The Odds of Successfully Serving Out the Set

Serving for the set is hard … or so they say. Like other familiar tennis conceits, this one is ripe for confirmation bias. Every time we see a player struggle to serve out a set, we’re tempted to comment on the particular challenge he faces. If he doesn’t struggle, we ignore it or, even worse, remark on how he achieved such an unusual feat.

My findings–based on point-by-point data from tens of thousands of matches from the last few seasons–follow a familiar refrain: If there’s an effect, it’s very minor. For many players, and for some substantial subsets of matches, breaks of serve appear to be less likely at these purportedly high-pressure service games of 5-4, 5-3 and the like.

In ATP tour-level matches, holds are almost exactly as common when serving for the set as at other stages of the match. For each match in the dataset, I found each player’s hold percentage for the match. If serving for the set were more difficult than serving in other situations, we would find that those “average” hold percentages would be higher than players’ success rates when serving for the set.

That isn’t the case. Considering over 20,000 “serving-for-the-set” games, players held serve only 0.7% less often than expected–a difference that shows up only once every 143 attempts. The result is the same when we limit the sample to “close” situations, where the server has a one-break advantage.

Only a few players have demonstrated any notable success or lack thereof. Andy Murray holds about 6% more often when serving for the set than his average rate, making him one of only four players (in my pool of 99 players with 1,000 or more service games) to outperform his own average by more than 5%.

On the WTA tour, serving for the set appears to be a bit more difficult. On average, players successfully serve out a set 3.4% less often than their average success rate, a difference that would show up about once every 30 attempts. Seven of the 85 players with 1,000 service games in the dataset were at least 10% less successful in serving-for-the-set situations than their own standard.

Maria Sharapova stands out at the other end of the spectrum, holding serve 3% more often than her average when serving for the set, and 7% more frequently than average when serving for the set with a single-break advantage. She’s one of 30 players for whom I was able to analyze at least 100 single-break opportunities, and the only one of them to exceed expectations by more than 5%.

Given the size of the sample–nearly 20,000 serving-for-the-set attempts, with almost 12,000 of them single-break opportunities–it seems likely that this is a real effect, however small. Strangely, though, the overall finding is different at the lower levels of the women’s game.

For women’s ITF main draw matches, I was able to look at another 30,000 serving-for-the-set attempts, and in these, players were 2.4% more successful than their own average in the match. In close sets, where the server held a one-break edge, the server’s advantage was even greater: 3.5% better than in other games.

If anything, I would have expected players at lower levels to exhibit greater effects in line with the conventional wisdom. If it’s difficult to serve in high-pressure situations, it would make sense if lower-ranked players (who, presumably, have less experience with and/or are less adept in these situations) were not as effective. Yet the opposite appears to be true.

Lower-level averages from the men’s tour don’t shed much light, either. In main draw matches at Challengers, players hold 1.4% less often when serving for the set, and 1.8% less often with a single-break advantage. In futures main draws, they are exactly as successful when serving for the set as they are the rest of the time, regardless of their lead. In all of the samples, there are only a handful of players whose record is 10% better or worse when serving for the set, and a small percentage who over- or underperform by even 5%.

The more specific situations I analyze, the more the evidence piles up that games and points are, for the most part, independent–that is, players are roughly as effective at one score as they are at any other, and it doesn’t matter a great deal what sequence of points or games got them there. There are still plenty of situations that haven’t yet been analyzed, but if the ones that we talk about the most don’t exhibit the strong effects that we think they do, that casts quite a bit of doubt on the likelihood that we’ll find notable effects elsewhere.

If there is any truth to claims like those about the difficulty of serving for the set, perhaps it is the case that the pressure affects both players equally. After all, if a server needs to hold at 5-4, it is equally important for the returner to seize the final break opportunity. Maybe the level of both players drops, something we might be able to determine by analyzing how these points are played.

For now, though, we can conclude that players–regardless of gender or level–serve out the set about as often as they successfully hold at 1-2, or 3-3, or any other particular score.

How Important is the Seventh Game of the Set?

Few nuggets of tennis’s conventional wisdom are more standard than the notion that the seventh game of each set is particularly crucial. While it’s often difficult to pin down such a well-worn conceit, it seems to combine two separate beliefs:

  1. If a set has reached 3-3, the pressure is starting to mount, and the server is less likely to hold serve.
  2. The seventh game is somehow more important than its immediate effect on the score, perhaps because the winner gains momentum by taking such a pivotal game.

Let’s test both.

Holding at 3-3

Drawing on my database of over 11,000 ATP tour-level matches from the last few years, I found 11,421 sets that reached three-all. For each, I calculated the theoretical likelihood that the server would hold (based on his rate of service points won throughout the match) and his percentage of service games won in the match. If the conventional wisdom is true, the percentage of games won by the server at 3-3 should be noticeably lower.

It isn’t. Using the theoretical model, these servers should have held 80.5% of the time. Based on their success holding serve throughout these matches, they should have held 80.2% of the time. At three-all, they held serve 79.5% of the time. That’s lower, but not enough lower that a human would ever notice. The difference between 80.2% and 79.5% is roughly one extra break at 3-3 per Grand Slam. Not Grand Slam match–an entire tournament.

None of that 0.7% discrepancy can be explained by the effect of old balls [1]. Because new balls are introduced after the first seven games of each match, the server at three-all in the first set is always using old balls, which should, according to another bit of conventional wisdom, work against him. However, the difference between actual holds and predicted holds at 3-3 is slightly greater after the first set: 78.9% instead of the predicted 79.8%. Still, this difference is not enough to merit the weight we give to the seventh game.

The simple part of our work is done: Servers hold at three-all almost as often as they do at any other stage of a match.

Momentum from the seventh game

At 3-3, a set is close, and every game matters. This is especially true in men’s tennis, where breaks are hard to come by. Against many players, getting broken so late in the set is almost the same as losing the set.

However, the focus on the seventh game is a bit odd. It’s important, but not as important as serving at 3-4, or 4-4, or 4-5, or … you get the idea. The closer a game to the end of the set, the more important it is–theoretically, anyway. If 3-3 is really worth the hoopla, it must grant the winner some additional momentum.

To measure the effect of the seventh game, I took another look at that pool of 11,000-plus sets that reached three-all. For each set, I calculated the two probabilities–based on each player’s service points won throughout the match–that the server would win the set:

  1. the 3-3 server’s chance of winning the set before the 3-3 game
  2. his chance of winning the set after winning or losing the 3-3 game

In this sample of matches, the average server at three-all had a 48.1% chance of winning the set before the seventh game. The servers went on to win 49.4% of the sets [2].

In over 9,000 of our 3-3 sets, the server held at 3-3. These players had, on average, a 51.3% chance of winning the set before serving at 3-3, which rose to an average of a 57.3% chance after holding. In fact, they won the set 58.6% of the time.

In the other 2,300 of our sets, the server failed to hold. Before serving at three-all, these players had a 35.9% chance of winning the set, which fell to 12.6% after losing serve. These players went on to win the set 13.7% of the time. In all of these cases, the model slightly underestimates the likelihood that the server at 3-3 goes on to win the set.

There’s no evidence here for momentum. Players who hold serve at three-all are slightly more likely to win the set than the model predicts, but the difference is no greater than that between the model and reality before the 3-3 game. In any event, the difference is small, affecting barely one set in one hundred.

When a server is broken at three-all, the evidence directly contradicts the momentum hypothesis. Yes, the server is much less likely to win the set–but that’s because he just got broken! The same would be true if we studied servers at 3-4, 4-4, 4-5, or 5-5. Once we factor in the mathematical implications of getting broken in the seventh game, servers are slightly more likely to win the set than the model suggests. Certainly the break does not swing any momentum in the direction of the successful returner.

There you have it. Players hold serve about as often as usual at three-all (whether they’re serving with new balls or not), and winning or losing the seventh game doesn’t have any discernible momentum effect on the rest of the set [3]. Be sure to tell your friendly neighborhood tennis pundits.

Continue reading How Important is the Seventh Game of the Set?

The Effects (and Maybe Even Momentum) of a Long Rally

In yesterday’s quarterfinal between Simona Halep and Victoria Azarenka, a highlight early in the third set was a 25-shot rally that Vika finished off with a forehand winner. It was the longest point of the match, and moved her within a point of holding serve to open the set.

As very long rallies often do, the point seemed like it might represent a momentum shift. Instead, Halep sent the game back to deuce after a 10-stroke rally on the next point. If there was any momentum conferred by these two points, it disappeared as quickly as it arose. It took eight more points before Azarenka finally sealed the hold of serve.

Does a long rally tell us anything at all? Does it have predictive value for the next point, or even the entire game, or is it just highlight-reel fodder that is forgotten as soon as the umpire announces the score?

To answer those questions, I delved into the shot-by-shot data of the Match Charting Project, which now contains point-by-point accounts of nearly 1,100 matches. I identified the longest 1% of points–17 shots or longer for women, 18 shots for men–and analyzed what happened afterwards, looking for both fatigue and momentum effects.

The next point

There’s one clear effect of a long rally: The next point will be shorter than average. The 10-shot rally contested by Vika and Simona yesterday was an outlier: Women average 4.45 shots on the point after a long rally, while the overall average (controlled for server and first or second serve) is 4.85. Men average 4.03 shots on the following point, compared to an average of 4.64.

For women, fatigue is also a factor for the server. Following a long rally, women land only 61.3% of first serves, compared to an average of 64.6%. Men don’t exhibit the same fatigue effect; the equivalent numbers are 62.3% and 62.2%.

There’s more evidence of an immediate fatigue factor for women, as well. The players who win those long rallies are slightly better than their opponents, winning 50.7% of points on average. Immediately after a long rally, however, players win only 49% of points.  It’s not obvious to me why this should be the case. Perhaps the player who won the long rally worked a bit harder than her opponent, maybe putting all of her remaining effort into a groundstroke winner, or finishing the point with a couple of athletic shots at the net.

In any case, there’s no equivalent effect for men.  After winning a long rally, players win 51.1% of their next points, compared to an expected 50.8%. That’s either a very small momentum effect or, more likely, a bit of statistical noise.

Both men and women double fault more often than usual after a long rally, though the effect is much greater for women. Immediately following these points, women double fault 4.7% of the time, compared to an average of 3.3%. Men double fault 4.5% of the time after a long rally, compared to an expected rate of 4.2%.

Longer-term momentum

Beyond a slightly effect on the characteristics of the next point, does a long rally influence the outcome of the game? The evidence suggests that it doesn’t.

For each long rally, I identified whether the winner of the rally went on to win the game, as Vika did yesterday. I also combined the score after the long rally with the average rate of points won on the appropriate player’s serve to calculate the odds that, from such a score, the player who won the rally would go on to win the game. To use yesterday’s example, when Azarenka held game point at AD-40, her chances of winning the game were 77.6%.

For both men and women, there is no significant effect. Women who won long rallies went on to win 66.2% of those games, while they would have been expected to win 65.7%. Men won 64.4% of those games, compared to an expected rate of 64.1%.

With a much larger dataset, these findings might indicate a very slight momentum effect. But limited to under 1,000 long-rally points for each gender, the differences represent only a few games that went the way of the player who won the long point.

For now, we’ll have to conclude that the aftereffects of a long rally have a very short lifespan: barely one point for women, perhaps not even that long for men. These points may well have a greater effect on fans than they do on the players themselves.