Jeff, Author at Heavy Topspin

Measuring the Best Smashes in Tennis

Italian translation at settesei.it: part 1, part 2

How can we identify the best shots in tennis? At first glance, it seems like a simple problem. Thanks to the shot-by-shot data collected for over 3,500 matches by the Match Charting Project, we can look at every instance of the shot in question and see what happened. If a player hits a lot of winners, or wins most of the ensuing points, he or she is probably pretty good at that shot. Lots of unforced errors would lead us to conclude the opposite.

A friend recently posed a more specific question: Who has the best smash in the men’s game? Compared to other shots such as, say, slice backhands, smashes should be pretty easy to evaluate. A large percentage of them end the point–in the contemporary men’s game (I discuss the women’s game later on), 69% are winners or induce forced errors–which reduces the problem to a straightforward one.

The simplest algorithm to answer my friend’s question is to determine how often each player ends the point in his favor when hitting a smash–that is, with a winner or by inducing a forced error. Call the resulting ratio “W/SM.” The Match Charting Project (MCP) dataset has at least 10 tour-level matches for 80 different men, and the W/SM ratio for those players ranges from 84% (Jeremy Chardy) all the way down to 30% (Paolo Lorenzi). Both of those extremes are represented by players with relatively small samples; if we limit our scope to men with at least 90 recorded smashes, the range isn’t quite as wide. The best of the bunch is Jo-Wilfried Tsonga, at 79%, and the “worst” is Ivan Lendl, at 57%. That isn’t quite fair to Lendl, since smash success rates have improved quite a bit over the years, and Lendl’s rate is only a couple percentage points below the average for the 1980s. Among active players with at least 90 smashes in the books, Stan Wawrinka brings up the rear, with a W/SM of 65%.

We can look at the longer-term effects of a player’s smashes without adding much complexity. It’s ideal to end the point with a smash, but most players would settle for winning the point. When hitting a smash, ATPers these days end up winning the point 81% of the time, ranging from 97% (Chardy again) down to 45% (Lorenzi again). Once again, Tsonga leads the pack of the bigger-sample-size players, winning the point 90% of the time after hitting a smash, and among active players, Wawrinka is still at the bottom of that subset, at 77%.

Here is a list of all players with at least 90 smashes in the MCP dataset, with their winners (and induced forced errors) per smash (W/SM), errors per smash (E/SM), and points won per smash (PTS/SM):

PLAYER              W/SM  E/SM  PTS/SM  
Jo-Wilfried Tsonga   78%    6%     90%  
Tomas Berdych        76%    6%     88%  
Pete Sampras         75%    7%     86%  
Roger Federer        73%    7%     86%  
Rafael Nadal         69%    7%     84%  
Milos Raonic         73%    9%     82%  
Andy Murray          67%    6%     82%  
Kei Nishikori        68%   11%     81%  
David Ferrer         71%    9%     81%  
Andre Agassi         67%    8%     80%  
Novak Djokovic       66%    9%     80%  
Stefan Edberg        62%   12%     78%  
Stan Wawrinka        65%   10%     77%  
Ivan Lendl           57%   13%     71%

These numbers give us a pretty good idea of who you should back if the ATP ever hosts the smash-hitting equivalent of baseball’s Home Run Derby. Best of all, it doesn’t commit any egregious offenses against common sense: We’d expect to see Tsonga and Roger Federer near the top, and we’d know something was wrong if Novak Djokovic were too far from the bottom.

Smash opportunities

Still, we need to do better. Almost every shot made in a tennis match represents a decision made by the player hitting it: topspin or slice? backhand or run-around forehand? approach or stay back? Many smashes are obvious choices, but a large number are not. Different players make different choices, and to evaluate any particular shot, we need to subtly reframe the question. Instead of vaguely asking for “the best,” we’d be better served looking for the player who gets the most value out of his smash. While the two questions are similar, they are not the same.

Let’s expand our view to what we might call “smash opportunities.” Once again, smashes make our task relatively straightforward: We can define a smash opportunity simply as a lob hit by the opponent.* In the contemporary ATP, roughly 72% of lobs result in smashes–the rest either go for winners or are handled with a different shot. Different players have very different strategies: Federer, Pete Sampras, and Milos Raonic all hit smashes in more than 84% of opportunities, while a few other men come in under 50%. Nick Kyrgios, for instance, tried a smash in only 20 of 49 (41%) of recorded opportunities. Of those players with more available data, Juan Martin Del Potro elected to go for the overhead in 61 of 114 (54%) of chances, and Andy Murray in 271 of 433 (62.6%).

* Using an imperfect dataset, it’s a bit more complicated; sometimes the shots that precede smashes are coded as topspin or slice groundstrokes. I’ve counted those as smash opportunities as well.

Not all lobs are created equal, of course. With a large number of points, we would expect them to even out, but even then, a player’s overall style may effect the smash opportunities he sees. That’s a more difficult issue for another day; for now, it’s easiest to assume that each player’s mix of smash opportunities are roughly equal, though we’ll keep in mind the likelihood that we’ve swept some complexity under the rug.

With such a wide range of smashes per smash opportunities (SM/SMO), it’s clear that some players’ average smashes are more difficult than others. Federer hits about half again as many smashes per opportunity as del Potro does, suggesting that Fed’s attempts are more difficult than Delpo’s; on those more difficult attempts, Delpo is choosing a different shot. The Argentine is very effective when he opts for the smash, winning 84% of those points, but it seems likely that his rate would not be so high if he hit smashes as frequently as Federer does.

This leads us to a slightly different question: Which players are most effective when dealing with smash opportunities? The smash itself doesn’t necessarily matter–if a player is equally effective with, say, swinging volleys, the lack of a smash would be irrelevant. The smash is simply an effective tool that most players employ to deal with these situations.

Smash opportunities don’t offer the same level of guarantee that smashes themselves do: In the ATP these days, players win 72% of points after being handed a smash opportunity, and 56% of the shots they hit result in winners or induced forced errors. Looking at these situations takes us a bit off-track, but it also allows us to study a broader question with more impact on the game as a whole, because smash opportunities represent a larger number of shots than smashes themselves do.

Here is a list of all the players with at least 99 smash opportunities in the MCP dataset, along with the rate at which they hit smashes (SM/SMO), the rate at which they hit winners or induced forced errors in response to smash opportunites (W/SMO), hit errors in those situations (E/SMO), and won the points when given lobs (PTW/SMO). Like the list above, players are ranked by the rightmost column, points won.

PLAYER              SM/SMO  W/SMO  E/SMO  PTW/SMO  
Jo-Wilfried Tsonga     80%    68%    13%      80%  
Roger Federer          84%    66%    13%      78%  
Pete Sampras           86%    68%    15%      78%  
Tomas Berdych          75%    66%    16%      76%  
Milos Raonic           85%    67%    14%      76%  
Novak Djokovic         81%    60%    13%      75%  
Kevin Anderson         66%    57%    12%      74%  
Rafael Nadal           74%    57%    16%      73%  
Andre Agassi           77%    62%    17%      73%  
Boris Becker           85%    59%    18%      72%  
Stan Wawrinka          79%    58%    15%      72%  
Kei Nishikori          72%    57%    17%      70%  
Andy Murray            63%    52%    15%      70%  
Dominic Thiem          66%    52%    11%      70%  
David Ferrer           71%    57%    17%      69%  
Pablo Cuevas           73%    54%    14%      67%  
Stefan Edberg          81%    52%    23%      65%  
Bjorn Borg             81%    41%    20%      63%  
JM del Potro           54%    48%    19%      60%  
Ivan Lendl             74%    45%    28%      59%  
John McEnroe           74%    43%    24%      56%

The order of this list has much in common with the previous one, with names like Federer, Sampras, and Tsonga at the top. Yet there are key differences: Djokovic and Wawrinka are particularly effective when they respond to a lob with something other than an overhead, while del Potro is the opposite, landing near the bottom of this ranking despite being quite effective with the smash itself.

The rate at which a player converts opportunities to smashes has some impact on his overall success rate on smash opportunities, but the relationship isn’t that strong (r^2 = 0.18). Other options, such as swinging volleys or mid-court forehands, also give players a good chance of winning the point.

Smash value

Let’s get back to my revised question: Who gets the most value out of his smash? A good answer needs to combine how well he hits it with how often he hits it. Once we can quantify that, we’ll be able to see just how much a good or bad smash can impact a player’s bottom line, measured in overall points won, and how much a great smash differs from an abysmal one.

As noted above, the average current-day ATPer wins the point 81% of the time that he hits a smash. Let’s reframe that in terms of the probability of winning a point: When a lob is flying through the air and a player readies his racket to hit an overhead, his chance of winning the point is 81%–most of the hard work is already done, having generated such a favorable situation. If our player ends up winning the point, the smash improved his odds by 0.19 points (from 0.81 to 1.0), and if he ends up losing the point, the smash hurt his odds by 0.81 (from 0.81 to 0.0). A player who hits five successful smashes in a row has a smash worth about one total point: 5 multiplied by 0.19 equals 0.95.

We can use this simple formula to estimate how much each player’s smash is worth, denominated in points. We’ll call that Point Probability Added (PPA). Finally, we need to take into account how often the player hits his smash. To do so, we’ll simply divide PPA by total number of points played, then multiply by 100 to make the results more readable. The metric, then, is PPA per 100 points, reflecting the impact of the smash in a typical short match. Most players have similar numbers of smash opportunities, but as we’ve seen, some choose to hit far more overheads than others. When we divide by points, we give more credit to players who hit their smashes more often.

The overall impact of the smash turns out to be quite small. Here are the 1990s-and-later players with at least 99 smash opportunities in the dataset along with their smash PPA per 100 points:

PLAYER                 SM PPA/100  
Jo-Wilfried Tsonga           0.17  
Pete Sampras                 0.11  
Tomas Berdych                0.11  
Roger Federer                0.10  
Rafael Nadal                 0.05  
Milos Raonic                 0.04  
Juan Martin del Potro        0.02  
Andy Murray                  0.01  
Kevin Anderson               0.01  
Kei Nishikori                0.00  
David Ferrer                 0.00  
Andre Agassi                 0.00  
Novak Djokovic              -0.02  
Stan Wawrinka               -0.07  
Dominic Thiem               -0.07  
Pablo Cuevas                -0.10

Tsonga reigns supreme, from the most basic measurement to the most complex. His 0.17 smash PPA per 100 points means that the quality of his overhead earns him about one extra point (compared to an average ATPer) every 600 points. That doesn’t sound like much, and rightfully so: He hits fewer than one smash per 50 points, and as good as Tsonga is, the average player has a very serviceable smash as well.

The list gives us an idea of the overall range of smash-hitting ability, as well. Among active players, the laggard in this group is Pablo Cuevas, at -0.1 points per 100, meaning that his subpar smash costs him one point out of every thousand he plays. It’s possible to be worse–in Lorenzi’s small sample, his rate is -0.65–but if we limit our scope to these well-studied players, the difference between the high and low extremes is barely 0.25 points per 100, or one point out of every 400.

I’ve excluded several players from earlier generations from this list; as mentioned earlier, the average smash success rate in those days was lower, so measuring legends like McEnroe and Borg using a 2010s-based point probability formula is flat-out wrong. That said, we’re on safe ground with Sampras and Agassi; the rate at which players convert smashes into points won has remained fairly steady since the early 1990s.

Lob-responding value

We’ve seen the potential impact of smash skill; let’s widen our scope again and look at the potential impact of smash opportunity skill. When a player is faced with a lob, but before he decides what shot to hit, his chance of winning the point is about 72%. Thus, hitting a shot that results in winning the point is worth 0.28 points of point probability added, while a choice that ends up losing the point translates to -0.72.

There are more smash opportunities than smashes, and more room to improve on the average (72% instead of 81%), so we would expect to see a bigger range of PPA per 100 points. Put another way, we would expect that lob-responding skill, which includes smashes, is more important than smash-specific skill.

It’s a modest difference, but it does look like lob-responding skill has a bigger range than smash skill. Here is the same group of players, still showing their PPA/100 for smashes (SM PPA/100), now also including their PPA/100 for smash opportunities (SMO PPA/100):

PLAYER                 SM PPA/100  SMO PPA/100  
Jo-Wilfried Tsonga           0.17         0.18  
Roger Federer                0.10         0.16  
Pete Sampras                 0.11         0.16  
Milos Raonic                 0.04         0.12  
Tomas Berdych                0.11         0.09  
Kevin Anderson               0.01         0.08  
Novak Djokovic              -0.02         0.07  
Rafael Nadal                 0.05         0.03  
Andre Agassi                 0.00         0.01  
Stan Wawrinka               -0.07         0.00  
Kei Nishikori                0.00        -0.03  
Andy Murray                  0.01        -0.03  
Dominic Thiem               -0.07        -0.05  
David Ferrer                 0.00        -0.06  
Pablo Cuevas                -0.10        -0.12  
Juan Martin del Potro        0.02        -0.19

Djokovic and Delpo draw our attention again as the players whose smash skills do not accurately represent their smash opportunity skills. Djokovic is slightly below average with smashes, but a few notches above the norm on opportunities; Delpo is a tick above average when he hits smashes, but dreadful when dealing with lobs in general.

As it turns out, we can measure the best smashes in tennis, both to compare players and to get a general sense of the shot’s importance. What we’ve also seen is that smashes don’t tell the entire story–we learn more about a player’s overall ability when we widen our view to smash opportunities.

Smashes in the women’s game

Contemporary women hit far fewer smashes than men do, and they win points less often when they hit them. Despite the differences, the reasoning outlined above applies just as well to the WTA. Let’s take a look.

In the WTA of this decade, smashes result in winners (or induced forced errors) 63% of the time, and smashes result in points won about 75% of the time. Both numbers are lower than the equivalent ATP figures (69% and 81%, respectively), but not dramatically so. Here are the rates of winners, errors, and points won per smash for the 14 women with at least 80 smashes in the MCP dataset:

PLAYER               W/SM  E/SM  PTS/SM  
Jelena Jankovic       73%    9%     83%  
Serena Williams       72%   13%     81%  
Steffi Graf           61%    9%     81%  
Svetlana Kuznetsova   70%   10%     79%  
Simona Halep          66%   11%     76%  
Caroline Wozniacki    61%   16%     74%  
Karolina Pliskova     62%   18%     74%  
Agnieszka Radwanska   54%   13%     74%  
Angelique Kerber      57%   15%     72%  
Martina Navratilova   54%   13%     71%  
Monica Niculescu      50%   15%     70%  
Garbine Muguruza      63%   19%     70%  
Petra Kvitova         59%   22%     68%  
Roberta Vinci         58%   14%     68%

Historical shot-by-shot data is less representative for women than for men, so it’s probably safest to assume that trends in smash success rates are similar for men than for women. If that’s true, Steffi Graf’s era is similar to the present, while Martina Navratilova’s prime saw far fewer smashes going for winners or points won.

Where the women’s game really differs from the men’s is the difference between smash opportunities (lobs) and smashes. As we saw above, 72% of ATP smash opportunities result in smashes. In the current WTA, the corresponding figure is less than half that: 35%. Some of the single-player numbers are almost too extreme to be believed: In 12 matches, Catherine Bellis faced 41 lobs and hit 3 smashes; in 29 charted matches, Jelena Ostapenko saw 103 smash opportunities and tried only 10 smashes. A generation ago, the gender difference was tiny: Graf, Martina Hingis, Arantxa Sanchez Vicario, and Monica Seles all hit smashes in at least three-quarters of their opportunities. But among active players, only Barbora Strycova comes in above 70%.

Here are the smash opportunity numbers for the 17 women with at least 150 smash opportunities in the MCP dataset. SM/SMO is smashes per chance, W/SMO is winners (and induced forced errors) per smash opportunity, E/SMO is errors per opportunity, and PTS/SMO is points won per smash opportunity:

PLAYER                SM/SMO  W/SMO  E/SMO  PTW/SMO  
Maria Sharapova          12%    57%    11%      76%  
Serena Williams          55%    58%    18%      72%  
Steffi Graf              82%    52%    17%      71%  
Karolina Pliskova        47%    52%    16%      70%  
Simona Halep             14%    41%    11%      69%  
Carla Suarez Navarro     25%    33%     9%      69%  
Eugenie Bouchard         29%    50%    18%      68%  
Victoria Azarenka        35%    52%    17%      67%  
Angelique Kerber         39%    42%    14%      66%  
Garbine Muguruza         43%    51%    18%      66%  
Monica Niculescu         57%    41%    19%      65%  
Petra Kvitova            48%    50%    19%      65%  
Agnieszka Radwanska      44%    42%    18%      65%  
Johanna Konta            30%    47%    21%      64%  
Caroline Wozniacki       36%    44%    18%      64%  
Elina Svitolina          14%    38%    14%      63%  
Martina Navratilova      67%    42%    26%      58%

It’s clear from the top of this list that women’s tennis is a different ballgame. Maria Sharapova almost never opts for an overhead, but when faced with a lob, she is the best of them all. Next up is Serena Williams, who hits almost as many smashes as any active player on this list, and is nearly as successful. Recall that in the men’s game, there is a modest positive correlation between smashes per opportunity and points won per smash opportunity; here, the relationship is weaker, and slightly negative.

Because most women hit so few smashes, there isn’t quite as much to be gained by using point probability added (PPA) to measure WTA smash skill. Graf was exceptionally good, comparable to Tsonga in the value she extracted from her smash, but among active players, only Serena and Victoria Azarenka can claim a smash that is worth close to one point per thousand. At the other extreme, Monica Niculescu is nearly as bad as Graf was good, suggesting she ought to figure out a way to respond to more smash opportunities with her signature forehand slice.

Here is the same group of women (minus Navratilova, whose era makes PPA comparisons misleading), with their PPA per 100 points for smashes (SM PPA/100) and smash opportunities (SMO PPA/100):

PLAYER                SM PPA/100  SMO PPA/100  
Maria Sharapova             0.03         0.21  
Serena Williams             0.09         0.15  
Steffi Graf                 0.15         0.14  
Karolina Pliskova          -0.01         0.09  
Carla Suarez Navarro        0.04         0.08  
Simona Halep                0.00         0.07  
Eugenie Bouchard           -0.02         0.03  
Victoria Azarenka           0.08         0.00  
Angelique Kerber           -0.03        -0.02  
Garbine Muguruza           -0.07        -0.03  
Petra Kvitova              -0.07        -0.04  
Monica Niculescu           -0.13        -0.06  
Caroline Wozniacki         -0.01        -0.07  
Agnieszka Radwanska        -0.02        -0.07  
Johanna Konta              -0.12        -0.08  
Elina Svitolina             0.01        -0.09

The table is sorted by smash opportunity PPA, which tells us about a much more relevant skill in the women’s game. Sharapova’s lob-responding ability is well ahead of the pack, worth better than one point above average per 500, with Serena and Graf not far behind. The overall range among these well-studied players, from Sharapova’s 0.21 to Elina Svitolina’s -0.09, is somewhat smaller than the equivalent range in the ATP, but with such outliers as Sharapova here and Delpo on the men’s side, it’s tough to draw firm conclusions from small subsets of players, however elite they are.

Final thought

The approach I’ve outlined here to measure the impact of smash and smash-opportunity skills is one that could be applied to other shots. Smashes are a good place to start because they are so simple: Many of them end points, and even when they don’t, they often virtually guarantee that one player will win the point. While smashes are a bit more complex than they first appear, the complications involved in applying a similar algorithm to, say, backhands and backhand opportunities, are considerably greater. That said, I believe this algorithm represents a promising entry point to these more daunting problems.

Forecasting the Laver Cup

Italian translation at settesei.it

This weekend brings us the first edition of the Laver Cup, a star-studded three-day affair that pits Europe against the rest of the world. The European team features Roger Federer and Rafael Nadal, and even though several other elites from the continent are missing due to injury, the European team is still much stronger on paper.

Here are the current rosters, along with each competitor’s weighted hard court Elo rating and rank among active players:

EUROPE                  Elo Rating  Elo Rank  
Roger Federer                 2350         2  
Rafael Nadal                  2225         4  
Alexander Zverev              2127         7  
Tomas Berdych                 2038        14  
Marin Cilic                   2029        15  
Dominic Thiem                 1995        17  
                                              
WORLD                   Elo Rating  Elo Rank  
Nick Kyrgios                  2122         8  
John Isner                    1968        22  
Jack Sock                     1951        23  
Sam Querrey                   1939        25  
Denis Shapovalov              1875        36  
Frances Tiafoe                1574       153  
Juan Martin del Potro*        2154         5

*del Potro has withdrawn. I’ve included his singles Elo rating and rank to emphasize how damaging his absence is to the World squad.

“Weighted” surface Elo is the average of overall (all-surface) Elo and surface-specific Elo. The 50/50 split is a much better predictor of match outcomes than either number on its own.

Nick Kyrgios can hang with anybody on a hard court. But despite some surface-specific skills represented by the American contingent, every other member of the World team rates lower than every member of team Europe. This isn’t a good start for the rest of the world.

What about doubles? Here are the D-Lo (Elo for doubles) ratings and rankings for all twelve participants, plus Delpo:

EUROPE                  D-Lo rating  D-Lo rank  
Rafael Nadal                   1895          4  
Tomas Berdych                  1760         28  
Marin Cilic                    1676         76  
Roger Federer**                1650         90  
Alexander Zverev               1642         99  
Dominic Thiem                  1521        185  
                                                
WORLD                   D-Lo rating  D-Lo rank  
Jack Sock                      1866          8  
John Isner                     1755         29  
Nick Kyrgios                   1723         45  
Sam Querrey                    1715         49  
Denis Shapovalov**             1600        130  
Frances Tiafoe                 1546        166  
Juan Martin del Potro*         1711         55

** Federer hasn’t played tour-level doubles since 2015, and Shapovalov hasn’t done so at all. These numbers are my best guesses, nothing more.

Here, the World team has something of an edge. While both sides feature an elite doubles player–Rafa and Jack Sock–the non-European side is a bit deeper, especially if they keep Denis Shapovalov and last-minute Delpo replacement Frances Tiafoe on the sidelines. Only one-quarter of Laver Cup matches are doubles (plus a tie-breaking 13th match, if necessary), so it still looks like team Europe are the heavy favorite.

The format

The Laver Cup will take place in Prague over three days (starting Friday, September 22nd), and consist of four matches each day: three singles and one doubles. Every match is best-of-three sets with ad scoring and a 10-point super-tiebreak in place of the third set.

On the first day, the winner of each match gets one point; on the second day, two points, and on the third day, three points. That’s a total of 24 points up for grabs, and if the twelve matches end in a 12-12 deadlock, the Cup will be decided with a single doubles set.

All twelve participants must play at least one singles match, and no one can play more than two. At least four members of each squad must play doubles, and no doubles pairing can be repeated, except in the case of a tie-breaking doubles set.

Got it? Good.

Optimal strategy

The rules require that three players on each side will contest only one singles match while the other three will enter two each. A smart captain would, health permitting, use his three best players twice. Since matches on days two and three count for more than matches on day one, it also makes sense that captains would use their best players on the final two days.

(There are some game-theoretic considerations I won’t delve into here. Team World could use better players on day one in hopes of racking up each points against the lesser members of team Europe, or could drop hints that they will do so, hoping that the European squad would move its better players to day one. As far as I can tell, neither team can change their lineup in response to the other side’s selections, so the opportunities for this sort of strategizing are limited.)

In doubles, the ideal roster deployment strategy would be to use the team’s best player in all three matches. He would be paired with the next-best player on day three, the third-best on day two, and the fourth-best on day one. Again, this is health permitting, and since all of these guys are playing singles, fatigue is a factor as well. My algorithm thus far would use Nadal five times–twice in singles and three times in doubles–and I strongly suspect that isn’t going to happen.

The forecast

Let’s start by predicting the outcome of the Cup if both captains use their roster optimally, even if that’s a longshot. I set up the simulation so that each day’s singles competitors would come out in random order–if, say, Querrey, Shapovalov, and Tiafoe play for team World on day one, we don’t know which of them will play first, or which European opponent each will face. So each run of the simulation is a little different.

As usual, I used Elo (and D-Lo) to predict the outcome of specific matchups. Because of the third-set super-tiebreak, and because it’s an exhibition, I added a bit of extra randomness to every forecast, so if the algorithm says a player has a 60% chance of winning, we knock it down to around 57.5%. When I dug into IPTL results last winter, I discovered that exhibition results play surprisingly true to expectations, and I suspect players will take Laver Cup a bit more seriously than they do IPTL.

Our forecast–again, assuming optimal player usage–says that Europe has an 84.3% chance of winning, and the median point score is 16-8. There’s an approximately 6.5% chance that we’ll see a 12-12 tie, and when we do, Europe has a slender 52.4% edge.

If Delpo were participating, he would increase the World team’s chances by quite a bit, reducing Europe’s likelihood of victory to 75.5% and narrowing the most probable point score to 15-9.

What if we relax the “optimal usage” restriction? I have no idea how to predict what captains John McEnroe and Bjorn Borg will do, but we can randomize which players suit up for which matches to get a sense of how much influence they have. If we randomize everything–literally, just pick a competitor out of a hat for each match–Europe comes out on top 79.7% of the time, usually winning 15-9. There’s a 7.6% chance of a tie-breaking 13th match, and because the World team’s doubles options are a bit deeper, they win a slim majority of those final sets. (When we randomize everything, there’s a slight risk that we violate the rules, perhaps using the same doubles pairing twice or leaving a player on the bench for all nine singles matches. Those chances are very low, however, so I didn’t tackle the extra work required to avoid them entirely.)

We can also tweak roster usage by team, in case it turns out that one captain is much savvier than the other. (Or if a star like Nadal is unable to play as much as his team would like.) The best-case scenario for our World team underdogs is that McEnroe chooses the best players for each match and Borg does not. Assuming that only European players are chosen from a hat, the probability that the favorites win falls all the way to 63.1%, and the typical gap between point totals narrows all the way to 13-11. The chance of a tie rises to 10%.

On the other hand, it’s possible that Borg is better at utilizing his squad. After all, it doesn’t take an 11-time grand slam winner to realize that Federer and Nadal ought to be on court when the stakes are the highest. This final forecast, with random roster usage from team World and ideal choices from Borg, gives Europe a whopping 92.3% chance of victory, and median point totals of 17 to 7. The World team would have only a 4% shot at reaching a deadlock, and even then, the Europeans win two-thirds of the tiebreakers.

There we have it. The numbers bear out our expectation that Europe is the heavy favorite, and they give us a sense of the likely margin of victory. Tiafoe and Shapovalov might someday be part of a winning Laver Cup side, but it looks like they’ll have to wait a few years before that happens.

–

Update: One more thing… What about doubles specialists? Both captains have two discretionary picks to use on players regardless of ranking. Most great doubles players are much worse at singles, but as we’ve seen, a player can be relegated to a lone one-point singles match on day one, and as a doubles player, he can have an effect on three different matches, totaling six points.

Sure enough, swapping out Dominic Thiem (a very weak doubles player for whom indoor hard courts are less than ideal) for Nicolas Mahut would have increased Europe’s chances of winning from 84.3% to 88.5%. On the slight chance that the Cup stayed tight through the final doubles match and into a tiebreaker, the doubles team of Mahut-Nadal (however unorthodox that sounds) would be among the best that any captain could put on the court.

There’s even more room for improvement on the World side, especially with del Potro out. At the moment, the third-highest rated hard court player by D-Lo is Marcelo Melo, who would be a major step down in singles but a huge improvement on most of the potential partners for Sock in doubles. If we give him a singles Elo of 1450 and put him on the roster in place of Tiafoe and pit the resulting squad against the original Europe team (with Thiem, not Mahut), it almost makes up for the loss of Delpo–World’s chances of winning increase from 15.7% to 19.3%.

Unfortunately, Borg and McEnroe may have missed their chance to eke out extra value from their six-man rosters–this is a trick that will only work once. If both teams made this trade, Mahut-for-Thiem and Melo-for-Tiafoe, each side’s win probability goes back to near where it started: 85.8% for Europe. That’s a boost over where we started (84.3%), just because Mahut is better suited for the competition than Melo is, as an elite doubles specialist who is also credible on the singles court. No one available to the World team (except for Sock, who is already on the roster) fits the same profile on a hard court. Vasek Pospisil comes to mind, though he has taken a step back from his peaks in both singles and doubles. And on clay, Pablo Cuevas would do nicely, but on a faster surface, he would represent only a marginal improvement over the doubles players already playing for team World.

Maybe next year.

A Preface to All GOAT Arguments

Italian translation at settesei.it

Earlier this week, The Economist published my piece about Rafael Nadal’s and Roger Federer’s grand slam counts. I made the case that, because Nadal’s paths to major titles had been more difficult (the 2017 US Open notwithstanding), his 16 slams are worth more–barely!–than Federer’s.

Inevitably, some readers reduced my conclusion to something like, “stats prove that Nadal is the greatest ever.” Whoa there, kiddos. It may be true that Nadal is better than Federer, and we could probably make a solid argument based on the stats. But a rating of 18.8 to 18.7, based on 35 tournaments, can’t quite carry that burden.

There are two major steps in settling any “greatest ever” debate (tennis or otherwise). The first is definitional. What do we mean by “greatest?” How much more important are slams than non-slams? What about longevity? Rankings? Accomplishments across different surfaces? How much weight do we give a player’s peak? How much does the level of competition matter? What about head-to-head records? I could go on and on. Only when we decide what “greatest” means can we even attempt to make an argument for one player over another.

The second step–answering the questions posed by the first–is more work-intensive, but much less open to debate. If we decide that the greatest male tennis player of all time is the one who achieved the highest Elo rating at his peak, we can do the math. (It’s Novak Djokovic.) If you pick out ten questions that are plausible proxies for “who’s the greatest?” you won’t always get the same answer. Longevity-focused variations tend to give you Federer. (Or Jimmy Connors.) Questions based solely on peak-level accomplishments will net Djokovic (or maybe Bjorn Borg). Much of the territory in between is owned by Nadal, unless you consider the amateur era, in which case Rod Laver takes a bite out of Rafa’s share.

Of course, many fans skip straight to the third step–basking in the reflected glory of their hero–and work backwards. With a firm belief that their favorite player is the GOAT, they decide that the most relevant questions are the ones that crown their man. This approach fuels plenty of online debates, but it’s not quite at my desired level of rigor.

When the big three have all retired, someone could probably write an entire book laying out all the ways we might determine “greatest” and working out who, by the various definitions, comes out on top. Most of what we’re doing now is simply contributing sections of chapters to that eventual project. Now or then, one blog post will never be enough to settle a debate of this magnitude.

In the meantime, we can aim to shed more light on the comparisons we’re already making. Grand slam titles aren’t everything, but they are important, and “19 is more than 16” is a key weapon in the arsenal of Federer partisans. Establishing that this particular 19 isn’t really any better than that particular 16 doesn’t end the debate any more than “19 is more than 16” ever did. But I hope that it made us a little more knowledgeable about the sport and the feats of its greatest competitors.

At the one-article, 1,000-word scale, we can achieve a lot of interesting things. But for an issue this wide-ranging, we can’t hope to settle it in one fell swoop. The answers are hard to find, and choosing the right question is even more difficult.

Fun With Service Point Ratios

Italian translation at settesei.it

In Rafael Nadal’s comprehensive victory over Kevin Anderson in the 2017 US Open final, Nadal didn’t face a single break point. Anderson didn’t even earn very many deuces. Nadal, on the other hand, constantly challenged in his opponent’s service games.

This produced an unusual ratio: Anderson had to play way more service points than Nadal did, even though they served the same number of games. Rafa toed the line only 72 times to the South African’s 108, for a ratio of 2/3 or, rounded, 0.67. In this week’s podcast, I speculated that this service point ratio is a handy way of spotting winners–if one man is getting through his service games much quicker than the other, it’s probably because he is holding easily and his opponent is not.

It wasn’t the best hypothesis I’ve ever put forward. It’s true, but not by an overwhelming margin. In the average ATP match, the ratio of the winner’s service points played to the loser’s service points played is 0.96 — equivalent to Rafa serving 88 times to Anderson’s 92. The winnner plays fewer service points in 57% of contests. We’ve hardly discovered the next IBM Key to the Match here.

Instead of discovering a useful proxy for success in the most basic of match stats, we’ve come upon yet another item to add to the list of Nadal’s extreme accomplishments. Of nearly 13,000 completed grand slam singles matches since 1991, only 147 of the winners–barely one percent–had service point ratios below 0.67. Out of 106 major finals with stats available, Rafa’s ratio on Sunday was the lowest on record. He just edged out Roger Federer‘s 0.68 ratio from the 2007 Australian Open final against Fernando Gonzalez.

It turns out that the service point ratio is as fluky for Rafa as it is for men as a whole. Of his 16 victories in grand slam finals, he has posted a ratio below 1.0 in eight of them, equal to 1.0 once, and above 1.0 seven times. His average is an uninteresting 0.98.

There you have it: Over the course of a single week, we’ve seen an oddity, devised a stat to capture it, and determined that it doesn’t tell us much. Analytics, anyone?

—

For a more serious look at Rafa’s career accomplishments after bringing home his 16th major title, check out my analysis posted yesterday at The Economist’s Game theory blog.

Denis Shapovalov and Fast ATP Starts

Italian translation at settesei.it

18-year-old Canadian lefty Denis Shapovalov has had one heck of a summer. In Montreal, he defeated Juan Martin del Potro and Rafael Nadal in back-to-back matches, and at the US Open, he qualified for the main draw, upset Jo Wilfried Tsonga, and reached the fourth round in only his second appearance at a major.

Thanks to those wins and the big stages on which he achieved them, he has cracked the ATP top 60, despite playing fewer than 20 tour-level matches. The Elo rating system, which awards points based on opponent quality, is even more optimistic. By that measure, with his win over Tsonga, Shapovalov improved to 1950–good for 34th on tour–before losing about 25 Elo points in his loss to Pablo Carreno Busta.

While an Elo score of 1950 is an arbitrary number–there’s nothing magical about any particular Elo threshold; it’s just a mechanism to compare players to each other–it gives us a way to compare Shapovalov’s hot start with other players who made quick impacts at tour level. Since the early 1980s, only 13 players have reached a 1950 Elo score in fewer matches than the Canadian needed. As usual with early-career accomplishments, there are a few unexpected names in the mix, but overall, it’s very promising company for an 18-year-old:

Player               Matches   Age  
Lleyton Hewitt             7  16.9  
Jarkko Nieminen            7  20.2  
Juan Carlos Ferrero       10  19.4  
David Ferrer              12  20.4  
Kenneth Carlsen           12  19.4  
Tommy Haas                13  19.1  
Peter Lundgren            13  20.7  
John Van Lottum           14  21.8  
Sergi Bruguera            14  18.4  
Julian Alonso             15  20.0

Player               Matches   Age   
Xavier Malisse            16  18.6  
Jan Siemerink             16  20.9  
Ivo Minar                 16  21.2  
Florian Mayer             17  20.7  
Cristiano Caratti         17  20.7  
Nick Kyrgios              17  19.3  
Denis Shapovalov          17  18.4  
Martin Strelba            17  22.1  
Jay Berger                17  20.2  
Andy Roddick              18  18.6

I identified just over 350 players who, at some point in their careers, peaked with an Elo score of at least 1950. On average, these players needed 75 matches to reach that level (the median is 59), and two active tour-regulars, Gilles Muller and Albert Ramos, needed almost 300 matches to achieve the threshold.

Shapovalov’s record so far is equally impressive when we consider it in terms of age. Again, he’s among the top 20 players in modern tennis history: Only 11 players got to 1950 before their 18th birthday. The Canadian is only a few months beyond his. And many of the other ATPers who reached that score at an early age needed much more tour experience. I’ve included the top 30 on this list to show how Shapovalov compares to so many of the game’s greats:

Player                  Matches   Age  
Aaron Krickstein             25  16.4  
Michael Chang                32  16.5  
Lleyton Hewitt                7  16.9  
Boris Becker                 27  17.5  
Mats Wilander                27  17.5  
Guillermo Perez Roldan       26  17.6  
Andre Agassi                 46  17.6  
Pat Cash                     66  17.6  
Goran Ivanisevic             35  17.7  
Andrei Medvedev              22  17.8  

Player                  Matches   Age
Rafael Nadal                 44  17.9  
Sammy Giammalva              21  18.0  
Horst Skoff                  19  18.1  
Jimmy Arias                  61  18.2  
Kent Carlsson                56  18.3  
Sergi Bruguera               14  18.4  
Denis Shapovalov             17  18.4  
Andy Murray                  22  18.4  
Juan Martin del Potro        31  18.4  
Fabrice Santoro              59  18.5  

Player                  Matches   Age
John McEnroe                 28  18.5  
Roger Federer                40  18.5  
Stefan Edberg                40  18.5  
Andy Roddick                 18  18.6  
Pete Sampras                 56  18.6  
Thomas Enqvist               28  18.6  
Xavier Malisse               16  18.6  
Novak Djokovic               33  18.8  
Jim Courier                  51  18.8  
Yannick Noah                 41  18.8

There are no guarantees when it comes to tennis prospects, but this is very good company. On average, the 23 other players to reach the 1950 Elo threshold at age 18 improved their Elo ratings to 2100 before age 20, and rose to 2250 at some point in their careers. The first number would be good for 12th on today’s list, and the second would merit 5th place, just behind the Big Four. Nadal and del Potro were the first of Shapovalov’s high-profile victims, and judging from this sharp career trajectory, they won’t be the last.

Podcast Episode 17: US Open in Review

Episode 17 of the Tennis Abstract Podcast, with Carl Bialik, is our US Open recap. We start with a discussion of Sloane Stephens–her performance here as well as what we expect from her, and we delve into possible explanations of her impressive performance in particular against aggressive players.

We then talk Nadal and his no-nonsense strategy to defeat Kevin Anderson, and consider best-case scenarios for the rest of Kev’s career. We finish up with some doubles and a bit on this weekend’s Davis Cup ties. As always, thanks for listening!

Click to listen, subscribe on iTunes, or use our feed to get updates on your favorite podcast software.

Podcast Episode 16: The Second Half of our Second Week Chat

Episode 16 of the Tennis Abstract Podcast, with Carl Bialik, didn’t quite work out as planned — my microphone malfunctioned for much of the first half of the recording — but the second half of our conversation could be salvaged. Thus, this episode is missing the big news of the second week, the all-American women’s semifinals, but we still touched on a variety of burning US Open topics, like the youngsters making news in New York and the inevitable hypotheticals of the players who weren’t able to participate in Flushing this year.

This is a shorter-than usual episode, clocking in at 34 minutes; we hope to return on Friday with another mid-Slam update. Thanks for listening!

Click to listen, subscribe on iTunes, or use our feed to get updates on your favorite podcast software.

Quantifying Cakewalks, or The Time Rafa Finally Got Lucky

Italian translation at settesei.it

During this year’s US Open, much has been made of some rather patchy sections of the draw. Many great players are sitting out the tournament with injury, and plenty of others crashed out early. Pablo Carreno Busta reached the quarterfinals by defeating four straight qualifiers, and Rafael Nadal could conceivably win the title without beating a single top-20 player.

None of this is a reflection on the players themselves: They can play only the draw they’re dealt, and we’ll never know how they would’ve handled a more challenging array of opponents. The weakness of the draw, however, could affect how we remember this tournament. If we are going to let the quality of the field color our memories, we should at least try to put this year’s players in context to see how they compare with majors in the past.

How to measure draw paths

There are lots of ways to quantify draw quality. (There’s an entire category on this blog devoted to it.) Since we’re interested in the specific sets of opponents faced by our remaining contenders, we need a metric that focuses on those. It doesn’t really matter that, say, Nick Kyrgios was in the draw, since none of the semifinalists had to play him.

Instead of draw difficulty, what we’re after is what I’ll call path ease. It’s a straightforward enough concept: How hard is it to beat the specific set of guys that Rafa (for instance) had to play?

To get a number, we’ll need a few things: The surface-weighted Elo ratings of each one of a player’s opponents, along with a sort of “reference Elo” for an average major semifinalist. (Or finalist, or title winner.) To determine the ease of Nadal’s path so far, we don’t want to use Nadal’s Elo. If we did that, the exact same path would look easier or harder depending on the quality of the player who faced it.

(The exact value of the “reference Elo” isn’t that important, but for those of you interested in the numbers: I found the average Elo rating of every slam semifinalist, finalist, and winner back to 1988 on each of the three major surfaces. On hard courts, those numbers are 2145, 2198, and 2233, respectively. When measuring the difficulty of a path to the semifinal round, I used the first of those numbers; for the difficulty of a path to the title, I used the last.)

To measure path ease, then, we answer the question: What are the odds that an average slam semifinalist (for instance) would beat this particular set of players? In Rafa’s case, he has yet to face a player with a weighted-hard-court Elo rating above 1900, and the typical 2145-rated semifinalist would beat those five players 71.5% of the time. That’s a bit easier than Kevin Anderson‘s path the semis, but a bit harder than Carreno Busta’s. Juan Martin del Potro, on the other hand, is in a different world altogether. Here are the path ease numbers for all four semifinalists, showing the likelihood that average contenders in each round would advance, giving the difficulty of the draws each player has faced:

Semifinalist   Semi Path  Final Path  Title Path  
Nadal              71.5%       49.7%       51.4%  
del Potro           9.1%        7.5%       10.0%  
Anderson           69.1%       68.9%       47.1%  
Carreno Busta      74.3%       71.2%       48.4%

(We don’t yet know each player’s path to the title, so I averaged the Elos of possible opponents. Anderson and Carreno Busta are very close, so for Rafa and Delpo, their potential final opponent doesn’t make much difference.)

There’s one quirk with this metric that you might have noticed: For Nadal and del Potro, their difficulty of reaching the final is greater than that of winning the title altogether! Obviously that doesn’t make logical sense–the numbers work out that way because of the “reference Elos” I’m using. The average slam winner is better than the average slam finalist, so the table is really saying that it’s easier for the average slam winner to beat Rafa’s seven opponents than it would be for the average slam finalist to get past his first six opponents. This metric works best when comparing title paths to title paths, or semifinal paths to semifinal paths, which is what we’ll do for the rest of this post.

Caveats and quirks aside, it’s striking just how easy three of the semifinal paths have been compared to del Potro’s much more arduous route. Even if we discount the difficulty of beating Roger Federer–Elo thinks he’s the best active player on hard courts but doesn’t know about his health issues–Delpo’s path is wildly different from those of his semifinal and possible final opponents.

Cakewalks in context

Semifinalist path eases of 69% or higher–that is, easier–are extremely rare. In fact, the paths of Anderson, Carreno Busta, and Nadal are all among the ten easiest in the last thirty years! Here are the previous top ten:

Year  Slam             Semifinalist               Path Ease  
1989  Australian Open  Thomas Muster                  84.1%  
1989  Australian Open  Miloslav Mecir                 74.2%  
1990  Australian Open  Ivan Lendl                     73.8%  
2006  Roland Garros    Ivan Ljubicic                  73.7%  
1988  Australian Open  Ivan Lendl                     72.2%  
1988  Australian Open  Pat Cash                       70.1%  
2004  Australian Open  Juan Carlos Ferrero            69.2%  
1996  US Open          Michael Chang                  68.8%  
1990  Roland Garros    Andres Gomez                   68.4%  
1996  Australian Open  Michael Chang                  66.2%

In the last decade, the easiest path to the semifinal was Stan Wawrinka‘s route to the 2016 French Open final four, which rated 59.8%. As we’ll see further on, Wawrinka’s draw got a lot more difficult after that.

Del Potro’s draw so far isn’t quite as extreme, but it is quite difficult in the historical context. Of the nearly 500 major semifinalists since 1988, all but 15 are easier than his 9.1% path difficulty. Here are the top ten, all of whom faced draws that would have given the average slam semifinalist less than an 8% chance of getting that far:

Year  Slam             Semifinalist              Path Ease  
2009  Roland Garros    Robin Soderling                1.6%  
1988  Roland Garros    Jonas Svensson                 1.9%  
2017  Wimbledon        Tomas Berdych                  3.7%  
1996  Wimbledon        Richard Krajicek               6.4%  
2011  Wimbledon        Jo Wilfried Tsonga             6.6%  
2012  US Open          Tomas Berdych                  6.8%  
2017  Roland Garros    Dominic Thiem                  6.9%  
2014  Australian Open  Stan Wawrinka                  7.0%  
1989  Roland Garros    Michael Chang                  7.1%  
2017  Wimbledon        Sam Querrey                    7.5%

Previewing the history books

In the long term, we’ll care a lot more about how the 2017 US Open champion won the title than how he made it through the first five rounds. As we saw above, three of the four semifinalists have a path ease of around 50% to win the title–again, meaning that a typical slam winner would have a roughly 50/50 chance of getting past this particular set of seven opponents.

No major winner in recent memory has had it so easy. Nadal’s path would rate first in the last thirty years, while Carreno Busta’s or Anderson’s would rate in the top five. (If it comes to that, their exact numbers will depend on who they face in the final.) Here is the list that those three men have the chance to disrupt:

Year  Slam             Winner                  Path Ease  
2002  Australian Open  Thomas Johansson            48.1%  
2001  Australian Open  Andre Agassi                47.6%  
1999  Roland Garros    Andre Agassi                45.6%  
2000  Wimbledon        Pete Sampras                45.3%  
2006  Australian Open  Roger Federer               44.5%  
1997  Australian Open  Pete Sampras                44.4%  
2003  Australian Open  Andre Agassi                43.9%  
1999  US Open          Andre Agassi                41.5%  
2002  Wimbledon        Lleyton Hewitt              39.9%  
1998  Wimbledon        Pete Sampras                39.1%

At the 2006 Australian Open, Federer lucked into a path that was nearly as easy as Rafa’s this year. His 2003 Wimbledon title just missed the top ten as well. By comparison, Novak Djokovic has never won a major with a path ease greater than 18.7%–harder than that faced by more than half of major winners.

Nadal has hardly had it easy as he has racked up his 15 grand slams, either. Here are the top ten most difficult title paths:

Year  Slam             Winner                Path Ease  
2014  Australian Open  Stan Wawrinka              2.2%  
2015  Roland Garros    Stan Wawrinka              3.1%  
2016  Us Open          Stan Wawrinka              3.2%  
2013  Roland Garros    Rafael Nadal               4.4%  
2014  Roland Garros    Rafael Nadal               4.7%  
1989  Roland Garros    Michael Chang              5.0%  
2012  Roland Garros    Rafael Nadal               5.2%  
2016  Australian Open  Novak Djokovic             5.4%  
2009  US Open          J.M. Del Potro             5.9%  
1990  Wimbledon        Stefan Edberg              6.2%

As I hinted in the title of this post, while Nadal got lucky in New York this year, it hasn’t always been that way. He appears three times on this list, facing greater challenges than any major winner other than Wawrinka the giant-killer.

On average, Rafa’s grand slam title paths haven’t been quite as harrowing as Djokovic’s, but compared to most other greats of the last few decades, he has worked hard for his titles. Here are the average path eases of players with at least three majors since 1988:

Player           Majors        Avg Path Ease  
Stan Wawrinka         3                 2.8%  
Novak Djokovic       12                11.3%  
Rafael Nadal         15                13.6%  
Stefan Edberg         4                14.6%  
Andy Murray           3                18.8%  
Boris Becker          4                18.8%  
Mats Wilander         3                19.8%  
Gustavo Kuerten       3                22.0%  
Roger Federer        19                23.5%  
Jim Courier           4                26.4%  
Pete Sampras         14                28.9%  
Andre Agassi          8                32.3%

If Rafa adds to his grand slam haul this weekend, his average path ease will take a bit of a hit. Still, he’ll only move one place down the list, behind Stefan Edberg. After more than a decade of battling all-time greats in the late rounds of majors, it’s fair to say that Nadal deserved this cakewalk.

Update: This post reads a bit differently than when I first wrote it: I’ve changed the references to “path difficulty” to “path ease” to make it clearer what the metric is showing.

Nadal and Anderson advanced to the final, so we can now determine the exact path ease number for whichever one of them wins the title. Rafa’s exact number remains 51.4%, and should he win, his career average across 16 slams will increase to about 15%. Anderson’s path ease to the title is “only” 41.3%, which would be good for ninth on the list shown above, and just barely second easiest of the last 30 US Opens.

Measuring the Impact of the Serve in Men’s Tennis

By just about any measure, the serve is the most important shot in tennis. In men’s professional tennis, with its powerful deliveries and short points, the serve is all the more crucial. It is the one shot guaranteed to occur in every rally, and in many points, it is the only shot.

Yet we don’t have a good way of measuring exactly how important it is. It’s easy to determine which players have the best serves–they tend to show up at the top of the leaderboards for aces and service points won–but the available statistics are very limited if we want a more precise picture. The ace stat counts only a subset of those points decided by the serve, and the tally of service points won (or 1st serve points won, or 2nd serve points won) combines the effect of the serve with all of the other shots in a player’s arsenal.

Aces are not the only points in which the serve is decisive, and some service points won are decided long after the serve ceases to have any relevance to the point. What we need is a method to estimate how much impact the serve has on points of various lengths.

It seems like a fair assumption that if a server hits a winner on his second shot, the serve itself deserves some of the credit, even if the returner got it back in play. In any particular instance, the serve might be really important–imagine Roger Federer swatting away a weak return from the service line–or downright counterproductive–think of Rafael Nadal lunging to defend against a good return and hitting a miraculous down-the-line winner. With the wide variety of paths a tennis point can follow, though, all we can do is generalize. And in the aggregate, the serve probably has a lot to do with a 3-shot rally. At the other extreme, a 25-shot rally may start with a great serve or a mediocre one, but by the time by the point is decided, the effect of the serve has been canceled out.

With data from the Match Charting Project, we can quantify the effect. Using about 1,200 tour-level men’s matches from 2000 to the present, I looked at each of the server’s shots grouped by the stage of the rally–that is, his second shot, his third shot, and so on–and calculated how frequently it ended the point. A player’s underlying skills shouldn’t change during a point–his forehand is as good at the end as it is at the beginning, unless fatigue strikes–so if the serve had no effect on the success of subsequent shots, players would end the point equally often with every shot.

Of course, the serve does have an effect, so points won by the server end much more frequently on the few shots just after the serve than they do later on. This graph illustrates how the “point ending rate” changes:

On first serve points (the blue line), if the server has a “makeable” second shot (the third shot of the rally, “3” on the horizontal axis, where “makeable” is defined as a shot that results in an unforced error or is put back in play), there is a 28.1% chance it ends the point in the server’s favor, either with a winner or by inducing an error on the next shot. On the following shot, the rate falls to 25.6%, then 21.8%, and then down into what we’ll call the “base rate” range between 18% and 20%.

The base rate tells us how often players are able to end points in their favor after the serve ceases to provide an advantage. Since the point ending rate stabilizes beginning with the fifth shot (after first serves), we can pinpoint that stage of the rally as the moment–for the average player, anyway–when the serve is no longer an advantage.

As the graph shows, second serve points (shown with a red line) are a very different story. It appears that the serve has no impact once the returner gets the ball back in play. Even that slight blip with the server’s third shot (“5” on the horizontal axis, for the rally’s fifth shot) is no higher than the point ending rate on the 15th shot of first-serve rallies. This tallies with the conclusions of some other research I did six years ago, and it has the added benefit of agreeing with common sense, since ATP servers win only about half of their second serve points.

Of course, some players get plenty of positive after-effects from their second serves: When John Isner hits a second shot on a second-serve point, he finishes the point in his favor 30% of the time, a number that falls to 22% by his fourth shot. His second serve has effects that mirror those of an average player’s first serve.

Removing unforced errors

I wanted to build this metric without resorting to the vagaries of differentiating forced and unforced errors, but it wasn’t to be. The “point-ending” rates shown above include points that ended when the server’s opponent made an unforced error. We can argue about whether, or how much, such errors should be credited to the server, but for our purposes today, the important thing is that unforced errors aren’t affected that much by the stage of the rally.

If we want to isolate the effect of the serve, then, we should remove unforced errors. When we do so, we discover an even sharper effect. The rate at which the server hits winners (or induces forced errors) depends heavily on the stage of the rally. Here’s the same graph as above, only with opponent unforced errors removed:

The two graphs look very similar. Again, the first serve loses its effect around the 9th shot in the rally, and the second serve confers no advantage on later shots in the point. The important difference to notice is the ratio between the peak winner rate and the base rate, which is now just above 10%. When we counted unforced errors, the ratio between peak and base rate was about 3:2. With unforced errors removed, the ratio is close to 2:1, suggesting that when the server hits a winner on his second shot, the serve and the winner contributed roughly equally to the outcome of the point. It seems more appropriate to skip opponent unforced errors when measuring the effect of the serve, and the resulting 2:1 ratio jibes better with my intuition.

Making a metric

Now for the fun part. To narrow our focus, let’s zero in on one particular question: What percentage of service points won can be attributed to the serve? To answer that question, I want to consider only the server’s own efforts. For unreturned serves and unforced errors, we might be tempted to give negative credit to the other player. But for today’s purposes, I want to divvy up the credit among the server’s assets–his serve and his other shots–like separating the contributions of a baseball team’s pitching from its defense.

For unreturned serves, that’s easy. 100% of the credit belongs to the serve.

For second serve points in which the return was put in play, 0% of the credit goes to the serve. As we’ve seen, for the average player, once the return comes back, the server no longer has an advantage.

For first-serve points in which the return was put in play and the server won by his fourth shot, the serve gets some credit, but not all, and the amount of credit depends on how quickly the point ended. The following table shows the exact rates at which players hit winners on each shot, in the “Winner %” column:

Server's…  Winner %  W%/Base  Shot credit  Serve credit  
2nd shot      21.2%     1.96        51.0%         49.0%  
3rd shot      18.1%     1.68        59.6%         40.4%  
4th shot      13.3%     1.23        81.0%         19.0%  
5th+          10.8%     1.00       100.0%          0.0%

Compared to a base rate of 10.8% winners per shot opportunity, we can calculate the approximate value of the serve in points that end on the server’s 2nd, 3rd, and 4th shots. The resulting numbers come out close to round figures, so because these are hardly laws of nature (and the sample of charted matches has its biases), we’ll go with round numbers. We’ll give the serve 50% of the credit when the server needed only two shots, 40% when he needed three shots, and 20% when he needed four shots. After that, the advantage conferred by the serve is usually canceled out, so in longer rallies, the serve gets 0% of the credit.

Tour averages

Finally, we can begin the answer the question, What percentage of service points won can be attributed to the serve? This, I believe, is a good proxy for the slipperier query I started with, How important is the serve?

To do that, we take the same subset of 1,200 or so charted matches, tally the number of unreturned serves and first-serve points that ended with various numbers of shots, and assign credit to the serve based on the multipliers above. Adding up all the credit due to the serve gives us a raw number of “points” that the player won thanks to his serve. When we divide that number by the actual number of service points won, we find out how much of his service success was due to the serve itself. Let’s call the resulting number Serve Impact, or SvI.

Here are the aggregates for the entire tour, as well as for each major surface:

         1st SvI  2nd SvI  Total SvI  
Overall    63.4%    31.0%      53.6%  
Hard       64.6%    31.5%      54.4%  
Clay       56.9%    27.0%      47.8%  
Grass      70.8%    37.3%      61.5%

Bottom line, it appears that just over half of service points won are attributable to the serve itself. As expected, that number is lower on clay and higher on grass.

Since about two-thirds of the points that men win come on their own serves, we can go even one step further: roughly one-third of the points won by a men’s tennis player are due to his serve.

Player by player

These are averages, and the most interesting players rarely hew to the mean. Using the 50/40/20 multipliers, Isner’s SvI is a whopping 70.8% and Diego Schwartzman‘s is a mere 37.7%. As far from the middle as those are, they understate the uniqueness of these players. I hinted above that the same multipliers are not appropriate for everyone; the average player reaps no positive after-effects of his second serve, but Isner certainly does. The standard formula we’ve used so far credits Isner with an outrageous SvI, even without giving him credit for the “second serve plus one” points he racks up.

In other words, to get player-specific results, we need player-specific multipliers. To do that, we start by finding a player-specific base rate, for which we’ll use the winner (and induced forced error) rate for all shots starting with the server’s fifth shot on first-serve points and shots starting with the server’s fourth on second-serve points. Then we check the winner rate on the server’s 2nd, 3rd, and 4th shots on first-serve points and his 2nd and 3rd shots on second-serve points, and if the rate is at least 20% higher than the base rate, we give the player’s serve the corresponding amount of credit.

Here are the resulting multipliers for a quartet of players you might find interesting, with plenty of surprises already:

                   1st serve              2nd serve       
                    2nd shot  3rd  4th     2nd shot  3rd  
Roger Federer            55%  50%  30%           0%   0%  
Rafael Nadal             31%   0%   0%           0%   0%  
John Isner               46%  41%   0%          34%   0%  
Diego Schwartzman        20%  35%   0%           0%  25%  
Average                  50%  30%  20%           0%   0%

Roger Federer gets more positive after-effects from his first serve than average, more even than Isner does. The big American is a tricky case, both because so few of his serves come back and because he is so aggressive at all times, meaning that his base winner rate is very high. At the other extreme, Schwartzman and Rafael Nadal get very little follow-on benefit from their serves. Schwartzman’s multipliers are particularly intriguing, since on both first and second serves, his winner rate on his third shot is higher than on his second shot. Serve plus two, anyone?

Using player-specific multipliers makes Isner’s and Schwartzman’s SvI numbers more extreme. Isner’s ticks up a bit to 72.4% (just behind Ivo Karlovic), while Schwartzman’s drops to 35.0%, the lowest of anyone I’ve looked at. I’ve calculated multipliers and SvI for all 33 players with at least 1,000 tour-level service points in the Match Charting Project database:

Player                 1st SvI  2nd SvI  Total SvI  
Ivo Karlovic             79.2%    56.1%      73.3%  
John Isner               78.3%    54.3%      72.4%  
Andy Roddick             77.8%    51.0%      71.1%  
Feliciano Lopez          83.3%    37.1%      68.9%  
Kevin Anderson           77.7%    42.5%      68.4%  
Milos Raonic             77.4%    36.0%      66.0%  
Marin Cilic              77.1%    34.1%      63.3%  
Nick Kyrgios             70.6%    41.0%      62.5%  
Alexandr Dolgopolov      74.0%    37.8%      61.3%  
Gael Monfils             69.8%    37.7%      60.8%  
Roger Federer            70.6%    32.0%      58.8%  
                                                    
Player                 1st SvI  2nd SvI  Total SvI  
Bernard Tomic            67.6%    28.7%      58.5%  
Tomas Berdych            71.6%    27.0%      57.2%  
Alexander Zverev         65.4%    30.2%      54.9%  
Fernando Verdasco        61.6%    32.9%      54.3%  
Stan Wawrinka            65.4%    33.7%      54.2%  
Lleyton Hewitt           66.7%    32.1%      53.4%  
Juan Martin Del Potro    63.1%    28.2%      53.4%  
Grigor Dimitrov          62.9%    28.6%      53.3%  
Jo Wilfried Tsonga       65.3%    25.9%      52.7%  
Marat Safin              68.4%    22.7%      52.3%  
Andy Murray              63.4%    27.5%      52.0%  
                                                    
Player                 1st SvI  2nd SvI  Total SvI  
Dominic Thiem            60.6%    28.9%      50.8%  
Roberto Bautista Agut    55.9%    32.5%      49.5%  
Pablo Cuevas             57.9%    28.9%      47.8%  
Richard Gasquet          56.0%    29.0%      47.5%  
Novak Djokovic           56.0%    26.8%      47.3%  
Andre Agassi             54.3%    31.4%      47.1%  
Gilles Simon             55.7%    28.4%      46.7%  
Kei Nishikori            52.2%    30.8%      45.2%  
David Ferrer             46.9%    28.2%      41.0%  
Rafael Nadal             42.8%    27.1%      38.8%  
Diego Schwartzman        39.5%    25.8%      35.0%

At the risk of belaboring the point, this table shows just how massive the difference is between the biggest servers and their opposites. Karlovic’s serve accounts for nearly three-quarters of his success on service points, while Schwartzman’s can be credited with barely one-third. Even those numbers don’t tell the whole story: Because Ivo’s game relies so much more on service games than Diego’s does, it means that 54% of Karlovic’s total points won–serve and return–are due to his serve, while only 20% of Schwartzman’s are.

We didn’t need a lengthy analysis to show us that the serve is important in men’s tennis, or that it represents a much bigger chunk of some players’ success than others. But now, instead of asserting a vague truism–the serve is a big deal–we can begin to understand just how much it influences results, and how much weak-serving players need to compensate just to stay even with their more powerful peers.

Podcast Episode 15: Return of the Podcast

The long-awaited Episode 15 of the Tennis Abstract Podcast is Carl Bialik’s and my lightning-round update on the summer of tennis and a midway-point checkup of the US Open. I boldly select Diego Schwartzman and Anastasija Sevastova as my picks to win it all, while Carl says much more sensible things.

This is a shorter-than usual episode, clocking in at 34 minutes; we hope to return on Friday with another mid-Slam update. Thanks for listening!

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