Measuring the Clutchness of Everything

Matches are often won or lost by a player’s performance on “big points.” With a few clutch aces or un-clutch errors, it’s easy to gain a reputation as a mental giant or a choker.

Aside from the traditional break point stats, which have plenty of limitations, we don’t have a good way to measure clutch performance in tennis. There’s a lot more to this issue than counting break points won and lost, and it turns out that a lot of the work necessary to quantify clutchness is already done.

I’ve written many times about win probability in tennis. At any given point score, we can calculate the likelihood that each player will go on to win the match. Back in 2010, I borrowed a page from baseball analysts and introduced the concept of volatility, as well. (Click the link to see a visual representation of both metrics for an entire match.) Volatility, or leverage, measures the importance of each point–the difference in win probability between a player winning it or losing it.

To put it simply, the higher the leverage of a point, the more valuable it is to win. “High leverage point” is just a more technical way of saying “big point.”  To be considered clutch, a player should be winning more high-leverage points than low-leverage points. You don’t have to win a disproportionate number of high-leverage points to be a very good player–Roger Federer’s break point record is proof of that–but high-leverage points are key to being a clutch player.

(I’m not the only person to think about these issues. Stephanie wrote about this topic in December and calculated a full-year clutch metric for the 2015 ATP season.)

To make this more concrete, I calculated win probability and leverage (LEV) for every point in the Wimbledon semifinal between Federer and Milos Raonic. For the first point of the match, LEV = 2.2%. Raonic could boost his match odds to 50.7% by winning it or drop to 48.5% by losing it. The highest leverage in the match was a whopping 32.8%, when Federer (twice) had game point at 1-2 in the fifth set. The lowest leverage of the match was a mere 0.03%, when Raonic served at 40-0, down a break in the third set. The average LEV in the match was 5.7%, a rather high figure befitting such a tight match.

On average, the 166 points that Raonic won were slightly more important, with LEV = 5.85%, than Federer’s 160, at LEV = 5.62%. Without doing a lot more work with match-level leverage figures, I don’t know whether that’s a terribly meaningful difference. What is clear, though, is that certain parts of Federer’s game fell apart when he needed them most.

By Wimbledon’s official count, Federer committed nine unforced errors, not counting his five double faults, which we’ll get to in a minute. (The Match Charting Project log says Fed had 15, but that’s a discussion for another day.) There were 180 points in the match where the return was put in play, with an average LEV = 6.0%. Federer’s unforced errors, by contrast, had an average LEV nearly twice as high, at 11.0%! The typical leverage of Raonic’s unforced errors was a much less noteworthy 6.8%.

Fed’s double fault timing was even worse. Those of us who watched the fourth set don’t need a fancy metric to tell us that, but I’ll do it anyway. His five double faults had an average LEV of 13.7%. Raonic double faulted more than twice as often, but the average LEV of those points, 4.0%, means that his 11 doubles had less of an impact on the outcome of the match than Roger’s five.

Even the famous Federer forehand looks like less of a weapon when we add leverage to the mix. Fed hit 26 forehand winners, in points with average LEV = 5.1%. Raonic’s 23 forehand winners occurred during points with average LEV = 7.0%.

Taking these three stats together, it seems like Federer saved his greatness for the points that didn’t matter as much.

The bigger picture

When we look at a handful of stats from a single match, we’re not improving much on a commentator who vaguely summarizes a performance by saying that a player didn’t win enough of the big points. While it’s nice to attach concrete numbers to these things, the numbers are only worth so much without more context.

In order to gain a more meaningful understanding of this (or any) performance with leverage stats, there are many, many more questions we should be able to answer. Were Federer’s high-leverage performances typical? Does Milos often double fault on less important points? Do higher-leverage points usually result in more returns in play? How much can leverage explain the outcome of very close matches?

These questions (and dozens, if not hundreds more) signal to me that this is a fruitful field for further study. The smaller-scale numbers, like the average leverage of points ending with unforced errors, seem to have particular potential. For instance, it may be that Federer is less likely to go for a big forehand on a high-leverage point.

Despite the dangers of small samples, these metrics allow us to pinpoint what, exactly, players did at more crucial moments. Unlike some of the more simplistic stats that tennis fans are forced to rely on, leverage numbers could help us understand the situational tendencies of every player on tour, leading to a better grasp of each match as it happens.

How Much Is a Challenge Worth?

When the Hawkeye line-calling system is available, tennis players are given the right to make three incorrect challenges per set. As with any situation involving scarcity, there’s a choice to make: Take the chance of getting a call overturned, or make sure to keep your options open for later?

We’ve learned over the last several years that human line-calling is pretty darn good, so players don’t turn to Hawkeye that often. At the Australian Open this year, men challenged fewer than nine calls per match–well under three per set or, put another way, less than 1.5 challenges per player per set. Even at that low rate of fewer than once per thirty points, players are usually wrong. Only about one in three calls are overturned.

So while challenges are technically scarce, they aren’t that scarce.  It’s a rare match in which a player challenges so often and is so frequently incorrect that he runs out. That said, it does happen, and while running out of challenges is low-probability, it’s very high risk. Getting a call overturned at a crucial moment could be the difference between winning and losing a tight match. Most of the time, challenges seem worthless, but in certain circumstances, they can be very valuable indeed.

Just how valuable? That’s what I hope to figure out. To do so, we’ll need to estimate the frequency with which players miss opportunities to overturn line calls because they’ve exhausted their challenges, and we’ll need to calculate the potential impact of failing to overturn those calls.

A few notes before we get any further.  The extra challenge awarded to each player at the beginning of a tiebreak would make the analysis much more daunting, so I’ve ignored both that extra challenge and points played in tiebreaks. I suspect it has little effect on the results. I’ve limited this analysis to the ATP, since men challenge more frequently and get calls overturned more often. And finally, this is a very complex, sprawling subject, so we often have to make simplifying assumptions or plug in educated guesses where data isn’t available.

Running out of challenges

The Australian Open data mentioned above is typical for ATP challenges. It is very similar to a subset of Match Charting Project data, suggesting that both challenge frequency and accuracy are about the same across the tour as they are in Melbourne.

Let’s assume that each player challenges a call roughly once every sixty points, or 1.7%. Given an approximate success rate of 30%, each player makes an incorrect challenge on about 1.2% of points and a correct challenge on 0.5% of points. Later on, I’ll introduce a different set of assumptions so we can see what different parameters do to the results.

Running out of challenges isn’t in itself a problem. We’re interested in scenarios when a player not only exhausts his challenges, but when he also misses an opportunity to overturn a call later in the set. These situations are much less common than all of those in which a player might want to contest a call, but we don’t care about the 70% of those challenges that would be wrong, as they wouldn’t have any effect on the outcome of the match.

For each possible set length, from 24-point golden sets up to 93-point marathons, I ran a Monte Carlo simulation, using the assumptions given above, to determine the probability that, in a set of that length, a player would miss a chance to overturn a later call. As noted above, I’ve excluded tiebreaks from this analysis, so I counted only the number of points up to 6-6. I also excluded all “advantage” fifth sets.

For example, the most common set length in the data set is 57 points, which occured 647 times. In 10,000 simulations, a player missed a chance to overturn a call 0.27% of the time. The longer the set, the more likely that challenge scarcity would become an issue. In 10,000 simulations of 85-point sets, players ran out of challenges more than three times as often. In 0.92% of the simulations, a player was unable to challenge a call that would have been overturned.

These simulations are simple, assuming that each point is identical. Of course, players are aware of the cap on challenges, so with only one challenge remaining, they may be less likely to contest a “probably correct” call, and they would be very unlikely to use a challenge to earn a few extra seconds of rest. Further, the fact that players sometimes use Hawkeye for a bit of a break suggests that what we might call “true” challenges–instances in which the player believes the original call was wrong–are a bit less frequent that the numbers we’re using. Ultimately, we can’t address these concerns without a more complex model and quite a bit of data we don’t have.

Back to the results. Taking every possible set length and the results of the simulation for each one, we find the average player is likely to run out of challenges and miss a chance to overturn a call roughly once every 320 sets, or 0.31% of the time. That’s not very often–for almost all players, it’s less than once per season.

The impact of (not) overturning a call

Just because such an outcome is infrequent doesn’t necessarily mean it isn’t important. If a low-probability event has a high enough impact when it does occur, it’s still worth planning for.

Toward the end of a set, when most of these missed chances would occur, points can be very important, like break point at 5-6. But other points are almost meaningless, like 40-0 in just about any game.

To estimate the impact of these missed opportunities, I ran another set of Monte Carlo simulations. (This gets a bit hairy–bear with me.) For each set length, for those cases when a player ran out of challenges, I found the average number of points at which he used his last challenge. Then, for each run of the simulation, I took a random set from the last few years of ATP data with the corresponding number of points, chose a random point between the average time that the challenges ran out and the end of the set, and measured the importance of that point.

To quantify the importance of the point, I calculated three probabilities from the perspective of the player who lost the point and, had he conserved his challenges, could have overturned it:

  1. his odds of winning the set before that point was played
  2. his odds of winning the set after that point was played (and not overturned)
  3. his odds of winning the set had the call been overturned and the point awarded to him.

(To generate these probabilities, I used my win probability code posted here with the assumption that each player wins 65% of his service points. The model treats points as independent–that is, the outcome of one point does not depend on the outcomes of previous points–which is not precisely true, but it’s close, and it makes things immensely more straightforward. Alert readers will also note that I’ve ignored the possibility of yet another call that could be overturned. However, the extremely low probability of that event convinced me to avoid the additional complexity required to model it.)

Given these numbers, we can calculate the possible effects of the challenge he couldn’t make. The difference between (2) and (3) is the effect if the call would’ve been overturned and awarded to him. The difference between (1) and (2) is the effect if the point would have been replayed. This is essentially the same concept as “leverage index” in baseball analytics.

Again, we’re missing some data–I have no idea what percentage of overturned calls result in each of those two outcomes. For today, we’ll say it’s half and half, so to boil down the effect of the missed challenge to a single number, we’ll average those two differences.

For example, let’s say we’re at five games all, and the returner wins the first point of the 11th game. The server’s odds of winning the set have decreased from 50% (at 5-all, love-all) to 43.0%. If the server got the call overturned and was awarded the point, his odds would increase to 53.8%. Thus, the win probability impact of overturning the call and taking the point is 10.8%, while the effect of forcing a replay is 7.0%. For the purposes of this simulation, we’re averaging these two numbers and using 8.9% as the win probability impact of this missed opportunity to challenge.

Back to the big picture. For each set length, I ran 1,000 simulations like what I’ve described above and averaged the results. In short sets under 40 points, the win probability impact of the missed challenge is less than five percentage points. The longer the set, the bigger the effect: Long sets are typically closer and the points tend to be higher-leverage. In 85-point sets, for instance, the average effect of the missed challenge is a whopping 20 percentage points–meaning that if a player more skillfully conserved his challenges in five such sets, he’d be able to reverse the outcome of one of them.

On average, the win probability effect of the missed challenge is 12.4 percentage points. In other words, better challenge management would win a player one more set for every eight times he didn’t lose such an opportunity by squandering his challenges.

The (small) big picture

Let’s put together the two findings. Based on our assumptions, players run out of challenges and forgo a chance to overturn a later call about once every 320 matches. We now know that the cost of such a mistake is, on average, a 12.4 percentage point win probability hit.

Thus, challenge management costs an average player one set out of every 2600. Given that many matches are played on clay or on courts without Hawkeye, that’s maybe once in a career. As long as the assumptions I’ve used are in the right ballpark, the effect isn’t even worth talking about. The mental cost of a player thinking more carefully before challenging might be greater than this exceedingly unlikely benefit.

What if some of the assumptions are wrong? Anecdotally, it seems like challenges cluster in certain matches, because of poor officiating, bad lighting, extreme spin, precise hitting, or some combination of these. It seems possible that certain scenarios would arise in which a player would want to challenge much more frequently, and even though he might gain some accuracy, he would still increase the risk.

I ran the same algorithms for what seems to me to be an extreme case, almost doubling the frequency with which each player challenges, to 3.0%, and somewhat increasing the accuracy rate, to 40%.

With these parameters, a player would run out of challenges and miss an opportunity to overturn a call about six times more often–once every 54 sets, or 1.8% of the time. The impact of each of these missed opportunities doesn’t change, so the overall result also increases by a factor of six. In these extreme case, poor challenge management would cost a player the set 0.28% of the time, or once every 356 sets. That’s a less outrageous number, representing perhaps one set every second year, but it also applies to unusual sets of circumstances which are very unlikely to follow a player to every match.

It seems clear that three challenges is enough. Even in long sets, players usually don’t run out, and when they do, it’s rare that they miss an opportunity that a fourth challenge would have afforded them. The effect of a missed chance can be enormous, but they are so infrequent that players would see little or no benefit from tactically conserving challenges.

The Difficulty (and Importance) of Finding the Backhand

One disadvantage of some one-handed backhands is that they tend to sit up a little more when they’re hit crosscourt. That gives an opponent more time to prepare and, often, enough time to run around a crosscourt shot and hit a forehand, which opens up more tactical possibilities.

With the 700 men’s matches in the Match Charting Project database (please contribute!), we can start to quantify this disadvantage–if indeed it has a negative effect on one-handers. Once we’ve determined whether one-handers can find their opponents’ backhands, we can try to answer the more important question of how much it matters.

The scenario

Let’s take all baseline rallies between right-handers. Your opponent hits a shot to your backhand side, and you have three choices: drive (flat or topspin) backhand, slice backhand, or run around to hit a forehand. You’ll occasionally go for a winner down the line and you’ll sometimes be forced to hit a weak reply down the middle, but usually, your goal is to return the shot crosscourt, ideally finding your opponent’s backhand.

Considering all righty-righty matchups including at least one player among the last week’s ATP top 72 (I wanted to include Nicolas Almagro), here are the frequency and results of each of those choices:

SHOT    FREQ  FH REP  BH REP    UFE  WINNER  PT WON  
ALL             9.9%   68.1%  10.8%    5.8%   43.1%  
SLICE  11.9%   34.1%   49.5%   7.1%    0.6%   40.2%  
FH     44.9%    2.8%   69.0%  13.0%    9.8%   42.1%  
BH     43.3%   10.7%   72.2%   9.5%    3.1%   45.0%  
                                                     
1HBH   42.6%   12.0%   69.5%   9.3%    3.8%   44.2%  
2HBH   43.5%   10.0%   73.4%   9.6%    2.8%   45.4%

“FH REP” and “BH REP” refer to a forehand or backhand reply, and we can see just how much shot selection matters in keeping the ball away from your opponent’s forehand. A slice does a very poor job, while an inside-out forehand almost guarantees a backhand reply, though it comes with an increased risk of error.

The differences between one- and two-handed backhands aren’t as stark. One-handers don’t find the backhand quite as frequently, though they hit a few more winners. They hit drive backhands a bit less often, but that doesn’t necessarily mean they are hitting forehands instead. On average, two-handers hit a few more forehands from the backhand corner, while one-handers are forced to hit more slices.

One hand, many types

Not all one-handed backhands are created equal, and these numbers bear that out. Stanislas Wawrinka‘s backhand is as effective as the best two-handers, while Roger Federer‘s is typically the jumping-off point for discussions of why the one-hander is dying.

Here are the 28 players for whom we have at least 500 instances (excluding service returns) when the player responded to a shot hit to his backhand corner. For each, I’ve shown how often he chose a drive backhand or forehand, and the frequency with which he found the backhand–excluding his own errors and winners.

Player                 BH  BH FRQ  FIND BH%  FH FRQ  FIND BH%  
Alexandr Dolgopolov     2   45.7%     94.2%   43.3%     98.7%  
Kei Nishikori           2   51.1%     94.0%   38.9%     98.1%  
Andy Murray             2   41.0%     92.4%   46.5%     98.6%  
Stanislas Wawrinka      1   48.6%     92.1%   37.5%     98.0%  
Bernard Tomic           2   33.8%     91.7%   43.8%     97.9%  
Novak Djokovic          2   47.2%     91.7%   41.4%     98.5%  
Kevin Anderson          2   41.0%     91.5%   45.8%     96.6%  
Borna Coric             2   46.5%     90.7%   44.2%     96.9%  
Pablo Cuevas            1   41.9%     90.6%   54.5%     96.5%  
Marin Cilic             2   45.4%     89.7%   43.3%     97.2%  
                                                               
Player                 BH  BH FRQ  FIND BH%  FH FRQ  FIND BH%  
Tomas Berdych           2   41.6%     89.3%   44.2%     97.5%  
Pablo Carreno Busta     2   55.4%     87.8%   41.1%     93.5%  
Fabio Fognini           2   46.0%     87.4%   47.0%     96.1%  
Richard Gasquet         1   57.2%     87.3%   32.1%     96.8%  
Andreas Seppi           2   40.3%     87.2%   50.0%     93.9%  
Nicolas Almagro         1   53.6%     86.5%   39.3%     98.0%  
Dominic Thiem           1   38.5%     86.2%   50.0%     96.5%  
Gael Monfils            2   48.0%     85.3%   46.3%     85.3%  
David Ferrer            2   48.2%     84.9%   40.4%     97.1%  
Roger Federer           1   42.7%     84.8%   43.6%     94.5%  
                                                               
Player                 BH  BH FRQ  FIND BH%  FH FRQ  FIND BH%  
Gilles Simon            2   46.9%     84.6%   46.5%     94.6%  
David Goffin            2   45.4%     84.6%   45.7%     94.9%  
Roberto Bautista Agut   2   39.6%     83.3%   46.7%     98.4%  
Jo Wilfried Tsonga      2   43.5%     82.0%   44.5%     96.3%  
Grigor Dimitrov         1   41.4%     78.6%   39.4%     92.8%  
Milos Raonic            2   31.5%     63.5%   56.5%     94.3%  
Jack Sock               2   27.0%     62.5%   62.9%     96.3%  
Tommy Robredo           1   26.6%     56.1%   62.3%     88.4%

One-handers Wawrinka, Pablo Cuevas, and Richard Gasquet (barely) are among the top half of these players, in terms of finding the backhand with their own backhand. Federer and his would-be clone Grigor Dimitrov are at the other end of the spectrum.

Taking all 60 righties I included in this analysis (not just those shown above), there is a mild negative correlation (r^2 = -0.16) between a player’s likelihood of finding the opponent’s backhand with his own and the rate at which he chooses to hit a forehand from that corner. In other words, the worse he is at finding the backhand, the more inside-out forehands he hits. Tommy Robredo and Jack Sock are the one- and two-handed poster boys for this, struggling more than any other players to find the backhand, and compensating by hitting as many forehands as possible.

However, Federer–and, to an even greater extent, Dimitrov–don’t fit this mold. The average one-hander runs around balls in their backhand corner 44.6% of the time, while Fed is one percentage point under that and Dimitrov is below 40%. Federer is perceived to be particularly aggressive with his inside-out (and inside-in) forehands, but that may be because he chooses his moments wisely.

Ultimate outcomes

Let’s look at this from one more angle. In the end, what matters is whether you win the point, no matter how you get there. For each of the 28 players listed above, I calculated the rate at which they won points for each shot selection. For instance, when Novak Djokovic hits a drive backhand from his backhand corner, he wins the point 45.4% of the time, compared to 42.3% when he hits a slice and 42.4% when he hits a forehand.

Against his own average, Djokovic is about 3.6% better when he chooses (or to think of it another way, is able to choose) a drive backhand. For all of these players, here’s how each of the three shot choices compare to their average outcome:

Player                 BH   BH W   SL W   FH W  
Dominic Thiem           1  1.209  0.633  0.924  
David Goffin            2  1.111  0.656  0.956  
Grigor Dimitrov         1  1.104  0.730  1.022  
Gilles Simon            2  1.097  0.922  0.913  
Tomas Berdych           2  1.085  0.884  0.957  
Pablo Carreno Busta     2  1.081  0.982  0.892  
Kei Nishikori           2  1.070  0.777  0.965  
Roberto Bautista Agut   2  1.055  0.747  1.027  
Stanislas Wawrinka      1  1.050  0.995  0.936  
Borna Coric             2  1.049  1.033  0.941  
                                                
Player                 BH   BH W   SL W   FH W  
Bernard Tomic           2  1.049  1.037  0.943  
Jack Sock               2  1.049  0.811  1.010  
Gael Monfils            2  1.048  1.100  0.938  
Fabio Fognini           2  1.048  0.775  0.987  
Milos Raonic            2  1.048  0.996  0.974  
Nicolas Almagro         1  1.046  0.848  0.964  
Kevin Anderson          2  1.038  1.056  0.950  
Novak Djokovic          2  1.036  0.966  0.969  
Andy Murray             2  1.031  1.039  0.962  
Roger Federer           1  1.023  1.005  0.976  
                                                
Player                 BH   BH W   SL W   FH W  
Richard Gasquet         1  1.020  0.795  1.033  
Andreas Seppi           2  1.019  0.883  1.008  
David Ferrer            2  1.018  0.853  1.020  
Alexandr Dolgopolov     2  1.010  1.010  0.987  
Marin Cilic             2  1.006  1.009  0.991  
Pablo Cuevas            1  0.987  0.425  1.048  
Jo Wilfried Tsonga      2  0.956  0.805  1.095  
Tommy Robredo           1  0.845  0.930  1.079

In this view, Dimitrov–along with his fellow one-handed flame carrier Dominic Thiem–looks a lot better. His crosscourt backhand doesn’t find many backhands, but it is by far his most effective shot from his own backhand corner. We would expect him to win more points with a drive backhand than with a slice (since he probably opts for slices in more defensive positions), but it’s surprising to me that his backhand is so much better than the inside-out forehand.

While Dimitrov and Thiem are more extreme than most, almost all of these players have better results with crosscourt drive backhands than with inside-out (or inside-in forehands). Only five–including Robredo but, shockingly, not including Sock–win more points after hitting forehands from the backhand corner.

It’s clear that one-handers do, in fact, have a slightly more difficult time forcing their opponents to hit backhands. It’s much less clear how much it matters. Even Federer, with his famously dodgy backhand and even more famously dominant inside-out forehand, is slightly better off hitting a backhand from his backhand corner. We’ll never know what would happen if Fed had Djokovic’s backhand instead, but even though Federer’s one-hander isn’t finding as many backhands as Novak’s two-hander does, it’s getting the job done at a surprisingly high rate.

Toward a Better Understanding of Return Effectiveness

The deeper the return, the better, right? That, at least, is the basis for many of the flashy graphics we see these days on tennis broadcasts, indicating the location of every return, often separated into “shallow,” “medium,” and “deep” zones.

In general, yes, deep returns are better than shallow ones. But return winners aren’t overwhelmingly deep, since returners can achieve sharper angles if they aim closer to the service line. There are plenty of other complicating factors as well: returns to the sides of the court are more effective than those down the middle, second-serve returns tend to be better than first-serve returns, and topspin returns result in more return points won than chip or slice returns.

While most of this is common sense, quantifying it is an arduous and mind-bending task. When we consider all these variables–first or second serve, deuce or ad court, serve direction, whether the returner is a righty or lefty, forehand or backhand return, topspin or slice, return direction, and return depth–we end up with more than 8,500 permutations. Many are useless (righties don’t hit a lot of forehand chip returns against deuce court serves down the T), but thousands reflect some common-enough scenario.

To get us started, let’s set aside all of the variables but one. When we analyze 600+ ATP matches in the Match Charting Project data, we have roughly 61,000 in-play returns coded in one of nine zones, including at least 2,000 in each.  Here is a look at the impact of return location, showing the server’s winning percentage when a return comes back in play to one of the nine zones:rzones1show

(“Shallow” is defined as anywhere inside the service boxes, while “Medium” and “Deep” each represent half of the area behind the service boxes. The left, center, and right zones are intended to indicate roughly where the return would cross the baseline, so for sharply angled shots, a return might bounce shallow near the middle of the court but be classified as a return to the forehand or backhand side.)

As we would expect, deeper returns work in favor of the returner, as do returns away from the center of the court. A bit surprisingly, returns to the server’s forehand side (if he’s a right-hander) are markedly more effective than those to the backhand. This is probably because right-handed returners are most dangerous when hitting crosscourt forehands, although left-handed returners are also more effective (if not as dramatically) when returning to that side of the court.

Let’s narrow things down just a little and see how the impact of return location differs on first and second serves. Here are the server’s chances of winning the point if a first-serve return comes back in each of the nine zones:

rzones2showF

And the same for second-serve returns:

rzones3showF

It’s worth emphasizing just how much impact a deep return can have. So many points are won with unreturnable serves–even seconds–that simply getting the ball back in play comes close to making the point a 50/50 proposition. A deep second-serve return, especially to a corner, puts the returner in a very favorable position. Consistently hitting returns like that is a big reason why Novak Djokovic essentially turns his opponents’ second serves against them.

The final map makes it clear how valuable it is to move the server away from the middle of the court. Think of it as a tactical first strike, forcing the server to play defensively instead of dictating play with his second shot. Among second-serve returns put in play, any ball placed away from the middle of the court–regardless of depth–gives the returner a better chance of winning the point than does a deep return down the middle.

For today, I’m going to stop here. This is just the tip of the iceberg, as there are so many variables that play some part in the effectiveness of various service returns. Ultimately, understanding the potency of each return location will give us additional insight into what players can achieve with different kinds of serve, which players are deadliest with certain types of returns, and how best to handle different returns with the server’s crucial second shot.

Measuring the Effectiveness of Backhand Returns

One-handed backhands can be beautiful, but they aren’t always the best tools for the return of serve. Some of the players with the best one-handers in the game must often resort to slicing backhand returns–Stanislas Wawrinka, for example, slices 68% of backhand first serve returns and 40% of backhand second serve returns, while Andy Murray uses the slice 41% and 3%, respectively.

Using the 650 men’s matches in the Match Charting Database, I looked at various aspects of backhand serve returns to try to get a better sense of the trade-offs involved in using a one-handed backhand. Because the matches in the MCP aren’t completely representative of the ATP tour, the numbers are approximate. But given the size and breadth of the sample, I believe the results are broadly indicative of men’s tennis as a whole.

At the most general level, players with double-handed backhands are slightly better returners, putting roughly the same number of returns in play (about 56%) and winning a bit more often–46.9% to 45.7%–when they do so. The gap is a bit wider when we look at backhand returns put in play: 46.5% of points won to 44.7%. While the favorable two-hander numbers are influenced by the historically great returning of Novak Djokovic, two-handers still have an edge if we reduce his weight in the sample or remove him entirely.

Unsurprisingly, players realize that two-handed backhands are more effective returns, and they serve accordingly. The MCP divides serves into three zones–down the tee, body, and wide–and I’ve re-classified those as “to the forehand,” “to the body,” and “to the backhand” depending on the returner’s dominant hand and whether the point is in the deuce or ad court. While we can’t identify exactly where servers aimed those to-the-body serves, we can determine some of their intent from serves aimed at the corners.

Against returners with two-handed backhands, servers went for the backhand corner on 44.2% of first serves and 34.8% of second serves. Against one-handers, they aimed for the same spot on 47.3% of first serves and 40.9% of second serves. Looking at the same question from another angle, backhands make up 61.7% of the returns in play hit by one-handers compared to 59.0% for double-handers. It seems likely that one-handers more aggressively run around backhands to hit forehand returns, so this last comparison probably understates the degree to which servers aim for single-handed backhands.

When servers do manage to find the backhand side of a single-hander, they’re often rewarded with a slice return. On average, one-handers (excluding Roger Federer, who is overrepresented in this dataset) use the slice on 53.9% of their backhand first-serve returns and 32.3% of their backhand second-serve returns. Two-handers use the slice 20.5% of the time against firsts and only 2.5% of the time against seconds.

For both types of players, against first and second serves, slice returns are less effective than flat or topspin backhand returns. This isn’t surprising, either–defensive shots are often chosen in defensive situations, so the difference in effectiveness is at least partly due to the difference in the quality of the serves themselves. Still, since one-handers choose to go to the slice so much more frequently, it’s valuable to know how the types of returns compare:

Return Type   BH in play W% SL in play W% 
1HBH vs Firsts        43.3%         37.6% 
1HBH vs Seconds       46.0%         44.1% 
                        
2HBH vs Firsts        46.8%         36.2% 
2HBH vs Seconds       48.6%         41.9%

(Again, I’ve excluded Fed from the 1HBH averages.)

In three of the four rows, there’s a difference of several percentage points between the effectiveness of slice returns and flat or topspin returns, as measured by the ultimate outcome of the point. The one exception–second-serve returns by one-handers–reminds us that the slice can be an offensive weapon, even if it’s rarely used as one in the modern game. Some players–including Federer, Feliciano Lopez, Grigor Dimitrov, and Bernard Tomic–are more effective with slice returns than flat or topspin returns against either first or second serves.

However, these players are the exceptions, and in the theoretical world where we can set all else equal, a slice return is the inferior choice. All players have to hit slice returns sometimes, and many of those seem to be forced by powerful serving, but the fact remains: one-handers hit slices much more than two-handers do, and despite the occasional offensive opportunity, slice returns are more likely to hand the point to the server.

These differences are real, but they are still modest. A good returner with a one-handed backhand is considerably better than a bad returner with a two-hander, and it’s even possible to have a decent return game while hitting mostly slices. All that said, in the aggregate, a one-handed backhand is a bit of a liability on the return. It will take further research to determine whether other benefits–such as the sizzling down-the-line winners we’ve come to expect from the likes of Wawrinka and Richard Gasquet–outweigh the costs.

The Dreaded Deficit at the Tiebreak Change of Ends

Some of tennis’s conventional wisdom manages to be both blindingly self-evident and obviously wrong. Give pundits a basic fact (winning more points is good), add a dash of perceived momentum, and the results can be toxic.

A great example is the tiebreak change of ends. The typical scenario goes something like this: Serving at 2-3 in a tiebreak, a player loses a point on serve, going down a minibreak to 2-4. As the players change sides, a commentator says, “You really don’t want to go into this change of ends without at least keeping the score even.”

While the full rationale is rarely spelled out, the implication is that losing that one point–going from 2-3 to 2-4–is somehow worse than usual because the point precedes the changeover. Like the belief that the seventh game of the set is particularly important, this has passed, untested, into the canon.

Let’s start with the “blindingly self-evident” part. Yes, it’s better to head into the change of ends at 3-3 than it is at 2-4. In a tiebreak, every point is crucial. Based on a theoretical model and using sample players who each win 65% of service points, here are the odds of winning a tiebreak from various scores at the changeover:

Score  p(Win)  
1*-5     5.4%  
2*-4    21.5%  
3*-3    50.0%  
4*-2    78.5%  
5*-1    94.6%

It’s easy to sum that up: You really want to win that sixth point. (Or, at least, several of the points before the sixth.) On the other hand, compare that to the scenarios after eight points:

Score  p(Win)  
2*-6     2.6%  
3*-5    17.6%  
4*-4    50.0%  
5*-3    82.4%  
6*-2    97.4%

At the risk of belaboring the obvious, when the score is close, points become more important later in the tiebreak. The outcome at 4-4 matters more than at 3-3, which matters more than at 2-2, and so on. If players changed ends after eight points, we’d probably bestow some magical power on that score instead.

Real-life outcomes

So far, I’ve only discussed what the model tells us about win probabilities at various tiebreak scores. If the pundits are right, we should see a gap between the theoretical likelihood of winning a tiebreak from 2-4 and the number of times that players really do win tiebreaks from those scores. The model says that players should win 21.5% of tiebreaks from 2*-4; if the conventional wisdom is correct, we would find that players win even fewer tiebreaks when trying to come back from that deficit.

By analyzing the 20,000-plus tiebreaks in this dataset, we find that the opposite is true. Falling to 2-4 is hugely worse than reaching the change of ends at 3-3, but it isn’t worse than the model predicts–it’s a bit better.

To quantify the effect, I determined the likelihood that the player serving immediately after the changeover would win the tiebreak, based on each player’s service points won throughout the match and the model I’ve referred to above. By aggregating all of those predictions, together with the observed result of each tiebreak, we can see how real life compares to the model.

In this set of tiebreaks, a player serving at 2-4 would be expected to win 20.9% of the time. In fact, these players go to win the tiebreak 22.0% of the time–a small but meaningful difference. We see an even bigger gap for players returning at 2-4. The model predicts that they would win 19.9% of the time, but they end up winning 22.1% of these tiebreaks.

In other words, after six points, the player with more points is heavily favored, but if there’s any momentum–that is, if either player has more of an advantage than the mere score would suggest–the edge belongs the player trailing in the tiebreak.

Sure enough, we see the same effect after eight points. Serving at 3-5, players in this dataset have a 16.3% (theoretical) probability of winning the tiebreak, but they win 19.0% of the time. Returning at 3-5, their paper chance is 17.2%, and they win 19.5%.

There’s nothing special about the first change of ends, and there probably isn’t any other point in a tiebreak that is more crucial than the model suggests. Instead, we’ve discovered that underdogs have a slightly better chance of coming back than their paper probabilities indicate. I suspect we’re seeing the effect of front-runners getting tight and underdogs swinging more freely–an aspect of tennis’s conventional wisdom that has much more to recommend itself than the idea of a magic score after the first six points of a tiebreak.

Does Serving First in a Tiebreak Give You an Edge?

Tiebreaks are so balanced, with frequently alternating servers and sides of the court, that it seems they must be fair. As far as I know, there is no commonly-cited conventional wisdom to the effect that the first server (or second server) in a tiebreak has any kind of advantage.

Let’s check. In a dataset of over 5,200 tiebreaks at ATP tour events, the first server won 50.8% of the time. Calculating each player’s service points won for the entire match and using those numbers to determine the likelihood that the first server would win a tiebreak, we get an estimate that those first servers should have won only 48.8% of them.

Two percentage points is a small gap, but here, it’s a meaningful one. It’s persistent across each of the three years most heavily represented in the dataset (2013-15), and it holds regardless of the set. While there might be some bias in the results of first-set tiebreaks, since better servers often choose to serve first and lesser servers choose to receive, the effect in each set favors the first server, and the impact of serving first is greater in the third set than in the first.

However, this effect–at least in its magnitude–is limited to ATP results. A survey of 2,500 recent WTA tiebreaks shows that first servers have won 49.7% of tiebreaks, compared to 49.4% that they should have won. Women’s ITF matches and men’s futures matches return similar results. Running the same algorithm on 6,200 men’s Challenger-level tiebreaks confuses the issue even further: Here, first servers won 48.1% of tiebreaks, while they should have won 48.7%.

A byproduct of this research is the discovery that, for both genders and at multiple levels of the game, the first server in a tiebreak is, on average, the weaker player. At first glance, that doesn’t make a lot of sense: We think of tiebreaks as deciding sets when the two players are equal. And since the effect is present for the second and third sets as well as the first, this finding isn’t biased by players choosing who will serve first.

As it turns out, this result can be at least partially explained by another byproduct of my recent research. In my attempt to determine whether it’s particularly difficult to hold when serving for the set, I calculated the odds of holding serve at every score throughout a set, compared to how frequently players should have held. At most holds–including those with the set on the line–there aren’t any major discrepancies between actual hold rates and expected hold rates.

But I did find some small effects that are relevant here. In general, it is a bit harder to hold serve as the second server, at scores such as 3-4, 4-5, and 5-6, than as the first, at scores like 3-3, 4-4, and 5-5. For instance, in the ATP data, players hold serve at 4-4 exactly as often as we would expect them to, based on their rate of service points won throughout the match. But at 4-5, their performance drops to 1.4% below expectations. In the WTA data, while players underperform at 5-5 by 1.4%, they are far worse at 5-6, winning 5.2% less often than they should.

In other words, if two players of equal abilities stay on serve for the first several games of a set, the second server is a little more likely to crack, getting broken and losing the set. Thus, if neither player is broken (or the number of breaks is equal), the second server is likely to be just a little bit better.

That explains, at least in part, why second servers are favored on paper going into tiebreaks. What it doesn’t account for is the discovery that on the ATP tour, first servers overcome that paper advantage and win more than half of tiebreaks. For that, I don’t have a good answer.