What Does the “Hot Hand” Mean in Tennis?

Italian translation at settesei.it

In sports analytics, the topic of streakiness–the “hot hand“–is a popular one. Nearly everyone believes it exists, that players (or even teams) can go on a hot or cold streak, during which they temporarily play above or below their true level.

To a certain extent, streakiness is inevitable–if you flip a coin 100 times, you’ll see segments of 5 or 10 flips in which most of the flips are heads. That’s not because the coin suddenly got “better,” it’s a natural occurence over a long enough time span. So if you watch an entire tennis match, there are bound to be games where one player seems to be performing better than usual, perhaps stringing together several aces or exceptional winners.

The question, then, is whether a player is more streaky than would occur purely at random. To take just one example, let’s say a player hits aces on 10% of service points. If he did occasionally serve better than usual, we would observe that after he hits one ace, he is more likely (say, 15% or 20%) to hit another ace. A missed first or second serve might make it more likely than he misses his next try.

My last couple of topics–differences in the deuce/ad court, and the “reverse hot hand” at 30-40–have hinted that tennis may be structured in a way that prevents players from getting hot.

One of the most popular subjects for hot hand research is basketball free throw shooting. Researchers like it because it’s as close as basketball players get to a laboratory: every shot is from the same distance, there’s no defensive quality to consider, and even better, players usually get two tries, one right after the other. There’s nothing like it in tennis.

The one thing that seems a bit akin to free throw shooting is serving, especially for more dominant players. John Isner, Roger Federer, and Milos Raonic seem to go on serving streaks; certainly they can play game after game and control play with unreturnable serving. But when we look closer, their experience is much more nuanced. As we’ve seen, players generally are better in the deuce or ad court. It would be as if basketball player shot one free throw, then took two steps to the left and one step forward before attempting his next shot.

And, of course, there’s another player on the court. If Federer uses a relatively slow serve out wide in the deuce court for a service winner at 15-15, he is much less likely to use the same tactic at 30-30 or 40-15. Even if he was capable of hitting 50 perfect serves of that nature, he would never do so in a match. If it has any relevance for professional tennis, the hot hand must refer to something broader than a single skill.

On a more general level, the rules of tennis involve alternation more than more sports. Sure, most sports give the ball to the other team after a goal, but the length of possession–or in baseball, the length of an inning–can vary widely. In tennis, you can only add one game to your tally before handing the ball to your opponent. And even within that game, you are constantly moving from your stronger court to your weaker court; your opponent might be doing the same.

My question to you is this: If there is a hot hand in tennis, where would you expect to find it? Consecutive aces? Aces specifically in the deuce court? Service winners? Short service points? Points won? Return points won? Games won? First serves in? Point-ending winners? Avoidance of unforced errors? It’s possible that any or all of these things could occur in bunches, but which of them would indicate what we think of as a tennis player on a hot streak?

Point-by-Point Profile: Novak Djokovic

In the last few weeks, we’ve seen some overall serving trends–how righties and lefties perform in the deuce and ad courts, and how successful they are at specific point scores.

The tour-wide results are interesting enough, but there’s much more to discover at the individual player level. Because point-by-point data is only available for 2011 grand slam matches, only a few players have had enough points tracked to allow us to make meaningful conclusions. Fortunately, those are the best players in the game, and there’s plenty to discover.

Let’s start with Novak Djokovic. Much of his success seems to stem from rock-solid consistency: he can attack when returning almost as much as most players do on serve; he is strong on both forehand and backhand, and he rarely shows signs of mental weakness. If there is a player who doesn’t display the typical differences between deuce and ad courts and various point scores, it would seem to be Djokovic.

The first table shows the frequency of different outcomes in the deuce court, in the ad court, and on break point, relative to Djokovic’s average. For instance, the 1.018 in the upper left corner means that Djokovic wins 1.8% more points than average in the deuce court.

OUTCOME       Deuce     Ad  Break  
Point%        1.018  0.980  0.975  
                                   
Aces          1.117  0.869  1.046  
Svc Wnr       1.101  0.886  0.865  
Dbl Faults    1.176  0.802  1.102  
1st Sv In     1.028  0.968  1.081  
                                   
Server Wnr    1.027  0.970  0.815  
Server UE     0.973  1.030  0.941  
                                   
Return Wnr    0.972  1.031  2.125  
Returner Wnr  0.832  1.189  1.487  
Returner UE   0.927  1.082  1.092  
                                   
Rally Len     0.938  1.070  1.184

There are some huge differences here. Given the gap between deuce and ad results for many types of outcomes, it’s surprising that Novak wins so many ad-court points. He hits nearly 12% more aces in the deuce court, suggesting that even when he doesn’t hit an ace or service winner, he better sets up the point. Returners are much more likely to hit winners against him in the ad court, and the point requires more shots.

There are even more extreme numbers on break point. It’s unclear from the numbers whether Djokovic consistently goes for more on the serve on break point–more aces, fewer service winners, more double faults, but more first serves in–but it appears he plays much more gingerly, hitting far fewer winners and allowing opponents to hit more than twice as many return winners than average.

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    360  70.0%  70.2%  1.00  
g0-15   107  65.4%  68.5%  0.96  
g0-30    37  59.5%  66.5%  0.89  
g0-40    15  66.7%  64.8%  1.03  
                                 
g15-0   248  66.5%  71.0%  0.94  
g15-15  153  69.9%  69.9%  1.00  
g15-30   68  67.6%  68.1%  0.99  
g15-40   32  68.8%  66.7%  1.03  
                                 
g30-0   165  67.3%  71.3%  0.94  
g30-15  161  70.2%  70.4%  1.00  
g30-30   94  77.7%  67.9%  1.14  
g30-40   43  62.8%  66.4%  0.95  
                                 
g40-0   111  73.9%  72.0%  1.03  
g40-15  142  76.8%  71.3%  1.08  
g40-30  106  67.0%  68.6%  0.98  
g40-40  104  72.1%  66.8%  1.08  
                                 
g40-AD   29  69.0%  66.2%  1.04  
gAD-40   75  70.7%  67.0%  1.05

It appears that Djokovic’s caution on break point isn’t hurting him; despite losing a point or two more than expected at 30-40, he gets it back at 40-AD. Novak excels most in the pressure points: 30-30 and 40-40, with strong showings at nearly every point from 30-30 on, with the exception of 30-40–which may just be a fluke–we only have 43 points to work with.

We can go through the same exercises for Djokovic’s return points. The next two tables are trickier to read. Look at them as Serving against Djokovic. Thus, the number in the upper-left corner means that when serving against Djokovic, players win 1% more points than average in the deuce court.

(I’ve excluded return points against lefty servers, including Nadal. 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%        1.010  0.989  1.023  
                                   
Aces          1.024  0.974  1.091  
Svc Wnr       0.998  1.002  1.105  
Dbl Faults    0.994  1.007  0.986  
1st Sv In     1.055  0.940  0.957  
                                   
Server Wnr    1.002  0.998  1.091  
Server UE     0.987  1.015  0.964  
                                   
Return Wnr    1.123  0.867  0.849  
Returner Wnr  0.895  1.114  1.124  
Returner UE   0.858  1.153  1.069  
                                   
Rally Len     0.992  1.009  0.959

It seems that Novak goes big on the return in the deuce court, but tries to do more later in ad-court points. The break point tendencies may speak to other players’ fear of Djokovic’s return game: They go bigger with their serve, hitting more aces and service winners, and severely limiting Novak’s return winners. In the end, though, it doesn’t matter: he converts the break points anyway.

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

SCORE   Pts   Win%    Exp  Rate  
g0-0    346  58.7%  56.3%  1.04  
g0-15   143  53.1%  55.1%  0.96  
g0-30    67  52.2%  53.9%  0.97  
g0-40    32  53.1%  53.0%  1.00  
                                 
g15-0   198  62.1%  57.2%  1.09  
g15-15  151  54.3%  56.0%  0.97  
g15-30  104  45.2%  54.7%  0.83  
g15-40   74  59.5%  53.4%  1.11  
                                 
g30-0   123  60.2%  58.1%  1.04  
g30-15  131  51.9%  56.9%  0.91  
g30-30  110  60.0%  56.0%  1.07  
g30-40   88  54.5%  54.3%  1.00  
                                 
g40-0    74  64.9%  59.0%  1.10  
g40-15   94  61.7%  57.5%  1.07  
g40-30  102  53.9%  57.2%  0.94  
g40-40  189  50.8%  55.4%  0.92  
                                 
g40-AD   93  54.8%  54.3%  1.01  
gAD-40   96  55.2%  56.4%  0.98

While Djokovic excels at deuce (servers should win 55.4% of those points; they manage to win only 50.8%), the reverse happens at 30-30. There aren’t many clear trends here, which may simply attest to Djokovic’s return dominance, regardless of point score.

The Hot Hand in Reverse at 30-40

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.

Point Outcomes for Righties and Lefties in the Deuce and Ad Courts

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

Server Strength, Point by Point

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

Win Probability Tables for Righties and Lefties

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%

Righties and Lefties in the Deuce and Ad Courts

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

Point Outcomes by Game Score

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.

Server’s Advantage: First and Second Serves

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%

The Effect of One More MPH

Italian translation at settesei.it

All else equal, increasing your first serve speed is a good thing … so how useful is it?  Earlier this week, I published some generic numbers, but those are far too crude to answer this question.

To get a better answer, we need to see what happens when specific players serve a little faster or slower.  Sometimes, players dramatically mix up serve speed (as with slice serves wide), but most of the time, each player stays within a fairly limited range defined by his own power and skill.

The algorithm I’ve employed is  fairly complicated, so I’ll give you the results first.

It appears that most players, if they increased their average serve speed by one mile per hour, would win 0.2% more first service points.  That’s not many–it’s not even one point in every match.  But every little bit helps, and according to my win probability models, winning 0.2% more first serve points can increase your chance of winning an even match from 50% to just short of 51%.  Except possibly at the extremes, that continues to be the case for 2 MPH, 3 MPH, and greater increases–so a 5 MPH increase takes that 50/50 match and turns it into a 54/46 contest.

(One assumption here is that all players respond to increases in serve speed the same way.  I’m sure that’s not true, but at this stage it’s a necessary assumption.)

The effect of a speed increase is even greater on ace and service winner rates.  Each additional MPH on a player’s serve increases his ace rate by about 0.4%, and his service winner rate by about 0.5%.

Now for the algorithm and some caveats.

Process

The algorithm was designed to control (to the extent possible) for different types of serving and playing styles, as well as the different average speeds to the deuce and ad court, as well as to different directions (wide, body, and T).

I used only US Open data, to avoid differences between surfaces and between the speed guns used at different events.  I used data only from the 18 players who had more than 150 first-serve points tracked by Pointstream.  For each of those players, I found their average first-serve speed for each of six directions: wide, body, and T to the deuce and ad courts.  Then, I randomly selected 150 of their first-serve points, and for each point, noted the difference between the point’s serve speed and the player’s average in the relevant court/direction.

Thus, every one of 2700 points was labeled 0 (average for that player/court/direction), or +1 (one mph above average), or -4, and so on.  That results in large pools of points with each label.  Many of the pools were too small for useful analysis, so I grouped them in sets of five: (-2, -1, 0, +1, +2), (-1, 0, +1, +2, +3), and so on.  The pools, then, were useful from about -15 to +15.

From there, I looked at  each of several stats (points won, aces, service winners) for each pool, and compared the rates from one pool to the next.  The results were somewhat erratic–in some instances, an additional mph results in aces or points won going down, but over the set of 31 pools, they generally went up.  The numbers presented above are the averages of each one-mph change.

Caveats

It’s not a very big sample, especially when separating serves into pools of 0, +1, +2, and so on.

One issue with the dataset is that the 18 servers were usually winning–that’s how they got enough first serves to merit inclusion.  Thus, the average returner in the dataset is below average.  That isn’t necessarily a bad thing–perhaps below-average returners respond to changes in serve speed the way above-average returners do–but without more data, it’s tough to know.

Another concern is what the numbers really tell us below about 5 mph slower than average.  The algorithm operates on the assumption that a 120 mph serve is the same as a 121 mph serve, only slower.  Comparing 120 and 121, that’s probably true.   But comparing 120 and 108–for the same player, serving in the same direction–it probably isn’t.  The 108 mph isn’t a simulation of what would happen if the player wasn’t as good; it’s probably a strategic choice, likely accompanied by some spin.

That said, the algorithm doesn’t directly compare 120 and 108, it compares 108 and 109, and perhaps in the aggregate, there is something useful to be gleaned from comparing a strategic spin first serve to an identical serve one mph faster.  In any event, limiting the range to between -10 and 10, or even -7 and 7, doesn’t change the results much.

Finally, the sample is completely inadequate to tell us what happens at the extremes.  The average player appears to improve his chances by adding another bit of speed, but does John Isner?  There may be a ‘sweet spot’ where a player can get maximum gains from an additional 1, 5, or 10 mph on his first serves, but beyond which, the gain is more limited.