Elo Ratings – Page 5 – Heavy Topspin

The All-Time ATP Masters Race Is Even Closer Than You Think

As I write this, Novak Djokovic has earned himself a place in the semi-finals at the Shanghai Masters, putting him two wins away from his 32nd career ATP Masters title. A championship this week would leave him one behind Rafael Nadal, the all-time leader with 33 Masters victories. Roger Federer, also in action today in Shanghai, is not far back, with a career total of 27.

Masters tallies aren’t as important as grand slam counts, but they make up an important part of an elite player’s resume. For one thing, there are more of them, and the mix of surfaces–more clay, no grass, and an indoor event–adds to our knowledge of a player’s range of skills. It’s no surprise that Nadal, Djokovic, and Federer are miles ahead of the pack on this list, as on so many others.

But all Masters aren’t created equal. At last year’s Madrid Masters, Nadal had to defeat Djokovic, as well as clay-court giant Dominic Thiem and the ever-threatening Nick Kyrgios. Six months later in Paris, Jack Sock earned the same number of Masters titles by coasting down a path that included only one player ranked in the top 35. Like major titles, Masters championships are heavily influenced by luck, and when we focus on raw totals, we trust that fortune tends to even out.

It doesn’t even out, even for the top players who have played Masters events for more than a decade and racked up dozens of titles. To account for opponent quality and the difficulty of each title, I applied the same algorithm I’ve used in the past to rate slam titles. [1] The formula spits out a number for each Masters title, where 1 is average, less than 1 is easier than the norm, and greater than 1 is more difficult. Sock’s Paris trophy was the luckiest in recent years, clocking in at 0.39, while David Nalbandian’s 2007 Madrid title was the most difficulty, rating 1.92. These extremes nothwithstanding, almost every championship comes in between 0.5 and 1.5.

The all-time tally

Let’s start by looking at the top ten in “adjusted Masters.” The table below shows the results of my formula, alongside each player’s actual Masters count, and the average rating of the tournaments he has won:

Player          Adj Masters  Masters  Average  
Rafael Nadal           35.4       33     1.07  
Novak Djokovic         35.0       31     1.13  
Roger Federer          28.0       27     1.04  
Andre Agassi           15.0       17     0.88  
Andy Murray            15.0       14     1.07  
Pete Sampras           11.2       11     1.02  
Thomas Muster           7.5        8     0.94  
Michael Chang           6.4        7     0.91  
Boris Becker            5.4        5     1.08  
Jim Courier             5.0        5     1.00

Boris Becker and Jim Courier aren’t the only men to have recorded five Masters titles, but they are only ones who have done so against average or better competition. Andy Roddick won five, but the algorithm gives him credit for just under four, and it is even more harsh on Marat Safin, whose five victories translate into only 3.2 adjusted Masters.

The real story is at the top of the list, where adjusting for competition almost eliminates the gap between Nadal and Djokovic. Both men have won their titles against more difficult than average competition (often, by beating each other), but Djokovic has faced the tougher paths. If he wins on Sunday in Shanghai, he’ll overtake Nadal’s adjusted tally.

Also of note is the near-tie between Andre Agassi and Andy Murray. Agassi holds three more trophies, but won them against the weakest competition of anyone in the top ten. Murray has dealt with much of the same field that Nadal and Djokovic have, so it’s no surprise to see his difficulty rating well above 1.0.

The Paris swoon

Sock’s title last year was unquestionably weak, but not entirely out of character for Bercy. With the exception of the short-lived Essen Masters, Paris titles have come the cheapest of any other tour stop at this level:

Tournament            Years  Average Rating  
Madrid (clay)            10            1.18  
Rome Masters             29            1.09  
Indian Wells Masters     29            1.07  
Stuttgart Masters         6            1.05  
Stockholm Masters         5            1.04  
Hamburg Masters          19            1.02  
Miami Masters            29            1.01  
Monte Carlo Masters      29            0.98  
Canada Masters           29            0.97  
Cincinnati Masters       29            0.97  
Madrid (hard)             7            0.97  
Shanghai Masters          9            0.95  
Paris Masters            28            0.84  
Essen Masters             1            0.80

Paris was played on carpet until 2006, and that may be a factor. When I first ran the numbers, I used carpet-specific Elo ratings, which are limited by a relatively small sample. I tried again using hard-court ratings for carpet events, and while individual numbers shifted up and down, the overall results were about the same. Bercy was particularly weak during the carpet era, and it has gotten stronger, but I’m confident this is a feature of the results in the 1990s and early 2000s, not merely an artifact of Elo rating quirks.

Still, broadly speaking, fast courts seem to result in lower ratings. I suspect that’s because at predominantly best-of-three events, early-round upsets are more likely to occur on the quickest surfaces. Fast courts, then, effectively gut the field for the eventual champion. It certainly worked for Sock last year. But it’s no guarantee–the five most difficult Masters titles all came on hard courts, and one of them took place in Paris.

Peak Nalbandian

At the end of 2007, Nalbandian enjoyed two of the most glorious weeks of tennis ever played. In Madrid, he defeated Nadal in the quarters, Djokovic in the semis, and Federer in the final, not to mention Tomas Berdych and Juan Martin del Potro in the early rounds. Two weeks later, he beat Federer and Nadal again in Paris, along with wins over David Ferrer, Richard Gasquet, and Carlos Moya. Those two titles rate 1.92 and 1.70, respectively, and are two of the three most difficult since the Masters series began.

(Oddly enough, the only man who could stop Nalbandian that fall was Stan Wawrinka, who beat him both in Vienna and Basel. Wawrinka’s slam titles rate as the most difficult in that category.)

Here are the 20 most difficult Masters titles, along with their ratings:

Year  Event         Surface  Champion            Rating  
2007  Madrid        Hard     David Nalbandian      1.92  
2014  Canada        Hard     Jo Wilfried Tsonga    1.78  
2007  Paris         Hard     David Nalbandian      1.70  
2007  Canada        Hard     Novak Djokovic        1.68  
2009  Indian Wells  Hard     Rafael Nadal          1.61  
2009  Madrid        Clay     Roger Federer         1.52  
2017  Madrid        Clay     Rafael Nadal          1.52  
2016  Madrid        Clay     Novak Djokovic        1.51  
2011  Indian Wells  Hard     Novak Djokovic        1.50  
2013  Indian Wells  Hard     Rafael Nadal          1.50  
                                                         
Year  Event         Surface  Champion            Rating  
2010  Canada        Hard     Andy Murray           1.48  
2011  Rome          Clay     Novak Djokovic        1.48  
2012  Rome          Clay     Rafael Nadal          1.47  
2010  Indian Wells  Hard     Ivan Ljubicic         1.45  
2004  Hamburg       Clay     Roger Federer         1.44  
2015  Cincinnati    Hard     Roger Federer         1.44  
2013  Rome          Clay     Rafael Nadal          1.43  
2015  Canada        Hard     Andy Murray           1.43  
2008  Monte Carlo   Clay     Rafael Nadal          1.42  
2015  Madrid        Clay     Andy Murray           1.42

Nalbandian and Jo Wilfried Tsonga stand out at the top, but after that, it’s a whole lot of Big Four. Even in the next ten toughest paths, Djokovic, Nadal, and Federer account for seven.

Like just about any adjustment to high-profile counting stats, tweaking Masters titles for difficulty doesn’t exactly clear up the debate over the greatest players of all time. This is only one small part of that conversation. However, seeing the wide range of challenges faced by Masters champions is a necessary reminder that not all titles are the same, even if they all count for one thousand ATP ranking points.

—

[1] Here’s how I first described the algorithm:

To evaluate the overall difficulty of grand-slam titlists’ draws, I used Elo—a rating system that assesses a player’s strength based on his won-lost record and the quality of his opponents—to measure the skill of an average major champion. I then estimated the probability that such a player would have won all seven matches against the opponents that each tournament’s victor had to face. For each win, I credit the champion with the difference between one and the Elo forecast: If an average slam champion on the tournament’s surface had a 90% chance of winning the match, the player gets 0.1 points (1 – 0.9); if a typical major winner would have gone in with a 20% shot, he’s assigned 0.8. Summing all the matches for each winner and applying the algorithm to the last several decades of grand slams results in an average credit of 1.23 per titlist, so I then divided each sum by 1.23 to normalise the results.

For Masters events, it’s five or six matches instead of seven, and the final step involves dividing by 1.34 instead of 1.23.

Forecasting the 2018 Laver Cup

Embed from Getty Images

Italian translation at settesei.it

It’s that time of year again: group selfies in suits, dodgy Davis Cup excuses, and a reminder that it takes more than six continents just to equal Europe. That’s right, it’s Laver Cup.

Last year, I worked out a forecast of the event, walking through a variety of ways in which captains Bjorn Borg and John McEnroe could use their rosters and ultimately predicting a 16-8 win for Team Europe. As it happened, both captains intelligently deployed their stars, and the result was 15-9. This year, the competitors are a little different and the home court has moved from Prague to Chicago, but the format remains the same.

Let’s start with a look at the rosters. I’ve included two additional players for reference: Juan Martin del Potro, scheduled to play for Team World, but withdrew; and Pierre Hugues Herbert, the doubles specialist Borg hasn’t realized he needs. Each player is shown alongside his surface-weighted singles Elo rating and surface-weighted doubles “D-Lo” rating:

EUROPE                       Singles Elo  Doubles D-Lo  
Novak Djokovic                      2137          1667  
Roger Federer*                      2097          1700  
Alexander Zverev                    1971          1690  
David Goffin                        1960          1582  
Grigor Dimitrov                     1928          1719  
Kyle Edmund                         1780          1542  
                                                        
WORLD                        Singles Elo  Doubles D-Lo  
Kevin Anderson                      1914          1692  
Nick Kyrgios                        1910          1668  
John Isner                          1887          1800  
Diego Sebastian Schwartzman         1814          1540  
Frances Tiafoe                      1772          1544  
Jack Sock                           1724          1925  
                                                        
ALSO                                                    
Juan Martin Del Potro               2062          1678  
Pierre Hugues Herbert               1691          1890

* Federer has played very little tour-level doubles for a long time. Last year I estimated his D-Lo at 1650; he played rather well last year, so I’ll bump him up to 1700 this time around.

Especially with Delpo on the sidelines, Europe looks to dominate the singles. The doubles leans in World’s favor, largely because Jack Sock is so good, especially in comparison with guys who have focused on singles.

Format review

Let’s do a quick refresher on the format. Laver Cup takes place over three days, each of which has three singles matches and one doubles match. Each player must play singles at least once, and no doubles pairing can repeat itself. Day 1 matches are worth one point each, day 2 matches are worth 2 points each, and day 3 matches are worth 3 points each. If there’s a 12-12 tie at the end of day 3, a single doubles set–in which a previously-used team may compete–will decide it all.

Given that format, the best way for the captains to use their rosters is to stick their three worst singles players on day 1 duty, then use their best three on both day 2 and day 3. For doubles, they should use their best doubles player every day, with the best partner on day 3, next best on day 2, and third best on day 1. As I’ve mentioned, Borg and McEnroe came close to this last year, although Borg didn’t use Rafael Nadal (his best doubles player) in day 3 doubles, and he generally overused Tomas Berdych. Both decisions are understandable, as Nadal may not have been physically able to play every possible match, and Berdych was in front of a Czech crowd.

Now that we know the captains will act in a reasonably savvy way, we can forecast the second edition with a little more confidence than the inaugural one.

The forecast

Nadal’s absence this year will hurt the Europe squad on both singles and doubles. Combined with a small step backward for Federer’s singles game, this year’s Laver Cup figures to be closer than last year. Recall that my forecast a year ago called for a 16-8 Europe victory, and the result was 15-9.

Assuming optimal usage, the 2018 forecast gives Europe a 67.6% chance of winning, with a most likely final score of 14-10. There’s a nearly one-in-ten shot that we’ll see a 12-12 tie, in which the superior doubles capabilities of Team World give them the edge, with a 70.7% probability of winning the tie-breaking set.

Were del Potro not so fragile, this could get even more interesting. Swap out Frances Tiafoe for the Tower of Tandil, and Europe’s chances fall to 56.8%, with a most likely final score of 13-11.

Nothing McEnroe could have done, short of going to medical school a few decades ago, could have put the Argentine on his team this week. But Borg has less of an excuse for failing to maximize the potential of his team. Unlike World, with its world-beating doubles specialist, Europe has a stunning singles roster that rarely takes to the doubles court. As we’ve seen, one doubles player can take the court three times, plus the potential 12-12 tie-breaking set. The specialist would need to play singles only once, on the low-leverage first day.

The obvious choice is Pierre Hugues Herbert, a top-five doubles player with the ability to play respectable singles as well. The Frenchman would be considerably more valuable than Kyle Edmund, who is a better singles player, but not good enough to be of much help to an already loaded side. (I made a similar point last year and illustrated it with Herbert’s partner, Nicolas Mahut. Since then, Herbert has taken the lead over his Mahut in both singles and doubles Elo ratings.)

When we sub in Herbert for Edmund, the simulation spits out the best result yet for Europe. Against the actual World team (that is, no Delpo), the hypothetical Europe squad would have a 74.6% chance of winning, with the likely final score between 14-10 and 15-9. Herbert and a mediocre partner would still be the underdogs in a tie-breaking final set against Sock and John Isner, but the presence of a legitimate doubles threat would narrow the odds to about 58/42.

We won’t get to see either Delpo or Herbert in Chicago this year, but we can expect a slightly more competitive Laver Cup than last year. Add in home court advantage, and the result is even less of a foregone conclusion. It’s no match for last week’s Davis Cup World Group play-offs, but I suspect it’ll make for more compelling viewing this weekend than the final rounds in Metz and St. Petersburg.

Jack Sock, Doubles King Once Again

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Italian translation at settesei.it

A couple of years ago, I wrote an article introducing D-Lo, an Elo-like rating system for doubles, which crowned Jack Sock as the best doubles player on the men’s tour. Sock grabbed the top spot in October 2016 and hung on for about nine months, largely by not playing very much. A couple of first-round losses in Washington and Montreal last summer sent him tumbling, landing at 8th after the US Open and as low as 14th going into this year’s Australian Open.

Despite his preference for singles, Sock has rocketed back into the lead, first pairing with John Isner for the Indian Wells title, and then partnering Mike Bryan (replacing injured brother Bob) to win both Wimbledon and the US Open. With the exception of one week immediately after Indian Wells, Sock sits at the head of the D-Lo table for the first time in more than a year. Here are the current top ten, along with their ratings:

Rank  Player                 D-Lo  
1     Jack Sock              1949  
2     Bob Bryan              1930  
3     Mike Bryan             1917  
4     Pierre Hugues Herbert  1906  
5     Nicolas Mahut          1893  
6     Jamie Murray           1886  
7     Bruno Soares           1883  
8     Oliver Marach          1867  
9     Robert Farah           1863  
10    Nikola Mektic          1863

Yes, that’s the injured Bob Bryan in the second spot. More on that in a moment.

A quick refresher on the D-Lo system: It mostly works like a standard Elo algorithm, in that players gain points for winning matches and lose points for losing them, based on the quality of the opponent and the amount of prior information already baked into their ratings. A big upset earns more points than a victory over an equal, and for players with fewer prior matches, the effect of each match is greater. Thus, Sock got a few more points than Mike did for winning the 12 matches at the last two slams, because we knew relatively less about him before those tournaments.

D-Lo assumes that the quality of each team is equal to the average of the two players. If a team wins, each member of the partnership gains points, with one tweak: If the two players have different ratings, their ratings slightly move toward the average of the two. This is because it’s impossible to know how much each player contributes to a win. The system is designed so that, after a year or so of playing together, the two mens’ ratings will meet in the middle. It’s an imperfect system, but it does a reasonably good job of forecasting results, which means it usually provides a solid representation of each player’s skill level.

Back to the matter at hand: Doubles ratings have been particularly volatile this year, with five different men (Sock, Bob, Pierre Hugues Herbert, Mate Pavic, and Henri Kontinen) holding the #1 spot, and two more (Nicolas Mahut and John Peers) peaking at #2. This parity means that no player has a particularly high rating. Two years ago, Sock’s mark of 1949 would have been good for only fourth (behind himself, Herbert, and Mahut), and several players (the Bryans, Herbert, and Daniel Nestor among them) have peaked with ratings above 2000.

Take a look at how much the rank order has fluctuated since the beginning of 2018:

For clarity’s sake, I’ve left off Oliver Marach (whose rating tracks closely with that of his partner, Pavic, and whose season hasn’t lived up to its early promise) and Peers (ditto, with Kontinen). Herbert has reached the highest level of anyone this season, but a rough second half so far has left him behind the American trio of Jack, Bob, and Mike.

Back to the curious case of Bob Bryan. The Bryan brothers’ title at the Madrid Masters this year gave the twins their highest D-Lo ratings in nearly two years. “Standard” Elo doesn’t penalize players for absence, so Bob’s mark has remained at 1930 ever since. (I’ve added an injury/absence penalty in my singles Elo ratings, but haven’t done so for D-Lo. I suspect there is less of an effect, but still a measureable one, in doubles.) Mike’s rating has slipped because of some bad results apart from the pair of majors, and only Sock has caught up.

If Bob is healthy enough to play this fall, the twins are expected to pair up for the World Tour Finals, once again leaving the best doubles player in the world out of the field. In that case, Sock, down to 157th in the ATP singles race, could end up spending that week playing the new ATP Challenger event in Houston. Without their young compatriot in the way, the Bryans will be back in familiar territory, headed to London as the favorites for another year-end title.

Handling Injuries and Absences With Tennis Elo

Italian translation at settesei.it

For the last year or so, every mention of my ATP and WTA Elo ratings has required some sort of caveat. Ratings don’t change while players are absent from the tour, so Serena Williams, Novak Djokovic, Andy Murray, Maria Sharapova, and Victoria Azarenka were all stuck at the top of their tour’s Elo rankings. When their layoffs started, they were among the best, and even a smattering of poor results (or a near season’s worth, in the case of Sharapova) isn’t enough to knock them too far down the list.

This is contrary to common sense, and it’s very different from how the official ATP and WTA rankings treat these players. Common sense says that returning players probably aren’t as good as they were before a long break. The official rankings are harsher, removing players entirely after a full year away from the tour. Serena probably isn’t the best player on tour right now (as Elo insisted during her time off), but she’s also much more of a threat than her WTA ranking of No. 454 implies. We must be able to do better.

Before we fix the Elo algorithm, let’s take a moment to consider what “better” means. Fans tend to get worked up about rankings and seedings, as if a number confers value on the player. The official rankings are, by design, backward-looking: They measure players based on their performance over the last 52 weeks, weighted by how the tour prioritizes events. (They are used in a forward-looking way, for tournament seedings, but the system is not designed to be predictive of future results.) In this way, the official rankings say, “this is how good she has played for the last year.” Whatever her ability or potential, Serena (along with Vika, Murray, and Djokovic) hasn’t posted many positive results this year, and her ranking reflects that.

Elo, on the other hand, is designed to be predictive. Out of necessity, it can only use past results, but it uses those results in a way to best estimate how well a player is competing right now–our best proxy for how someone will play tomorrow, or next week. Elo ratings–even the naive ones that said Serena and Novak are your current No. 1s–are considerably better at predicting match outcomes than are the official rankings. For my purposes, that’s the definition of “better”–ratings that offer more accurate forecasts and, by extension, the best approximation of each player’s level right now.

The time-off penalty

When players leave the tour for very long, they return–at least on average, and at least temporarily–at a lower level. I identified every layoff of eight weeks or longer in ATP history, taken by a player with an Elo rating of 1900 or above*. In their first matches back on tour, their pre-break Elo overestimated their chances of winning by about 25%. It varies a bit by the amount of time off: eight- to ten-week breaks resulted in an overestimation around 17%, while 30- to 52-week breaks meant Elo overestimated a player’s chances by nearly 50% upon return. There are exceptions to every rule, like Roger Federer at the 2017 Australian Open, and Rafael Nadal, who won 14 matches in a row after his two-month break this season, but in general, players are worse when they come back.

* I used the cutoff of 1900 because, below that level, some players are alternating between the ATP and Challenger tours. My Elo algorithm doesn’t include challenger results, so for lower-rated players, it’s not clear which timespans are breaks, and which are series of challenger events. Also, the eight-week threshold doesn’t count the offseason, so an eight-week layoff might really mean ~16 weeks between events, with the break including the offseason.

Translated into Elo terms, an eight-week break results in a drop of 100 Elo points, and a not-quite-one-year break, like Andy Murray’s current injury layoff, means a drop of 150 points. Making that adjustment results in an immediate improvement in Elo’s predictiveness for the first match after a layoff, and a small improvement in predictiveness for the first 20 matches after a break.

Incorporating uncertainty

Elo is designed to always provide a “best estimate”–when a player is new on tour, we give him a provisional rating of 1500, and then adjust the rating after each match, depending on the result, the quality of the opponent, and how many matches our player has contested. That provisional 1500 is a completely ignorant guess, so the first adjustment is a big one. Over time, the size of a player’s Elo adjustments goes down, because we learn more about him. If a player loses his first-ever match to Joao Sousa, the only information we have is that he’s probably not as good as Sousa, so we subtract a lot of points. If Alexander Zverev loses to Sousa after more than 150 career matches, including dozens of wins over superior players, we’ll still dock Zverev a few points, but not as many, because we know so much more about him.

But after a layoff, we are a bit less certain that what we knew about a player is still relevant. Djokovic a great example right now. If he lost six out of nine matches (as he did between the Australian Open fourth round and Madrid) without missing any time beforehand, we’d know it was a slump, but most of us would expect him to snap out of it. Elo would reduce his rating, but he’d remain near the top. Since he missed the second half of last season, however, we’re more skeptical–perhaps he’ll never return to his former level. Other cases are even more clear-cut, as when a player returns from injury without being fully healed.

Thus, after a layoff, it makes sense to alter how much we adjust a player’s Elo ratings. This isn’t a new idea–it’s the core concept behind Glicko, another chess rating system that expands on Elo. Over the years, I’ve tinkered with Glicko quite a bit, looking for improvements that apply to tennis, without much success. Changing the multiplier that determines rating adjustments (known as the k factor) doesn’t improve the predictiveness of tennis Elo on its own, but combined with the post-layoff penalties I described above, it helps a bit.

The nitty-gritty: After a layoff, I increase the multiplier by a factor of 1.5, and then gradually reduce it back to 1x over the next 20 matches. The flexible multiplier slightly improves the accuracy of Elo ratings for those 20 matches, though the difference is minor compared to the effect of the initial penalty.

No more caveats*

* I thought it would be funny to put an asterisk after “no more caveats.”

Post-layoff penalties and flexible multipliers end up bringing down the current Elo ratings of the players who are in the middle of long breaks or have recently come back from them, giving us ranking tables that come closer to what we expect–and should do a better job of predicting the outcome of upcoming matches. These changes to the algorithm also have minor effects on the ratings of other players, because everyone’s rating depends on the rating of all of his or her opponents. So Taro Daniel’s Elo bounce from defeating Djokovic in Indian Wells doesn’t look quite as good as it did before I implemented the penalty.

On the ATP side, the new algorithm knocks Djokovic down to 3rd in overall Elo, Murray to 6th, Jo-Wilfried Tsonga to 21st, and Stan Wawrinka to 24th. That’s still quite high for Novak considering what we’ve seen this year, but remember that the Elo algorithm only knows about his on-court performances: A six-month break followed by a half-dozen disappointing losses. The overall effect is about a 200-point drop from his pre-layoff level; the “problem” is that his Elo a year ago reflected how jaw-droppingly good he had recently been.

The WTA results match my intuition even better than I hoped. Serena falls to 7th, Sharapova to 18th, and Azarenka to 23rd. Because of the flexible multiplier, a few early wins for Williams will send her quickly back up the rankings. Like Djokovic, she rates so high in part because of her stratospheric Elo rating before her time off. For her part, Sharapova still rates higher by Elo than she does in the official rankings. Despite the penalty for her one-year drug suspension, the algorithm still treats her prior success as relevant, even if that relevance fades a bit more every week.

Elo is always an approximation, and given the wide range of causes that will sideline a player, not to mention the spectrum of strategies for returning to the tour, any rating/forecasting system is going to have a harder time with players in that situation. That said, these improvements give us Elo ratings that do a better job of representing the current level of players who have missed time, and they will allow us to make superior predictions about matches and tournaments involving those players.

Under the hood

If you’re interested in some technical details, keep reading.

Before making these adjustments, the Brier score for Elo-based predictions of all ATP matches since 1972 was about 0.20. For all matches that involved at least one player with an Elo of 1900 or better, it was 0.17. (Not only are 1900+ players better, their ratings tend to be based on more data, which at least partly explains why the predictions are better. The lower the Brier score, the better.)

For the population of about 500 “first matches” after layoffs for qualifying players, the Brier score before these changes was 0.192. After implementing the penalty, it improved to 0.173.

For the 2nd through 20th post-comeback matches, the Brier score for the original algorithm was 0.195. After adding the penalty, it was 0.191, and after making the multiplier flexible, it fell a bit more to 0.190. (Additional increases to the post-layoff multiplier had negative results, pushing the Brier score back to about 0.195 when the 2nd-match multiplier was 2x.) I realize that’s a tiny change, and it very possibly won’t hold up in the future. But in looking at various notable players over the course of their comebacks, that’s the option that generated results that looked the most intuitively accurate. Since my intuition matched the best Brier score (however miniscule the difference), it seems like the best option.

Finally, a note on players with multiple layoffs. If someone misses six months, plays a few matches, then misses another two months, it doesn’t seem right to apply the penalty twice. There aren’t a lot of instances to use for testing, but the limited sample confirms this. My solution: If the second layoff is within two years of the previous comeback, combine the length of the two layoffs (here: eight months), find the penalty for a break of that length, and then apply the difference between that penalty and the previous one. Usually, that results in second-layoff penalties of between 10 and 50 points.

Rafael Nadal and the Greatest Single-Tournament Performances

Italian translation at settesei.it

In the last two weeks, Rafael Nadal recorded his 11th titles in both Monte Carlo and Barcelona. His career records at those two events, along with his ten Roland Garros championships, reflect a level of dominance never before seen on a single surface. They have to be considered among the greatest achievements in tennis history, and perhaps in all of sport.

The tennis fan in me is content to speculate about whether anyone will ever stop him. The analyst wants to dig deeper: Has Nadal’s performance at one of the tournaments been even better than the rest? How do these single-event records compare to other exploits, such as Roger Federer’s trophy haul at Wimbledon, or Bjorn Borg’s nearly-undefeated career at the French Open?

Barcelona by the numbers

Let’s start with Barcelona. Since 2005–we’ll ignore his 2003 appearance as a 16-year-old wild card–he has played the event 13 times, winning 11 of them. That’s a won-loss record of 57-2.

Usually, I would calculate the probability of a player winning so many tournaments in that many chances, then come up with a tiny percentage that would represent his odds of achieving such a feat. That would miss the mark here. Instead, I want to look at the problem from the opposite perspective: In order to win so many titles, how good must Nadal be?

We already know that Rafa is the best of all time on clay, in general. Using the Elo rating system, his peak surface-specific rating–that is, Elo calculated using only results on clay courts–is over 2,500, better than anyone else on clay … or anyone else on any surface. (Nadal’s current clay-specific Elo is around 2,400, and the closest things he has to rivals on the surface right now, Dominic Thiem and Kei Nishikori, sit at about 2,190 and 2,150. Stefanos Tsitsipas’s rating is 1865.) Since Rafa has posted his best results at these three events, it stands to reason that his tournament-specific levels are even higher.

Here, then, is the method we can use to figure that out. First, for each year he entered Barcelona, determine his path to the title. (For the 11 titles, that’s easy; for the other two, we use the players he would have faced had he kept winning.) Using each opponent’s clay court Elo rating at the time of the match, we can determine the odds that various hypothetical (and dominant) players would have progressed through the draw and won the title.

Here is Nadal’s path to the 2018 title, showing each player’s pre-match clay court Elo*, along with the odds that Rafa (given his own current rating) would beat him:

Round  Opponent                 Opp Elo  p(Rafa W)  
R32    Roberto Carballes Baena     1767      97.3%  
R16    Guillermo Garcia Lopez      1769      97.2%  
QF     Martin Klizan               1894      94.5%  
SF     David Goffin                2079      84.5%  
F      Stefanos Tsitsipas          1900      94.3%

* from this point on, the clay court Elos I use are 50/50 blends of clay-specific Elo–that is, a rating calculating only with clay court results–and overall Elo. The blended rating is the one that has proven best at predicting match outcomes. Nadal is the all-time leader in this category as well, with a 50/50 clay Elo that peaked around 2,510.

Given those five single-match probabilities, the odds that Nadal would win the tournament were just over 70%. That’s dominant, but it’s not 11-out-of-13 dominant.

What if Rafa were underrated by Elo, at least in Barcelona? Here is the probability that a player at various Elo ratings would have beaten the five opponents that he faced last week:

Clay Elo  p(2018 Title)  
2200              41.2%  
2250              50.4%  
2300              59.1%  
2350              66.9%  
2400              73.6%  
2450              79.3%  
2500              83.9%  
2550              87.6%  
2600              90.5%

It turns out that this year’s title path was one of the weakest since 2005. It is roughly equivalent to the players Nadal needed to defeat in 2006 (with Nicolas Almagro in the semis and Tommy Robredo in the final), and a bit tougher than last year’s route, which didn’t feature a top-50 player until Thiem in the final. The toughest was his hypothetical path in 2015, when he lost to Fabio Fognini in the second round. Had he progressed, he would have faced David Ferrer in the semis and Nishikori in the final.

Once we figure out the quality of Rafa’s opponents (and would-have-been opponents, for the two years he lost early), we can work out the odds that any player–given those paths–would have won the tournament each year.

If we assume that Rafa’s average level since 2005 is the same as his current level–a clay Elo of around 2,400–the odds that he would have won 11 Barcelona titles in 13 tries is 13.0%. We don’t have the luxury of replaying those 13 tournaments in a few thousand alternate universes, so it’s not entirely clear what to make of that number–was Rafa lucky? would he do it again, given the chance? is he actually way better than an Elo level of 2,400 in Barcelona?

I don’t know the answer to those questions; all we know is what happened. To compare (un)decimas (and related accomplishments by other players), we’re going to look at the Elo level that would have resulted in the achievement at least 50% of the time. In other words, how good would Nadal have to have been to give himself a 50/50 chance at winning 11 Barcelona titles in 13 tries?

At various clay Elo levels, here are the odds that Rafa would have completed the Barcelona undécima:

Clay Elo  p(11 of 13)  
2300             1.0%  
2350             4.6%  
2400            13.0%  
2450            28.0%  
2500            47.2%  
2550            64.2%  
2600            77.7%  
2650            87.3%  
2700            93.1%

Thus, a player with a clay Elo of about 2,505 would have had a 50% chance of matching Nadal’s feat at his home tournament. To put it another way: At this event, over a span of 14 years, he has played at a level roughly equal to his career peak which, incidentally, is the all-time best clay Elo rating ever achieved by an ATP player.

Comparing las (un)decimas

I hope that my method makes sense and seems like a reasonable way of quantifying a rare feat. Algorithm in hand, we can compare Nadal’s Barcelona record with his efforts in Monte Carlo and Paris.

Monte Carlo

Rafa has entered 14 times since 2005 (again, excluding his 2003 appearance) and won 11. That’s a bit less impressive than 11-of-13, but the competition level is much higher. Only last year’s tournament, in which the opposing finalist was Albert Ramos, is in the same league as most of the Barcelona draws.

Sure enough, the Monte Carlo undécima is lot more impressive. To have a 50% chance of winning 11 titles in 14 attempts, a player would need a clay Elo of about 2,595, almost 100 points higher than the comparable number for Barcelona, and well above the level any player has ever achieved, even at their peak.

Roland Garros

At the French Open, Nadal has entered 13 times, winning 10. The field is even more challenging than in Monte Carlo, but on the other hand, the five-set format gives a greater edge to favorites, lessening the chance of an underdog scoring an upset with two magical sets.

The Roland Garros 10-of-13 is not quite as eye-popping as the record at Monte Carlo. The clay Elo required to give a player a 50% chance of matching Nadal’s French Open feat is “only” around 2,570–still better than any player has ever attained, but a bit short of the comparable mark for Monte Carlo.

But wait … what about 2016? Rafa won two rounds and then withdrew from his third-rounder against Marcel Granollers. I don’t know whether that should count, but at least for argument’s sake, we should run the numbers without it, treating Nadal’s French Open record as 10 titles in 12 appearances, not 13. In that case, the clay Elo that would give a player a 50/50 shot at matching the record is 2,595–the same as the Monte Carlo number.

At the moment, Monte Carlo appears to be the tournament where Nadal has played his very best. With another French Open a few weeks away, though, that answer may be temporary.

Rafa vs other record holders

A few other players have racked up impressive totals at single events. Wikipedia has a convenient list, and a few accomplishments stand out: Federer’s tallies at Wimbledon, Basel, and Halle, Guillermo Vilas’s eight titles in Buenos Aires, and Borg’s six French Open titles in only eight appearances.

Let’s have a look at how they compare, ranked by the surface-specific Elo rating that would give a player a 50% chance of equaling the feat:

Player   Tourney          Wins  Apps  50% Elo  
Nadal    Monte Carlo        11    14     2595  
Nadal    French Open*       10    12     2595  
Nadal    French Open        10    13     2570  
Borg     French Open**       6     7     2550  
Nadal    Barcelona          11    13     2505  
Borg     French Open         6     8     2475  
Vilas    Buenos Aires***     8    10     2285  
Federer  Wimbledon           7    18     2285  
Federer  Halle               8    15     2205  
Federer  Basel               8    15     2180

* excluding 2016

** excluding 1973, when Borg was 16 years old, and lost in the fourth round

*** excluding 1969-71, both because Vilas was very young, and due to sketchy data

The only single-event achievement that ranks with Nadal’s is Borg’s record at Roland Garros–and even then, only when we don’t consider Borg’s loss there as a 16-year-old. Federer’s records in Wimbledon, Halle, and Basel are impressive, but fail to rate as highly because he has entered those tournaments so many times. Federer didn’t appear on tour ready to win everything on his chosen surface, the way Rafa did, and those early losses are part of the reason that his records at these tournaments are so low.

We never needed any numbers to know that Nadal’s accomplishments at his three favorite tournaments are among the best of all time. With these results, though, we can see just how dominant he has been, and how few achievements in tennis history can even compare. The scary thing: A month from now, I may need to come back and update this post with even more eye-popping numbers. The greatest show on clay courts isn’t over yet.

Big Four Losing Streaks

Italian translation at settesei.it

This is a guest post by Peter Wetz.

Novak Djokovic’s loss against Benoit Paire in his first match at this year’s Miami Masters caused a lot of head scratching. Not only did Benoit equalize his head to head against Novak–next to Hyeon Chung he is now the only active player with a balanced record against Novak; four active players hold positive records–but this was also the Serbian’s third consecutive loss.

Novak immediately made some changes, announcing the end of his partnership with his coach Andre Agassi and part-time coach Radek Stepanek after having worked with them just a few months.

A losing streak of this length by such a dominant player must be rare, and it prompted me to look for similar instances among the big four. The following table shows all three (or more) match losing streaks of the big four after they cracked the top ten in reverse chronological order. The last column shows the Elo-based probability (Prob) of having such a streak. This is simply the product of the probabilities of losing the matches that made up the streak.

Player    Start	        End	Length	Prob
Djokovic  2018-01-15	-*	3	0.002%  (0.027%**)
Murray	  2011-01-17	03-23	4	0.02%
Murray	  2010-03-11	04-11	3	0.63%
Nadal	  2009-11-08	11-22	4	1.89%
Djokovic  2007-10-15	11-12	5	0.07%
Federer	  2002-07-08	08-19	4	0.66%

* Streak still active

** Probability when adjusting Elo ratings due to absence from the tour

The table shows that since August 2002 Roger Federer never lost more than two matches in a row. Even his four match losing streak is the second most likely due to the strong competition he had to face. In November 2009 Rafael Nadal lost four matches in a row, but with a probability far higher than the other streaks. The reason is that three of the four matches occurred at the World Tour Finals, increasing the likelihood of a loss.

A number that stands out is the probability of Novak’s current streak: 0.002%. However, this number is based on traditional Elo ratings which do not take into account player absence, for instance, due to injury. Before this season Novak took a six month break suffering from a shoulder injury.

As has already been discussed, there are ways to adjust Elo ratings for players coming back on the tour. In the case of Maria Sharapova, who stayed absent for 15 months, a 200 point drop in her first five matches after the break was more in line with her level of play than simply assuming that she remained as competitive as before. For this analysis I used a drop of 150 rating points for Novak, which results in a more realistic streak probability of 0.027%, still the second lowest in the list.

This brings us to Andy Murray‘s losing streak of 2011, which most of us probably have already forgotten. After losing the Australian Open final to Novak, Andy lost against Marcos Baghdatis (#20) in Rotterdam, Donald Young (#143) in Indian Wells, and Alex Bogomolov (#118) in Miami. This looks very similar to Novak’s current situation, but Murray bounced back to achieve a 50-9 record for the remainder of the season. It remains to be seen whether Djokovic can do the same.

—

Peter Wetz is a computer scientist interested in racket sports and data analytics based in Vienna, Austria.

Dominic Thiem, Old-School Clay Court Specialist

Italian translation at settesei.it

With a tennis calendar tilted heavily toward hard court events, we don’t see many true clay court specialists these days. The best male players who excel on clay are forced to adapt their games to hard courts, as well: Rafael Nadal has won six majors off of clay, while Pablo Carreno Busta and Diego Schwartzman have both hoisted trophies at tour level hard court events. It’s possible to play a mostly-clay schedule at the Challenger level, but it’s nearly impossible to establish yourself as an ATP regular without winning some matches on hard courts.

Dominic Thiem is capable enough on fast surfaces, but more than any other tour player, he is considerably better on the dirt than off it. In the last 52 weeks, he has won 25 of 31 matches on clay, compared to only 24 of 42 on other surfaces. Against the top ten, he is a respectable 7-9 on clay (more impressive when you consider that 12 of the 16 matches were against the Big Four, seven of them facing Nadal, and two of the others came against Stan Wawrinka), but a dismal 2-15 on hard courts. If you, like me, had settled into thinking of Thiem as a solid but not particuarly threatening member of the top ten, you probably didn’t realize quite how bad he is on hard courts–or just how good he has become on clay.

When only clay court results are taken into consideration, Thiem rates as the second-best player on the surface. According to clay court Elo, the Austrian outranks everyone on tour except for Rafa and Novak Djokovic, whose rating reflects his skill level when he last regularly played and very likely will overstate his ability when he returns. Thiem trails Nadal by about 180 points, 2410 to 2235, implying that in a head-to-head matchup, we’d except the Austrian to win only 26% of the time. But when we compare Thiem to the rest of the pack and exclude the walking wounded–Djokovic, Wawrinka, Andy Murray, and Kei Nishikori–along with clay-skipper Roger Federer, his position looks much better. The next best clay courter, Alexander Zverev, trails Thiem by about the same margin, nearly 170 points.

A clay court Elo rating over 2200 is a useful marker of elite status. In the professional era, only 29 players have reached that level, 22 of whom can count at least one Grand Slam title to their names. Among active players, only the Big Four, Nishikori, Juan Martin del Potro, David Ferrer, and Thiem belong to the club.

Where Thiem really stands out is the juxtaposition of his clay court success and his hard court mediocrity. After his title last week in Buenos Aires, his Elo rating based only on clay court results was 2234, compared to a hard court rating of 1869. The first number, as we’ve seen, is good for third overall, second if we exclude Djokovic’s increasingly stale results; the second puts him 31st on tour, behind Schwartzman, Damir Dzumhur, and Fabio Fognini.

No one active today is more of a clay specialist–in the sense that his results on clay exceed his results on hard–than Thiem. (There are some even more extreme differences between grass and either hard or clay, but the brevity of the grass season means that many of those contrasts are due only to small samples.) The ratio of Thiem’s clay court Elo rating to his hard court rating–again, 2234 to 1869–is 1.20, far beyond any of the 44 other active players with a clay court Elo rating of 1800 or higher. Simone Bolelli comes in second, at 1.12, and a handful of players, including Nadal, register at 1.10. Here is the entire top 20:

Player                 Clay Elo  Hard Elo  Ratio
Dominic Thiem              2234      1869   1.20
Simone Bolelli             1834      1634   1.12
Rafael Nadal               2410      2182   1.10
Albert Ramos               1873      1696   1.10
Federico Delbonis          1869      1696   1.10
Pablo Carreno Busta        1921      1746   1.10
Pablo Cuevas               1873      1709   1.10
Nicolas Almagro            1903      1755   1.08
Karen Khachanov            1838      1701   1.08
Leonardo Mayer             1878      1741   1.08
Aljaz Bedene               1826      1695   1.08
David Ferrer               2017      1894   1.07
Philipp Kohlschreiber      1951      1845   1.06
Stan Wawrinka              2138      2027   1.06
Martin Klizan              1800      1709   1.05
Guido Pella                1825      1744   1.05
Borna Coric                1830      1760   1.04
Fernando Verdasco          1863      1794   1.04
Alexander Zverev           2067      1997   1.04
Feliciano Lopez            1830      1772   1.03

A few decades ago, when it was possible for top players to spend more than two or three months per year racking up points on clay courts, such lopsided ratings were a bit more common. Of the 29 men who have ever topped 2200 in clay court Elo rating, 11 have at some point recorded a ratio of 1.20 or higher. That includes Nadal, whose clay rating was 20% higher than his hard court number early in 2008, and Sergi Bruguera, whose ratio topped out at 1.29. Four other major titlists–Bjorn Borg, Juan Carlos Ferrero, Thomas Muster, and Guillermo Vilas–also exceeded 1.20 at some point during their career. To put Thiem’s specialization in context, though, consider that Guillermo Coria maxed out at 1.19 and Gustavo Kuerten peaked at 1.16. Even Ferrer–the epitome of the clay court specialist to a generation of fans–never exceeded 1.15 once his clay court Elo rating had passed the 2000-point threshold.

The category into which Thiem fits most neatly–specialists who are decidedly middle-of-the-pack on hard courts–largely belongs to an earlier era. When we lower our clay court Elo standard to a career peak of 2000 points, a mark equal to about 15th on tour right now, we’re left with a group of 145 players in the professional era. Of those, 65 (45%) were at some point as lopsided as Thiem is now, with a clay-to-hard rating ratio of at least 1.20. Yet only five of those belong to active players (Nadal, Thiem, Fognini, Pablo Cuevas, and Nicolas Almagro) and two-thirds of them came before 1995.

In some cases, players with substantially better clay court results learn to compete at a higher level on faster surfaces. Thiem is 24, and Nadal had a similar specialist’s ratio at age 22. Other former greats enjoyed early success on clay and quickly figured out hard courts as well. The Austrian may prove to be a late bloomer in that regard. That’s unlikely, but when Nadal retires or (improbable as it seems) fades, Thiem is poised to rack up titles and emerge as the greatest clay court player of his generation, regardless of whether his hard court game improves.

Roger Federer’s 20th, Easiest Grand Slam Title

Italian translation at settesei.it

After Rafael Nadal’s US Open title last fall, I wrote a piece for the Economist that attempted to measure each Grand Slam title by difficulty. If you’re interested in the methodology, you can review it there. The conclusion was intriguing: Nadal’s opponents en route to his 16 major titles were considerably more difficult than the routes Roger Federer took to his first 19. By “difficulty-adjusted” Slam titles, Rafa led by a whisker, 18.8 to 18.7.

Since then, Federer won the 2018 Australian Open, incrementing his major tally by one. Even though he faced rather weak competition, surely the additional title nudged his difficulty-adjusted total above Rafa’s, right?

It did, but not by much. Adjusted for difficulty, Roger’s seven wins in Melbourne were worth only 0.42 majors. By comparison, his previous low was the 2006 Australian, worth 0.61, and Rafa’s lowest was last year’s US Open, at 0.62. Federer’s previous average was 0.98, Nadal’s was 1.18, and Rafa’s route to the 2013 French Open was worth a whopping 1.65.

Fed’s draw was historically weak. Only a handful of majors in the professional era were easier for their champion, and they all came before 1985–most of them in Melbourne, which didn’t yet attract the best talent in the world. This year’s Australian Open path to the title was even weaker when put in the context of the current decade: The average major title from 2010-17 was worth 1.23, largely because the Big Four usually needed to overcome each other.

According to surface-specific Elo, the toughest challenge Federer faced last month was Tomas Berdych, closely followed by Marin Cilic. Even after deep runs in Australia, neither player even ranks in the current Elo top ten. The algorithm that adjusts slam titles considers how the average major champion would fare against a particular set of competition; against Berdych and Cilic, that hypothetical average champ is expected to win 88% and 89% of the time, respectively. Even Nadal had to get past Juan Martin del Potro in New York last year.

Still, Federer can claim the top spot on yet another list, as his 19.1 difficulty-adjusted Grand Slam titles exceed Rafa’s 18.8 as well as the 15.3 of Novak Djokovic. It doesn’t have quite the same ring that “20 majors” does, and it’s in considerably more immediate danger. If Nadal stays healthy and wins the French Open, he is virtually guaranteed to reclaim the difficulty-adjusted crown, and by a wider margin than Roger currently holds. Roland Garros has traditionally been tough: With the exception of 2010, all of Rafa’s trophies in Paris have been tougher than average. Unlike the traditional Grand Slam tally, the difficulty-adjusted ranking could yo-yo between the two rivals for as long as they remain competitive.

Overperforming in Davis Cup

This is a guest post by Peter Wetz.

With the help of weighted surface specific Elo ratings we have a powerful new tool to measure player performance. The traditional conclusion of the tennis season, the Davis Cup final, provides us with an opportunity once again to examine which players thrive when competing for their nation and which players seem to suffer from the pressure. While we are at it, I don’t like the sound of the word offseason. After all, there are still ITF tournaments, not to mention the Australian Open Asia-Pacific Wildcard Play-offs.

As already hinted, Elo ratings have proven to represent a better picture of player quality than traditional ATP rankings. Hence, comparing expected wins based on Elo with actual wins provides us with a clearer picture of who outperforms expectations and who does not.

In this evaluation, I consider completed World Group and Group 1 Davis Cup live rubbers played since 1980. The data set contains around 5000 matches through this year’s World Group Quarterfinals, and I’ve limited my focus to players having played 15 or more matches.

Let’s first take a glance at the obvious stat, win-loss percentage. The following table shows the top ten win-loss records of all players under consideration. (The Active column denotes if the player is still an active player).

Name	        W	L	Perc	Active
Rafael Nadal	20	1	95%	1
Boris Becker	31	2	94%	0
Andy Murray	25	3	90%	1
Balazs Taroczy	23	3	89%	0
David Ferrer	20	3	87%	1
Andre Agassi	23	4	85%	0
Roger Federer	40	7	85%	1
Novak Djokovic	27	5	84%	1
Guillermo Vilas	16	3	84%	0
Andrei Medvedev	16	3	84%	0

As one would expect, the big four and other all time greats are included. However, this obviously does not tell the whole story. Rafael Nadal is expected to win most of the time and that is what he does. For such a player, it is hard to outperform expectations.

If we compute how much a player outperforms his expectations, we get a clearer picture, given we want to know who does especially well in Davis Cup. Expected wins are calculated based on a half-and-half mix of surface specific Elo and overall Elo as this, in general, provides close to the best results, as pointed out in a previous article.

The tables below show the top and bottom five among all (first table) and active (second table) players in terms of over and underperforming expected wins. It shows actual wins (W), expected wins (eW), the percentage of over or underperformance (+/-), and if a player is still active.

Name	         W	eW	+/-	active
Francisco Maciel 11	6	72%	0
Slobodan Zi'vic  20	11	72%	0
Vasek Pospisil	 9	5	71%	1
Adrian Ungur	 6	3	56%	1
Mahesh Bhupathi	 5	3	55%	0
...
Wally Masur	 7	10     -31%	0
Sebastien Lareau 7	10     -31%	0
James Blake	 7	10     -36%	0
Nicolas Kiefer	 6	10     -40%	0
Aqeel Khan	 2	4      -57%	0

Name	        W	eW	+/-	Active
Vasek Pospisil	9	5	71%	1
Adrian Ungur	6	3	56%	1
Andrey Golubev	13	8	46%	1
Di Wu	        14	9	45%	1
Steve Darcis	15	11	35%	1
...
Florian Mayer	7	8      -14%	1
Gilles Muller	9	10     -15%	1
Alejandro Falla	8	9      -17%	1
John Isner	9	11     -19%	1
Jurgen Melzer	20	25     -22%	1

The tables seem to overlap with some conventional wisdom floating through the tennis sphere. Namely, that Steve Darcis, despite his recent losses at the Davis Cup final, plays above expectations. Also, Jurgen Melzer is known for regularly disappointing Austrian Davis Cup fans. (In his defense, he created several moments of joy, too).

If we were to pick a Davis Cup hero for the active and inactive group of players, Slobodan Zivojinovic and Andrey Golubev seem to be good choices. Golubev has a record of 13-6 (68%) and outperforms expected wins by 46%. He provides a good combination of consistently beating players he should beat and scoring more than his share of exceptional upsets (Wawrinka 2014, Goffin 2014, Melzer 2013 and Berdych 2011).

Zivojinovic provides a similar pattern with a record of 20-8 (71%), 72% better than expected. He tallied six wins out of ten matches in which Elo assigned him a win probability of less than 25%. Further, he only lost one match in when his pre-match odds of winning were greater than 35%.

This post provides insight into how Elo ratings help in quantifying a player’s performance. We identified players who have (not) shown great improvement on what the algorithm expected based on results from the regular tour. For future research it would be interesting to delve into Davis Cup doubles heroes: Where there are no dead rubbers, stakes are always high.

—

Peter Wetz is a computer scientist interested in racket sports and data analytics based in Vienna, Austria.

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.

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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.