Research – Page 5 – Heavy Topspin

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:

If a set has reached 3-3, the pressure is starting to mount, and the server is less likely to hold serve.
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:

the 3-3 server’s chance of winning the set before the 3-3 game
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?

Should Andy Murray Skip the Tour Finals to Prepare for Davis Cup?

After advancing to the Davis Cup final, Andy Murray floated the idea that he might skip the World Tour Finals to prepare. The Belgian hosts are likely to choose clay for November’s Davis Cup tie (in part to make Murray less comfortable), and if Murray reached the final round in London the week before, he would have only four days off to recover and adjust to the different surface.

A lot of factors will go into Murray’s ultimate decision: how much importance he gives each event, how much he thinks fatigue will affect him, and how likely it is that the ATP would sanction him for skipping a required event. For today, I’ll have to ignore all of those and focus on the one most amenable to analysis: The effect of switching surfaces right before a Davis Cup tie.

Shifting from one surface to another immediately before Davis Cup is common. From 2009 to the present, there have been just over 2,000 World Group, Group 1, and Group 2 Davis Cup singles rubbers, and almost 450 of those involved at least one player who had played the previous week [1] on a different surface. It’s very rare that both players switched surfaces, so we have a sample of 432 matches in which one player changed surfaces from the previous week, and the other player either played or (presumably) prepared on the same surface.

At the simplest level of analysis, the switchers have been surprisingly effective. In those 432 matches between switchers and non-switchers, the switchers won 275, or 63.6% of the time. When we narrow the sample to the 130 times the switcher reached at least the round of 16 the week before Davis Cup (and, thus, had even less time to adjust), the results are surprisingly similar: 82 wins, or 63.1% in favor of the switchers.

Of course, there are all sorts of biases that could be working in favor of the switchers. The better the player, the less likely he can change his schedule to better prepare for Davis Cup, leaving him stuck on the “wrong” surface the week before a tie. And the better the player, the more likely he was a switcher in the smaller sample, one of those who reached the round of 16 the week before.

To evaluate the effect of switching, then, we must proceed with more subtlety. If switchers are more likely to be the favorites, we need to consider each player’s skill level and estimate how often switchers should have won. To do that, we can use JRank, my player rating system with surface-specific estimates for each competitor.

Immediately, we lose about 15% of our sample due to matches involving at least one player who didn’t have a rating at the time [2]. These are almost all Group 2 matches, so its doubtful that we lose very much. In the slightly smaller pool of 361 matches, the switcher won 62.0%, and when the switcher reached the round of 16 the previous week, he won 60.0%.

JRank confirms that the sample is strongly biased toward switchers. The player changing surfaces was favored in 69.8% of these contests. To take an extreme example, Murray went from hard courts at the 2013 US Open to clay courts in the World Group playoff against Croatia. Against Borna Coric, who hadn’t played the week before, Murray was a 99.1% favorite, and of course he won the match.

Once we calculate the probability that the switcher won each of the 361 matches, it turns out that the switchers “should have” won 227, or 62.8% of the time. That’s almost indistinguishable from the historical record, when the switchers won 224 matches. In the smaller sample of 120 matches when the switcher reached the round of 16 the previous week, switchers “should have” won 72 matches. As it happened, they won exactly 72.

In other words, it doesn’t appear to be a disadvantage to play Davis Cup matches on an unfamiliar surface. JRank-based predictions are primarily based on “regular” matches, so if switchers are performing at the level that JRank forecasts for them, they’re playing as well as they would at, say, the third round of a Slam, when the surface is familiar.

This isn’t a clear answer to Murray’s dilemma, of course. If he plays, say, Roger Federer and Novak Djokovic in back-to-back three-setters on Saturday and Sunday, then travels to a different venue, handles tons of press, and practices with a different set of coaches and fellow players before a big match the following Friday, he faces more of a challenge than your typical surface-switcher in our dataset.

However, there’s little evidence that surface-switching alone is a good reason to skip the Tour Finals. If history is any guide, Murray will play very well on the Belgian clay–just as well as he would at the same venue in the middle of the clay season.

Continue reading Should Andy Murray Skip the Tour Finals to Prepare for Davis Cup?

The Pivotal Point of 15-30

According to nearly every tennis commentator I’ve ever heard, 15-30 is a crucial point, especially in men’s tennis, where breaks of serve are particularly rare. One reasonable explanation I’ve heard is that, from 15-30, if the server loses either of the next two points, he’ll face break point.

Another way of looking at it is with a theoretical model. A player who wins 65% of service points (roughly average on the ATP tour) has a 62% chance of winning the game from 15-30. If he wins the next point, the probability rises to 78% at 30-all, but if he loses the next point, he will only have a 33% chance of saving the game from 15-40.

Either way, 15-30 points have a lot riding on them. In line with my analysis of the first point of each game earlier this week, let’s take a closer look at 15-30 points–the odds of getting there, the outcome of the next point, and the chances of digging out a hold, along with a look at which players are particularly good or bad in these situations.

Reaching 15-30

In general, 15-30 points come up about once every four games, and no more or less often than we’d expect. In other words, games aren’t particularly likely or unlikely to reach that score.

On the other hand, some particular players are quite a bit more or less likely. Oddly enough, big servers show up at both extremes. John Isner is the player who–relative to expectations–ends up serving at 15-30 the most often: 13% more than he should. Given the very high rate at which he wins service points, he should get to 15-30 in only 17% of service games, but he actually reaches 15-30 in 19% of service games.

The list of players who serve at 15-30 more often than they should is a very mixed crew. I’ve extended this list to the top 13 in order to include another player in Isner’s category:

Player                 Games  ExpW  ActW  Ratio  
John Isner             3166    537   608   1.13  
Joao Sousa             1390    384   432   1.12  
Janko Tipsarevic       1984    444   486   1.09  
Tommy Haas             1645    368   401   1.09  
Lleyton Hewitt         1442    391   425   1.09  
Tomas Berdych          3947    824   894   1.08  
Vasek Pospisil         1541    361   390   1.08  
Rafael Nadal           3209    661   713   1.08  
Pablo Andujar          1922    563   605   1.08  
Philipp Kohlschreiber  2948    652   698   1.07  
Gael Monfils           2319    547   585   1.07  
Lukasz Kubot           1360    381   405   1.06  
Ivo Karlovic           1941    299   318   1.06

(In all of these tables, “Games” is the number of service games for that player in the dataset, minimum 1,000 service games. “ExpW” is the expected number of occurences as predicted by the model, “ActW” is the actual number of times it happened, and “Ratio” is the ratio of actual occurences to expected occurences.)

While getting to 15-30 this often is a bit of a disadvantage, it’s one that many of these players are able to erase. Isner, for example, not only remains the favorite at 15-30–his average rate of service points won, 72%, implies that he’ll win 75% of games from 15-30–but from this score, he wins 11% more often than he should.

To varying extents, that’s true of every player on the list. Joao Sousa doesn’t entirely make up for the frequency with which he ends up at 15-30, but he does win 4% more often from 15-30 than he should. Rafael Nadal, Tomas Berdych, and Gael Monfils all win between 6% and 8% more often from 15-30 than the theoretical model suggests that they would. In Nadal’s case, it’s almost certainly related to his skill in the ad court, particularly in saving break points.

At the other extreme, we have players we might term “strong starters” who avoid 15-30 more often than we’d expect. Again, it’s a bit of a mixed bag:

Player                 Games  ExpW  ActW  Ratio  
Dustin Brown           1013    249   216   0.87  
Victor Hanescu         1181    308   274   0.89  
Milos Raonic           3050    514   462   0.90  
Dudi Sela              1066    297   270   0.91  
Richard Gasquet        2897    641   593   0.93  
Juan Martin del Potro  2259    469   438   0.93  
Ernests Gulbis         2308    534   500   0.94  
Kevin Anderson         2946    610   571   0.94  
Nikolay Davydenko      1488    412   388   0.94  
Nicolas Mahut          1344    314   297   0.94

With some exceptions, many of the players on this list are thought to be weak in the clutch. (The Dutch pair of Robin Haase and Igor Sijsling are 12th and 13th.) This makes sense, as the pressure is typically lowest early in games. A player who wins points more often at, say, 15-0 than at 40-30 isn’t going to get much of a reputation for coming through when it counts.

The same analysis for returners isn’t as interesting. Juan Martin del Potro comes up again as one of the players least likely to get to 15-30, and Isner–to my surprise–is one of the most likely. There’s not much of a pattern among the best returners: Novak Djokovic gets to 15-30 2% less often than expected; Nadal 1% less often, Andy Murray exactly as often as expected, and David Ferrer 3% more often.

Before moving on, one final note about reaching 15-30. Returners are much less likely to apply enough pressure to reach 15-30 when they are already in a strong position to win the set. At scores such as 0-4, 0-5, and 1-5, the score reaches 15-30 10% less often than usual. At the other extreme, two of the games in which a 15-30 score is most common are 5-6 and 6-5, when the score reaches 15-30 about 8% more often than usual.

The high-leverage next point

As we’ve seen, there’s a huge difference between winning and losing a 15-30 point. In the 290,000 matches I analyzed for this post, neither the server or returner has an advantage at 15-30. However, some players do perform better than others.

Measured by their success rate serving at 15-30 relative to their typical rate of service points won, here is the top 11, a list unsurprisingly dotted with lefties:

Player             Games  ExpW  ActW  Ratio  
Donald Young       1298    204   229   1.12  
Robin Haase        2134    322   347   1.08  
Steve Johnson      1194    181   195   1.08  
Benoit Paire       1848    313   336   1.08  
Fernando Verdasco  2571    395   423   1.07  
Thomaz Bellucci    1906    300   321   1.07  
John Isner         3166    421   449   1.07  
Xavier Malisse     1125    175   186   1.06  
Vasek Pospisil     1541    243   258   1.06  
Rafael Nadal       3209    470   497   1.06  
Bernard Tomic      2124    328   347   1.06

There’s Isner again, making up for reaching 15-30 more often than he should.

And here are the players who win 15-30 points less often than other service points:

Player                  Games  ExpW  ActW  Ratio  
Carlos Berlocq          1867    303   273   0.90  
Albert Montanes         1183    191   173   0.91  
Kevin Anderson          2946    377   342   0.91  
Guillermo Garcia-Lopez  2356    397   370   0.93  
Roberto Bautista-Agut   1716    264   247   0.93  
Juan Monaco             2326    360   338   0.94  
Matthew Ebden           1088    186   176   0.94  
Grigor Dimitrov         2647    360   341   0.95  
Richard Gasquet         2897    380   360   0.95  
Andy Murray             3416    473   449   0.95

When we turn to return performance at 15-30, the extremes are less interesting. However, returning at this crucial score is something that is at least weakly correlated with overall success: Eight of the current top ten (all but Roger Federer and Milos Raonic) win more 15-30 points than expected. Djokovic wins 4% more than expected, while Nadal and Tomas Berdych win 3% more.

Again, breaking down 15-30 performance by situation is instructive. When the server has a substantial advantage in the set–at scores such as 5-1, 4-0, 3-2, and 3-0–he is less likely to win the 15-30 point. But when the server is trailing by a large margin–0-3, 1-4, 0-4, etc.–he is more likely to win the 15-30 point. This is a bit of evidence, though peripheral, of the difficulty of closing out a set–a subject for another day.

Winning the game from 15-30

For the server, getting to 15-30 isn’t a good idea. But compared to our theoretical model, it isn’t quite as bad as it seems. From 15-30, the server wins 2% more often than the model predicts. While it’s not a large effect, it is a persistent one.

Here are the players who play better than usual from 15-30, winning games much more often than the model predicts they would:

Player             Games  ExpW  ActW  Ratio  
Nikolay Davydenko  1488    194   228   1.17  
Steve Johnson      1194    166   190   1.14  
Donald Young       1298    163   185   1.13  
John Isner         3166    423   470   1.11  
Nicolas Mahut      1344    172   188   1.09  
Benoit Paire       1848    266   288   1.08  
Lukas Lacko        1162    164   177   1.08  
Rafael Nadal       3209    450   484   1.08  
Martin Klizan      1534    201   216   1.08  
Feliciano Lopez    2598    341   367   1.07  
Tomas Berdych      3947    556   597   1.07

Naturally, this list has much in common with that of the players who excel on the 15-30 point itself, including many lefties. The big surprise is Nikolay Davydenko, a player who many regarded as weak in the clutch, and who showed up on one of the first lists among players with questionable reputations in pressure situations. Yet Davydenko–at least at the end of his career–was very effective at times like these.

Another note on Nadal: He is the only player on this list who is also near the top among men who overperform from 15-30 on return. Rafa exceeds expectations in that category by 7%, as well, better than any other player in the last few years.

And finally, here are the players who underperform from 15-30 on serve:

Player               Games  ExpW  ActW  Ratio  
Dustin Brown         1013    122   111   0.91  
Tommy Robredo        2140    289   270   0.93  
Alexandr Dolgopolov  2379    306   288   0.94  
Federico Delbonis    1110    157   148   0.94  
Juan Monaco          2326    304   289   0.95  
Simone Bolelli       1015    132   126   0.96  
Paul-Henri Mathieu   1083    155   148   0.96  
Gilles Muller        1332    179   172   0.96  
Carlos Berlocq       1867    256   246   0.96  
Grigor Dimitrov      2647    333   320   0.96  
Richard Gasquet      2897    352   339   0.96

Tentative conclusions

This is one subject on which the conventional wisdom and statistical analysis agree, at least to a certain extent. 15-30 is a very important point, though in context, it’s no more important than some of the points that follow.

These numbers show that some players are better than others at certain stages within each game. In some cases, the strengths balance out with other weaknesses; in others, the stats may expose pressure situations where a player falters.

While many of the extremes I’ve listed here are significant, it’s important to keep them in context. For the average player, games reach 15-30 about one-quarter of the time, so performing 10% better or worse in these situations affects only one in forty games.

Over the course of a career, it adds up, but we’re rarely going to be able to spot these trends during a single match, or even within a tournament. While outperforming expectations on 15-30 points (or any other small subset) is helpful, it’s rarely something the best players rely on. If you play as well as Djokovic does, you don’t need to play even better in clutch situations. Simply meeting expectations is enough.

How Important is the First Point of Each Game?

Italian translation at settesei.it

A common belief among players, coaches, and commentators is that the first point of each game is of particular importance. It’s often suggested that the first point sets the tone for the entire game.

Of course, winning the first point is better than losing it, but that’s not what I’m talking about. Winning any point is better than losing it. If the first point is more important than the others, winning it would have to give a player even more of an advantage than the simple fact of having reached 15-0 instead of 0-15.

The difference between 15-0 and 0-15–apart from any momentum it generates–is a substantial one. Using a theoretical model that treats each point as independent, a player who typically wins 60% of service points will hold about 74% of the time, meaning that at love-all, they have a 74% of winning the game. At 15-0, that probability jumps to 84%. At 0-15, it’s only 58%.

To say that the first point is particularly important, then, is to say that the gap between winning and losing it is even greater than that. On the evidence of over 20,000 recent ATP and WTA matches, covering nearly half a million games, though, the first point is no more important than it should be. Except for, possibly, a few players and a few in-match situations, it gives no momentum to either player.

The basics

The broadest finding is perhaps the most surprising. Winning the first point fails to give the server any extra advantage, but losing the first point does. The results for ATP matches and WTA matches are the same. If the server loses the first point, he or she is then about one percent more likely to win the game than if points were truly independent of each other.

Naturally, this is not a recommendation that a server should lose the first point of any game! For our 60% server, winning the first point still improves her odds of a hold to 84%. But instead of the 58% chance at 0-15 that the theoretical model predicts, it’s really between 58.5% and 59%.

An effect of this size is not something that one would ever notice simply watching tennis matches. It probably doesn’t have any practical import, either. But over multiple very large samples of recent professional matches, the effect demonstrates that winning the first point of a game does not endow a player with any additional benefits.

Situations where it matters

In general, the first point is only as valuable as its immediate effect on the score. However, there are certain situations where winning it seems to give the server a bit more of an edge, or where losing it isn’t the disadvantage that it should be.

The latter situation is most pronounced. In both men’s and women’s tennis, servers outperform the theoretical model when serving down two breaks, at scores such as 0-4, 0-5, and 1-5. They beat the model to a much lesser, but still real, extent when serving down one break. This could be due to their acknowledgement that these games are “must wins,” or in the double-break situations, to a lack of effort on the part of the returner.

Regardless of the reason, with a double-break disadvantage, the effect of going down 0-15 is much less than in the model. Our 60% server, instead of facing a choice between an 84% chance of winning at 15-0 or 58% at 0-15, is looking at a 91% chance of winning at 15-0 or a 71% chance of winning at 0-15.

When serving with the break advantage, the situation is reversed, but it is much less pronounced. At scores such as 6-5 and 3-2, the model is a good predictor of win probability from 15-0, but servers underperform against the model from 0-15. The difference, though only a few percentage points, could be due to more aggression or focus on the part of the returner, or to the server feeling nerves.

At the majority of the most common scores, though, the effect of the first point is no different than the aggregate numbers, with the first point having almost no effect beyond the score.

Susceptible servers

There are a few players for whom the first point does seem to have an extra effect. These fall into two categories: players who fit the conventional wisdom, doing much better (compared to the model) from 15-0 than from 0-15, and those who are the opposite, reducing the gap between the likely outcomes from 15-0 and 0-15.

Among the 38 ATPers for whom I have more than 2,000 recorded service games, the player in the first category who sees the greatest first-point effect is Richard Gasquet. From 15-0, he beats the model by about one percent, but from 0-15, he underperforms by five percent. He’s the only male player whose gap between these two figures is more than five percent.

At the other end of the spectrum is Santiago Giraldo, who from 15-0 underperforms against the model by two percent, but from 0-15, beats the model by seven percent.

The rest of Giraldo’s category is where things get interesting. The other four players with a gap of four percent or greater are Feliciano Lopez, John Isner, Juan Martin del Potro, and Rafael Nadal. It’s no surprise to see two lefties here, as left-handers typically win more points in the ad court. Every other lefty in the dataset fits the same pattern, though their gaps are smaller.

The presence of big servers at this end of the list is a bit tougher to explain. Because they are so likely to hold in any given service game, perhaps they are sometimes unfocused on the first point of a game and become more serious after falling to 0-15.

Among WTA players, the distribution is about the same. The most extreme effect is on the serve of Sorana Cirstea, who, like Giraldo, is much more effective (compared to the model) from 0-15 than from 15-0. The other women in this category with more than a five percent gap are Flavia Pennetta, Ekaterina Makarova, and Ana Ivanovic.

At the other extreme, in Gasquet’s category, are Francesca Schiavone, Li Na, Julia Goerges, and Eugenie Bouchard, all of whom are about two percent more effective than expected from 15-0, and four percent less effective than expected from 0-15.

Conventional overstatement

As is so often the case, the conventional wisdom proves to have a grain of truth in it … sometimes, maybe, and to a much lesser extent than is generally claimed. Even the most extreme effect on tour, like that of Gasquet or Cirstea, doesn’t change the result of a game more than once every two or three matches.

The first point of a game is quite meaningful, because 15-0 is so much better than 0-15. But except for a few players and a few situations–some of which actually shrink the gap between 15-0 and 0-15–there’s little truth to the common claim that the first point is more important than its mere effect on the scoreline.

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

Italian translation at settesei.it

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.

Break Point Persistence: Why Venus is Better Than Her Ranking

Some points matter a lot more than others. A couple of clutch break point conversions or a well-played tiebreak make it possible to win a match despite winning fewer than half of the points. Even when such statistical anomalies don’t occur, one point won at the right time can erase the damage done by several other points lost.

Break points are among the most important points, and because tennis’s governing bodies track them, we can easily study them. I’ve previously looked at break point stats, with a special emphasis on Federer, here and here. Today we’ll focus on break points in the women’s game.

The first step is to put break points in context. Rather than simply looking at a percentage saved or converted, we need to compare those rates to a player’s serve or return points won in general. Serena Williams is always going to save a higher percentage of break points than Sara Errani does, but that has much more to do with her excellent service game than any special skills on break points.

Once we do that, we have two results for each player: How much better (or worse) she is when facing break point on serve, and how much better (or worse) she is with a break point on return.

For instance, this year Serena has won 2.8% more service points than average when facing break point, and 7.5% more return points than average with a break point opportunity. The latter number is particularly good–not only compared to other players, but compared to Serena’s own record over the last ten years, when she’s converted break points exactly as often as she has won other break points.

Serena’s experience isn’t unusual. From one year to the next, these rates aren’t persistent, meaning that most players don’t consistently win or lose many more break points than expected. Since 2006, Maria Sharapova has converted 1% fewer break points than expected. Caroline Wozniacki has recorded exactly the same rate, while Victoria Azarenka has converted 2% fewer break points than expected.

On serve, the story is similar, with a slight twist. Inexperienced players seem to perform a little worse when trying to convert a break point against a more experienced opponent, so most top players save break points about 4% more often than they win other service points. Serena, Sharapova, Wozniacki, Azarenka, and Petra Kvitova all have career rates at about this level.

Unlike in the men’s game, there’s little evidence that left-handers have a special advantage saving break points on serve. Angelique Kerber is a few percentage points above average, but Kvitova, Lucie Safarova, and Ekaterina Makarova are all within one percentage point of neutral.

While a few marginal players are as much as ten percentage points away from neutral saving break points or converting them, the main takeaway here is that no one is building a great career on the back of consistent clutch performances on break points. Among women with at least 250 tour-level matches in the last decade, only Barbora Strycova has won more than 3% more break points (serve and return combined) than expected. Maria Kirilenko is the only player more than 3% below expected.

This analysis doesn’t tell us anything very interesting about the intrinsic skills of our favorite players, but that doesn’t mean it’s without value. If we can count on almost all players posting average numbers over the long term, we can identify short-term extremes and predict that certain players will return to normal.

And that (finally) brings us to Venus Williams. Since 2006, Venus has played break points a little bit worse than average, saving 2% more break points than typical serve points (compared to +4% for most stars) and winning break points on return 3% less often than other return points.

But this year, Venus has saved break points 17% less often than typical service points, the lowest single-season number from someone who played more than 20 tour-level matches. That’s roughly once per match this year that Venus has failed to save a break point that–in an average year–she would’ve saved.

There’s no guarantee that saving those additional break points would’ve changed many of Venus’s results this year, but given the usual strength of her service game, holding serve even a little bit more would make a difference.

This type of analysis can’t say whether a rough patch like Venus’s is due to bad luck, mental lapses, or something else entirely, but it does suggest very strongly than she will bounce back. In fact, she already has. In her successful US Open run, she’s won about 66% of service points while saving 63% of break points. That’s not nearly as good as Serena’s performance this year, but it’s much closer to her own career average.

Like so many tennis stats that fluctuate from match to match or year to year, this is another one that evens out in the end. A particularly good or bad number probably isn’t a sign of a long-term trend. Instead, it’s a signal that the short-term streak is unlikely to last.

A Closer Look at the Winner-Unforced Error Ratio

Italian translation at settesei.it

Few tennis statistics are more frequently cited than winners and unforced errors. Nearly every broadcast displays them, and the ratio between the two numbers is discussed during matches as much as any other metric in the game.

If we set aside the problems with unforced errors, the winner-unforced error (W/UFE) ratio does appear to have some value. Winners are unquestionably good, so more winners must be better than fewer winners. Errors are definitely bad, so fewer is better.

It’s one small step from those anodyne assumptions to the conventional wisdom that a player should aim to tally more winners than unforced errors, resulting in a ratio of 1.0 or more.

Like any metric, this one isn’t perfect. With the help of detailed stats from over 1,000 matches in Match Charting Project data, we can take a closer look.

Is the W/UFE ratio all it’s cracked up to be?

If you compare two players’ W/UFE ratio, you’ll find that the player with the better ratio almost always wins. No surprise there, since winners and unforced errors directly represent points won and lost.

It isn’t perfect, though. In both men’s and women’s matches, the player with the lower W/UFE ratio wins the match 11% of the time. Winners and unforced errors only represent about 70% of total points, so if the remaining 30% of points tilt heavily in one direction–especially in a close match–we’ll see an unexpected result.

Things get a little messier when we test the magic W/UFE ratio of 1.0. That’s the number commentators cite all the time, as if it is the line between winning and losing. W/UFE ratios differ quite a bit by gender, so we’ll need to look at men and women separately.

In the 512 men’s matches logged by the Match Charting Project, players recorded a ratio of 1.0 or better only 41.3% of the time. In over a quarter of those “successes,” though, they lost the match. That means we have plenty of false positives and false negatives: losers who beat the target ratio as well as plenty of winners who failed to meet it.

Players who met or exceeded a 1.0 ratio won 74% of men’s matches. But the range just above the target–from 1.0 to 1.1–only resulted in wins about 60% of the time.

There’s no clear line separating a good ratio from a bad one: Even at 1.2 W/UFE, men only win about 70% of matches. As low as 0.8, they win nearly half.

Much of the problem here is that players influence each others’ numbers. Against a defensive baseliner, an average player will see his winners decrease and his unforced error count rise. In that hypothetical match, both players will have ratios below 1.0. Against an aggressive, big server, that same player will hit more winners, and because rallies end sooner, will tally fewer unforced errors. That scenario will often give you two ratios above 1.0.

A different story for women

In the sample of 552 women’s matches, players only recorded W/UFE ratios of 1.0 or better 26% of the time. Because the average ratio is so low–about 0.7–there aren’t very many false positives. Players who met the 1.0 standard won 89% of matches.

For women, a more reasonable target is in the 0.85 range. It’s roughly equivalent to 1.2 for men, in that a ratio at that level translates into about a 70% chance of winning.

There’s certainly no magic number. Even if we settle on revised targets like 0.85, winner and unforced error counts leave out too much data. In yesterday’s up-and-down match between Sara Errani and Jelena Ostapenko, Errani tallied 11 winners against 24 unforced. Ostapenko struck 54 winners against 49 unforced. A 0.46 ratio, like Errani’s, results in a win only 29% of the time, while a 1.1 ratio, like Ostapenko’s, is good for a victory 87% of the time. Yet, Errani is the one still standing.

Targeting the components

The Errani-Ostapenko match suggests another way of looking at the subject. Errani’s ratio was dreadful, but by keeping her unforced error rate low, she achieved at least half of the goal, leading to more Ostapenko errors. And while Ostapenko hit tons of winners, her own unforced error count was high enough to keep Errani in the match.

Looking at winners and unforced errors independently still doesn’t give us any magic numbers, but it does tell us more than the W/UFE ratio reveals by itself. Errani committed unforced errors on only 14% of points, which–taken by itself–results in a win about 70% of the time. Ostapenko’s error rate of 28% translates into success only 20% of the time.

By isolating the two components of the ratio, we can come up with clear targets for each. In women’s tennis, an error rate between about 14% and 16%–taken by itself–results in a 70% chance of winning. Consider winners independently, and we see that a winner rate of 19% to 20% also implies a 70% chance of victory.

These findings also cast a bit of light on another frequent question: Which is more important, increasing winners or decreasing errors? Based on this evidence, the answer is decreasing errors, but only by a whisker–and only in women’s matches. The player with more winners claims 68% of contests, while the player with fewer errors wins 73% of matches. A more sophisticated look, in which I separated all matches into buckets based on winner rate and error rate, suggests an even narrower margin. The relationship between error rate and winning percentage was very slightly stronger (r^2 = 0.92) than the relationship between winner rate and winning percentage (r^2 = 0.90).

Men’s components

For men, the 70% thresholds are different. Taken alone, a winner rate of about 22% will get you a 70% chance of winning. An unforced error percentage of 15% will achieve the same goal.

The relative importance of winners and unforced errors is different on the ATP tour, perhaps because aces–which are counted as winners–are such a large part of the game. Again, the difference is minor, but here, the relationship between winner rate and winning percentage is a bit stronger (r^2 = 0.94) than the relationship between error rate and winning percentage (r^2 = 0.92).

I’m almost done

Most men play plenty of matches in which they meet the W/UFE target of 1.0 and still lose. Most women fail to reach the 1.0 standard much of the time, and some players, like Errani, put together excellent careers despite almost never reaching it. We could do a lot better.

For a generic rule-of-thumb, the W/UFE target ratio of 1.0 isn’t horrible. But as we’ve seen, a slightly more nuanced view–one that takes into account the differences between men and women, as well as the independent value of winner rate and error rate–would be considerably more valuable.

The Myth of the Tricky First Meeting

Italian translation at settesei.it

Today, both Roger Federer and Stan Wawrinka will play opponents they’ve never faced before. In Federer’s case, the challenger is Steve Darcis, a 31-year-old serve-and-volleyer playing in his 22nd Grand Slam event. Wawrinka will face Hyeon Chung, a 19-year-old baseliner in only his second Slam draw.

For all those differences, both Federer and Wawrinka will need to contend with a new opponent–slightly different spins, angles, and playing styles than they’ve seen before. In the broadcast introduction to each match, we can expect to hear about this from the commentators. Something along the lines of, “No matter what the ranking, it’s never easy to play someone for the first time. He’s probably watched some video, but it’s different being out there on the court.”

All true, as even rec players can attest. But does it matter? After all, both players are facing a new opponent. While Darcis, for example, has surely watched a lot more video of Federer than Roger has of him, isn’t it just as different being out on the court facing Federer for the first time?

Attempting to apply common sense to the cliche will only get us so far. Let’s turn to the numbers.

Math is tricky; these matches aren’t

Usually, when we talk about “tricky first meetings,” we’re referring to these sorts of star-versus-newcomer or star-versus-journeyman battles. When two newcomers or two journeymen face off for the first time, it isn’t so notable. So, looking at data from the last fifteen years, I limited the view to matches between top-ten players and unseeded opponents.

This gives us a pretty hefty sample of nearly 7,000 matches. About 2,000 of those were first meetings. Even though the sample is limited to matches since 2000, I checked 1990s data–including Challengers–to ensure that these “first meetings” really were firsts.

Let’s start with the basics. Top-tenners have won 86.4% of these first meetings. The details of who they’re facing doesn’t matter too much. Their record when the new opponent is a wild card is almost identical, as is the success rate when the new opponent came through qualifying.

The first-meeting winning percentage is influenced a bit by age. When a top-tenner faces a player under the age of 24 for the first time, he wins 84.6% of matches. Against 24-year-olds and up, the equivalent rate is 88.0%. That jibes with what we’d expect: a newcomer like Chung or Borna Coric is more likely to cause problems for a top player than someone like Darcis or Joao Souza, Novak Djokovic‘s first-round victim.

The overall rate of 86.4% doesn’t do justice to guys like Federer. As a top-tenner, Roger has won 95% of his matches against first-time opponents, losing just 8 of 167 meetings. Djokovic, Rafael Nadal, and Andy Murray are all close behind, each within rounding distance of 93%.

By every comparison I could devise, the first-time meeting is the easiest type of match for top players.

The most broad (though approximate) control group consists of matches between top-tenners and unseeded players they have faced before. Favorites won 76.9% of those matches. Federer and Djokovic win 91% of those matches, while Nadal wins 89% and Murray 86%. In all of these comparisons, first-time meetings are more favorable to the high-ranked player.

A more tailored control group involves first-time meetings that had at least one rematch. In those cases, we can look at the winning percentage in the first match and the corresponding rate in the second match, having removed much of the bias from the larger sample.

Against opponents they would face again, top-tenners won their first meetings 85.1% of the time. In their second meeting, that success rate fell to 80.2%. It’s tough to say exactly why that rate went down–in part, it can be explained by underdogs improving their games, or learning something in the first match–but to make a weak version of the argument, it certainly doesn’t provide any evidence that first matches are the tough ones.

It may be true that first matches–no matter the quality of the opponent–feel tricky. It’s possible it takes more time to get used to first-time opponents, and that those underdogs are more likely to take a first set, or at least push it to a tiebreak. That’s a natural thing to think when such a match turns out closer than expected.

Whether or not any of that is true, the end result is the same. Top players appear to be generally immune to whatever trickiness first meetings hold, and they win such contests at a rate higher than any comparable set of matches.

Certainly, Fed fans have little to worry about. Most of his first-meeting losses were against players who would go on to have excellent careers: Mario Ancic, Guillermo Canas, Gilles Simon, Tomas Berdych, and Richard Gasquet.

His last loss facing a new opponent was his three-tiebreak heartbreaker to Nick Kyrgios in Madrid, only his third first-meeting defeat in a decade. As a rising star, Kyrgios fits the pattern of Fed’s previous first-meeting conquerors. Darcis, however, looks like yet another opponent that Federer will find distinctly not tricky.

Do Players Get Broken More Often After Failing to Convert Break Point?

The headline is a bit unwieldy, but it refers to one of the most common nuggets of conventional wisdom in tennis. When a player has the opportunity to break and doesn’t do so, this viewpoint holds that they are more likely to get broken in their following service game.

Like so much conventional wisdom, this assumes that momentum plays a role. Break points are crucial moments, and if a player doesn’t capitalize, the momentum will turn against him. That momentum then carries into the following game, and the player who failed to convert gets broken himself.

Or so the story goes.

However, data from almost 3,000 2013 tour-level and qualifying-round matches suggests the opposite. The likelihood that a player holds serve has almost nothing to do with what happened in the previous game.

Let’s start with some general numbers. To make sure we’re comparing apples to apples, I’ve ignored the first game of every set. This way, we compare “games after missed break point chances” to “games after breaks” to “games after holds.” In other words, we’re only concerned with “games after something.” I’ve also limited our view to sequences of games within the same set, since the long break between sets (not to mention other psychological factors) seem to put those multi-set sequences of games in a different category altogether.

Once those exclusions are made, this set of several thousand ATP matches showed that players got broken in 21.7% of their service games. Compare that to break rates after various events:

after a hold of serve: 22.6%
after a break of serve: 19.3%
after a hold including a missed break point chance: 21.2%
after a hold including three missed bp chances: 20.9%
after a hold including four or more missed bp chances: 19.4%

These are aggregate numbers, not adjusted for specific players, so they don’t tell the whole story. But they already suggest that the conventional wisdom is overstating its case. After failing to convert a break point, players hold serve almost exactly as often as they do in general. In fact, they get broken a bit less frequently in those situations (21.2%) than they do following a more conventional hold without any break points (22.6%).

Let’s see what happens when we adjust these numbers on a match-by-match basis. For example, if Tomas Berdych gets broken by Novak Djokovic 6 times in 15 tries, we can use that 40% break rate as a benchmark by which to measure more specific scenarios. If Berdych fails to convert break point twice, we would “expect” that he gets broken in 40% of his following service games, or 0.8 times in the two games. Of course, no one can get broken a fractional amount of a game, but by summing those “expected” breaks, we can see what the aggregate numbers look like with a much lesser chance of particular players or matchups biasing the numbers.

Once that cumbersome step is out of the way, we discover that–again, but more confidently–there is virtually no difference between average service games and service games that follow unconverted break points.

In my sample of 2013 ATP matches, there were 5,701 service games that followed missed break point opportunities. Players held 4,493 of those games (78.8%). That’s almost precisely the rate at which they held in other games. Had those specific players performed at their usual level within those matches, they would’ve held 4,488 times (78.7%).

We see the same findings when we focus on the most high-pressure games, ones with three or more break points. This sample contained 722 games in which the server held despite three break points. Servers held the following game 571 times. Had they performed at their usual, average-momentum rate, they would’ve held 570 times. After holds with four or more break points (206 in all), servers held 166 times instead of an “expected” 162.

There’s no evidence here that these particular service games have different results than other service games do.

Envoi

Momentum, the basis for so many of the beliefs that make up tennis’s conventional wisdom, is surely a factor in the game, but my research has shown, over and over again, that it isn’t nearly as influential as fans and pundits tend to think.

Once we hear a claim like this one, we tend to notice when events confirm it, reinforcing our mostly-baseless belief. When we see something that doesn’t match the belief, we’re surprised, often leading to a discussion that takes for granted the truth of the original claim. Our brains are wired to understand and tell stories, not to recognize the difference between something that happens 77% of the time and 79% of the time.

It may turn out that some players are unusually likely or unlikely to get broken after failing to convert a break point. Or perhaps this particular sequence of events is more common at certain junctures in a match. But barring research that establishes that sort of thing, there is simply no evidence that momentum plays any role in the service game following unconverted break points.

A Quick Look at the Odds of Three-Setters

In the comments to my match-fixing post earlier this week, Elihu Feustel commented:

There are almost no situations where a best of 3 match is a favorite to go to three sets. If the market priced a player as greater than 50% to win in exactly 3 sets, that alone is compelling evidence of match fixing.

In Monday’s questionable Challenger match, not only did the betting markets believe that the match was likely to go three sets, it picked a specific winner in three sets.

It takes only a bit of arithmetic to see why Elihu’s point is correct. Let’s say two players, A and B, are exactly evenly matched. Each one has a 50% chance of winning the match and a 50% chance of winning each set. Thus, the odds that A wins the match in straight sets are 25% (50% for the first set multiplied by 50% for the second). The odds that B wins in straights are the same. The probability that the match finishes in straight sets, then, is 50% (25% for an A win + 25% for B), meaning that the odds of a three-set match are also 50%.

As soon as one player has an edge, the probability of a three-set match goes down. Consider the scenario in which A has a 70% chance of winning each set. The odds that player A wins in straight sets are 49% (70% times 70%) and the odds that B wins in straight sets are 9% (30% times 30%). Thus, there’s a 58% chance of a straight-set victory, leaving a 42% chance of a three-setter.

This simple approach makes one major assumption: each player’s chances don’t change from one set to another. That probably isn’t true. It seems most likely to me that the player who wins the first set gets stronger relative to his opponent, perhaps because he gains confidence, or because his opponent loses confidence, or because he figures he doesn’t have much chance of winning. (I’m sure this isn’t true in all matches, but I suspect it applies often enough.)

If it’s true that the probability of the second set is dependent–even slightly–on the outcome of the first set, the likelihood of a three-setter decreases even further.

Probability in practice

As expected, far fewer than half of tour-level matches go three sets. (I’m considering only best-of-three matches.) So far this year, 36% of ATP best-of-threes have gone the distance, while only 32% of Challenger-level matches have done so.

In fact, men’s tennis has even fewer three-set matches than expected. For every match, I used a simple rankings-based model to estimate each player’s chances of winning a set and, as shown above, the odds that the match would go three sets. For 2014 tour-level matches, the model–which assumes that set probabilities are independent–predicts that 44% of matches would go three sets. That’s over 20% more third sets that we see in practice.

There are two factors that could account for the difference between theory and practice. I think both play a part:

Sets aren’t independent. If winning the first set makes a player more likely to win the second, there would be fewer three-setters than predicted.
There’s usually a bigger gap between players than aggregate numbers suggest. On paper, one player might have a 60% chance of winning the match, but on the day, one player might be tired, under the weather, unhappy with his racquets, uncomfortable with the court … or playing his best tennis, in a honeymoon period with a new coach, enjoying friendly calls from home line judges. The list of possible factors is endless. The point is that for any matchup, there are plenty of effectively random, impossible to predict variables that affect each player’s performance. I suspect that those variables are more likely to expand the gap between players–and thus lower the likelihood of a three-setter–than shrink it.

A note on outliers

Despite the odds against three-setters, some players are more likely go three than others. Among the 227 players who have contested 100 or more ATP best-of-threes since 1998, 20 have gone the distance in 40% of more of their matches. John Isner, tennis’s most reliable outlier, tops the list at 47.4%.

Big servers don’t dominate the list, but Isner’s presence at the top isn’t entirely by chance. After John, Richard Fromberg is a close second at 46.7%, while Goran Ivanisevic is not far behind at 43.0%. Mark Philippoussis and Sam Querrey also show up in the top ten.

It’s no surprise to see these names come up. One-dimensional servers are more likely to play tiebreaks, and tiebreaks are as close to random as a set of tennis can get. Someone who plays tiebreaks as often as Isner does will find himself losing first sets to inferior opponents and winning first sets against players who should beat him.

That randomness not only makes it more likely the match will go three sets, it’s also something the players are aware of. If Isner drops a first-set tiebreak, he realizes that he still has a solid chance to win the match–losing the breaker doesn’t mean he’s getting outplayed. If there is a mental component that partially explains the likelihood of the first-set winner taking the second set, it doesn’t apply to players like him.

Still, even Big John finishes sets in straights more than half the time. Every other tour regular does so as well, so it would take a very unusual set of circumstances for a betting market–or common sense–to favor a three-set outcome.