A Quick Look at the Odds of Three-Setters

In the comments to my match-fixing post earlier this week, Elihu Feustel commented:

There are almost no situations where a best of 3 match is a favorite to go to three sets. If the market priced a player as greater than 50% to win in exactly 3 sets, that alone is compelling evidence of match fixing.

In Monday’s questionable Challenger match, not only did the betting markets believe that the match was likely to go three sets, it picked a specific winner in three sets.

It takes only a bit of arithmetic to see why Elihu’s point is correct. Let’s say two players, A and B, are exactly evenly matched. Each one has a 50% chance of winning the match and a 50% chance of winning each set. Thus, the odds that A wins the match in straight sets are 25% (50% for the first set multiplied by 50% for the second). The odds that B wins in straights are the same. The probability that the match finishes in straight sets, then, is 50% (25% for an A win + 25% for B), meaning that the odds of a three-set match are also 50%.

As soon as one player has an edge, the probability of a three-set match goes down. Consider the scenario in which A has a 70% chance of winning each set. The odds that player A wins in straight sets are 49% (70% times 70%) and the odds that B wins in straight sets are 9% (30% times 30%). Thus, there’s a 58% chance of a straight-set victory, leaving a 42% chance of a three-setter.

This simple approach makes one major assumption: each player’s chances don’t change from one set to another. That probably isn’t true. It seems most likely to me that the player who wins the first set gets stronger relative to his opponent, perhaps because he gains confidence, or because his opponent loses confidence, or because he figures he doesn’t have much chance of winning. (I’m sure this isn’t true in all matches, but I suspect it applies often enough.)

If it’s true that the probability of the second set is dependent–even slightly–on the outcome of the first set, the likelihood of a three-setter decreases even further.

Probability in practice

As expected, far fewer than half of tour-level matches go three sets. (I’m considering only best-of-three matches.) So far this year, 36% of ATP best-of-threes have gone the distance, while only 32% of Challenger-level matches have done so.

In fact, men’s tennis has even fewer three-set matches than expected. For every match, I used a simple rankings-based model to estimate each player’s chances of winning a set and, as shown above, the odds that the match would go three sets. For 2014 tour-level matches, the model–which assumes that set probabilities are independent–predicts that 44% of matches would go three sets. That’s over 20% more third sets that we see in practice.

There are two factors that could account for the difference between theory and practice. I think both play a part:

  1. Sets aren’t independent. If winning the first set makes a player more likely to win the second, there would be fewer three-setters than predicted.
  2. There’s usually a bigger gap between players than aggregate numbers suggest. On paper, one player might have a 60% chance of winning the match, but on the day, one player might be tired, under the weather, unhappy with his racquets, uncomfortable with the court … or playing his best tennis, in a honeymoon period with a new coach, enjoying friendly calls from home line judges. The list of possible factors is endless. The point is that for any matchup, there are plenty of effectively random, impossible to predict variables that affect each player’s performance. I suspect that those variables are more likely to expand the gap between players–and thus lower the likelihood of a three-setter–than shrink it.

A note on outliers

Despite the odds against three-setters, some players are more likely go three than others. Among the 227 players who have contested 100 or more ATP best-of-threes since 1998, 20 have gone the distance in 40% of more of their matches. John Isner, tennis’s most reliable outlier, tops the list at 47.4%.

Big servers don’t dominate the list, but Isner’s presence at the top isn’t entirely by chance. After John, Richard Fromberg is a close second at 46.7%, while Goran Ivanisevic is not far behind at 43.0%. Mark Philippoussis and Sam Querrey also show up in the top ten.

It’s no surprise to see these names come up. One-dimensional servers are more likely to play tiebreaks, and tiebreaks are as close to random as a set of tennis can get. Someone who plays tiebreaks as often as Isner does will find himself losing first sets to inferior opponents and winning first sets against players who should beat him.

That randomness not only makes it more likely the match will go three sets, it’s also something the players are aware of. If Isner drops a first-set tiebreak, he realizes that he still has a solid chance to win the match–losing the breaker doesn’t mean he’s getting outplayed. If there is a mental component that partially explains the likelihood of the first-set winner taking the second set, it doesn’t apply to players like him.

Still, even Big John finishes sets in straights more than half the time. Every other tour regular does so as well, so it would take a very unusual set of circumstances for a betting market–or common sense–to favor a three-set outcome.

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The Untapped Potential of Umpire Scorecard Data

There’s a lot more that can be done with tennis data. Everyone knows this. Even the ATP and WTA tours–along with their rather prominent partners–know this.

Both tours are sitting on a mountain of information that they’ve barely exploited: umpire scorecard data. It’s not cutting edge–there are no cameras, no courtside loggers counting unforced errors and winners. It’s just a log of every point, along with first or second serve, aces, and double faults. Despite those limits, there are many untapped advantages.

First: There’s an umpire scorecard for every match. Not every match on a TV court, not every match on a Hawkeye court, not every main draw match.  Every match. If a ATP, WTA, or ITF umpire is officiating the match, to the best of my knowledge, there is a scorecard–when you see a chair umpire tap on a screen, this is what they’re recording. That means data on thousands of matches and players every year, from Novak Djokovic to Djordje Djokovic.

It’s tough to overstate how valuable that is. The main drawback of most tennis stats is context. For instance, when Hawkeye puts a graphic on your TV screen, it’s often based on data from a single match or the present tournament. IBM’s much-publicized analytics are based on Grand Slam matches only. Umpire scorecards have no such problem.

Second, there’s a ton of information lurking in this low-tech tracking system. The basics of first and second serves, aces, and double faults may not sound like much, but as we’ll see below, they open the door to a huge array of stats. ATP and WTA “Match Stats” are compiled from these scorecards, but they only scratch the surface.

How to do more with scorecards

In a minute, I’ll make specific suggestions for additional totals and rates that the tours could compile from the data they already have. Before that, let me explain why simply expanding the contents of “Match Stats” should be Plan B.

More and more journalism is data-based, and more and more avid fans are, to some extent or other, analyzing tennis for publication. In other words, there is a rapidly growing base of analysts who don’t need data pre-packaged for them. Every match is different, and the numbers needed to illustrate any match report are different as well. For broader analysis, like comparing players over the course of a season, the need for customized data is more important still.

So: Release the point-by-point data from the scorecards.

Another benefit of the simplicity of umpire scorecard data is that more analysts can easily manage it. No organization could foresee everything that might be interesting about a match, so why even try? Not every journalist will want to dig into a point-by-point spreadsheet to see how often Julien Benneteau missed his first serve of a game, or how Rafael Nadal responded every time he fell behind 0-30. But some will do just that. When they do, their work benefits, their readers have more ways to engage with the next match they watch, and the sport ultimately wins.

A not-so-brief wish list

I have a sneaking feeling that no one’s going to release point-by-point data for every ATP or WTA match. I hope that’s not the case, but if it is, the tours should still consider vastly expanding the stats they compile for each match–including past matches for as far back as their databases go.

  1. Deuce/ad comparisons. Some players serve much more effectively in one than the other. For all deuce-court service points, I would like: (a) total points, (b) aces, (c) double faults, and (d) first serves in. Same for ad-court service points.
  2. Break point stats. Same as the above: For both servers facing break point: (a) aces, (b) double faults, and (c) first serves in.
  3. Break point games. In how many games did each player earn a break point?
  4. Stats for other important point scores. Break points are key, but other scenarios are important as well. If I have to pick only a few, let’s start with 0-30, 15-30, deuce, and ad-in (including 40-30). For all service points at each of those scores, I’d like (a) total points, (b) aces, (c) double faults, and (d) first serves in.
  5. Set points and match points: Same as above. Fans love match point stats.
  6. The game sequence–at what points did breaks of serve occur? This would allow us to answer many oft-posed questions: Do players hold serve more early in sets? Do breaks of serve more frequently follow breaks than holds? (And if so, how much more often?) Are players more like to drop serve immediately after winning a tight set?
  7. Set-by-set breakdowns of all stats that are currently kept, plus all of the above. The live scoring app separates stats by set, but there is no official archive with set-by-set breakdowns. This is particularly key for journalists attempting to tell the story of a match, when a small change in approach can turn the tide.
  8. Tiebreak breakdowns. Tiebreaks–especially long ones–have a life of their own, and analysts should be able to see all of the same stats for each tiebreak as for each set as a whole. For example, it would be interesting to see if a player’s ace or double fault rates (or even his or her first-serve percentage) changed between the first twelve games of a set and the breaker.
  9. A list of the score when each double fault occurred. (Aces would be nice, too.) Especially in men’s tennis, DFs are quite rare, and they often loom large in match narratives.
  10. Longest streaks for each player: consecutive aces, consecutive double faults, consecutive points won on serve, consecutive points won overall, and the score at the beginning and end of each of those streaks.
  11. For doubles matches, a separation of all of the above service stats by server. For the Samuel Groth/Leander Paes partnership, aggregate serve stats f (as they are presented now) aren’t going to tell you anything useful about either player’s performance at the line.

To reiterate, all of this stuff is in the scorecards. Most of the above are no more difficult to compile than the Match Stats that the tours already publish.

If the tours added everything on my list, that would be one big step out of the dark ages for tennis. Certainly, tennis writers would be able to file more intelligent stories and fans would have a much better way to experience the performances of their favorite players.

If the tours published current and archived raw point-by-point data, tennis would go one better: it would become an example for many other individual sports to follow. We would see an boom in fan engagement as every follower of the sport would have the opportunity to learn much more about tennis and relive matches–whether last week or late last century–in detail.

We’re not talking about a multi-million dollar infrastructure investment. To achieve all this, the tours need only do a little bit more with what they already have.

 

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Jo-Wilfried Tsonga and the (Extremely Specific) Post-Masters Blues

Two days after winning a Masters title in Toronto, Jo-Wilfried Tsonga played his first match in Cincinnati. Betting odds had the Frenchman as a heavy favorite over the unseeded Mikhail Youzhny, but a sluggish Tsonga dropped the match in straight sets.

An explanatory narrative springs to mind immediately: After playing six matches in a week, it’s tough to keep a winning streak going. Losing the match, even against a lesser opponent, is predictable. (Of course, it’s more predictable in hindsight.)

As usual with such “obvious” storylines, it’s not quite so straightforward. On average, ATP title winners who enter a tournament the following week perform exactly as well as expected in their first match of the next event. The same results hold for finalists, who typically played as many matches the previous week, and must also bounce back from a high-stakes loss.

To start with, I looked at the 1,660 times since the 2000 season that an ATP finalist took the court again within three weeks. Those players won, on average, 1.93 matches in their post-title event, losing their first matches 29% of the time. Their 71% next-match winning percentage is virtually identical to what a simple ranking-based model would predict. In other words, there’s no evidence here of a post-final letdown.

More relevant to Tsonga’s situation is the set of 1,055 finalists who played the following week. Those men won 1.7 matches in their next event, losing 31% of the their first matches at the follow-up event. That’s about 1% worse than expected–not nearly enough to allow us to draw any conclusions. Narrowing the set even further to the 531 tournament winners who played the next week, we find that they won 2.0 matches in their next event, losing 26.3% of their first matches, just a tick better than the simple model would predict.

Some of these numbers–1.7 match wins in a tournament; a 70% winning percentage–don’t sound particularly impressive. But we need to keep in mind that the majority of ATP tournaments don’t feature Masters-level fields, and plenty of finalists are well outside the top ten. Plus, the players who play an event the week after winning a title tend to be lower ranked. Masters events occupy back-to-back weeks on the schedule only a couple of times each season.

If we limit our scope to the more uncommon back-to-back tourneys for Masters winners, a bit of a trend emerges. The week after winning a Masters tournament, players win an average of 2.9 matches, losing their first match only 20.4% of the time. That sounds pretty good, except that, in the last 15 years, the group of Masters winners has been extremely good. That 80% first-match winning percentage is 5% below what a simple model would’ve predicted for them.

If we limit the sample even further, to Masters winners ranked outside the top five–a group that includes Tsonga–we finally find more support for the “obvious” narrative. Since 2000, 17 non-top-fivers have shown up for a tournament the week after winning a Masters event. They’ve won only 1.8 matches in their next events, losing their first match more than 40% of the time. That’s 20% worse than expected.

This small set of non-elite Masters winners is the only group I could find that fit the narrative of a post-title or post-final blues. (I tested a lot more than I’ve mentioned here, and nearly all showed players performing within a couple percent of expectations.)

Tsonga cited low energy in his post-match press conference, but we shouldn’t forget that there are plenty of other reasons the Frenchman might lose a first-round match. He’s split his six career matches against Youzhny, and 7 of his 19 losses in the past year have come to players ranked lower than the Russian. Losses don’t always need precedents, and in this case, the precedents aren’t particularly predictive.

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Westerhof, Van Der Duim, and a Strong Whiff of Match Fixing

This afternoon at the ATP Meerbusch Challenger in Germany, all eyes were definitely not on a first-round match between Dutchmen Boy Westerhof and Antal Van Der Duim. Both are ranked outside the top 250, neither has ever cracked the top 200, and both are in their late twenties.

It appears that the two players assumed no one would be watching. Before the match, the markets on Betfair were very suspicious:

For those of you not accustomed to parsing betting markets, here’s a summary of what the market thought was going to happen:

  • Van Der Duim’s chances of winning the match were between 75% and 80%.
  • Van Der Duim’s odds of winning the first set were roughly 35%.
  • There was a better than 50/50 chance that Van Der Duim would win the match in three sets. The odds of any other specific outcome (e.g. Westerhof wins in three) were minuscule in comparison.

The match odds in themselves might have raised a few eyebrows, but could be written off as owing to Westerhof’s recent run of poor play, or perhaps some information gathered on site about a nagging injury. When combined with the other markets, however, it’s clear that something very fishy was going on.

The match went precisely according to script. After Westerhof took the first set, 6-4, the market got more and more confident about Van Der Duim winning the second set:

Van Der Duim remained the favorite even after going down an early break in the second set. Shortly thereafter, with no cameras watching, Westerhof seems to have decided not to waste any more time:

https://twitter.com/baselinebetting/status/498823405282807809

In the end, Van Der Duim beat Westerhof, 4-6 6-3 6-3. No one following the betting markets was at all surprised.

Nor should we be shocked that this sort of thing happens. With the middling prize money on offer at Challenger events–Westerhof will get about $500, and if Van Der Duim loses in the next round, he’ll be awarded about $800–there’s more money to be made by losing matches than winning them.

While we don’t know how often matches are fixed, something was very wrong about this one. Because the markets so blatantly telegraphed the fix, it poses an important question to the sport’s governing bodies. If they don’t take this opportunity to act, it will send a very clear message to Challenger-level players that match-fixing is acceptable practice.

(Thanks to the three Twitter users quoted above, who brought this match to my attention.)

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The Dream of the Nineties is Alive

Last weekend, the four finalists in ATP events were David Goffin, Dominic Thiem, Vasek Pospisil, and Milos Raonic. All were born in the 1990s, making the Kitzbuhel and Washington finals the first all-nineties championship matches in tour history.

It’s about time. The first half of the 1990-born cohort is already 24 years old, an age that used to suggest a tennis player was approaching his prime. Of the four finalists, only Thiem wasn’t born in 1990. (He was born in 1993, making him the youngest finalist of the season so far.)

It has never taken so long for a single-year-or-younger group of ATP players to play each other in a final. For the thirty-one years between 1960 births and 1990 births, it has, on average, taken less than 21 years before youngsters in each cohort face off for a title. It took 24 years and seven months before the 1990 group–with the help of Thiem–finally reached this milestone.

Here are the breakthrough finals for each age group in five-year intervals, to put the 1990 group in perspective:

The age of these milestone finals has been steadily creeping up over the last few years. The class of 1987 was a good one, giving us Novak Djokovic and Andy Murray, but even those two stars didn’t meet in a final until the 2008 Cincinnati Masters, when both had passed their 21st birthdays.

There’s a sharp downturn after that 1987 class. The ATP didn’t see a 1988-or-younger final until three years later, when Alexandr Dolgopolov faced Marin Cilic in the 2011 Umag title match. In the three years since then, there have been only six more 1988-or-younger finals, including the two last weekend.

Thiem, along with a few other young players, offers hope that the tides are beginning to turn. This week, for the first time since 2005 (when, as we’ve seen, Nadal and Berdych played the last all-teenage final), the ATP top 200 features four teenagers, two of whom–Borna Coric and Alexander Zverev–are not yet 18. Then again, neither Goffin nor Pospisil reached a final until they had been inside the top 200 for three years. We may need to keep looking to 23-year-olds for ATP firsts.

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Margin of Error Podcast: Episode 25

A new podcast is (finally) out!

Fresh from a week at the Citi Open in Washington DC, Amy and I run down the previous week’s events, along with some young American prospects worth watching, and preview the WTA Premier and ATP Masters tournaments getting underway in Canada.

You can find the podcast and subscribe with iTunes here. For other subscription methods, here’s an XML feed. Otherwise, keep an eye out for a new episode early each week, which I’ll post here on the blog.

Click here to listen.

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ATP Single-Country Finals in the Open Era

Today’s Washington final between Milos Raonic and Vasek Pospisil is the first all-Canadian final in the Open era. Here’s a list of the other 20 countries that have represented both sides of an ATP final, along with the total number of such finals and the most recent such match:

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