Which points in a service game actually matter
Every point counts the same on the scoreboard and nothing like the same in the game. A point’s importance is how far it moves the server’s chance of holding: the chance they hold if they win this point, minus the chance they hold if they lose it. That is a question the scoring system answers exactly, once you say how strong the server is.
Importance of the next point, in percentage points of the chance of holding
v returner
At 0-15, serving at 60%
34.6
percentage points: how far this one point swings the server’s chance of holding
The server holds 57.6% of the time from here. Win this point and it is 71.4%; lose it and it is 36.9%. That makes it the 6th most important of 15 positions in a service game.
What this does not say about real players
This is what the scoring system does to a player who plays every point the same. Real servers don’t, quite. At Wimbledon from 1992 to 1995, servers won fewer points on important points than on ordinary ones. By the paper’s estimates, at 30-40 in the first game of a match the chance of winning the point fell by 0.8%, and at 5-5, 30-40 in a deciding set by 4.6% for men and 4.8% for women. The weaker the player, the bigger the effect (Klaassen and Magnus, Journal of the American Statistical Association, 2001). That is one tournament, on one surface, and the authors say themselves that it may limit how far their conclusions carry.
So the table above tells you which points swing a service game, not which points players handle badly. The paper draws its own lesson from finding that they do, and it is the only advice this page offers: “players should be trained to ‘play every point as it comes.’” That is advice to become the player this model already describes.
The biggest point is 30-40, and it is worth exactly the deuce
Save a break point at 30-40 and you are at deuce; lose it and the game is gone. So its importance is exactly the server’s chance of winning from deuce: an importance of 69.2 percentage points, where the chance from deuce at 60% is 69.2% — the same number twice. It is the most important point in a service game at every serve strength above even. At 60% the next biggest is 30-30, at 46.2. What the deuce loop is worth is on the deuce page.
Which points matter depends on who is serving
The surprise is lower down. For a server winning fewer than 65.0% of points on serve, the point at 0-15 matters more than the point at 0-40 — a triple break point. Above that the order flips.
At 60%, 0-15 is worth 34.6 percentage points of the hold and 0-40 only 24.9. At 0-40 this server holds just 15.0% of the time, so the game is mostly gone whatever happens next. At 0-15 it is still in the balance: win the point and they hold 71.4%, lose it and 36.9%.
The same instability shows up if you try to list the “big points”. Count the scores at least as important as the least important break point: at 55% there are 13 of the 15 positions in a game; at 80% there are 3 — the break points themselves, and nothing else. Which points count as big is not a fixed list; it is a property of the server.
| Server wins | 0-15 | 0-40 | Scores as big as a break point |
|---|---|---|---|
| 50% | 31.3 | 12.5 | 15 of 15 |
| 55% | 34.0 | 18.1 | 13 of 15 |
| 60% | 34.6 | 24.9 | 11 of 15 |
| 65% | 32.7 | 32.8 | 6 of 15 — on the crossover |
| 70% | 28.7 | 41.4 | 5 of 15 |
| 75% | 23.2 | 50.6 | 3 of 15 |
| 80% | 16.9 | 60.2 | 3 of 15 |
“Hasn’t faced a break point all match” — is that unusual?
Commentators say it as if it were rare. Whether it is depends almost entirely on who is serving. A server winning 60% of points gets through a single service game without facing a break point 61.0% of the time — but through a whole set only 9.9%, and through a best-of-three match between equals 0.93%, about one match in 108.
| Server wins | One game | A set | Best of three | Best of five |
|---|---|---|---|---|
| 50% | 39.6% | 2.9% | 0.08% | under 0.01% |
| 55% | 50.2% | 5.5% | 0.30% | 0.02% |
| 60% | 61.0% | 9.9% | 0.93% | 0.09% |
| 65% | 71.1% | 17.5% | 2.56% | 0.40% |
| 70% | 80.2% | 29.8% | 6.63% | 1.53% |
| 75% | 87.6% | 46.9% | 16.63% | 5.74% |
| 80% | 93.2% | 66.2% | 36.53% | 19.42% |
Scroll the table sideways for every column.
The same statistic is routine for a big server and remarkable for an ordinary one. Through a best-of-three match it happens about once in 15 matches at 70% and once in 108 at 60% — 7 times as often, from a ten-point difference in serve. The set and match figures count the service games a set actually contains rather than assuming six, and a tiebreak has no break points in it at all. Match figures are between two players of the same strength.
Important for the game, or for the match?
Everything above is importance for the game. When the research literature says importance it means importance for the match — the same difference, taken on the chance of winning the whole thing — a definition due to Morris (1977). The two are not close in size, because one game is a small part of a match. Klaassen and Magnus report an average importance across all the points they studied of 0.028 for men and 0.034 for women — about 3 percentage points of the chance of winning the match — and the distribution is skewed, which they read as “important points are rare”.
Here are the three scores their figures are quoted at, worked out under the scoring rules Wimbledon used at the time (no tiebreak in a deciding set) and at the serve strengths the paper gives for two average players, 65% for men and 56% for women.
| 30-40, at | For the game (pp) | For the match (pp) | Server’s drop, as estimated |
|---|---|---|---|
| First game of the match (men) | 77.5 | 9.1 | 0.8% |
| 5-5 in the first set (men) | 77.5 | 14.5 | 1.5% |
| 5-5 in the deciding set (men) | 77.5 | 38.8 | 4.6% |
| First game of the match (women) | 61.8 | 7.9 | 0.8% |
| 5-5 in the first set (women) | 61.8 | 15.5 | 2.1% |
| 5-5 in the deciding set (women) | 61.8 | 30.9 | 4.8% |
Scroll the table sideways for every column.
To the game, all three are the same point: 30-40 is 30-40, worth 77.5 percentage points of the hold to a 65% server wherever it falls. To the match they are nothing alike. At 30-40 in the first game of a men’s match it is worth 9.1 percentage points of the match; at 5-5 in the fifth set, 38.8 — 4 times as much. The paper’s estimated drop rises with it in both tours — but that is how its model is built, with the drop rising in a straight line with a point’s importance, so the column restates the model rather than adding evidence. The evidence is the model’s fit to 86,298 real points.
How this was checked: the paper treats the server’s drop as growing in a straight line with a point’s match importance, and prints enough figures to work backwards from each drop to the importance it implies. For all six scores, the importance computed here sits inside the range those figures allow once their rounding is accounted for — 7.9 percentage points against an implied 7.9 for a women’s first game, for example. It is the one check against outside numbers that tennis offers, and it is asserted every time the data is built. It checks this site’s engine, not the paper’s conclusions about players.
What the model assumes
Every figure assumes each point is an independent trial at a fixed strength for the server. That is an assumption, not a fact about tennis. A study of 86,298 points at Wimbledon from 1992 to 1995 (Klaassen and Magnus, "Are Points in Tennis Independent and Identically Distributed?", Journal of the American Statistical Association, 2001) found that real points are not quite like this, and that the divergence is small. Read these figures as exact for players who serve every point at the same strength.