Category: Research

Continuing with our point-by-point player profiles, let’s look at Andy Murray. The Scot finished strong and performed up to expectations at the grand slams despite a dreadful stretch following the Australian Open.

Using all of his grand slam matches from 2011, we can begin to analyze his tendencies on serve and return.

The first table shows the frequency of different outcomes in the deuce court, in the ad court, and on break point, relative to Murray’s average. For instance, the 1.014 in the upper left corner means that Murray wins 1.7% more points than average in the deuce court.

OUTCOME       Deuce     Ad  Break  
Point%        1.017  0.981  1.020  
                                   
Aces          1.034  0.963  1.048  
Svc Wnr       1.036  0.960  1.043  
Dbl Faults    1.104  0.886  0.872  
1st Sv In     1.007  0.993  0.957  
                                   
Server Wnr    1.009  0.990  0.860  
Server UE     0.968  1.035  1.013  
                                   
Return Wnr    0.775  1.246  0.558  
Returner Wnr  1.019  0.979  1.012  
Returner UE   0.988  1.013  1.003  
                                   
Rally Len     1.015  0.984  1.037  

Like most righties, Murray is a little better in the deuce court. The substantial difference in return winners hints at a larger issue: When he serves cautiously, he serves very cautiously, leading to horrible second-serve results. That’s a topic for another day.

What’s remarkable about the above table, though, is Andy’s results serving against break point. Sure, 2% better than average doesn’t sound like much, but keep in mind that when fighting off breakers, he’s generally playing his best opponents. As we’ve seen, both Nadal and Federer perform serve more than 10% worse than average on break point for this reason; Murray bucks that trend, all the more remarkable because most break points are in the ad court.

Next, this is how he performs on a point-by-point basis. Win% shows what percentage of points he wins at that score; Exp is how many he would be expected to win (given how he performs in each match), and Rate is the difference between the two. A rate above 1 means he plays better on those points; below 1 is worse.

SCORE   Pts   Win%    Exp  Rate  
g0-0    398  66.6%  65.5%  1.02  
g0-15   131  58.8%  64.5%  0.91  
g0-30    54  61.1%  63.3%  0.97  
g0-40    21  66.7%  61.0%  1.09  
                                 
g15-0   262  62.2%  66.0%  0.94  
g15-15  176  68.2%  65.1%  1.05  
g15-30   89  65.2%  63.5%  1.03  
g15-40   45  66.7%  61.5%  1.08  
                                 
g30-0   163  69.9%  66.7%  1.05  
g30-15  169  60.4%  65.5%  0.92  
g30-30  125  64.0%  64.7%  0.99  
g30-40   75  65.3%  63.0%  1.04  
                                 
g40-0   114  64.9%  68.0%  0.96  
g40-15  142  66.2%  66.5%  1.00  
g40-30  128  72.7%  65.0%  1.12  
g40-40  148  60.8%  62.0%  0.98  
                                 
g40-AD   58  58.6%  59.6%  0.98  
gAD-40   90  66.7%  63.5%  1.05  

None of the numbers in this table are that extreme, but the overall picture they paint is of a player with better clutch serving abilities than Murray gets credit for. He serves better than expected at both 15-40 and 30-40, and he is barely below average at 30-30, 40-40, or 40-AD. According to these numbers, his game doesn’t change much according to the score–at least at the slams this year.

Serving Against Murray

We can go through the same exercises for Murray’s return points. The next two tables are trickier to read. Look at them as Serving against Murray. Thus, the number in the upper-left corner means that when serving against him, players win 1.5% more points than average in the deuce court; he is a better returner in the ad court. That’s mostly attributable to the fact that righties serve better in the deuce court, regardless of who is returning.

(I’ve excluded return points against lefty servers. Since lefties and righties have such different serving tendencies, limiting the sample to righty servers gives us clearer results, even as the sample shrinks a bit.)

OUTCOME       Deuce     Ad  Break  
Point%        1.015  0.984  0.977  
                                   
Aces          1.018  0.980  0.741  
Svc Wnr       0.993  1.008  0.979  
Dbl Faults    0.956  1.049  1.811  
1st Sv In     0.998  1.003  0.974  
                                   
Server Wnr    1.066  0.927  0.974  
Server UE     1.016  0.982  1.148  
                                   
Return Wnr    0.704  1.324  1.287  
Returner Wnr  0.885  1.126  0.883  
Returner UE   0.917  1.091  1.170  
                                   
Rally Len     0.999  1.001  0.920  

These numbers continue to challenge the conventional wisdom on Murray. What sticks out is the rally length on break points: 8% shorter than usual. I would have expected that Murray plays extremely cautiously in converting break points, but instead, he hits more return winners, makes more unforced errors, and keeps points shorter.

Here’s more on Murray’s return game, again with numbers from the perspective of players serving against him.

SCORE   Pts   Win%    Exp  Rate  
g0-0    388  59.8%  57.0%  1.05  
g0-15   152  52.0%  55.9%  0.93  
g0-30    73  49.3%  55.6%  0.89  
g0-40    37  51.4%  54.8%  0.94  
                                 
g15-0   231  60.6%  57.7%  1.05  
g15-15  170  53.5%  56.6%  0.95  
g15-30  115  54.8%  55.2%  0.99  
g15-40   71  53.5%  54.5%  0.98  
                                 
g30-0   140  57.9%  58.2%  0.99  
g30-15  150  60.0%  58.0%  1.04  
g30-30  123  56.1%  55.8%  1.01  
g30-40   92  56.5%  54.1%  1.04  
                                 
g40-0    81  56.8%  58.8%  0.97  
g40-15  125  59.2%  58.6%  1.01  
g40-30  120  50.0%  57.3%  0.87  
g40-40  209  56.5%  55.9%  1.01  
                                 
g40-AD   91  53.8%  54.8%  0.98  
gAD-40  118  59.3%  56.8%  1.05  

Murray’s results when returning at 40-30 are the only ones that really stick out. He returns much better than expected, winning exactly half of those points. He also appears to string together more streaks than expected at 0-15 and 0-30. Beyond that, he is fairly steady, much like Djokovic in the return game.

Continuing with our point-by-point player profiles, let’s look at Roger Federer. Despite a down year, his service game remains one to be envied and emulated. His more conservative return game also provides a contrast to the styles of Djokovic and Nadal.

Using all of his grand slam matches from 2011, we can begin to analyze his tendencies on serve and return.

The first table shows the frequency of different outcomes in the deuce court, in the ad court, and on break point, relative to Federer’s average. For instance, the 1.014 in the upper left corner means that Fed wins 1.4% more points than average in the deuce court.

OUTCOME       Deuce     Ad  Break  
Point%        1.014  0.984  0.875  
                                   
Aces          1.107  0.883  0.738  
Svc Wnr       1.052  0.943  0.861  
Dbl Faults    1.019  0.979  0.453  
1st Sv In     1.030  0.967  0.872  
                                   
Server Wnr    0.989  1.012  0.845  
Server UE     0.941  1.065  1.405  
                                   
Return Wnr    1.109  0.880  0.877  
Returner Wnr  0.955  1.049  1.341  
Returner UE   0.974  1.028  1.020  
                                   
Rally Len     0.957  1.047  1.278  

For a big-serving right-hander, we might expect to see more success in the deuce court. Yet the difference isn’t that large, except in aces and service winners. Federer counterbalances his lack of aces in the ad court by preventing return winners.

Some of his break point tendencies are striking. As with most players, he wins fewer break points than average points (because opponents who push him to break point are better). He serves much more conservatively, particularly the second serve, which he almost never misses when down break point.

Next, this is how he performs on a point-by-point basis. Win% shows what percentage of points he wins at that score; Exp is how many he would be expected to win (given how he performs in each match), and Rate is the difference between the two. A rate above 1 means he plays better on those points; below 1 is worse.

SCORE   Pts   Win%    Exp  Rate  
g0-0    398  74.1%  71.0%  1.04  
g0-15   102  64.7%  68.0%  0.95  
g0-30    36  63.9%  66.9%  0.95  
g0-40    13  61.5%  66.6%  0.92  
                                 
g15-0   291  74.6%  72.1%  1.03  
g15-15  140  69.3%  70.1%  0.99  
g15-30   66  68.2%  68.0%  1.00  
g15-40   29  75.9%  67.3%  1.13  
                                 
g30-0   217  72.8%  72.3%  1.01  
g30-15  156  66.0%  70.9%  0.93  
g30-30   98  70.4%  68.7%  1.02  
g30-40   51  52.9%  68.1%  0.78  
                                 
g40-0   158  73.4%  72.7%  1.01  
g40-15  145  66.2%  71.4%  0.93  
g40-30  118  76.3%  69.7%  1.09  
g40-40   95  72.6%  68.6%  1.06  
                                 
g40-AD   26  53.8%  65.6%  0.82  
gAD-40   69  62.3%  69.8%  0.89  

You don’t have to watch Federer much to realize he likes his service games quick–and often, he has no problem putting another one on the board with only four or five serves. But when he fails to do that, his results aren’t very good.

His performance at both 40-AD and AD-40 might be a clue as to why he didn’t win a grand slam this year; in both cases, he should be winning between 65 and 70% of points, but he failed to do so by a large margin. (Though at 40-AD, the sample size is small enough to be discarded altogether.) The success rate at 30-40 is even worse. At least when he reached deuce, he performed better than average.

Serving Against Federer

We can go through the same exercises for Fed’s return points. The next two tables are trickier to read. Look at them as Serving against Federer. Thus, the number in the upper-left corner means that when serving against Roger, players win 1.9% more points than average in the deuce court; he is a better returner in the ad court.

(I’ve excluded return points against lefty servers. Since lefties and righties have such different serving tendencies, limiting the sample to righty servers gives us clearer results, even as the sample shrinks a bit.)

OUTCOME       Deuce     Ad  Break  
Point%        1.019  0.979  0.961  
                                   
Aces          1.073  0.920  0.772  
Svc Wnr       1.027  0.971  1.019  
Dbl Faults    0.912  1.097  0.525  
1st Sv In     1.030  0.967  1.006  
                                   
Server Wnr    1.079  0.914  0.630  
Server UE     0.945  1.060  1.046  
                                   
Return Wnr    1.003  0.997  0.393  
Returner Wnr  1.022  0.976  0.988  
Returner UE   0.954  1.050  1.085  
                                   
Rally Len     0.969  1.034  1.095  

What sticks out here is Roger’s dearth of return winners on break point. He often seems to be an aggressive player–sometimes even too aggressive, but apparently he doesn’t do a lot with his first shot. Considered in light of “The Shot” that Djokovic scored against him, it provides another window into the small differences that kept Federer from winning the biggest matches this year.

Here’s more on Federer’s return game, again with numbers from the perspective of players serving against him.

SCORE   Pts   Win%    Exp  Rate  
g0-0    387  57.9%  58.8%  0.98  
g0-15   161  57.8%  56.9%  1.02  
g0-30    68  51.5%  56.4%  0.91  
g0-40    33  66.7%  55.3%  1.20  
                                 
g15-0   221  57.9%  60.1%  0.96  
g15-15  186  62.9%  57.8%  1.09  
g15-30  104  55.8%  56.4%  0.99  
g15-40   68  61.8%  55.1%  1.12  
                                 
g30-0   128  58.6%  61.4%  0.96  
g30-15  170  59.4%  59.6%  1.00  
g30-30  127  66.9%  57.5%  1.16  
g30-40   84  54.8%  55.5%  0.99  
                                 
g40-0    75  61.3%  61.7%  0.99  
g40-15  130  60.0%  60.9%  0.99  
g40-30  137  59.1%  58.7%  1.01  
g40-40  201  53.2%  56.4%  0.94  
                                 
g40-AD   94  52.1%  55.4%  0.94  
gAD-40  107  53.3%  57.3%  0.93  

Apparently Fed returns under pressure better than he serves under pressure. He wins more points than expected when returning at both 40-AD and AD-40. Oddly, he had a tough time when returning at 30-30 but considerably more success at deuce.

Moving on with our point-by-point player profiles, let’s look at Rafael Nadal. Perhaps more than anyone else on tour, he is one of a kind, exhibiting many tendencies that reflect his left-handedness, but not consistently so.

Using all of his grand slam matches from 2011, we can begin to quantify those tendencies.

The first table shows the frequency of different outcomes in the deuce court, in the ad court, and on break point, relative to Nadal’s average. For instance, the 0.990 in the upper left corner means that Djokovic wins 1.0% fewer points than average in the deuce court.

OUTCOME       Deuce     Ad  Break  
Point%        0.990  1.011  0.849  
                                   
Aces          0.875  1.139  0.884  
Svc Wnr       0.998  1.002  0.792  
Dbl Faults    1.076  0.916  1.421  
1st Sv In     0.980  1.022  1.011  
                                   
Server Wnr    0.942  1.064  1.057  
Server UE     1.028  0.969  1.119  
                                   
Return Wnr    0.771  1.254  1.038  
Returner Wnr  0.986  1.015  1.571  
Returner UE   1.006  0.993  0.950  
                                   
Rally Len     1.017  0.981  1.172  

There are plenty of differences between his deuce and ad-court performance, but they aren’t consistent. He hits far more aces in the ad court, but also allows way more return winners. The safest conclusion seems to be that his ad-court serving generates a different, more explosive kind of tennis. In the deuce court, he hits fewer aces, more second serves, fewer winners..but allows his opponent fewer winners. It’s almost as if he plays clay-court tennis in the deuce court and hard-court tennis in the ad court.

The break point tendencies are even more marked. These points generally go longer (17% longer rallies), which would seem to work in Nadal’s favor, but he doesn’t win points at anywhere near his average rate. To some extent, this is because his break points come against better players, but his break point numbers are generally much worse than Djokovic’s.

Next, this is how he performs on a point-by-point basis. Win% shows what percentage of points he wins at that score; Exp is how many he would be expected to win (given how he performs in each match), and Rate is the difference between the two. A rate above 1 means he plays better on those points; below 1 is worse.

SCORE   Pts   Win%    Exp  Rate  
g0-0    377  67.9%  67.2%  1.01  
g0-15   118  65.3%  64.8%  1.01  
g0-30    41  58.5%  61.8%  0.95  
g0-40    17  70.6%  61.0%  1.16  
                                 
g15-0   252  68.7%  68.3%  1.01  
g15-15  156  63.5%  66.5%  0.95  
g15-30   81  60.5%  64.0%  0.94  
g15-40   44  54.5%  62.0%  0.88  
                                 
g30-0   173  69.4%  69.1%  1.00  
g30-15  152  67.8%  67.5%  1.00  
g30-30   98  66.3%  65.4%  1.01  
g30-40   57  57.9%  63.4%  0.91  
                                 
g40-0   120  71.7%  69.8%  1.03  
g40-15  137  67.2%  68.1%  0.99  
g40-30  110  74.5%  67.0%  1.11  
g40-40  143  59.4%  60.9%  0.98  
                                 
g40-AD   58  72.4%  57.9%  1.25  
gAD-40   85  52.9%  62.9%  0.84  

From the past season, the lingering image I have of Rafa is of him fighting off a slew of break points. That is in evidence at 40-AD, where he wins a staggering 72.4% of points. That’s just remarkable: his 40-AD points come against his best opponents, and he performs considerably better at that score than he does at the logically equivalent 30-40.

But 40-AD is the exception. At almost every other crucial score, when Nadal is playing from behind, he plays worse than expected. 15-40 is the most marked, where he wins only 54.5% of points compared to the 62.0% of points he “should” win. Also worrisome is his performance at AD-40; it seems that Nadal is the best in the game when it comes to getting the score back to deuce.

Serving Against Nadal

We can go through the same exercises for Nadal’s return points. The next two tables are trickier to read. Look at them as Serving against Nadal. Thus, the number in the upper-left corner means that when serving against Nadal, players win 1.3% more points than average in the deuce court; Nadal is a better returner in the ad court.

(I’ve excluded return points against lefty servers. Since lefties and righties have such different serving tendencies, limiting the sample to righty servers gives us clearer results, even as the sample shrinks a bit.)

OUTCOME       Deuce     Ad  Break  
Point%        1.013  0.986  1.009  
                                   
Aces          1.150  0.837  0.994  
Svc Wnr       1.098  0.894  0.909  
Dbl Faults    0.860  1.152  0.998  
1st Sv In     1.008  0.991  1.004  
                                   
Server Wnr    1.040  0.956  0.946  
Server UE     0.994  1.006  0.935  
                                   
Return Wnr    0.791  1.227  0.874  
Returner Wnr  0.925  1.082  1.254  
Returner UE   0.918  1.090  1.073  
                                   
Rally Len     1.006  0.993  1.031  

As we might expect, Nadal is a monster returner in the ad court–and servers know it. Righties serve better in the deuce court, but not this much better; Nadal wins 16% more points than average when returning in the deuce court. His opponents help him out, double-faulting at a much higher rate when serving to Rafa’s forehand.

On break point, Nadal isn’t quite so dominant in shutting down the service game, but he does generate a lot more winners later in the point.

Here’s more on Nadal’s return game, again with numbers from the perspective of players serving against him.

SCORE   Pts   Win%    Exp  Rate  
g0-0    380  57.4%  57.2%  1.00  
g0-15   158  50.0%  55.5%  0.90  
g0-30    79  50.6%  54.6%  0.93  
g0-40    39  61.5%  53.6%  1.15  
                                 
g15-0   216  57.9%  58.5%  0.99  
g15-15  170  52.9%  56.6%  0.93  
g15-30  120  54.2%  55.5%  0.98  
g15-40   79  62.0%  52.9%  1.17  
                                 
g30-0   125  61.6%  59.7%  1.03  
g30-15  138  58.0%  57.8%  1.00  
g30-30  123  57.7%  57.5%  1.00  
g30-40  101  50.5%  56.6%  0.89  
                                 
g40-0    77  63.6%  60.7%  1.05  
g40-15  108  60.2%  59.2%  1.02  
g40-30  114  69.3%  57.4%  1.21  
g40-40  160  57.5%  55.9%  1.03  
                                 
g40-AD   68  47.1%  55.7%  0.85  
gAD-40   92  54.3%  56.0%  0.97  

Once again, Nadal loves 40-AD. This time, it’s his chance to convert a break point, and he does so at an alarming rate. And in the return game, he performs nearly as well at 30-40. At 40-AD, he wins more than half of points, a far better performance that we would expect, given the quality of his opponents and his performance on other points against the same players.

Italian translation at settesei.it

30-40 is the most common break point score in professional men’s tennis. It occurs about 15% more often than 40-AD, 30% more often than 15-40, and more than three times as often as 0-40.

It seems that all 30-40s are not created equal. Within the microcosm of a single game, the momentum can swing either way: 30-40 could be the result of a fight to 30-30 followed by a lapse by the server; it could emerge when the server fights back from 0-40.

Regardless of an individual game’s history, the outcome of all points at 30-40 should be created equal. At that score, the server has proven himself skilled enought to win two points against his opponent’s three. In theory, the sequence doesn’t matter any more than it would in a series of coin flips.

Yet anecdotally, it seems that the sequence does matter. Coming from 30-30, the server may feel that he just lost focus for a moment. From 0-40, the returner may feel that he’s due after missing his first two opportunities. (Or to support the opposite hypothesis, the server may have gained confidence by fighting off the first two breakers.)

Regardless of the conventional wisdom, this is now something we can test. If tennis players are completely consistent from one point to the next, the route to 30-40 shouldn’t matter. If they are susceptible to mental ebbs and flows (in predictable ways, anyway), the route to 30-40 should affect how often these break point chances are converted.

15-40 or 30-30?

Let’s start with the simplest possible question. Whenever a game reaches 30-40, the previous point was either 15-40 or 30-30. From 15-40, the server has regained the momentum, though the returner may feel he has a golden opportunity. From 30-30, the returner has the momentum, but the server may feel he can regain control with a single swing of the racquet.

It turns out that there isn’t much difference between the two. From 2011 grand slam men’s singles matches, we have 2136 games in which the score reached 30-40. (Not 40-AD, as 40-AD points must follow deuce.) 890 of those games went through 15-40, while the other 1246 went through 30-30.

In the 15-40 games, the break point at 30-40 was converted 41.2% of the time. In 30-30 games, the break point was converted 40.2% of the time. This gives a slight edge to the “returner sees a golden opportunity” hypothesis, but it is hardly overwhelming evidence.

Love-40

If we look further into each game’s history, two points back, we can compare 0-40 games to the alternatives. Of the 2136 games that reached 30-40, not even 10% passed through 0-40. In those 206 games that passed through 0-40 en route to 30-40, the third break point was converted a whopping 45.1% of the time.

There’s also a noticeable difference between the two other three-point scores. More than half of 30-40 games pass through 15-30; in those 1310 games, the 30-40 break point was converted 41% of the time. But when the game passed through 30-15 before the server lost two consecutive points, the break point was converted only 38.3% of the time.

While the evidence isn’t conclusive, it suggests a sort of reverse hot-hand effect: The player who won the most of the first three points has the best chance of winning at 30-40; the player who won the last two does not.

The same argument even extends to the first two points: If the server reached 30-0, then loses the next three points, the break point is converted only 34.9% of the time. In other words, if a game passes through 30-0 en route to 30-40, you’re better off betting on the guy who just lost the last three points.

If there is a qualitative explanation for this, it might be that fighting off break points requires more mental energy; after coming back from 0-40 (or even 15-40, maybe even 15-30) to 30-40, the server may not have much left. Alternatively, it may require more physical energy; perhaps a rush to 0-40 serves as a wake-up call to the server that he must fight harder to stay in the game. If he does (and if he succeeds in the staying in the game), he’s still competing against the superman who won the first three points of the game. I’m automatically skeptical of explanations of this sort, largely because it would be just as easy to generate stories to support the opposite conclusion. But in this case, at least they explain a quantitative finding.

Another possible explanation may not be as likely, but it is a bit more amusing. Economists and statisticians like to poke fun at the general populace and its innumeracy. Most people think that if you’ve flipped a coin ten times and it has come up heads every time, the odds are better than 50% that it will come up tails on the next flip. After all, it’s “due.”

Perhaps tennis players feel the same way. If a server falls to 0-40, then saves two break points, maybe the returner feels that he’s due. It’s true that the returner is very likely to break at 0-40, but by the time the server saves two breakers, both players start from a clean slate: it’s just as if a coin were flipped five times, with three consecutive heads followed by two tails. But if the coin thinks it’s due … all bets are off.

In the last couple of weeks, we’ve seen that righties and lefties are not equal, at least in their performances in the deuce and ad courts. The differences between them go beyond the rate at which they win points.

To recap: righties win more points in the deuce court and fewer in the ad court. Lefties are the opposite, and the gap between the average lefty’s deuce/ad performance is about twice the same gap for a righty. In the table below, you’ll see that righties win about 1.4% more points than average (1.014) in the deuce court, while lefties win 3.0% more in the ad court.

In every other type of point outcome, either righties, lefties, or both exhibit a noticeable difference in deuce and ad court performance. This extends to outcomes such as winners and unforced errors by the returner, suggesting that the relative strength of deuce and ad court serving extends beyond the first and second shots of each point.

Below, find the complete results for 10 different possible point outcomes. One of the most dramatic differences is in aces, where both righties and lefties hit at least 8% more than average in their stronger court. Both righties and lefties also have higher first-serve percentages in their stronger court.

The most substantial difference between deuce and ad performance in any of the categories comes as a surprise. When lefties are serving in the deuce court, returners are 11% more likely than average to end the point with a winner at some point in the rally. Compared to a mere 1% improvement in return winners (that is, winners on the second shot of the point), this is downright bizarre.

A few notes on my categories. “Svc Wnr” is an unreturned serve, whether an ace or not. “Server Wnr” is a winner hit by the server, not including service winners. “Server UE” and “Returner UE” refer to unforced errors on any shot, excepting the serve. Finally: “Return Wnr” is a winner on the second shot of the point, while “Returner Wnr” is any winner by the returner, including second shot winners.

It may be that the handedness of the returner has some bearing on the outcome, as well; that’s a project for another day.

OUTCOME       RH-Deuce  RH-Ad    LH-Deuce  LH-Ad  
Point%           1.014  0.984       0.972  1.030  

Aces             1.081  0.914       0.920  1.087  
Svc Wnr          1.037  0.960       0.945  1.060  
Dbl Faults       0.999  1.001       1.037  0.960  
1st Sv In        1.013  0.986       0.976  1.026  

Server Wnr       1.001  0.998       0.957  1.047  
Server UE        0.981  1.021       1.014  0.984  

Return Wnr       0.936  1.069       1.008  0.991  
Returner Wnr     0.956  1.048       1.110  0.880  
Returner UE      0.967  1.037       1.040  0.956

When watching a match, it seems that some points are more difficult for the server or returner. There is an oft-cited sense that “40-0 is the best time to break,” suggesting that servers may let up a bit given a big lead.

Building on the work from my last few posts, we can check some of that conventional wisdom. As we’ll see, servers perform about as well as expected at almost every juncture within a game–with the exception of 0-40, when they are at their weakest.

To determine how servers perform at various scores, we first need an estimate of how they “should” perform. Servers win more points at 30-0 than at 0-15, but not necessarily because reaching 30-0 makes you a better server; rather, better servers reach 30-0 more frequently, skewing the sample of 30-0 points.

Before going any further, we need to control for that bias. To do so, I looked at each 2011 grand slam match tracked by Pointstream and found each player’s percentage of service points won. That number, slightly adjusted for deuce/ad court and their handedness (because righties win more points in the deuce court, etc.), is the percentage of points they “should” win at each score.

For example, if a player won 68% of service points in a given match, I estimate that he should have won 68% of 0-0 points, 68% of 15-0 points, and so on (before adjusting for handedness and deuce/ad). This doesn’t account for ups and downs during a match, but it does take into account that players will have different success rates on serve depending on the surface and their opponent.

Across about 11,000 service games, we’ve got a good sample of how players performed at each point score, and we can compare that to how well they should have performed.

For instance, in close to 11,000 game-starting points, servers won 63.5% of points, while–accounting for the overall performance of those players, as well as the advantage of mostly righties serving in the deuce court–they should have won about 64.1% of those points. That’s a minor difference, and 0-0 is one of the nine scores at which players performed within about one percentage point of how we would expect them to.

Of the remaining seven scores, six of them see servers win only two percent more or less than they should. A few notable scores here are 40-0, 40-30, and AD-40. At 40-0, we might expect servers to let up or returners to loosen up, but instead, servers are more successful than ever. That is particularly impressive because the pool of servers who reach 40-0 is already skewed toward the most successful servers. (Though, oddly enough, not quite as much as 30-0. Both of the surprises here may be due to strong servers on mini-streaks.)

40-30 and AD-40 appear to be part of a larger trend where players player better on game point (or tighten up against game point). The server plays better than expected on 40-0, 40-30, and AD-40, and worse than expected (or the returner plays better than expected) at 0-40, 30-40, and AD-40. The only exception is 15-40.

The only point score at which the observed success rate deviates more than two percent from the predicted success rate is 0-40. Servers who get themselves in a 0-40 hole are expected to win only 58.2% of points, but they don’t come close, winning only 54.8% of 0-40 points. Given the results at 0-40 and 40-0, it seems that winning a point, building momentum, and returning to deuce is less common than we might in the professional men’s game.

(In my amateur game, it’s much more common, implying than my regular partners and I aren’t quite as mentally strong as the top 100 players in the world. No offense, regular partners.)

Finally, note that this is not yet an estimate of how players in general respond to the pressure of various moments. At, say, 30-40, the server may be feeling pressure to save break point, but the returner is under pressure, as well. These numbers reflect the outcome given both players’ response to the moment. The results of specific players, as well as stats like double faults and unforced errors, may give us a better idea of what happens when players feel the pressure.

Below, find the complete results. “Obs” is the rate at which players win points given specific scores. “Exp” is the rate at which they “should” have won those points, given their overall performance in each match.

If you’re curious, the “g” preceding each score means “game,” to differentiate 0-0 in a game and 0-0 in a tiebreak. Finally, eagle-eyed readers may note that the observed rates are a bit different than those I published a few days ago. Since then, I added in games with set scores of 6-6 and later, which changed a few of the numbers a bit.

Score     Obs    Exp  Rate  
g0-0    63.5%  64.1%  0.99  
g0-15   60.7%  61.2%  0.99  
g0-30   62.0%  60.8%  1.02  
g0-40   54.8%  58.3%  0.94  

g15-0   63.8%  63.9%  1.00  
g15-15  63.4%  63.3%  1.00  
g15-30  60.1%  60.5%  0.99  
g15-40  61.1%  59.9%  1.02  

g30-0   64.9%  66.0%  0.98  
g30-15  62.7%  63.2%  0.99  
g30-30  64.0%  62.6%  1.02  
g30-40  59.3%  59.7%  0.99  

g40-0   67.1%  65.8%  1.02  
g40-15  65.7%  65.4%  1.00  
g40-30  63.7%  62.5%  1.02  
g40-40  61.6%  61.4%  1.00  

g40-AD  57.9%  58.8%  0.98  
gAD-40  62.3%  61.2%  1.02

As we’ve seen, right-handers serve more effectively to the deuce court than to the ad court, and lefties do the opposite. Based on available data, righties win about 64.0% of points to the deuce court against 62.1% to the ad court, while lefties exhibit a bigger difference, winning 59.3% in the deuce court, 62.8% in the ad court.

(These numbers are different than those I originally published last week. There was a bug in my calculations; while it does not change any overall conclusions, it turns out that the lefty gap is considerably wider than the initial numbers showed.)

While the differences are minor, they have some strategic implications. My previously-published win probability tables for a single game assume that players are consistent from point to point, regardless of the direction they serve. It would be foolish to generate new tables for each player’s tendencies, but it is possible to do the math separately for the populations of righties and lefties.

Implications

We start with a paradox. Given a righty server and a lefty server who win equal percentage of service points, the lefty has a better chance of winning a service game. The paradox is compounded by the fact that slightly more points are played in the deuce court, thanks to games ending at 40-15 and (much more rarely) at 15-40.

Two things explain the lefty advantage. First, close games (those that reach 30-30 or deuce) always have equal numbers of deuce and ad points. When the balance between deuce and ad points reaches 50/50, a 63% lefty server is a bit better than 63% (63.07%, to be exact), while a 63% righty server is a bit worse (62.96%.)

Second, the wider difference in deuce/ad outcomes for lefties makes it more likely that a lefty will keep himself in a game, fighting off break points and giving himself another chance to string two points together. As we’ll see in a moment, the difference at break point is the most important aspect of this table.

The table below shows win probabilities for right-handed and left-handed servers who win 63% and 70% of service points. (63% is average for 2011 grand slam matches; 70% is a round number for a dominant serving performance.) Each row shows the likelihood of each type of server winning a game from the given point score.

The most dramatic difference is–as expected–on break point at 30-40 or 40-AD. At both the 63% and 70% levels, left-handedness confers a 2% advantage over right-handedness. There is a noticeable advantage at 40-30 (and AD-40) as well, where the lefty has a better chance of finishing the game immediately, but it is only about one-third the effect of 30-40.

Here is the full table for each type of server at each point. I expect that you’ll keep it handy each time you watch a match.

          63%     63%     70%     70%           
SCORE      RH      LH      RH      LH           
0-0    79.42%  79.65%  90.02%  90.26%           
0-15   64.09%  65.36%  78.51%  79.71%           
0-30   43.22%  43.52%  58.69%  58.88%           
0-40   18.21%  19.28%  28.46%  29.97%           

15-0   88.05%  88.63%  94.75%  95.15%           
15-15  76.91%  77.22%  87.49%  87.66%           
15-30  57.21%  58.72%  70.95%  72.64%           
15-40  29.45%  29.62%  41.28%  41.55%           

30-0   94.83%  94.92%  98.02%  98.04%           
30-15  88.09%  88.81%  94.17%  94.72%           
30-30  74.31%  74.46%  84.53%  84.62%           
30-40  46.04%  48.36%  58.22%  61.04%  (40-AD)  

40-0   98.66%  98.76%  99.56%  99.62%           
40-15  96.48%  96.55%  98.61%  98.63%           
40-30  90.23%  91.05%  95.17%  95.68%  (AD-40)  
40-40  74.31%  74.46%  84.53%  84.62%

Italian translation at settesei.it

Taking the next step beyond yesterday’s post about servers’ success rates in the deuce and ad courts, along with each in-game score, let’s look at the tendencies of righties and lefties.

As I speculated yesterday, righties are more successful in the deuce court (64.0% to 62.3% of points won), while lefties are better serving to the ad court (63.0% to 62.3%).  The difference for lefties is a little more dramatic (62.8% to 61.8%) if we remove Rafael Nadal from the sample.

(In all of the numbers today, I’ll present lefties in two forms: with Nadal, and without Nadal.  While Rafa is just one player, he makes up nearly one-third of the service points played by lefties in the dataset we’re working with of 2011 grand slam men’s singles matches tracked by Pointstream.  As we’ll see, Nadal appears to have some tendencies that separate him from the lefty pack.)

This would seem to give lefties a bit of a strategic edge; more than three-quarters of break points are played in the ad court, including all of the break points (30-40, 40-AD) that bring the server back to even.  If lefties are more likely to win those points, they would seem to be more likely to fend off such threats.  Of course, it might cut both ways: A weakness in the deuce court may lead to more break points needing to be fended off.

Oddly enough, lefties do not seem to employ their advantage at the most common break point score, 30-40.  Both righties and lefties win 30-40 points at about 6% less frequently than they win points in general.  The most marked difference is at 40-AD, where righties win 10% fewer points than average, but lefties win only 3% fewer points than average.  Rafa accounts for a big part of that difference, probably attesting to his mental strength.  Without Rafa, lefties win 6% fewer points than average at 40-AD; still quite a bit better than righties.

(Remember, there’s a bias inherent in this approach.  If a server reaches 40-AD, the score itself reflects a disadvantage.  Federer and Isner don’t serve many 40-AD points, precisely because their serves are so dominant.  Verdasco, Fognini, and whoever is playingDjokovic serve more 30-40 and 40-AD points, meaning that the sample of 40-AD is disproportionately full of men serving against the odds.)

There’s much more to do here, but in the meantime, these broad differences between righties and lefties give us plenty to think about.

The table below shows all the numbers described above.  The columns with numbers between 0.85 and 1.1 indicates, for each type of server (righty, lefty, etc.), how their performance at a specific point compares to their overall rate.  That allows us to better compare righties (winners of 63.1% of service points) with lefties-minus-Nadal (winners of 62.3%).

SCORE    WIN%       RH               LH            LH-xRN          
ALL     62.9%    63.1%            62.7%             62.3%          
DC CT   63.5%    64.0%   1.01     62.3%   0.99      61.8%   0.99   
AD CT   62.3%    62.3%   0.99     63.0%   1.00      62.8%   1.01   

SCORE    WIN%       RH               LH            LH-xRN          
g0-0    63.4%    63.5%   1.01     63.5%   1.01      63.1%   1.01   
g0-15   60.6%    60.8%   0.96     59.5%   0.95      59.5%   0.96   
g0-30   62.0%    62.6%   0.99     59.4%   0.95      60.6%   0.97   
g0-40   54.8%    54.1%   0.86     56.3%   0.90      54.6%   0.88   

g15-0   63.8%    63.7%   1.01     65.3%   1.04      64.8%   1.04   
g15-15  63.4%    63.8%   1.01     59.5%   0.95      59.6%   0.96   
g15-30  60.1%    60.2%   0.95     60.6%   0.97      61.4%   0.99   
g15-40  61.2%    61.5%   0.98     58.2%   0.93      58.3%   0.94   

SCORE    WIN%       RH               LH            LH-xRN          
g30-0   64.9%    65.0%   1.03     64.1%   1.02      63.0%   1.01   
g30-15  62.7%    62.5%   0.99     65.7%   1.05      65.7%   1.05   
g30-30  64.0%    64.3%   1.02     62.5%   1.00      60.9%   0.98   
g30-40  59.3%    59.2%   0.94     58.6%   0.94      59.3%   0.95   

g40-0   67.1%    66.8%   1.06     67.0%   1.07      65.1%   1.05   
g40-15  65.6%    66.0%   1.05     62.5%   1.00      61.6%   0.99   
g40-30  63.7%    63.6%   1.01     67.0%   1.07      65.6%   1.05   
g40-40  61.6%    61.9%   0.98     61.4%   0.98      62.2%   1.00   
g40-AD  57.8%    57.0%   0.90     60.5%   0.97      58.7%   0.94   
gAD-40  62.3%    62.4%   0.99     59.7%   0.95      61.4%   0.99

Italian translation at settesei.it

If tennis players were machines, each player would have the same probability of winning every point.  Winning the point at 40-15 would be equally likely as winning the point at 15-40.  It seems a safe bet that this isn’t the case, and today I’m going to start talking about the difference, and why it exists.

To begin with, let’s look at the outcome of every grand slam men’s singles point in 2011, sorted by the score before the point was played.  (I’ll explain some of this in a minute.)

SCORE     PTS    WON   WIN%   REL  
g0-0    10757   6820  63.4%  1.00  
g0-15    3941   2390  60.6%  0.97  
g0-30    1552    963  62.0%  0.98  
g0-40     591    324  54.8%  0.88 

g15-0    6823   4356  63.8%  1.02  
g15-15   4858   3081  63.4%  1.00  
g15-30   2741   1648  60.1%  0.97  
g15-40   1416    866  61.2%  0.96  

SCORE     PTS    WON   WIN%   REL  
g30-0    4355   2826  64.9%  1.02  
g30-15   4609   2890  62.7%  1.01  
g30-30   3366   2155  64.0%  1.01  
g30-40   2080   1234  59.3%  0.95 

g40-0    2824   1895  67.1%  1.08  
g40-15   3819   2507  65.6%  1.03  
g40-30   3466   2209  63.7%  1.02  
g40-40   4556   2806  61.6%  0.97 

g40-AD   1749   1011  57.8%  0.93  
gAD-40   2806   1748  62.3%  1.00  

SCORE     PTS    WON   WIN%        
ALL     66309  41729  62.9%        
DC CT   34679  22024  63.5%        
AD CT   31630  19705  62.3%

One thing that sticks out is that as players get closer to winning a game (30-0, 40-0), they are more likely to win the next point.  When facing (or approaching) break point, they have less success.

Much of that (and maybe all of it) is simply the bias of the sample.  If a player reaches 40-0, he’s more likely to be a player who is dominant on serve, or facing a returner who hasn’t found the range.  A disproportionate number of 40-0 points are served by players who are better-than-average servers.  Similarly, a disproportionate number of 0-40 points are served by players without dominant service games … or served against Novak Djokovic.

Deuce and ad courts

A more useful finding is that players win more points in the deuce court.  In this sample, the server won 63.5% of points in the deuce court and 62.3% of points in the ad court.  This may be because right-handers (who make up about 85% of this sample) are more successful when serving across their body, but I haven’t tested that yet.

(If it is true that players are better serving across their body, then the difference is even more stark.  Assuming that righties and lefties have the same difference in success rates, the “serve across your body” success rate–deuce for righties, ad for lefties–should be about 63.8%, while the “serve away from your body” rate–ad for righties, deuce for lefties–should be 62.1%.)

Thus, the difference between success rates at 0-0 and 0-15 isn’t as extreme as it looks at first; some of the 0-15 winning percentage is due to the difficulty of serving to the ad court.  That’s the purpose of the ‘REL’ column, which shows how the winning percentage on that point relates to the average winning percentage in the relevant court.

If this difference is universally true, it would require a change in win probability tables.  For instance, when the returner reaches break point–which is more often in the ad court, at 30-40 or 40-AD–his chance of winning the game is a percentage point or two higher than previously estimated.  As long as he’s playing a right-hander, anyway.

There’s plenty more to investigate here.  To determine whether players really raise or drop their performance levels (for instance, raising their game against break point, or taking it easy at 40-0), we’ll need to switch to a player-by-player basis, to reduce the skewing effect of dropping every player in the same bucket.

A couple of months ago I presented some research that showed that, in the average men’s grand slam match, the server’s advantage was neutralized somewhere between the 4th and 8th shot.

That research left a major question unanswered: How do the results differ between first and second serves?  Some second serves are hardly better than rallying shots, so it stands to reason that the server’s advantage is neutralized even faster on the second serve.

Using all of the Pointstream-tracked men’s matches from this year’s grand slams, we have an enormous population of points, in which 63.2% of points were won by the server.  When the first serve went in, the percentage jumped to 71.7%.  If the first serve went out, the server’s chance of winning fell to 50.2%, then rose again to 56.1% if he landed his second serve.

On first serve points, the server’s advantage was not neutralized until at least the 8th shot of the rally, and perhaps not until the 9th or 10th.  On second serve points, however, the advantage was gone (or very nearly so) as soon as the returner got the ball back in play.

Below, find the exact percentages of service points won for rallies that reach various lengths.  In the table below, the ‘1’ row refers to points with at least one stroke (the serve) that went in.  A one-stroke rally is defined as an ace, service winner, or return error.  A two-stroke rally is defined as a point in which the return landed in but the server doesn’t get his second shot back in play.  Note that each individual percentage is biased in favor of the player (server or returner) who has the chance to put the point away; the point can be considered neutralized when the biased even out (e.g. 55, then 45, then 55, and so on).

Rally    All     1sts     2nds
0      63.2%
1      66.1%    71.7%    56.1%
2      50.3%    53.7%    45.6%
3      59.9%    63.6%    54.9%
4      46.4%    47.8%    44.6%
5      58.1%    60.3%    55.6%
6      44.9%    45.9%    43.9%
7      57.0%    58.2%    55.6%
8      44.5%    45.1%    43.9%
9      55.9%    56.4%    55.5%
10     44.1%    44.1%    44.0%
11     55.8%    55.9%    55.8%
12     43.2%    43.1%    43.3%
13     55.6%    55.6%    55.5%
14     43.5%    43.5%    43.6%
15     55.5%    55.3%    55.6%