Category: Grand Slams

20 > 21 > 20

Rafael Nadal has finally nosed his way into the lead. With his Australian Open title yesterday, he became the first man to 21 major singles titles, breaking away from the three-way tie at 20 with Novak Djokovic and Roger Federer.

For some people, leading the all-time grand slam race is enough to cement a player as the greatest of all time. A different crowd considers this year’s Australian Open tainted because Djokovic was not allowed to play. Still others think that Federer played some beautiful tennis, and they considered the matter concluded at least five years ago.

I belong to a fourth camp, which I can summarize with two positions:

  1. The grand slam race isn’t everything.
  2. If you do focus on grand slams, you must adjust the major count for the quality of opponents each player faced.

I’ve written about this before, first at The Economist, and then here at the blog. When I checked in 18 months ago, Nadal’s 20 majors were worth a bit more than Djokovic’s 17, which were themselves more impressive than Federer’s 20. The margins have always been slim between these three, and properly adjusting for quality of opponents makes things even tighter.

The update

Here’s how the adjustment works. For each slam that a player won, we take the Elo rating of all of his opponents, and work out the probability that the average Open Era grand slam winner would beat all of them. Once we have that number–which centers around 23%–we normalize it so that the value of an “average” major is 1.0.

When a major title requires facing down a lot of tough opponents, its rating is higher than 1.0, while a relatively easy one rates below 1.0. In the last few years, the numbers have drifted downward, because while the familiar names keep winning quite a bit, they haven’t needed to face each other as often as they used to.

You might disagree with the methodology, and that’s fine. But I find that most people end up making some sorts of adjustments, even if they shy away from stats or only tweak the totals when it favors their idol. Some Djokovic fans want to downplay Nadal’s recent win, and it’s true that Novak’s absence lowered the quality of the draw. But surely Rafa’s title isn’t worth zero. He beat many excellent players, and there was no guarantee that Novak would advance through the draw–or that Rafa would lose if they met.

This approach allows us to avoid specific minefields and answer all the analogous questions about every slam. Considering the seven opponents that Nadal faced, his Melbourne title rates at 0.84, weaker than average, but more difficult than seven of his prior titles. Djokovic has not enjoyed as many “easy” paths to major titles, but his Wimbledon victory last summer rates at a mere 0.60, the second-weakest of his career and lower than all but one of Rafa’s. Sometimes players just get lucky, with or without a geopolitical brouhaha.

Nadal’s 21st title rates only a bit lower than Djokovic’s two other titles last year: 0.90 at the Australian and 0.93 at the French.

Here are the updated rankings for “adjusted slams,” along with a table showing how many easy, medium, and hard paths that the Big Three have endured:

Player    Slams  Avg Score  Total  
Nadal        21       0.95   19.9  
Djokovic     20       1.01   20.1  
Federer      20       0.89   17.9  
                                   
Player     Easy     Medium   Hard  
Nadal         8          8      5  
Djokovic      6          7      7  
Federer       9         10      1

As if 21 and 20 weren’t close enough, this approach gives Djokovic 20.1 adjusted slams to Nadal’s 19.9. Again, you don’t have to agree with every step of my approach here to accept that we often think in terms of these kind of adjustments, and that Djokovic has–on average–faced tougher roads to titles than Nadal, while Federer had it easier than both of them.

Players can’t control who they face, but as fans, we can appreciate who worked the hardest to achieve near-equivalent feats. Fingers crossed that both Novak and Rafa excel at Roland Garros, so they can fight it out on the court, not in some random guy’s spreadsheets.

How Much Does Naomi Osaka Raise Her Game?

You’ve probably heard the stat by now. When Naomi Osaka reaches the quarter-final of a major, she’s 12-0. That’s unprecedented, and it’s especially unexpected from a player who doesn’t exactly pile up hardware outside of the hard court grand slams.

It sure looks like Osaka finds another level as she approaches the business end of a major. Translated to analytics-speak, “she raises her game” can be interpreted as “she plays better than her rating implies.” That is certainly true for Osaka. She has won 16 of her 18 matches in the fourth round or later of a slam, often in matchups that didn’t appear to favor her. In her first title run, at the 2018 US Open, my Elo ratings gave her 36%, 53%, 46%, and 43% chances of winning her fourth-round, quarter-final, semi-final, and final-round matches, respectively.

Had Osaka performed at her expected level for each of her 18 second-week matches, we’d expect her to have won 10.7 of them. Instead, she won 16. The probability that she would have won 16 or more of the 18 matches is approximately 1 in 200. Either the model is selling her short, or she’s playing in a way that breaks the model.

Estimating lift

Osaka’s results in the second week of slams are vastly better than the other 93% or so of her tour-level career. It’s possible that it’s entirely down to luck–after all, things with a 0.5% chance of happening have a habit of occurring about 0.5% of the time, not never. When those rare events do take place, onlookers are very resourceful when it comes to explaining them. You might believe Osaka’s claims about caring more on the big stage, but we should keep in mind that whenever the unlikely happens, a plausible justification often follows.

Recognizing the slim possibility that Osaka has taken advantage of some epic good luck but setting it aside, let’s quantify how good she’d have to be for such a performance to not look lucky at all.

That’s a mouthful, so let me explain. Going into her 16 second-week slam matches, Osaka’s average surface-blended Elos have been 2,022. That’s good but not great–it’s a tick below Aryna Sabalenka’s hard-court Elo rating right now. Those modest ratings are how we come up with the estimate that Osaka should’ve won 10.7 of her 18 matches, and that she had a 1-in-200 shot of winning 16 or more.

2,022 doesn’t explain Osaka’s success, so the question is: What number does? We could retroactively boost her Elo rating before each of those matches by some amount so that her chance of winning 16-plus out of 18 would be a more believable 50%. What’s that boost? I used a similar methodology a couple of years ago to quantify Rafael Nadal’s feats at his best clay court events, another string of match wins that Elo can’t quite explain.

The answer is 280 Elo rating points. If we retroactively gave Osaka an extra 280 points before each of these 16 matches, the resulting match forecasts would mean that she’d have had a fifty-fifty chance at winning 14 or more of them. Instead of a pre-match average of 2,022, we’re looking at about 2,300, considerably better than anyone on tour right now. (And, ho hum, among the best of all time.) A difference of 280 Elo points is enormous–it’s the difference between #1 and #22 in the current hard-court Elo rating.

Osaka versus the greats

I said before that Osaka’s 12-0 is unprecedented. Her 16-2 in slam second weeks may not have quite the same ring to it, but compared to expectations based on Osaka’s overall tour-level performance, it is every bit as unusual.

Take Serena Williams, another woman who cranks it up a notch when it really matters. Her second-week record, excluding retirements, is 149-39, while the individual forecasts before each match would’ve predicted about 124-64. The chances of a player outperforming expectations to that extent are basically zero. I ran 10,000 simulations, and that’s how many times a player with Serena’s pre-match odds won 147 of the 185 matches. Zero.

For Serena to have had a 50% chance of winning 149 of the 188 second-week contests, her pre-match Elo ratings would’ve had to have been 140 points higher. That’s a big difference, especially on top of the already stellar ratings that she has maintained throughout her career, but it’s only half of the jump we needed to account for Osaka’s exploits. Setting aside the possibility of luck, Osaka raises her level twice as much as Serena does.

One more example. Monica Seles won 70 of her 95 second-week matches at slams, a marked outperformance of the 60 matches that Elo would’ve predicted for her. Like Osaka, her chances of having won 70 instead of 60 based purely on luck are about 1 in 100. But you can account for her actual results by giving her a pre-match Elo bonus of “only” 100 points.

The full context

I ran similar calculations for the 52 women who won a slam, made their first second-week appearance in 1958 or later, and played at least 10 second-week matches. They divide fairly neatly into three groups. 18 of them have career second-week performances that can easily be explained without recourse to good luck or level-raising. In some cases we can even say that they were unlucky or that they performed worse than expected. Ashleigh Barty is one of them: Of her 14 second-week matches, she was expected to win 9.9 but has tallied only 8.

Another 16 have been a bit lucky or slightly raised their level. To use the terms I introduced above, their performances can be accounted for by upping their pre-match Elo ratings by between 10 and 60 points. One example is Venus Williams, who has gone 84-43 in slam second weeks, about six wins better than her pre-match forecasts would’ve predicted.

That leaves 18 players whose second-week performances range from “better than expected” to “holy crap.” I’ve listed each of them below, with their actual wins (“W”), forecasted wins (“eW”), probability of winning their actual total given pre-match forecasts (“p(W)”), and the approximate number of Elo points (“Elo+”) which, when added to their pre-match forecasts, would explain their results by shifting p(W) up to at least 50%.

Player               M    W     eW   p(W)  Elo+  
Naomi Osaka         18   16   10.7   0.5%   280  
Billie Jean King   123   94   76.2   0.0%   160  
Sofia Kenin         10    7    4.7  10.6%   150  
Serena Williams    188  149  124.4   0.0%   140  
Evonne Goolagong    92   69   58.7   0.4%   130  
Jennifer Capriati   70   42   33.2   1.2%   110  
Monica Seles        95   70   60.2   1.2%   100  
Hana Mandlikova     75   49   41.7   3.1%   100  
Kim Clijsters       67   47   40.6   4.6%    90  
Justine Henin       74   55   48.9   6.3%    80  
Mary Pierce         55   28   22.4   6.9%    80  
Li Na               36   22   18.0  10.6%    80  
Steffi Graf        157  131  123.6   6.1%    70  
Maria Bueno         93   70   63.4   6.3%    70  
Garbine Muguruza    31   18   14.9  15.8%    70  
Mima Jausovec       32   18   15.0  15.9%    70  
Marion Bartoli      20   11    8.8  20.6%    70  
Sloane Stephens     24   12    9.7  20.8%    70

There are plenty of names here that we’d comfortably put alongside Williams and Seles as luminaries known for their clutch performances. Still, the difference between Osaka’s levels is on another planet.

Obligatory caveats

Again, of course, Osaka’s results could just be lucky. It doesn’t look that way when she plays, and the qualitative explanations add up, but … it’s possible.

Skeptics might also focus on the breakdown of the 52-player sample. In terms of second-week performance relative to forecasts, only one-third of the players were below average. That doesn’t seem quite right. The “average” woman outperformed expectations by about 30 Elo points.

There are two reasons for that. The first is that my sample is, by definition, made up of slam winners. Those players won at least four second-week matches, no matter how they fared in the rest of their careers. In other words, it’s a non-random sample. But that doesn’t have any relevance to Osaka’s case.

The second, more applicable, reason that more than half of the players look like outperformers is that any pre-match player rating is a measure of the past. Elo isn’t as much of a lagging indicator as, say, official tour rankings, but by its nature, it can only consider past results.

Any player who ascends to the top of the game will, at some point, need to exceed expectations. (If you don’t exceed expectations, you end up with a tennis “career” like mine.) To go from mid-pack to slam winner, you’ll have at least one major where you defy the forecasts, as Osaka did in New York in 2018. Osaka was an extreme case, because she hadn’t done much outside of the slams. If, for instance, Sabalenka were to win the US Open this year, she has done so well elsewhere that it wouldn’t be the same kind of shock, but it would still be a bit of a surprise.

In other words, almost every player to win a slam had at least one or two majors where they executed better than their previous results offered any reason to expect. That’s one reason why we find Sofia Kenin only two spots below Osaka on the list.

For Serena or Seles, the “rising star” effect doesn’t make much of a difference–those early tournaments are just a drop in the bucket of a long career. Yeah, it might mean they really only up their game by 110 Elo points instead of 130, but it doesn’t call their entire career’s worth of results into question. For Osaka or Kenin, the early results make up a big part of the sample, so this is something to consider.

It will be tougher to Osaka to outperform expectations as the expectations continue to rise. Much depends on whether she continues to struggle away from the big stages. If she continues to manage only one non-major title per year, she’ll keep her rating down and suppress those pre-match forecasts. (The predictions of major media pundits will be harder to keep under control.) Beating the forecasts isn’t necessarily something to aspire to–even though Serena does it, her usual level is so high that we barely notice. But if Osaka is going to alternate levels between world-class and merely very good, she could hardly do better than to bring out her best stuff when she does.

Serena’s 23 vs Margaret’s 24

Since 2017, Serena Williams has held 23 major titles, leaving her just one shy of Margaret Court’s 24. The Williams-Court comparison forces us to think across eras in the same way that Federer-vs-Laver does, with the additional complication that Court has earned herself extreme dislike among many fans and fellow champions.

Let’s set aside the off-court stuff and work this out. The pro-Court case is simple: 24 is greater than 23, and you have to evaluate players relative to their own eras. The pro-Serena side is equally straightforward: 11 of Court’s 24 titles came in Australia, before Melbourne was a mandatory tour stop. Regardless of the era, Court’s home event was weaker back then.

As much as possible, I’m going to try to hold to the “relative to their own era” assumption. Everyone seems to accept it when it comes to Laver-vs-Federer. Plus, if we drop that constraint, the whole exercise is meaningless. With improved technology, fitness, and coaching, of course today’s players are better. But that’s not what people are talking about when they pick a side of Serena-vs-Margaret or Rod-vs-Roger.

Attentive readers of this blog might recall I took a stab at this problem back in 2019. That attempt relied on some extreme approximating due to the lack of pre-Open Era women’s tennis data. Regular readers will also know that the state of pre-Open Era women’s tennis data has vastly improved in the last few months. Tennis Abstract, plus the associated GitHub repo, now contains thousands of match results back to the mid-1950s.

Adjusting Australia

Let’s be clear: I’m not about to settle whether Margaret Court or Serena Williams (or someone else) is the GOAT of women’s tennis. That debate depends on much more than grand slam titles.

Today’s question is: How do Williams’s 23 titles stack up against Court’s 24?

That boils down to an even simpler question: How do Court’s 11 Australian titles measure up against other slams, then and now?

The anecdotal evidence is strongly anti-Margaret. As I mentioned in this morning’s Expected Points, the 1960 Australian Championships–Court’s first major title–had a 32-player draw (strike one), and 30 of those players were Australian (strikes two and three). Yes, it was a strong era for Australian women’s tennis, especially a few years later, but the tournament was hardly a showcase of international superstars. As such, it isn’t what we think of as a “major” tournament these days.

I’ve done a lot of “slam adjustments,” mostly to track the difficulty of the majors won by Djokovic, Federer, and Nadal. (Here’s the most recent.) The basic approach is simple. For each tournament, take the winning player’s draw, and for each match, calculate the chance that an average slam winner on that surface would beat that set of opponents. (Odds are determined by my Elo ratings, which are based on results before the event.) Take the resulting probabilities–on average, around 14% between 1952 and 2020–and normalize them, so that a mid-range slam draw is 1.0. Tougher draws are higher than 1, and easier draws are lower.

Equalizing the eras

This type of adjustment gets us most of the way there, but it doesn’t directly confront the “relative to the era” issue. The field in general was more lopsided in the 1960s than it is now, with a handful of very strong players swatting away a pack of also-rans who struggled to win more than a game or two per set against the elites. That in itself is a point in favor of Serena (and modern players in general), but again, on the Laver-vs-Federer principle, that’s not what we’re talking about today.

The easiest way to express this idea that all eras are equivalent is to use as a standard each season’s Wimbledon, the one tournament that everybody always wanted to play, and almost everyone actually did play. To avoid year-to-year fluctuations based on short-term injuries, we’ll make things a bit more resilient and compare the strength of each year’s Australian draw to the average strength of that year’s Wimbledon and US draws.

For example, my slam adjustments consider 1960 to be a strong year. Maria Bueno’s Wimbledon title was 40% more difficult than the average slam draw, and Darlene Hard’s US victory was about 30% tougher than usual. Court’s Australian title that year comes out as exactly average, so we compare Australia’s 1.0 to the average of Wimbledon and the US ( (1.4 + 1.3) / 2 = 1.35), and the 1960 Australian title, relative to the era, measures as:

1 / 1.35 = 0.75

The mostly-Australian field wasn’t as weak as the caricature makes it out to be, but it was weaker than the marquee majors that year.

Here is how the strength of the Australian draw has evolved relative to the other grass- and hard-court slams from 1952 to the present:

Except for an outlier in 1965, when Bueno, Billie Jean King, and several other international stars turned up, the Australian Championships was a second class member of the grand slam club until around 1980. It’s had plenty of weak years since then, as well, partly because of players who skipped due to injury, and partly due to contenders losing early, giving the eventual winners easier paths.

The main event

Margaret Court won the Australian 11 times. By this measure of relative strength, those titles were worth 62% as much as the other majors in those years. The strenght of individual titles ranged from a low of 0.29 in 1961, when no international elites made the trip, to a high of 1.02 in 1965, when the field was positively star-studded.

Serena Williams has won the Australian seven times. It is tempting to leave that “7” as is, because Melbourne is now a mandatory tour stop and virtually every woman on tour considers it one of the top targets in her season. However, we should treat Serena’s seven the same way we adjusted Court’s 11. For all the era differences, some things remain the same, like jetlag and the difficulty of playing top-flight tennis only a few weeks into the season.

Williams’s seven were worth, on average, 88% as much as the other majors in their respective years. The weakest of the bunch was her last, in 2017. So many top players lost early that Serena never faced a top-eight opponent.

Court’s 11 titles, then, are equivalent to about 7 non-Australian majors–a penalty of four. Serena’s 7 are worth about 6 non-Australian majors–a penalty of one.

The final, adjusted tally: Williams 22, Court 20.

Margaret Court was one of the greatest players of all time, but her position the all-time grand slam singles list depends too much on the shifting status of her home event. When we properly account for the Australian tournament’s position for decade as the most minor major, Court loses her remaining claim to the top spot. Serena may yet win 24, but to match or exceed Court, she shouldn’t have to.

The Next Five Years, According To a (Dumb) Grand Slam Crystal Ball

Last year, I introduced a bare-bones model that predicts men’s grand slam results for the next five years. It takes a minimum of inputs: a player’s age, and his number of major semi-finals, finals, and titles in the last two years. Despite leaving out so much additional data, the model explains a lot of the variation among players, achieving most of what a more complex algorithm would, but with nothing more than basic arithmetic.

A bit further down, I’ll introduce a similar model for women’s grand slam results. First, let’s look at the revised numbers for the men. Keep in mind that these are not career slam forecasts, but only slams in the next five years. That’s good enough for the Big Three, but it probably doesn’t tell the whole story for, say, Stefanos Tsitsipas.

Player              Projected Slams  
Novak Djokovic                  2.5  
Rafael Nadal                    2.1  
Dominic Thiem                   2.0  
Alexander Zverev                0.9  
Stefanos Tsitsipas              0.6  
Daniil Medvedev                 0.6  
Matteo Berrettini               0.3  
Lucas Pouille                   0.1  
Diego Schwartzman               0.1

A few other players (notably Roger Federer) reached a semi-final in the last two years, but because of their age, the model forecasts zero slams. Also keep in mind that Wimbledon was not played this year, so there was a bit less data to work with.* The sum of the forecasts is a mere 9.2 slams, out of a possible 20. In some previous years, the model predicted as many as 15 titles for the players it took into consideration. Because today’s top players are so old, they aren’t expected to dominate much of the 2021-25 calendar, leaving room for new contenders to emerge.

* My original post describes the forecasting algorithm as counting results from “the last four slams” and “the previous four slams.” We could account for the three-slam 2020 season by following those steps literally, giving greater weights to the last four slams (the 2019 US Open plus the three 2020 slams), and giving lesser (but still non-zero) weights to the four slams before that. I rejected that approach because (a) it would give an awful lot of weight to the US Open, and (b) the relative lack of 2020 data reflects higher-than-usual uncertainty, which ought to show up in the forecasts, as well. Thus, only seven slams were taken into account for 2021-25 predictions, instead of the usual eight.

Interestingly, the 2020 season has barely budged the predicted career totals for the big three. Numbers I published immediately after last year’s US Open forecast Rafael Nadal for 22.5 career slams: his (then) 19 plus 3.5 more. Now he has 20, and the model pegs him for another 2.1. Novak Djokovic was slated for a career total of 19.5: 3.5 more on top his then-total of 16. He’s still penciled in for 19.5: 17 plus another 2.5 in the future. Federer didn’t have reason to expect much a year ago, and it’s no better now.

The women’s model

It turns out that a similar back-of-the-envelope approach gives good approximations of future slam totals for WTA stars, as well. The weights are a bit different, the average peak age is one year sooner, and the age adjustment is slightly smaller, but the idea is essentially the same.

Here’s how to calculate the number of expected major titles for your favorite player:

  • Start with zero points
  • Add 20 points for each slam semi-final reached in the last 12 months
  • Add 20 points for each slam final reached in the last 12 months
  • Add 80 points for each slam title won in the last 12 months
  • Add 10 points for each slam semi-final reached in the previous 12 months
  • Add 10 points for each slam final reached in the previous 12 months
  • Add 40 points for each slam title won in the previous 12 months
  • If the player is older than 26 (at the time of the next slam), subtract 7 points for each year she is older than 26
  • If the player is younger than 26, add 7 points for each year she is younger than 26
  • Divide the sum by 100

To take a simple example, consider Iga Swiatek. For her recent French Open title, she gets 20 points for the semi, 20 points for the final, and 80 points for the title. She will still be 19 when the Australian Open rolls around, so we add another 49 points: 7 years younger than 26, times 7 points per year. Her projected total is (20 + 20 + 80 + 49) / 100 = 1.69.

Here are the results for all of the women who reached a major semi-final in 2019 or 2020 and are projected to win more than zero slams between 2021 and 2025:

Player               Projected Slams  
Naomi Osaka                      2.0  
Sofia Kenin                      1.9  
Iga Swiatek                      1.7  
Bianca Andreescu                 1.0  
Ashleigh Barty                   0.9  
Amanda Anisimova                 0.6  
Simona Halep                     0.6  
Marketa Vondrousova              0.6  
Nadia Podoroska                  0.4  
Garbine Muguruza                 0.3  
Belinda Bencic                   0.3  
Jennifer Brady                   0.3  
Elina Svitolina                  0.2  
Petra Kvitova                    0.1  
Victoria Azarenka                0.1

These forecasts sum to 11.0 slams, more than the men’s total. That’s largely because so many of recent women’s champions are younger, giving the model more reason to be optimistic about them. It still leaves plenty of room for other players to earn some hardware in the next half-decade, which makes sense. The WTA has featured a non-stop succession of breakout young stars for the past few years, and with players like Aryna Sabalenka, Elena Rybakina, and Cori Gauff in the mix, there’s no shortage of talent to keep the carousel turning.

And then there’s Serena Williams. The model projects her for zero slams, despite her three semi-finals and two finals in the last two years. The reason is her age: The algorithm expects players to steadily decline from age 27 onwards, so the age penalty by age 39 is harsh. One one hand, that makes sense: we’re forecasting the results of events that will mostly take place when she’s in her 40s. On the other hand, a player who had so much success at age 37 is probably a good bet to break the mold at 39, as well. Were this a more fully-developed model, we’d probably be smart to tinker with the age adjustment to reflect the reality that Williams is a much better bet to win a major title than Nadia Podoroska.

We could go on all day. For every variable that these forecasts take into account, there are a dozen more than have some plausible claim to relevance. But this simple approach gets us surprisingly far in telling the future–a future in which the men’s all-time grand slam race keeps getting more complicated, and the women’s game continues to feature a wide array of promising young stars.

Not All Twenties Are Created Equal

The top of the all-time men’s grand slam ranking just got even more crowded. With his 13th Roland Garros title, Rafael Nadal has matched Roger Federer at the top of the list by securing his 20th major title. Novak Djokovic, Nadal’s final obstacle en route to the historic mark, remains within shouting distance with 17 slams.

The Roger-Rafa tie has spurred another (interminable, unresolvable) round of the (interminable, unresolvable) GOAT debate. Of course there’s much more to determining the best ever than the slam count. But the slam count is a big part of the conversation. If we’re going to keep doing this, we ought to at least recognize that not all major titles are created equal. And by extension, not all collections of twenty major titles are equivalent.

We all have intuitions about the difficulty of how a particular draw shakes out, with its typical mix of good and bad fortune. Nadal was lucky that he missed a few dangerous opponents in the early rounds, luckier still that he didn’t have to face Dominic Thiem in the semi-final, and unfortunate that he had to face down the next-best player in the draw, Djokovic, in the final. As it turned out, it didn’t really matter, but I think most of us would agree that Nadal’s achievement–staggering as it is–would look even better had he faced more than two more players ranked in the top 70.

Stop dithering and start calculating

I’ve written about this before, and I’ve established a metric to quantify those intuitions. Take the surface-weighted Elo rating of each of a player’s opponents, and determine the probability that an average slam champion would beat those players. After a couple of steps to normalize the results, we end up with a single number for the path to each slam title. The larger the result, the more difficult the path, and an average slam works out to 1.0.

Nadal’s path was easier than the historical average. Aside from Djokovic, none of his opponents would have had more than an 8% chance of knocking out an average slam champion on clay. The exact result is 0.64, which is easier than almost nine-tenths of majors in the Open Era. Rafa has had three easier paths to his major titles, including the 2017 US Open, which scored only 0.33. That’s the easiest US Open, Wimbledon, or Roland Garros in a half-century.

Of course, he’s had his share of difficult paths, such as 2012 Roland Garros (1.36), when he faced several clay specialists and a peak-level Djokovic. Federer and Djokovic have gotten their own shares of lucky and unlucky draws over the years–that’s why we need a metric. You might have a better memory for this kind of thing than I do, but I don’t think any of us can weigh 57 majors with 7 opponents each and work out any meaningful results in our heads.

The tally

Sum up the difficulty of the title paths for these 57 slams, and here are the results:

Player    Slams  Avg Score  Total  
Nadal        20       0.95   19.0  
Djokovic     17       1.06   18.1  
Federer      20       0.89   17.9  
                                   
Player     Easy     Medium   Hard  
Nadal         7          8      5  
Djokovic      5          5      7  
Federer       9         10      1

The first table shows each player’s average score for the paths to his major titles, and the total number of “adjusted slams” that gives them. Nadal is in the lead with 19, and Djokovic and Federer follow in a near-tie, just above and below 18.

You might be surprised to see the implication that this is a slightly weak era, with average scores a bit below 1.0. That wasn’t the case a few years ago, but there has only been one above-average title path since 2016. The Big Three-or-Four has generally stayed out of each other’s way since then, and even when they do clash, as they did yesterday, the leading contenders for quarter-final or semi-final challenges failed to make it that far. The average score of the last 15 slam title paths is a mere 0.73, while the 16 before that (spanning 2013-16) averaged 1.20.

The second table paints with a broader brush, classifying all Open Era slam titles into thirds: “easy,” “medium” and “hard” paths to the championship. Anything below 0.89 rates as “easy,” anything above 1.14 is marked as “hard,” with the remainder left as “medium.”

Djokovic is the leader in hard slams, with 7 of his 17 meriting that classification. Federer has racked up 10 medium slams, including several that score above 1.0, but only one that cleared the bar for the “hard” category. Nadal’s mix is more balanced.

Go yell at someone else

Hopefully these numbers have given you some new ammunition for your next twitter fight. Some of you will froth at the mouth while insisting that players can’t control who they play. You’re right, but it doesn’t really matter. We can’t start giving out GOAT points for things that players didn’t do, like beat Thiem in the 2020 French Open semi-finals. All three of these guys were or are good enough at various points to have beaten some of the opponents they didn’t have to face. There are other approaches we could take to the GOAT debate that incorporate peak Elo ratings and longevity at various levels, but that’s not what we’re talking about when we count slams.

If we are going to focus so much on the slam count, we might as well acknowledge that Nadal’s 20 is better than Federer’s 20, and Djokovic’s 17 is awfully close to both of them.

US Open Asterisk Talk is Premature. It Might be Flat-Out Wrong.

Many high-profile players will be missing from the 2020 US Open. Rafael Nadal opted out of the abbreviated North American swing, and Roger Federer will miss the rest of the season due to injury. More than half of the WTA top ten is skipping Flushing Meadows as well. The thinned-out fields increase the odds that a few remaining favorites, such as Novak Djokovic and Serena Williams, add another major trophy to their collection.

As a result, pundits and fans are discussing whether the 2020 US Open deserves an “asterisk.” The idea is that, because of the depleted fields, this slam is worth less than others, so much so that the history books* should note the relative meaninglessness of this year’s titles.

* Nobody buys history books anymore, so we’re really talking** about a page on the US Open website, and a never-ending edit war on Wikipedia.

** Yes, I see the irony.

From what I’ve seen, people are thinking about this the wrong way. Yes, a weak field makes it easier–in theory–to win the tournament. It’s certainly true that the 2020 champions won’t have to go through Nadal or Ashleigh Barty to get their hardware. But the field isn’t what matters.

The field isn’t what matters

I repeated that on purpose, because it’s that important. The winner of a grand slam must get through seven matches. The difficulty of securing the title depends almost entirely on his or her opponents in those seven matches. Each main draw consists of 128 players, but 120 of them are mostly irrelevant.

I say “mostly” because I can foresee some objections. Sometimes a player can compete so hard in a loss that they weaken their opponent for the next round. Take the 2009 Madrid Masters, in which Nadal needed four hours to defeat Djokovic in the semi-final, then lost to Federer in the final. We could say that Djokovic’s presence was relevant, even though Federer won the title without playing him. That sort of thing happens, though probably not as much as you think. Even when it does, it needn’t be a top tier player who wears out their opponent in an early round.

Another objection is that a depleted field affects seedings. For instance, Serena’s current WTA ranking is 9th, an unenviable position going into most slams. The 9th seed lines up for a fourth-round match with a top-eight player, meaning that she could face four top-eight players en route to the title. But with all the absences, Williams will instead be seeded third, behind only Karolina Pliskova and Sofia Kenin.

I’m not dismissing these concerns out of hand. They do matter a bit. But they only matter insofar as they affect the way the tournament plays out. The difference between the difficulties facing the 3rd and 9th seeds could be enormous … or it could be nothing, especially if the draw is riddled with early upsets.

Difficulty is a continuum

Even if you grant some credence to the objections above (or others that I haven’t mentioned), I hope you’ll agree that the most meaningful obstacles standing between a player and a grand slam title are the seven opponents he or she will need to overcome.

If those seven opponents are, on average, very strong, we would say that the player faced a particularly tough path to a slam title. Take Stan Wawrinka’s 2014 Australian Open title: he beat both Djokovic and Nadal at a time when those two were dominating the game. If the collective skill level of the seven opponents doesn’t amount to much–at least by grand slam standards–we’d say it was an easy path. For example, Federer clinched the 2006 Australian Open despite facing only a single player ranked in the top 20, and none in the top four.

We can quantify path difficulty in a variety of ways. One approach that will be useful here is to calculate the odds that an average slam champion would beat those seven opponents. The difference between easy and hard championships is enormous. The typical major titlist (that is, someone with an Elo rating around 2100) would have had a 3.3% chance of beating the seven men that Wawrinka drew in Melbourne the year that he won. Only two slam paths have ever been tougher: Mats Wilander’s routes to the 1982 and 1985 French Open titles. By contrast, the average slam champion would have had a 51% chance of going 7-0 when faced by Federer’s 2006 Australian Open draw.

The extreme “easy” draw is fifteen times easier than the extreme “hard” draw. Fifteen times! You can find plenty of champions for any approximate level of difficulty in between those extremes. The typical slam champ would’ve had a 10% chance of doing what Djokovic did in progressing through seven rounds at the 2011 US Open. Same in New York in 2012. Andy Murray’s 2016 Wimbledon path would have given the average champion a 20% chance. The 2018 Roland Garros draw was manageable for Rafael Nadal, and a typical major titlist would have had a 30% chance of securing those seven match wins.

None of this is to say that any of those players did or didn’t “deserve” their titles. Federer didn’t choose his 2006 Melbourne opponents any more than Wawrinka selected his foes eight years later. The trophy is the same, and in many important ways, their achievements are the same–both of the Swiss stars swept away all of their opponents, who in turn were the best performers (at least during those fortnights) of the players who showed up.

Asterisks for everybody

Here’s another thing 2006 Roger and 2014 Stan had in common: Almost all of the best players in the world participated in the tournaments that they ultimately won. (I say “almost” because defending champion Marat Safin was injured and missed the 2006 Aussie Open.) The “field” was effectively the same, but to win the titles, one player cruised through a two-week cakewalk and the other needed to put together one of the most impressive final weeks of the modern era.

Tennis fans have collectively decided that each major title counts as “one.” It doesn’t have to be that way: We could give more “slam points” for achievements like Wawrinka’s and grant fewer for the easy ones. Most people don’t like this idea, and I admit that it sounds a bit weird. I’m not advocating it for general use, though it is an interesting concept that I’ve pursued in a number of earlier articles, showing that Djokovic’s majors are–on average–more impressive than Nadal’s, which in turn have been tougher than Federer’s. Weighting majors by difficulty results in some changes in the order of the all-time grand slam list, ensuring that fans of all players hate me because I wrote some code and played with some spreadsheets.*

* With, I admit, malice aforethought.

Adjusting slam counts for difficulty is, in a sense, asterisking every slam title. The tricky draws get an acknowledgement of their difficult, and the ones that opened up get tweaked to account for their ease. It’s a continuum, not a simple up-and-down decision between normal slams and abnormal slams.

The 2020 US Open champions will probably have title paths that sit in the easier half of that continuum. But even that modest claim is far from guaranteed.

Let’s say Venus Williams recaptures her vintage form and wins the title, beating 3rd seed Serena in the quarter-finals, 2nd seed Kenin in the semis, and top seed Pliskova in the title match. (It doesn’t matter if the surprise winner is Venus–it could be any lower-ranked player, though Venus seems more plausible than most.) An average slam champion would beat those three players in succession about 37% of the time. 37% is already lower odds than about 20% of women’s slam draws in the last 45 years. (Kenin’s Australian Open title rated 39%.)

37% for Venus’s hypothetical title isn’t even the whole story–four more rounds of journeywomen would knock the number down to around 26%–harder than one-third of women’s slam draws. Add in another tricky opponent or two–maybe Cori Gauff, or Petra Kvitova in the fourth round–and suddenly the path to the 2020 US Open women’s championship is just as hard as the typical slam.

It’s even easier to illustrate how the 2020 US Open men’s title could be as difficult as many other slams. By the numbers, simply upsetting Djokovic (simply! ha!) is more difficult than it was to defeat all seven of Federer’s opponents at the 2006 Australian Open. That’s right: Six withdrawals and one win over Novak wouldn’t be the easiest slam victory in the last 15 years. Tack on six actual wins, including a few against strong opponents, and the result is a seven-match path that stands up against the typical non-pandemic slam.

Ironically, the player who could win the title with the weakest possible draw is Djokovic. It would be odd to claim that any of Novak’s accomplishments should be asterisked, but it does make things much simpler when he doesn’t have to beat himself.

Masked competitiveness

Once again, the field doesn’t really matter. When we focus on the players who are in New York instead of the few dozen who aren’t, we see that the ingredients are in place for a couple of respectable path to US Open titles. Wilander’s and Wawrinka’s marks are probably safe, but it’s more than possible that the winners will have faced competition equivalent to that of the average slam champ.

At the very least, we don’t know any better until the tail end of the second week. Until then, asterisk talk is premature. After that, it will probably be moot.

Grand Slam Prize Money Whack-a-Mole

Eagle-eyed Twitterer @juki_tennis noticed the following tweaks to the rules for the 2020 grand slams:

Let’s start with the first underlined section. I’ll get to the doubles tweak in a bit.

The ITF is learning that incentives are tricky. In the olden days, back when Adrian Mannarino still had hair, prize money was simple. If you played, you got some. If you didn’t, you got none. Players who get hurt right before one of the four biggest events of the season suffered in silence.

Except it’s never been quite that simple. The slams have spent the last decade taking turns breaking prize-money records, raising in particular the take for first-round losers. A spot in the main draw of the Australian Open is now worth $63,000 USD ($90,000 AUD). Some players in the qualifying draw barely make that much in an entire season. Whatever one’s hangups about honesty or fair play, if you have a chance to grab that check, you take it.

The same logic applies whether you’re healthy or injured. The last decade or so of grand slam tennis has been littered with first-round losers who weren’t really fit to compete. That’s bad for the tournaments, bad for the fans, and probably not that great for the players themselves, even if $63k does buy a lot of physiotherapy.

Paid withdrawals

Two years ago, the ITF took aim at the problem. Players with a place in the main draw could choose to withdraw and still collect 50% of first-round loser prize money. The ATP does something similar, giving on-site withdrawals full first-round loser prize money for up to two consecutive tournaments. The ATP’s initiative has been particularly successful, cutting first-round retirements at tour-level events from a 2015 high of 48 to only 20 in 2019. In percentage terms, that’s a decline from 4.4% of first-round matches to only 1.6%.

The results at slams are cloudier. On the men’s side, there were nine first-round retirements in 2010, and nine in 2019. The ITF’s incentives might not be sufficient: 50% of first-round prize money is still a substantial sum to forego. In fairness to the slams, retirements may not tell the whole story. A hobbled player can still complete a match, and perhaps the prize money adjustment has convinced a few more competitors to give up their places in the main draw.

None of this, however, keeps out players who consciously game the system. Both the ATP and WTA allow injured players to use their pre-injury rankings to enter a limited number of events upon their return. Savvy pros maximize those entries (“protected” in ATP parlance, and “special” in WTA lingo) by using them where the prize pots are richest and, if possible, bridging the gap with wild cards into smaller events.

Emblematic of such tactics is Dmitry Tursunov, who played (and lost) his last six matches at majors, all using protected rankings. Two of those, including his final grand slam match at the 2017 US Open against Cameron Norrie, ended in retirement. Three of the others were straight-set losses. In one sense, Tursunov “earned” those paydays. He was ranked 31st going into Wimbledon in 2014, then missed most of the following 18 months. Upon return, he followed ATP tour rules. But with the increasingly disproportionate rewards available at slams, protected rankings seem sporting only when used as part of a concerted comeback effort.

While the ITF’s late-withdrawal policy wasn’t in place for Tursunov, it’s easy to imagine a player in a similar situation taking advantage. And that’s the gap that the latest tweak aims to plug. The new rule is not limited to players on protected or special rankings, which typically require absences of six months, not just one. Yet the idea is similar. You can no longer enter, turn up on site, plead injury, and take home tens of thousands of dollars … unless you’ve competed recently. It’s a low bar, but it raises the standard a bit for players who want to take home a $30,000 check.

One of two prongs

The rule adjustment wouldn’t have affected Tursunov’s lucrative protected-ranking tour of 2016-17. However, had the Russian come back from injury a couple of years later, his income might not have gone uncontested.

In 2019, both Roland Garros and Wimbledon invoked another rarely-used clause in the rulebook. It requires that players “perform to a professional standard,” and a failure to do so can result in fines up to the amount of first-round prize money. Anna Tatishvili–using a special ranking–was docked her full paycheck at the French Open, and Bernard Tomic–a convenient whipping boy whenever this sort of thing comes up–lost his take-home from the All England Club. Both fines were appealed, and Tatishvili’s was overturned. (Tomic’s should have been, too.)

What matters for the purposes of today’s discussion isn’t the size of Tatishvili’s bank account, but the fact that the majors have dug the “professional standard” clause out of cold storage. It’s worth quoting the various factors that the rulebook spells out as possibly contributing to a violation of the standard:

  • the player did not complete the match
  • the player did not compete in the 2-3 week period preceding the Grand Slam
  • the player retired from the last tournament he/she played before the Grand Slam
  • the player was using a Protected or Special Ranking for entry
  • the player received a Code Violation for failure to use Best Efforts

Every major has a few players who are skirting the line, perhaps returning to action a bit sooner than they would have if the grand slam schedule were different. With the fines in 2019, the ITF has made clear that they expect to see credible performances from all 256 main draw players. And with the prize money adjustment for 2020, the governing body has closed the door on five-figure paydays for players who shouldn’t have been on the entry list, even if they never take the court.

I promised to talk about doubles

The second section of the rulebook quoted above is a bit problematic, because I believe it is missing a key “not” in the opening sentence. Unless the ITF has some bizarre and unprecedented goals, the intention of the doubles regulations is to discourage singles players from retiring in doubles unless they are truly injured, and to prevent singles players from even entering doubles unless they plan to take it seriously.

Doubles prize money pales next to the singles pot, but even first-round losers in men’s and women’s doubles will take home $17,500 USD per team, or $8,750 per player. That’s enough to convince most singles players to enter if their ranking makes the cut, no matter how little they care about doubles during the 44 non-slam weeks of the year.

The majors determine which teams make the doubles cut the same way that ATP and WTA tour events do. Teams are ordered by their combined singles or doubles ranking. Each player can use whichever is better. The tours allow pros to use their singles rankings to encourage superstars to play doubles, and at events like Indian Wells, many big names do take part. At the slams, the bigger effect is on the next rung of singles players, giving us oddball doubles teams such as Mackenzie McDonald/Yoshihito Nishioka and Lukas Lacko/John Millman at the 2018 US Open.

As with other details of the entry process, most fans couldn’t care less. But they should. Whenever the rules let one team in, they leave another team out. By including more singles players in the doubles draw, the standard for full-time doubles players is made almost impossibly strict. An up-and-coming men’s singles player can crack the top 100–and gain admission to grand slam main draws–with a solid season on the challenger tour, but even the best challenger-level doubles teams are often left scrambling for partners whose singles rankings are sufficient to gain entry.

This year’s rulebook edit should help matters, at least a bit. (As long as someone inserts the missing “not,” anyway.) Grand slam doubles is not an exhibition, and it shouldn’t be contested by players who treat it that way. The ATP and WTA should follow suit, penalizing players who withdraw from doubles only to prove their health by continuing to play singles.

Incentives and intentions

These rule changes, while technical, are aimed at something rather simple: to ensure that the players who enter slam main draws–both singles are doubles–are healthy and motivated to play. The latest tweaks won’t close every loophole, and we can expect more disputes over issues like the Tatishvili and Tomic fines.

The bigger issue, complicated by the on-site withdrawal adjustment, is the underlying purpose of the rise in first-round loser prize money. The slams represent a huge proportion of the season-long prize pool, especially for players between approximately 50th and 110th in the ATP and WTA rankings. These competitors miss the cut for many of the most prestigious Masters and Premier tournaments. Even in later rounds, they are usually playing for four-figure stakes–if that. Four times a year, pros with double-digit rankings get a guaranteed cash infusion, and the potential for much more.

The presence of the four majors effectively funds the rest of the season for many players. The slams have upped first-round prize money–both nominally and relative to increases in later-round awards–partly in recognition of that fact. It is expensive to be a touring pro, and without paydays from the majors, it can easily be a money-losing endeavor.

Salary, not prize money

The majors rely on the less-lucrative tours for year-round publicity and a pool of highly-skilled players to drive fans and media attention to their mega-events. Much of the first-round loser prize money is in recognition of that fact. No one really thinks that the 87th-best player in the world deserves $63k just for showing up and giving Serena Williams a mild 59-minute workout. But does the 87th-best player in the world deserve to collect annual revenue of $250k–a figure that will largely go to cover travel, training, coaching, and equipment expenses? I think so, it appears that the slams think so, and I suspect you do, too.

So, when the ITF closes loopholes like these, keep in mind that they are operating within the silly $63k-per-hour framework, not the more reasonable $250k-per-season model. It is an important goal to ensure the integrity and quality of play at slams, but it ought to be paired with an effort to support tennis’s rank-and-file, even when those journeymen are injured.

A more sensible policy would be to separate much of the first-round loser prize pool from the literal act of playing a first round match. Perhaps the slams could each contribute $7.5 million each year–that’s $30k per singles player–to a general fund that would disburse annual grants to players ranked outside the top fifty, and lower every singles award by the same amount. (The details would be devilish, starting with these few parameters.) Such an approach would come out in the wash for most players, who would simply receive the extra $30k per slam in a different guise. But it would help injured players return to top form, and it would leave plenty of money for high-stakes combat at the sport’s biggest stages. Such a solution, of course, would require a lot more than a few minor edits to the rulebook.

Monkeying Around With Rafael Nadal’s 19 Grand Slams

The gap is closing. With his marathon victory last night in the US Open final over Daniil Medvedev, Rafael Nadal is up to 19 career major titles, second only to Roger Federer, who holds 20. Lurking in third place is Novak Djokovic, with 16, who was favored at Flushing Meadows this year, but retired due to injury in the fourth round.

Just two weeks ago, Djokovic seemed to be the biggest threat to Federer’s place atop the leaderboard. Now, with Nadal only one back and Djokovic dealing with another round of physical problems, Rafa has the momentum. Federer, now 38 years old, appears increasingly unlikely to pad his own total.

In an attempt to foresee the future of the grand slam leaderboard, I built a straightforward algorithm last month to predict future major titles. In the spirit of baseball’s “Marcel” projection system, it aims to be so simple that a monkey could do it. It uses the bare minimum of inputs: final-four performance at the last two years’ worth of slams, and age. It trades some optimization in favor of simplicity and ease of understanding. The result is pretty darn good. You can review the algorithm itself and look at how it would have performed in the past in my earlier article here.

Solve for RN = 19 + x

Before the US Open, the algorithm seemed tailor-made to aggravate as many fanbases as possible. It predicted that, over the next five years, Djokovic would win four more majors, Nadal two more, and Federer none, leaving the big three in a tie.

One more slam in the books, and the numbers have changed. Here is the revised forecast, reflecting both Nadal’s 19th slam and his rosier outlook after adding another title to his list of recent results:

Player          Slams  Forecast  Total  
Rafael Nadal       19       3.5   22.5  
Roger Federer      20       0.3   20.3  
Novak Djokovic     16       3.5   19.5

Rafa is in line to improve his total by at least three slams. By the time he’s done, perhaps he will have left Djokovic and Federer in the dust, and we’ll be speculating about whether he’ll catch Serena Williams … or Margaret Court.

More forecasts

My basic algorithm allows us to generate future slam forecasts for any player with at least one major semi-final in the last two years. Keep in mind that I’m not forecasting career slam totals–I’m looking ahead to only the next five years. For the big three, I’m assuming we don’t need to worry about 2025 and beyond.

We have current projections for 18 players:

Player                 Forecast  
Novak Djokovic              3.5  
Rafael Nadal                3.5  
Daniil Medvedev             0.8  
Dominic Thiem               0.7  
Stefanos Tsitsipas          0.6  
Matteo Berrettini           0.5  
Hyeon Chung                 0.4  
Lucas Pouille               0.3  
Kyle Edmund                 0.3  
Roger Federer               0.3  
Grigor Dimitrov             0.1  
Marco Cecchinato            0.1  
Marin Cilic                 0.0  
Juan Martin del Potro       0.0  
Roberto Bautista Agut       0.0  
Kevin Anderson              0.0  
Kei Nishikori               0.0  
John Isner                  0.0

Most of these guys have only a single recent semi-final to their name, and the only thing to separate them is their age. It seems logical to be more optimistic about the future slam performance of Stefanos Tsitsipas (age 21) than that of Roberto Bautista Agut (age 31), even though the algorithm sees their results so far–one semi-final appearance in the last 12 months–as identical.

Five years means 20 slams, and you might notice that the above table doesn’t get close to accounting for all of them. The projections add up to 10.8 majors, leaving plenty of room for players who haven’t even qualified for the list–Alexander Zverev and Felix Auger-Aliassime come to mind. At the 2024 US Open, we’re sure to look back at our late-2019 prognostications and laugh.

Federer will keep his spot at the top of the game’s most important leaderboard for at least four more months. Djokovic will probably be the top pick in Melbourne, so Roger could well enjoy nine more months as the only 20-slam man. But you won’t need an algorithm–even a simple one–to identify the favorite at Roland Garros next year. Organized men’s tennis lasted over a century without a 20-time major champion. In less than a year, we could have two.

Are You There, Margaret? It’s Me, Serena

As I write this, Serena Williams is two matches away from winning her 24th grand slam. She’s been stuck on 23 since early 2017, which must be frustrating, since the all-time record is 24. Serena already holds the open-era record (for titles since 1968), one ahead of Steffi Graf’s 22. But Margaret Court is the leader across all eras, with 24 major championships between 1960 and 1973.

Williams is, of course, one of the greatest players of all time. Maybe the greatest. Court is also in the conversation, along with other luminaries such as Graf, Martina Navratilova, and Chris Evert. Cross-era comparisons in tennis are extremely difficult, because nearly everything about the game has changed. Serena’s technique, training, equipment, and tour schedule–not to mention wealth and celebrity status!–would all be extremely foreign to a 1960s or 70s superstar such as Court.

The challenge of cross-era comparisons hasn’t stopped fans from expressing opinions about where Williams should stand on the all-time leaderboard. Regardless of whose trophy cabinet numbers 23 or 24, Serena supporters tend to rely on three main arguments:

  1. The level of competition is way higher now than it was back then.
  2. Court won the Australian Open 11 times, back when it was the weakest of the four majors.
  3. Court is an obnoxious blowhard whose opinions are unacceptable.

Number one is probably true, but if we’re going to attempt cross-era comparisons, I think the only valid way to do so is to treat all eras as equal. We’ll never know how Williams would have fared with a wooden racket, or how Court’s body would’ve responded to today’s more physical game. You can make a logical case that today’s players are simply better than those of a couple generations ago, who were better than the ones before them, and so on. But the very idea of a “greatest of all time” implies something different than the “greatest of all time measured by today’s standards,” so we’re going to treat all eras as equal.

Number three is also popular, but my database isn’t able to shine much light on that line of argument.

That leaves number two, the relative weakness of the Australian Open.

Aussie ease

Court won the Australian Open 11 times, more than any other woman has claimed a single major title. In itself, that’s not a negative. Nobody counts it against Rafael Nadal that he’s won the French Open 12 times. But in the amateur era–and for some years after tennis went fully professional–the Australian Open wasn’t a mandatory stop for the best players in the world. It was a long trip, and it hadn’t yet gained the prestige that it holds today.

Thus, it’s fair to conclude that Court’s 1963 Wimbledon title was a more noteworthy accomplishment than her trophy 1963 Australian Championships. Most of us would agree that we should discount those Australian Opens. But by how much?

Difficulty-adjusted slam titles

In the past, I’ve compared men’s greatest-of-all-time candidates by major titles, adjusted for the level of competition. In the modern game, the field is almost exactly the same from one major to the next, but the draw can make one tournament considerably more difficult to win than another. The same technique allows us to compare draw difficulty and field quality for tournaments from the 1970s when both varied. For instance, the difficulty of Court’s path to the 1973 US Open title rated as average, in line with many of Williams’s title paths. But her 1973 Australian crown was only two-thirds as difficult–one of the easiest paths to a major title in the open era.

It’s no accident that I’m using Court’s last few major titles as examples. By analyzing performances from the 1970s, we’re pushing up against the edge of the weakness of historical tennis data. It’s well-nigh impossible to estimate the exact difficulty level of most of Court’s titles, because so little data is available from the amateur era. Instead, we’ll need to approximate using the limited information we have.

My difficulty adjustments rely on Elo ratings, which I have calculated as far back as 1972. (We have fairly complete results back to 1970 or so, but it takes a bit of time to amass a decent sample of match results for each player and for ratings to stabilize.) Let’s look at the relative difficulty of the four grand slams in the first five possible years, 1972-76:

Major            Difficulty  
Australian Open        0.60  
French Open            0.54  
Wimbledon              0.99  
US Open                0.85

The average major title, 1972-present, rates 1.0, with more difficult paths earning higher numbers. The fields weren’t as deep in the 1970s as they are now, so the typical path to a slam title then was lower than 1.0. In this first five-year period, we see that Wimbledon was in line with the historical average, the US Open was a bit easier, and the other two quarters of the grand slam were considerably less challenging. If we follow my suggestion above, to treat all eras as equal–except for the weakness of the Australian draws–we need to normalize these difficulties so that the other three slams average 1.0:

Major            Difficulty  
Australian Open        0.76  
French Open            0.68
Wimbledon              1.25  
US Open                1.07

Extrapolating backwards

We don’t know much about the field quality of the Australian majors in Court’s prime. For lack of a better option, then, we’ll use the 1972-76 average, since that’s as close as we can get. These probably overstate the quality of the Australian draws relative to the other slams, but if we’re inching toward calling Serena the all-time leader at Court’s expense, we should make conservative assumptions, to give us more confidence in our end result.

Here’s what happens to Court’s career totals if we apply the normalized adjustments:

Major            Difficulty  Slams  Adj Slams  
Australian Open        0.76     11        8.3  
French Open            0.68      5        3.4  
Wimbledon              1.25      3        3.7  
US Open                1.07      5        5.4  
Total                           24       20.8

The same process–adjusting each slam for difficulty, and normalizing for era–makes milder tweaks to Williams’s and Graf’s totals. Serena ends up with 23.3, and Graf with 21.9. Neither is enough to give us reason to change how we view those players’ accomplishments. And both are better than Court’s modified tally.

The small herd of GOATs

Remember that this is not an era adjustment. To the contrary, this calculation is based on the simplifying assumption that all eras are equal, except for the fact that for many years, some of the best players didn’t travel to Australia, making that major easier to win than the others.

These numbers also–obviously!–don’t tell us that Court wasn’t one of the best ever. Even if she had skipped her home slam, she still would’ve retired with 13 majors, plus a pile of doubles grand slam trophies and a long list of other career accomplishments. If Australia were less geographically remote, she probably wouldn’t have won those eleven titles–but she may well have won eight.

For all of Court’s accomplishments, she loses her top spot on the sport’s most hallowed list once we account for the weakness of the early Australian Open draws. At the very least, she falls behind Williams and Graf. Remember that my adjustments are conservative ones, so if we collect more data and discover that we should more aggressively discount her 1960’s Australian titles, her resulting total might leave her closer to 18, tied with Evert and Navratilova.

Serena may never equal or beat Court’s 24 titles. But even if she retires with 23, the modern level of competition–which showed up at every major, every year–means that she already deserves her place atop the leaderboard.

GOAT Races: Forecasting Future Slams With a Monkey

After Novak Djokovic won his 16th career major at Wimbledon this year, more attention than ever focused on the all-time grand slam race. Roger Federer has 20, Rafael Nadal has 18, and Djokovic is–by far–the best player in the world on the surface of the next two slams. This is anybody’s ballgame.

Forecasting tennis is hard, and that’s just if you’re trying to pick the results of tomorrow’s matches. Players improve and regress seemingly at random, making it difficult to predict what the ranking table will look like only a few months from now. Fans love to speculate about which of the big three will, in the end, win the most slams, but there are an awful lot of unknowns to contend with.

One can imagine some way to construct a crystal ball to get these numbers in a rigorous way. Consider each player’s age, his likely career length, his chances of injury, his recent performance at each of the four slams, his current ranking, the quality of the field on each surface, and probably more, and maybe you could come up with some plausible numbers. Or… what if we skip most of that, and build the simplest model possible?

Enter the monkey

Baseball statheads are familiar with the Marcel projection system, named after a fictional monkey because it “uses as little intelligence as possible.” Just three years of results and an age adjustment. It isn’t perfect, and there are plenty of “obvious” improvements that it leaves on the table. But as in tennis, baseball stats are noisy. For most purposes, a “basic” forecasting system is as good as a complicated one, and over the years, Marcel has outperformed a lot of models that are considerably more complex.

Let’s apply primate logic to slam predictions. First, I’m going to slightly re-cast the question to something a bit more straightforward. Instead of forecasting “career” slam results, we’re going to focus on major titles over the next five years. (That should cover the big three, anyway.) And in keeping with Marcel, we’ll use just a few inputs: slam semi-finals, finals, and titles for the last three years, plus age. Actually, we’re going to lop off a bit of the monkey’s brain right away, because slam results from three years ago aren’t that predictive. So our list of inputs is even shorter: two years of slam semi-finals, finals, and titles, plus age.

The resulting model is pretty good! For players who have reached a major semi-final in any of the last eight slams, it predicts 40% of the variation in next-five-years slam titles. Without building the hyper-complex, optimal model, we don’t know exactly how good that is, but for a forecast that extends so far into the future, capturing almost half of the player-to-player variation in slam results sounds good to me. Think of all the things we don’t know about the slams in 2022, let alone 2024: who is still playing, who gets hurt, who has improved enough to contend, which prospects have come out of nowhere, and so on. Point being, the best model is going to miss a lot, so we shouldn’t set our standards too high.

Follow the monkey

The two-years-plus-age algorithm is so simple that you can literally do it on the back of an envelope. For any player, count his semi-final appearances (won or lost), final appearances (won or lost), and titles at the last four slams, then do the same for the previous four. Then note his age at the start of the next major. Start with zero points, then follow along:

  • add 15 points for each semi-final appearance in the last four slams
  • add 30 points for each final appearance in the last four slams
  • add 90 points for each title in the last four slams
  • add 6 points for each semi-final appearance in the previous four slams
  • add 12 points for each final appearance in the previous four slams
  • add 36 points for each title in the previous four slams
  • if the player is older than 27, subtract 8 points for each year he is older than 27
  • if the player is younger than 27, add 8 points for each year he is younger than 27
  • divide the sum by 100

That’s it! Let’s try Djokovic. In the last four majors, he’s won three titles and made one more semi-final. In the four before that, he won one title. He’ll enter the US Open at 32 years of age. Here goes:

  • +60 (15 points for each of his four semi-finals in the last four slams)
  • +90 (30 points for each of his three finals in the last four slams)
  • +270 (90 points for each of his three titles in the last four slams)
  • +6 (6 points for his 2017 Wimbledon semi-final)
  • +12 (12 points for his 2017 Wimbledon final)
  • +36 (36 points for his 2017 Wimbledon title)
  • -40 (Novak is 32, so we subtract 8 points for each of the 5 years he is older than 27)

Add it all up, and you get 434. Divide by 100, and we’re predicting 4.34 more slams for Novak.

Next-level GOAT trolling

I promise, I went about this project solely as a disinterested analyst. I just wanted to know how accurate a bare-bones long-term slam forecast could be. My goal was not to make you tear your hair out. But hey, you were probably going to lose your hair anyway.

Here is the number of slams that the model predicts for the big three between the 2019 US Open and 2024 Wimbledon:

  • Djokovic: 4.34
  • Nadal: 2.22
  • Federer: 0.26

You probably don’t need me to do the math for the next step, but you know I can’t not do it. Projected career totals:

  • Djokovic: 20.34
  • Federer: 20.26
  • Nadal: 20.22

Or, since we live in a world where you can’t win fractional majors:

  • Djokovic: 20
  • Federer: 20
  • Nadal: 20

Ha.

Back to the model

Djokovic’s forecast of 4.34 is quite high, in keeping with a player who has won three of the last four majors. For each year since 1971, I calculated a slam prediction for every player who had made a major semi-final in the previous two years–a total of more than 800 forecasts. Only 14 of those forecasts were higher than 4.34, and several of those belonged to the big three. Here are the top ten:

Year  Player         Age   Predicted  Actual     
2008  Roger Federer   26        6.38       5     
2007  Roger Federer   25        5.86       7     
2016  Novak Djokovic  28        5.20       6  *  
2005  Roger Federer   23        4.91      11     
2011  Rafael Nadal    24        4.89       5     
2006  Roger Federer   24        4.86      10     
2017  Novak Djokovic  29        4.79       4  *  
2012  Novak Djokovic  24        4.68       8     
1989  Mats Wilander   24        4.65       0     
1988  Ivan Lendl      27        4.56       2

* actual slam counts that could still increase

All of these predictions are based on data available at the beginning of the named year. So the top row, 2008 Federer, is the forecast for Federer’s 2008-12 title count, based on his 2006-07 performance and his age entering the 2008 Australian. Had the model existed back then, it would have guessed he’d win a half-dozen slams in that time period. He came close, winning five.

There will be plenty of noise at the extreme ends of any model like this. At the beginning of 2005, the algorithm pegged Federer to win “only” five of the next twenty majors. Instead, he won 11. I can’t imagine any data-based system would have been so optimistic as to guess double digits. On the flip side, the 1989 edition of the monkey would’ve been nearly as hopeful for Mats Wilander, who was coming off a three-slam campaign. Sadly for the Swede, a gang of youngsters overtook him and he never made another major final.

Let’s also take a look at the next 10 rosiest forecasts, plus the current guesstimate for Djokovic:

Year  Player          Age  Predicted  Actual     
2010  Roger Federer    28       4.48       2     
1981  Bjorn Borg       24       4.47       1     
1996  Pete Sampras     24       4.47       6     
1975  Jimmy Connors    22       4.45       2     
Curr  Novak Djokovic   32       4.34       0  *  
1980  Bjorn Borg       23       4.28       3     
2013  Novak Djokovic   25       4.24       7     
2009  Roger Federer    27       4.20       4     
1995  Pete Sampras     23       4.16       7     
2009  Rafael Nadal     22       4.12       8     
1979  Bjorn Borg       22       4.09       5

Plenty more noise here, with outcomes between 0 and 8 slams. Still, the average result of the 10 other predictions on this list is 4.5 slams, right in line with our forecast for Novak.

Missing slams…

The model expects that the big three will win around seven of the next twenty slams. You might reasonably wonder: What about the other thirteen?

The monkey only considers players with a slam semi-final in the last eight majors, so the forecasts shouldn’t add up to 20. There’s a chance that the champions in 2023 and 2024 aren’t yet on our radar, and many young names of interest to pundits these days, like Alexander Zverev, Felix Auger Aliassime, and Daniil Medvedev, haven’t yet reached the final four of a major. Here are the players for whom we can make predictions:

Player                 Predicted Slams  
Novak Djokovic                    4.34  
Rafael Nadal                      2.22  
Dominic Thiem                     0.71  
Stefanos Tsitsipas                0.63  
Hyeon Chung                       0.38  
Lucas Pouille                     0.31  
Kyle Edmund                       0.30  
Roger Federer                     0.26  
Juan Martin del Potro             0.19  
Marco Cecchinato                  0.06  
----------------                  ----  
TOTAL                             9.40

(The five other players with semi-final appearances since the 2017 US Open are forecast to win zero slams.)

Yeah, I know, Lucas Pouille and Hyeon Chung aren’t really better bets to win a slam than Federer is. But they are (relatively) young, and the model recognizes that many players who reach slam semi-finals early in their careers are able to build on that success.

More to the point, we’re leaving a lot of majors on the table. If the overall forecast is correct, that list of players will win fewer than half of the next 20 slams, leaving at least ten championships to players who have yet to win a major quarter-final.

…and age

Remember, I retro-forecasted every five-year period back to 1971-75. Over the 44 five-year spans starting each season between 1971 and 2014, the model typically predicted that the players it knew about–the ones who had reached slam semi-finals in the last two years–would win 13 of the next 20 slams. In fact, those on-the-radar players combined to win an average of 12 majors in the ensuing five-year spans.

Only in the last few years has the total number of predicted slams fallen below 10. The culprit is age: Recall that every forecast has an age adjustment, and we subtract 8 points (0.08 slams) for each year a player is older than 27. That’s a 0.4-slam penalty for both Djokovic and Nadal, and it’s 0.8 slams erased from Federer’s future tally. Thus, the model predicts that the big three are fading, and there aren’t many youngsters (like Pouille and Chung) on the list to compensate.

How you interpret these big three forecasts in light of the “missing” slams depends on a couple of factors:

  • Has the aging curve for superstars has changed? Is 30 the new 25; 32 the new 27?
  • Will the next few generations of players soon be good enough to topple the big three?

There’s plenty of evidence that the aging curve has changed, that we should expect more from 30-somethings these days than we did in the 1980s and 1990s. That would close much of the gap. Let’s say we set the new peak age at 31, four years later than the men’s Open Era average of 27. That would add 0.32 slams to every player’s forecast, possibly adding one more slam to each of the big three’s forecasted total. Overall, it would add a bit more than an additional three slams to the total of the the previous table, putting that number close to the historical average of 13.

Shifting the age adjustment doesn’t disentangle the big three, though, because it affects them all equally. It just means a three-way tie at 21 is a bit more likely than a three-way tie at 20.

The second question is the more important–and less predictable–one. It’s hard enough to know how well a single player will be competing in three, four, or five years. (Or, sometimes, tomorrow.) But even if we could puzzle out that problem, we’d be left with the still more difficult task of predicting the level of competition. Entering the 2003 season, the monkey would have opined that the then-current crop of stars–men who made slam semis in 2001 and 2002–would account for a combined 13 majors between 2003 and 2007. That included 2.5 for Lleyton Hewitt, plus one apiece for Thomas Johansson, Albert Costa, Pete Sampras, Marat Safin, David Nalbandian, and Juan Carlos Ferrero. Those seven men won only two. The entire group of 20 players who merited forecasts entering the 2003 Australian Open won only three.

We’ll probably never establish exactly how strong that group was in comparison with other eras. What we know for sure is that none of those men were as good as Federer in 2003-05, and by the end of the five-year span, they’d been shunted aside by Nadal as well. (Only Nalbandian ranked in the 2007 year-end top ten.) The generation of Zverev/Tsitsipas/Auger-Aliassime/etc won’t be as good as peak Big Four, but the course of the next 20 slams will depend a lot more on those players that it will on the (relatively) more predictable career trajectories of Djokovic, Federer, and Nadal.

So we’re left with a stack of known unknowns and error bars wider than a shanked Federer backhand. But based on what we do know, the top of the all-time slam leaderboard is going to get even more crowded. At least, that’s what the monkey says.