets through to the later rounds, when she is more likely to meet a player who is exceptional on serve, the scoring system will give a boost of 3 percentage points in her favour.
Should we be willing to put up with the unfairness of the unique scoring system in tennis because of the excitement it generates? Maybe we should consider the scoring system’s bias in favour of the underdog a feature rather than a bug. Other sports might even learn from it. Basketball is sometimes boring, and cricket seems to be on a perpetual search for a livelier format. But it does make sense to be aware of just how unfair the present tennis scoring system can potentially be.
Notes:
1. It has been known for some time that in 5% of matches on the ATP tour the loser wins more points than the winner. That is not what is being investigated here, as it can easily come about through the winner conserving energy in games or sets that he can afford to lose.
2. Matches are simulated by assuming each player has a given percentage chance of winning a point on serve that remains constant throughout the match. Each match is simulated 100,000 times, giving 90% confidence intervals for the match results of about plus or minus 0.2%.
3. The points-based system has the players taking turns to serve 6 times (like an ‘over’ in cricket), up to 25 times for men, and 15 times for women. A running total of points scored is kept and the winner is the first to 151 points for men, or 91 points for women. If the scores are tied at 150-150 for men or 90-90 for women, the match is decided by a coin toss. The maximum match length of 300 points for men or 180 points for women is the same as the average length of an actual 5 or 3 set match, which average 60 points per set.
4. The categories on serve for men and women are shown below:
Win % on serve men women
moderate 62.5 55.0
good 67.5 60.0
strong 72.5 65.0
exceptional 77.5 70.0
These categories are informed by ATP and WTA statistics. In the last year on grass, Kevin Anderson was the most dominant man on serve, winning 78% of his service points. On average women win about 8% fewer points on serve than men. The probability of winning a point when serving is about 62% for women winners at all 4 Grand Slam tournaments, and about 50% for losers. Men do better, with 70% for winners and 58% for losers. In 2018 so far, Naomi Broady has been the most dominant woman on serve, winning 70% of her service points. What actually matters for the model is how good each of the two players is on serve against the other player, rather than overall. This is much harder to assess from past statistics. So the categories in the table are there to show the important trends. Any other percentages on serve can easily be simulated by the model.
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“If tennis had a more straightforward scoring system, like basketball or cricket, just totting up the points scored, the results would be fairer. But just how unfair is the present system?”
Not unfair at all. One of the things the game tests is the ability to perform better on the crucial points. Without that it would (a) be a different game and (b) be more boring, but it would not be more ‘fair’. Under the current system both players have the same opportunity to win, but they have to do so by winning the important points, rather than just more points. That makes it a much more interesting psychological as well as physical test. But it doesn’t make it unfair, just because it’s testing something different.
Uh, the whole point of the exercise was to show that the game is more random than a straight points contest would be. Are you seriously saying that a game whose outcome is more random is NOT unfair?
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Why do you do simulations, with as a consequence an error in the win probabilities. Sure, computers are good enough these days to simulate very quickly, so the sampling errors can be kept small, but with little effort you can perfectly calculate the exact win probabilities under both systems.
For the simple points based system the probabilities are just a binomial. So if the win-percentage on serve are p_1 (for player 1) and p_2 respectively, the probability that player 1 wins k points equals
P(k) = sum_{n=0}^k binom{150}{n} p_1^n (1-p_1)^(150-n) binom{150}{k-n} p_2^(150-k+n) (1-p_2)^(k-n)
Thus the probability of player 1 winning the match is
1/2 * P(150) + sum_{k=151}^{300} P(k).
(For women of course, replace 150 by 90, and 300 by 180).
Checking your entries are indeed off by small amounts (it does seem your confidence has the proper size).
For calculating the win probabilities in the actual scoring system is a bit harder, but can still be done without much trouble.
Too often I see people model things and simulate where we can easily calculate probabilities exactly. I realize many questions are too hard for this, but when we can explicitly calculate results we should strive to do so. As scientists we owe it to the people to always use the best, most accurate, tools there are. And here a direct calculation costs less resources (sure my expression means adding up 10’s of thousands of numbers, but that’s still quicker than simulating a 100,000 matches) and is more accurate.