Data – Heavy Topspin

100 Years of Women’s Tennis History

Exactly one year ago, I updated Tennis Abstract with some missing 1970s and 1980s WTA tournaments. I tweeted this progress report:

https://twitter.com/tennisabstract/status/1332072224858255363

I didn’t know it then, but it was the beginning of an all-engrossing project to massively increase the amount of historical women’s tennis data available–not just on TA, but in any organized, easily-accessible form.

In the last year, TA has gained nearly a quarter of a million women’s singles match results going back a full century, to 1921. We all now have the ability to browse through the results of players from the 1920s the same way that we do players of the 2020s. It’s incredibly cool, and it constitutes a huge step toward a better understanding of tennis history.

The state of play

Until last November, Tennis Abstract’s database of women’s results was built on a combination of what I was able to find from the WTA and ITF websites. For contemporary players and their predecessors from the last few decades, that was enough. But as my tweet indicates, it didn’t even encompass the 80 matches of the greatest rivalry in tennis history. The WTA site still doesn’t display records of many top-tier events from the 1970s.

With Evert-Navratilova squared away*, I went to work on the remainder of the Open Era. Thanks to the Blast From the Past forum and John Dolan’s book, Women’s Tennis 1968-84, I was able to add results for the entire Open Era, including qualifying rounds and challenger-level events.

* I now have 81 of the 80 Evert-Navratilova matches, including one exhibition.

Of course, top-flight women’s tennis didn’t begin out of nowhere in 1968, and once you can look at a few thousand matches from 1968 and 1969, curiosity begins to take hold. Margaret Court and Billie Jean King began their careers in the early 1960s, so wouldn’t it be nice to know exactly what they were up to for the better part of the decade?

The amateur era

However incomplete the historical record was for the 1970s, it was considerably worse before 1968. Wikipedia has grand slam draws and not much else. The heroes of the next phase are the contributors to tennisforum.com’s Blast From the Past section.

Blast contains extensive results for the entire history of women’s tennis, accumulated over two decades. It’s a truly incredible project, the sort of thing that no single person could’ve accomplished on their own. The year-by-year forum entries have complete singles draws for notable events (and many minor ones), and doubles and mixed doubles finals for most tournaments. To give you an idea of just how serious an undertaking this is, the forum topic for 1930 has over 5,000 singles match results from that season alone. A small group of tireless contributors typed all those up.

The downside of typed-up results is that they are very cumbersome to search. There are other issues, like inconsistent player names, since a single player might go by a maiden name, a married name, abbreviations or initials, and nicknames over the course of her career. (Not to mention typos!) To address those inherent limitations, you need a proper database.

247,000 singles matches

That database is what I’ve been doing for the last year. Working backwards one year at a time, I’ve pushed the dataset back to 1921, which–incidentally–gives us almost the entire career of Helen Wills. The project has involved hundreds of hours of proofing, player matching (all those name variations I mentioned), and lots of good old-fashioned data entry. While I’ve developed some automated tools to speed things up, there’s a limit to how much a process like this can be accelerated.

In the process, I’ve jumped over to the newspaper-research side of things, filling in the gaps of the Blast From the Past forum’s extensive coverage. My best estimate is that I’ve added about 20,000 results to the dataset, mostly for North American events before World War II. It’s fascinating if occasionally mind-numbing, and looking at old newspapers can be distracting enough to threaten my progress entirely.

All told, from 1921 to the mid-1990s, the Tennis Abstract database has gained almost a quarter of a million matches since that tweet last year, and it now encompasses a reasonably complete view of the final 47 years of the amateur era.

How you can dig in

Amateur-era players are shown on Tennis Abstract in a nearly identical manner to that of current players. In addition to Wills, here are links for Althea Gibson, Maureen Connolly, and Simonne Mathieu. You can find most of these players using the search box or via the exhaustive yearly summary pages, like these for 1925, 1945, or 1965.

Player and yearly summary pages show Elo ratings for women who played a certain number of matches. There’s a ton of information beyond the simple list of results.

For those of you who would like to do your own calculations, ratings, or other data exploration, I’m also releasing all the raw data on GitHub. Releases of new seasons usually happen several weeks later than the results first hit the TA website, so the GitHub repo currently goes back to 1927. The format is the same from 1927 to the present, so if you’ve worked with my data before, you’ll find the historical results to be in a familiar format.

Black tennis

An interest that has grown into a sizable side project is the history of segregated tennis. In most histories, Black tennis starts with Althea Gibson. Yet the American Tennis Association and various local outfits created a thriving tennis scene for Black players as early as the 1910s, long before the USLTA (now USTA) integrated their events.

Beyond contemporary newspaper writeups, results from Black tournaments have rarely been published. Using sources such as the Chicago Defender, the New York Amsterdam News, and the Baltimore Afro-American, I’ve been able to reconstruct draws, discover forgotten tournaments, and start to piece together career records for women who weren’t allowed to compete elsewhere.

One fascinating place to start is the player page for Ora Washington, the greatest Black player of the pre-Althea era. She spent her winters playing basketball so well that she’s now a member of that sport’s Hall of Fame. Based on her record as a tennis player, the folks in Newport ought to honor her tennis exploits as well.

Challenges and caveats

This is the sort of project that, quite simply, will never be finished. Yes, we can close the door on certain tournaments, such as most majors and certain other events with top-flight competition. But there’s no clear line between amateur era tournaments worth including and worth skipping, so there’s always more to hunt down. And even some of the events of the greatest historical interest–like the national tournaments of the aforementioned American Tennis Association–are poorly represented in the dataset, simply because I can’t find more than a few match results.

Another central challenge has to do with names, and it gets worse the further back we go. Newspapers often identified players only by their last name, sometimes including a first initial. Is this “M Smith” in a London-area draw in the 1920s the same as that “M Smith” in a different London-area draw in the 1920s? I have no idea! There are hundreds of questions like this, and I can’t imagine we’ll ever answer even a fraction of them. Newspapers also made lots of mistakes. Even an august publication like the New York Times would occasionally mix-and-match the first names of players. “Madelon Westervelt” is surely the same as “Madeleine Westervelt,” but is “Margaret Westervelt” the same person? (In this case, probably, but you get the idea.)

When you combine spotty source data, hand-made tools to help automate things, and the bleary-eyed researcher that I often am, you end up with bugs. Lots and lots of bugs. If you poke around the site for long, you’ll surely find some. When you do run across something that looks wrong, feel free to let me know, and please be patient. I want to resolve known bugs, but I also want a more exhaustive dataset. Balancing those two goals–along with other aims such as not alienating my family–often results in long wait times for bugfixes.

Thanks for reading all this far. I’ll be writing more about pre-Open Era topics in 2022, and when I’m not doing that, I’ll be pushing back in the 1910s and beyond.

Little Data, Big Potential

This is a guest post by Carl Bialik.

I had more data on my last 30 minutes of playing tennis than I’d gotten in my first 10 years of playing tennis — and it just made me want so much more.

Ben Rothenberg and I had just played four supertiebreakers, after 10 minutes of warmup and before a forehand drill. And for most of that time — all but a brief break while PlaySight staff showed the WTA’s Micky Lawler the system — 10 PlaySight cameras were recording our every move and every shot: speed, spin, trajectory and whether it landed in or out. Immediately after every point, we could walk over to the kiosk right next to the net to watch video replays and get our stats. The tennis sure didn’t look professional-grade, but the stats did: spin rate, net clearance, winners, unforced errors, net points won.

Later that night, we could go online and watch and laugh with friends and family. If you’re as good as Ben and I are, laugh you will: As bad as we knew the tennis was by glancing over to Dominic Thiem and Jordan Thompson on the next practice court, it was so much worse when viewed on video, from the kind of camera angle that usually yields footage of uberfit tennis-playing pros, not uberslow tennis-writing bros.

https://www.youtube.com/watch?v=xJ7AUcNVPoM

This wasn’t the first time I’d seen video evidence of my take on tennis, an affront to aesthetes everyone. Though my first decade and a half of awkward swings and shoddy footwork went thankfully unrecorded, in the last five years I’d started to quantify my tennis self. First there was the time my friend Alex, a techie, mounted a camera on a smartphone during our match in a London park. Then in Paris a few years later, I roped him into joining me for a test of Mojjo, a PlaySight competitor that used just one camera — enough to record video later published online, with our consent and to our shame. Last year, Tennis Abstract proprietor Jeff Sackmann and I demo-ed a PlaySight court with Gordon Uehling, founder of the company.

With PlaySight and Mojjo still only in a handful of courts available to civilians, that probably puts me — and Alex, Jeff and Ben — in the top 5 or 10 percent of players at our level for access to advanced data on our games. (Jeff may object to being included in this playing level, but our USPS Tennis Abstract Head2Head suggests he belongs.) So as a member of the upper echelon of stats-aware casual players, what’s left once I’m done geeking out on the video replays and RPM stats? What actionable information is there about how I should change my game?

Little data, modest lessons

After reviewing my footage and data, I’m still looking for answers. Just a little bit of tennis data isn’t much more useful than none.

Take the serve, the most common shot in tennis. In any one set, I might hit a few dozen. But what can I learn from them? Half are to the deuce court, and half are to the ad court. And almost half of the ones that land in are second serves. Even with my limited repertoire, some are flat while others have slice. Some are out wide, some down the T and some to the body — usually, for me, a euphemism for, I missed my target.

If I hit only five slice first serves out wide to the deuce court, three went in, one was unreturned and I won one of the two ensuing rallies, what the hell does that mean? It doesn’t tell me a whole lot about what would’ve happened if I’d gotten a chance to I try that serve once more that day against Ben — let alone what would happen the next time we played, when he had his own racquet, when we weren’t hitting alongside pros and in front of confused fans, with different balls on a different surface without the desert sun above us, at a different time of day when we’re in different frames of mind. And the data says even less about how that serve would have done against a different opponent.

That’s the serve, a shot I’ll hit at least once on about half of points in any match. The story’s even tougher for rarer shots, like a backhand drop half volley or a forehand crosscourt defensive lob, shots so rare they might come up once or twice every 10 matches.

More eyes on the court

It’s cool to know that my spinniest forehand had 1,010 RPM (I hit pretty flat compared to Jack Sock’s 3,337 rpm), but the real value I see is in the kind of data collected on that London court: the video. PlaySight doesn’t yet know enough about me to know that my footwork was sloppier than usual on that forehand, but I do, and it’s a good reminder to get moving quickly and take small steps. And if I were focusing on the ball and my own feet, I might have missed that Ben leans to his backhand side instead of truly split-stepping, but if I catch him on video I can use that tendency to attack his forehand side next time.

Video is especially useful for players who are most focused on technique. As you might have gathered, I’m not, but I can still get tactical edge from studying patterns that PlaySight doesn’t yet identify.

Where PlaySight and its ilk could really drive breakthroughs is by combining all of the data at its disposal. The company’s software knows about only one of the thousands of hours I’ve spent playing tennis in the last five years. But it has tens of thousands of hours of tennis in its database. Even a player as idiosyncratic as me should have a doppelganger or two in a data set that big. And some of them must’ve faced an opponent like Ben. Then there are partial doppelgangers: women who serve like me even though all of our other shots are different; or juniors whose backhands resemble mine (and hopefully are being coached into a new one). Start grouping those videos together — I’m thinking of machine learning, clustering and classifying — and you can start building a sample of some meaningful size. PlaySight is already thinking this way, looking to add features that can tell a player, say, “Your backhand percentage in matches is 11 percent below other PlaySight users of a similar age/ability,” according to Jeff Angus, marketing manager for the company, who ran the demo for Ben and me.

The hardware side of PlaySight is tricky. It needs to install cameras and kiosks, weatherproofing them when the court is outdoors, and protect them from human error and carelessness. It’s in a handful of clubs, and the number probably won’t expand much: The company is focusing more on the college game. Even when Alex and I, two players at the very center of PlaySight’s target audience among casual players, happened to book a PlaySight court recently in San Francisco, we decided it wasn’t worth the few minutes it would have taken at the kiosk to register — or, in my case, remember my password. The cameras stood watch, but the footage was forever lost.

Bigger data, big questions

I’m more excited by PlaySight’s software side. I probably will never play enough points on PlaySight courts for the company to tell me how to play better or smarter — unless I pay to install the system at my home courts. But if it gets cheaper and easier to collect decent video of my own matches — really a matter of a decent mount and protector for a smartphone and enough storage space — why couldn’t I upload my video to the company? And why couldn’t it find video of enough Bizarro Carls and Bizarro Carl opponents around the world to make a decent guess about where I should be hitting forehands?

There are bigger, deeper tennis mysteries waiting to be solved. As memorably argued by John McPhee in Levels of the Game, tennis isn’t so much as one sport as dozens of different ones, each a different level of play united only by common rules and equipment. And a match between two players even from adjacent levels in his hierarchy typically is a rout. Yet tactically my matches aren’t so different from the ones I see on TV, or even from the practice set played by Thiem and Thompson a few feet from us. Hit to the backhand, disguise your shots, attack short balls and approach the net, hit drop shots if your opponent is playing too far back. And always, make your first serve and get your returns in.

So can a tactic from one level of the game even to one much lower? I’m no Radwanska and Ben is no Cibulkova, but could our class of play share enough similarity — mathematically, is Carl : Ben :: Aga : Pome — that what works for the pros works for me? If so, then medium-sized data on my style is just a subset of big data from analogous styles at every level of the game, and I might even find out if that backhand drop half volley is a good idea. (Probably not.)

PlaySight was the prompt, but it’s not the company’s job to fulfill product features only I care about. It doesn’t have to be PlaySight. Maybe it’s Mojjo, maybe Cizr. Or maybe it’s some college student who likes tennis and is looking for a machine-learning class. Hawk-Eye, the higher-tech, higher-priced, older competitor to PlaySight, has been slow to share its data with researchers and journalists. If PlaySight has figured out that most coaches value the video and don’t care much for stats, why not release the raw footage and stats to researchers, anonymized, who might get cracking on the tennis classification question or any of a dozen other tennis analysis questions I’ve never thought to ask? (Here’s a list of some Jeff and I have brainstormed, and here are his six big ones.) I hear all the time from people who like tennis and data and want to marry the two, not for money but to practice, to learn, to discover, and to share their findings. And other than what Jeff’s made available on GitHub, there’s not much data to share. (Just the other week, an MIT grad asked for tennis data to start analyzing.)

Sharing data with outside researchers “isn’t currently in the road map for our product team, but that could change,” Angus said, if sharing data can help the company make its data “actionable” for users to improve to their games.

Maybe there aren’t enough rec players who’d want the data with enough cash to make such ventures worthwhile. But college teams could use every edge. Rising juniors have the most plastic games and the biggest upside. And where a few inches can change a pro career, surely some of the top women and men could also benefit from PlaySight-driven insights.

Yet even the multimillionaire ruling class of the sport is subject to the same limitations driven by the fractured nature of the sport: Each event has its own data and own systems. Even at Indian Wells, where Hawk-Eye exists on every match court, just two practice courts have PlaySight; the company was hoping to install four more for this year’s tournament and is still aiming to install them soon. Realistically, unless pros pay to install PlaySight on their own practice courts and play lots of practice matches there, few will get enough data to be actionable. But if PlaySight, Hawk-Eye or a rival can make sense of all the collective video out there, maybe the most tactical players can turn smarts and stats into competitive advantages on par with big serves and wicked topspin forehands.

PlaySight has already done lots of cool stuff with its tennis data, but the real analytics breakthroughs in the sport are ahead of us.

Carl Bialik has written about tennis for fivethirtyeight.com and The Wall Street Journal. He lives and plays tennis in New York City and has a Tennis Abstract page.

Match Charting Project: More Matches, More Data, New Spreadsheet

The Match Charting Project keeps growing, and starting today, even more of the data is available for anyone who wants it. Several new contributors have helped us pass the 750-match milestone, having added an average of two matches per day since I first published the raw data.

New spreadsheet

The Match Charting spreadsheet now does a lot more. As you chart each point, the document updates stats for the match–both total and set-by-set. You’ll find the same stats you see on television (aces, double faults, winners, unforced errors, etc) along with some that are a little less common, like winning percentage in different lengths of rallies, and most consecutive points won.

In other words, As you chart the match, you’ll have access to many of the same stats that commentators do. Here’s what it looks like:

If you’ve hesitated to try charting because you couldn’t see what was in it for you, I hope this changes the calculation a bit.

Click here to download the MatchChart template.

New data

About a month ago, I published the point-by-point data from all charted matches. In raw form, it’s a bit daunting, and it’s more than what’s necessary for many interesting research projects.

Today, I added 15 different aggregate stats files for men, and another 15 for women. These contain the data that is shown in each charted match report. For instance, if you find it interesting that Simona Halep hit 14% of her backhands down the line in the Indian Wells final, you can take a look in the ShotDirection stats file and compare that number with the results from Halep’s other charted matches, or all matches in the database as a whole.

You can find these files (along with the updated raw data for 760+ matches) by clicking here.

Chart some matches

If you haven’t already, now is a great time to start charting professional matches and contributing to the project. An enormous number of matches are televised and streamed, and as the database of charted matches grows, there’s more and more useful context to all the data we’re generating.

You can start by jumping into the ‘Instructions’ tab of the new MatchChart spreadsheet, or for other tips, you can start with my blog post introducing the project.

Free ATP and WTA Results and Stats Databases

The vast majority of my men’s and women’s tennis results and stats databases are now free for anyone who wants to use them.

ATP Results and Stats:

Tour-level results back to 1968, with tons of data on both players in each match (age, handedness, country, rank), and matchstats from 1991-present.
Almost a decade of tour-level qualifying matches, with matchstats for the last few years.
Challenger results back to 1991, with matchstats for almost the last ten years.
Futures (and Satellite) results back to 1991.
Linked biographical and rankings data (introduced here).

WTA Results:

Tour-level results back to 1968, with the same player data as in the ATP files.
Tour-level qualifying matches.
Over 220,000 ITF main-draw matches.

Click the links to access the files. Enjoy!

Free ATP and WTA Ranking Databases

More data!

Today I’ve made available my entire ATP and WTA ranking databases through the end of the 2014 season. In addition, you’ll find my complete player tables, which include birthdate, country, and handedness for every player who has ever been ranked or played a tour-level match. (Plus thousands more players, who are included in the database for other reasons.)

This is all the data you need to research all sorts of topics, like the rise and fall of certain countries in the rankings and the changing age of top 10s, 50s, and 100s.

This is the third major dataset I’ve published this week, and more is on the way.

ATP rankings are here, and WTA rankings are here. Enjoy!

Raw Data From The Match Charting Project

In the last year and a half, dozens of contributors and I have amassed detailed shot-by-shot records of nearly 700 professional matches. You can see the full list here, or a menu sorted by player here.

I refer to this as The Match Charting Project, and I hope you’ll consider contributing as well. Using a straightforward text notation system, we record shot type, shot direction, return depth, error types, and more. The more matches, the more interesting the results. The project made up part of my presentation at the Sloan Sports Analytics Conference last month, which included some very preliminary findings on player tendencies.

Now, you can dig into the raw data yourself. I’ve posted all of the user-submitted match charts in one place, in a standardized format for anyone who wants to mess around with it.

Enjoy!

Point-by-Point Data From the Last 17 Grand Slams

I’ve been doing a lot of griping lately about the state of tennis data, so I figured now was a good time to start doing something about it.

I’ve just released point-by-point data for most Grand Slam singles matches back to 2011. Beyond the basic point sequence–which is valuable in and of itself–you’ll find serve speed, winner type, and for a few of the slams, rally length for each point.

More detailed notes on the data are available at that link. Enjoy, and if working with it turns up any interesting findings, please let me know.

Analytics That Aren’t: Why I’m Not Excited about SAP in Tennis

It’s not analytics, it’s marketing.

The Grand Slams (with IBM) and now the WTA (with SAP) are claiming to deliver powerful analytics to tennis fans. And it’s certainly true that IBM and SAP collect way more data than the tours would without them. But what happens to that data? What analytics do fans actually get?

Based on our experience after several years of IBM working with the Slams and Hawkeye operating at top tournaments, the answers aren’t very promising. IBM tracks lots of interesting stats, makes some shiny graphs available during matches, and the end result of all this is … Keys to the Match?

Once matches are over and the performance of the Keys to the Match are (blessedly) forgotten, all that data goes into a black hole.

Here’s the message: IBM collects the data. IBM analyzes the data. IBM owns the data. IBM plasters their logo and their “Big Data” slogans all over anything that contains any part of the data. The tournaments and tours are complicit in this: IBM signs a big contract, makes their analytics part of their marketing, and the tournaments and tours consider it a big step forward for tennis analysis.

Sometimes, marketing-driven analytics can be fun. It gives some fans what they want–counts of forehand winners, or average first-serve speeds. But let’s not fool ourselves. What IBM offers isn’t advancing our knowledge of tennis. In fact, it may be strengthening the same false beliefs that analytical work should be correcting.

SAP: Same Story (So Far)

Early evidence suggests that SAP, in its partnership with the WTA, will follow exactly the same model:

SAP will provide the media with insightful and easily consumable post-match notes which offer point-by-point analysis via a simple point tracker, highlight key events in the match, and compare previous head-to-head and 2013 season performance statistics.

“Easily consumable” is code for “we decide what the narratives are, and we come up with numbers to amplify those narratives.”

Narrative-driven analytics are just as bad–and perhaps more insidious–than marketing-driven analytics, which are simply useless. The amount of raw data generated in a tennis match is enormous, which is why TV broadcasts give us the same small tidbits of Hawkeye data: distance run during a point, average rally hit point, and so on. So, under the weight of all those possibilities, why not just find the numbers that support the prevailing narrative? The media will cite those numbers, the fans will feel edified, and SAP will get its name dropped all over the place.

What we’re missing here is context. Take this SAP-generated stat from a writeup on the WTA site:

The first promising sign for Sharapova against Kanepi was her rally hit point. Sharapova made contact with the ball 76% of the time behind the baseline compared to 89% for her opponent. It doesn’t matter so much what the percentage is – only that it is better than the person standing on the other side of the net.

Is that actually true? I don’t think anyone has ever published any research on whether rally hit point correlates with winning, though it seems sensible enough. In any case, these numbers are crying out for more context. Is 76% good for Maria? How about keeping her opponent behind the baseline 89% of the time? Is the gap between 76% and 89% particularly large on the WTA? Does Maria’s rally hit point in one match tell us anything about her likely rally hit point in her next match? After all, the article purports to offer “keys to match” for Maria against her next opponent, Serena Williams.

Here’s another one:

There is a lot to be said for winning the first point of your own service game and that rung true for Sharapova in her quarterfinal. When she won the opening point in 11 of her service games she went on to win nine of those games.

Is there any evidence that winning your first point is more valuable than, say, winning your second point? Does Sharapova typically have a tough time winning her opening service point? Is Kanepi a notably difficult returner on the deuce side, or early in games? “There is a lot to be said” means, roughly, that “we hear this claim a lot, and SAP generated this stat.”

In any type of analytical work, context is everything. Narrative-driven analytics strip out all context.

The alternative

IBM, SAP, and Hawkeye are tracking a huge amount of tennis data. For the most part, the raw data is inaccessible to researchers. The outsiders who are most likely to provide the context that tennis stats so desperately need just don’t have the tools to evaluate these narrative-driven offerings.

Other sporting organizations–notably Major League Baseball–make huge amounts of raw data available. All this data makes fans more engaged, not less. It’s simply another way for the tours to get fans excited about the game. Statheads–and the lovely people who read their blogs–buy tickets too.

So, SAP, how about it? Make your branded graphics for TV broadcasts. Provide your easily consumable stats for the media. But while you’re at it, make your raw data available for independent researchers. That’s something we should all be able to get excited about.

US Open Draw Datasets

Earlier today, I published a thorough analysis of the last ten years of US Open draws, showing that while first and second seeds have had extremely easy first-round matchups, there is no other credible statistical evidence that suggests any nonrandom manipulation of the draw.

If you want to take a look at the draws yourself, I’ve made it easier. The following files not only have the full draws going back to 2001, but they also include each player’s ATP or WTA ranking at the time of the tournament, their ordinal ranking among the players in the draw, the ordinal ranking of their first-round opponent, and the ordinal ranking of their best-possible second round opponent.

Click to download the files:

Here’s a quick rundown of the columns you’ll find in each sheet:

Year — each file contains the entire draws for the last ten years.
Draw Pos[ition] — numbers 1 to 128, so you can always sort the sheet to show the players in draw order. (For instance, the #1 seed is 1, that player’s opponent is 2, and so on.)
Player
Country
Seed — the seeding assigned by the US Open
Rank [ATP/WTA] — the player’s official ranking the Monday that the tourney began.
Ordinal — the player’s rank among the 128 players in the field. Last year, Shelby Rogers’s WTA ranking was 344, which made her ordinal ranking 124 out of 128.
1stRdOpp — the ordinal ranking of the player’s first-round opponent.
Best2nd — the ordinal ranking of the player’s best possible second-round opponent.

Let me know if you find anything interesting!