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**del Potro has withdrawn.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.

]]>

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.

]]>

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.*

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.

]]>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!

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]]>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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]]>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.

]]>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.

]]>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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]]>While 5’7″ (170 cm) Diego Schwartzman‘s surprise run to the US Open quarterfinals has brought this issue to the forefront, pundits and fans talk about it all the time. This is a topic crying out for some basic data analysis, yet as is too often the case in tennis, some really simple work is missing from the conversation. Let’s try to fix that.

When I say “basic,” I really mean it. We all know that tall men hit more aces than short men. But how many? How strong is the relationship between height and, say, first serve points won? In this post, I’ll show the relationship between height and each of nine different stats, from overall records to serve- and return-specific numbers.

For my dataset, I took age-25 seasons from 1998 to 2017 in which the player completed at least 30 tour-level matches. (I used only one season per player so that the best players with the longest careers wouldn’t be weighted too heavily.) That gives us 156 player-seasons, from Hicham Arazi and Greg Rusedski in 1998 up to Schwartzman and Jack Sock in 2017. There aren’t very many players at the extremes, so I lumped together everyone 5’8″ (173 cm) and below and did the same with everyone 6’5″ (196 cm) and above. I also grouped players standing 5’10” with those at 5’9″, because there were only four 5’10” guys in the dataset.

That gives us nine “height levels”: one per inch from 5’8″ to 6’5″ with the exception of 5’10”. (The ATP website displays heights in meters, but its database must record and/or store them in inches, because every height translates to something close to an integer height in inches. For example, no player is listed at 174 cm, or 5’8.5″.) Some individual heights are certainly exaggerated, as male athletes and their organizations tend to do, but we have to make do with the information available, and we may assume that the exaggerations are fairly consistent.

Let’s start with the most basic building block of tennis, the match win. There is a reasonably strong relationship here, although the group of players at 6’1″ is nearly as good as the tallest subset. In each of these graphs, height is given on the horizontal axis in centimeters, from 173 (the 5’8″ and below group) up to 196 (the 6’5″ and higher group).

There is a similar, albeit slightly weaker, relationship when we look at the level of single points. Since a small difference in points results in a larger difference in matches won (at the extreme, winning 55% of points translates to nearly a 100% chance of winning the match) this isn’t a surprise. At the match level, r^2 = 0.38, and at the point level, below, r^2 = 0.27:

(If you’re wondering how all of the averages are above 50%, it’s because the sample is limited to player-seasons with at least 30 matches. A fair number of those matches are against players who aren’t tour regulars, and the regulars–the guys in this sample–win a hefty proportion of those matches.)

**Serve stats**

Now we get to confirm our main assumptions. Taller players are better servers, and the gap is enormous, ranging from 60% of service points won for the shortest players up to nearly 70% for the tallest:

As strong as that relationship is (r^2 = 0.81), the relationship between height and ace rate is stronger still, at r^2 = 0.83:

Aces don’t tell the whole story–the stat with the strongest correlation to height is first serve points won (r^2 = 0.92) as you can see here:

But this is where things start to get interesting. Nearly every inch makes a player more effective on the first serve, but opponents are able to negotiate tall players’ second serves much more successfully. There remains a modest relationship with height (r^2 = 0.18), but it is the weakest of all the stats presented here:

It’s nice to be tall, as anyone who has seen John Isner casually spin a second-serve ace out of the reach of an unlucky opponent. But except in the tallest category, height doesn’t confer much of a second-serve advantage. Players standing 6’4″ (193 cm) win about as many second-serve points as do players at 5’9″ (175 cm). That doesn’t mean that the second serves of the shorter players are just as good–they probably aren’t–but that shorter players tend to possess other skills that they can leverage in second-serve points, which usually last longer. For the purposes of today’s overview, it doesn’t really matter *why *short players are able to negate the advantage of height on second serve points, just that they are clearly able to do so.

**Return stats**

We wouldn’t be having this conversation–and David Ferrer wouldn’t be headed to a likely place in the Hall of Fame–if the inverse relationship between height and return effectiveness weren’t nearly as strong as the positive one between height and serving prowess. “Nearly” is the key word here. The relationship between height and overall return points won is almost as strong (r^2 = 0.74) as that of height and overall service points won, but not quite:

Schwartzman is doing more than his part to hold up the left side of that trendline: He is both the shortest player in the top 50 and the best returner. On first serve points, however, there’s only so much the returner can do, so while shorter players still have an advantage, it is less substantial. The relationship here is a bit weaker, at r^2 = 0.63:

It follows, then, that the relationship between height and second-serve return points won must be stronger, at r^2 = 0.77:

The overall and first-serve return point graphs make clear just how much worse the tallest players are than the rest of the pack. The graphs exaggerate it a bit, because I’ve grouped players from 6’5″ all the way up to 6’11”, and the Isners of the sport are considerably less effective than players such as Marin Cilic. Still, we find plenty of confirmation for the conventional wisdom that a height of 6’2″ or 6’3″ (188 cm to 190 cm) allows for players to remain effective on both sides of the ball, while a small increase from there can be a disadvantage.

**A note on selection bias**

It’s easy to lapse into shorthand and say something like, “shorter players are better returners.” More precisely, what we mean is, “of the players who have become tour regulars, shorter players are better returners.” They have to be, because it is nearly impossible for them to be top-tier servers. If they’ve cracked the top 50, they *must* have developed a world-class return game. The shorter the player, the more likely this is true.

The same logic is considerably weaker if we descend a couple rungs lower on the ladder of tennis skill. In collegiate tennis, it’s still an advantage to be tall–as Isner can attest–but a player such as 5’10” Benjamin Becker can serve as well as nearly all the competition he will face at that level.

**One more note on selection bias**

My choice to use each player’s age-25 season might understate the ability of either short or tall players. It is possible that certain playing styles result in earlier or later peaks, meaning that while tall players could be better at age 25, shorter players may be superior at age 28. There are anecdotes that support the argument in both directions, so I don’t think it’s a major issue, but it is one worthy of additional study.

**Further reading**

A guest post on this blog earlier this year posed the question, Are Taller Players the Future of Tennis?

I didn’t mention serve speed in the above, but here’s a quick study of the fastest serves and their correlation with height.

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