Roger Federer Wasn’t Clutch, But He Was Almost Clutch Enough

Italian translation at settesei.it

The stats from the Wimbledon final told a clear story. Over five sets, Roger Federer did most things slightly better than did his opponent, Novak Djokovic. Djokovic claimed a narrow victory because he won more of the most important points, something that doesn’t show up as clearly on the statsheet.

We can add to the traditional stats and quantify that sort of clutch play. A method that goes beyond simply counting break points or thinking back to obviously key moments is to use the leverage metric to assign a value to each point, according to its importance. After every point of the match, we can calculate an updated probability that each player will emerge victorious. A point such as 5-all in a tiebreak has the potential to shift the probability a great deal; 40-15 in the first game of the match does not.

Leverage quantifies that potential. The average point in a best-of-five match has a leverage of about 4%, and the most important points are several times that. Another way of saying that a player is “clutch” is that he is winning a disproportionate number of high-leverage points, even if he underwhelms at low-leverage moments.

Leverage ratio

In my match recap at The Economist, I took that one step further. While Djokovic won fewer points than Federer did, his successes mattered more. The average leverage of Djokovic’s points won was 7.9%, compared to Federer’s 7.2%. We can represent that difference in the form of a leverage ratio (LR), by dividing 7.9% by 7.2%, for a result of 1.1. A ratio of that magnitude is not unusual. In the 700-plus men’s grand slam matches in the Match Charting Project, the average LR of the more clutch player is 1.11. Djokovic’s excellence in key moments was not particularly rare, but in a close match such as the final, it was enough to make the difference.

Recording a leverage ratio above 1.0 is no guarantee of victory. In about 30% of these 700 best-of-five matches, a player came out on top despite winning–on average–less-important points than his opponent did. Some of the instances of low-LR winners border on the comical, such as the 2008 French Open final, in which Rafael Nadal drubbed Federer despite a LR of only 0.77. In blowouts, there just isn’t that much leverage to go around, so the number of points won matters a lot more than their timing. But un-clutch performances often translate to victory even in closer matches. Andy Murray won the 2008 US Open semi-final over Nadal in four sets despite a LR of 0.80, and in a very tight Wimbledon semi-final last year, Kevin Anderson snuck past John Isner with a LR of 0.88.

You don’t need a spreadsheet to recognize that tennis matches are decided by a mix of overall and clutch performance. The numbers I’ve shown you so far don’t advance our understanding much, at least not in a rigorous way. That’s the next step.

DR, meet BLR

Regular users of Tennis Abstract player pages are familiar with Dominance Ratio (DR), a stat invented by Carl Bialik that re-casts total points won. DR is calculated by dividing a player’s rate of return points won by his rate of service points lost (his opponent’s rate of return points won), so the DR for a player who is equal on serve and return is exactly 1.0.

Winners are usually above 1.0 and losers below 1.0. In the Wimbledon final, Djokovic’s DR was 0.87, which is extremely low for a winner, though not unheard of. DR balances the effect of serve performance and return performance (unlike total points won, which can skew in one direction if there are many more serve points than return points, or vice versa) and gives us a single-number summary of overall performance.

But it doesn’t say anything about clutch, except that when a player wins with a low DR, we can infer that he outperformed in the big moments.

To get a similarly balanced view of high-leverage performance, we can adapt leverage ratio to equally weight clutch play on serve and return points. I’ll call that balanced leverage ratio (BLR), which is simply the average of LR on serve points and LR on return points. BLR usually doesn’t differ much from LR, just as we often get the same information from DR that we get from total points won. Djokovic’s Wimbledon final BLR was 1.11, compared to a LR of 1.10. But in cases where a disproportionate number of points occur on one player’s racket, BLR provides a necessary correction.

We can capture leverage-adjusted performance by simply multiplying these two numbers. For example, let’s take Stan Wawrinka’s defeat of Djokovic in the 2016 US Open final. Wawrinka’s DR was 0.90, better than Djokovic at Wimbledon this year but rarely good enough to win. But win he did, thanks to a BLR of 1.33, one of the highest recorded in a major final. The product of Wawrinka’s DR and his BLR–let’s call the result DR+–is 1.20. That number can be interpreted on the same scale as “regular” DR, where 1.2 is often a close victory if not a truly nail-biting one. DR+ combines a measure of how many points a player won with a measure of how well-timed those points were.

Out of 167 men’s slam finals in the Match Charting Project dataset, 14 of the winners emerged triumphant despite a “regular” DR below 1.0. In every case, the winner’s BLR was higher than 1.1. And in 13 of the 14 instances, the strength of the winner’s BLR was enough to “cancel out” the weakness of his DR, in the sense that his DR+ was above 1.0. Here are those matches, sorted by DR+:

```Year  Major            Winner              DR   BLR   DR+
2019  Wimbledon        Novak Djokovic    0.87  1.11  0.97
1982  Wimbledon        Jimmy Connors     0.88  1.20  1.06
2001  Wimbledon        Goran Ivanisevic  0.95  1.16  1.10
2008  Wimbledon        Rafael Nadal      0.98  1.13  1.10
2009  Australian Open  Rafael Nadal      0.99  1.13  1.12
1981  Wimbledon        John McEnroe      0.99  1.16  1.15
1992  Wimbledon        Andre Agassi      0.97  1.19  1.16
1989  US Open          Boris Becker      0.96  1.22  1.18
1988  US Open          Mats Wilander     0.98  1.21  1.18
2015  US Open          Novak Djokovic    0.98  1.21  1.18
2016  US Open          Stan Wawrinka     0.90  1.33  1.20
1999  Roland Garros    Andre Agassi      0.98  1.25  1.23
1990  Roland Garros    Andres Gomez      0.94  1.34  1.26
1991  Australian Open  Boris Becker      0.99  1.30  1.29```

167 slam finals, and Djokovic-Federer XLVIII was the first one in which the player with the lower DR+ ended up the winner. (Some of the unlisted champions had subpar leverage ratios and thus DR+ figures lower than their DRs, but none ended up below the 1.0 mark.) While Federer was weaker in the clutch–notably in tiebreaks and when he held match points–his overall performance in high-leverage situations wasn’t as awful as those few memorable moments would suggest. More often than not, a player who combined Federer’s DR of 1.14 with his BLR of 0.90 would conclude the Wimbledon fortnight dancing with the Ladies’ champion.

Surprisingly, 1-out-of-167 might understate the rarity of a winner with a DR+ below 1.0. Only one other best-of-five match in the Match Charting Project database (out of more than 700 in total) fits the bill. That’s the controversial 2019 Australian Open fourth-rounder between Kei Nishikori and Pablo Carreno Busta. Nishikori won with a 1.06 DR, but his BLR was a relatively weak 0.91, resulting in a DR+ of 0.97. Like the Wimbledon final, that Melbourne clash could have gone either way. Carreno Busta may have been unlucky with more than just the chair umpire’s judgments.

What does it all mean?

We knew that the Wimbledon final was close–now we have more numbers to show us how close it was. We knew that Djokovic played better when it mattered, and now we have more context that indicates how much better he was, which is not a unusually wide margin. Federer has won five of his slams despite title-match BLRs below 1.0, and two others with DRs below 1.14. He’s never won a slam with a DR+ of 1.03 or lower, but then again, there had never before been a major final that DR+ judged to be that close. Roger is no one’s idea of a clutch master, but he isn’t that bad. He just should’ve saved a couple of doses of second-set dominance for more important junctures later on.

If you’re anything like me, you’ll read this far and be left with many more questions. I’ve started looking at several, and hope to write more in this vein soon. Is Federer usually less clutch than average? (Yes.) Is Djokovic that much better? (Yes.) How about Nadal? (Also better.) Is Nadal really better, or do his leverage numbers just look good because important points are more likely to happen in the ad court? (No, he really is better.) Does Djokovic have Federer’s number? (Not really, unless you mean his mobile number. Then yes.) Did everything change after Djokovic hit that return? (No.)

There are many interesting related topics beyond the big three, as well. I started writing about leverage for subsets of matches a few years ago, prompted by another match–the 2016 Wimbledon Federer-Raonic semi-final–in which Roger got outplayed when it mattered. Just as we can look at average leverage for points won and lost, we can also estimate the importance of points in which a player struck an ace, hit a backhand unforced error, or chose to approach the net.

Matches are decided by a combination of overall performance and high-leverage play. Commonly-available stats do a pretty good job at the former, and fail to shine much light on the latter. The clutch part of the equation is often left to the speculation of pundits. As we build out a more complete dataset and have access to more and more point-by-point data (and thus leverage numbers for each point and match), we can close the gap, enabling us to better quantify the degree to which situational performance affects every player’s bottom line.

Did Rafael Nadal Almost Lose a Set to David Ferrer?

Italian translation at settesei.it

In David Ferrer’s final grand slam, the draw gods handed him a doozy of a first-round assignment in Rafael Nadal. Ferrer has struggled all year, and no one seriously expected him to improve on his 6-24 career record against the King of Clay. In the end, he didn’t: Ferrer was forced to retire midway through the second set with a calf injury. But before his final Flushing exit, he gave Rafa a bit of a scare.

Nadal won the first set, 6-3. The second set was a bit messier: Ferrer broke to love in the opening game, Rafa broke him back in the next, and a bit later, Ferrer broke again to take a 3-2 lead. He maintained that one break advantage until he physically couldn’t continue. Leading 4-3 and serving the next game, he was been two holds away from leveling the match.

Does that mean Nadal “almost” lost the set? People on the internet argue about these things, and while I don’t understand why, I do love a good probability question. If it overlaps with semantics (yay sematics!), that’s a bonus.

Let’s forget the word choice for now and reframe the question: Ignoring the injury, what were Ferrer’s chances of winning the set? If we assume that both players were equal, it’s a simple thing to plug into my win probability model and–ta da!–we find that from 4*-3, Ferrer had a roughly 85% chance of winning the set.

But wait: I can already hear the Rafa fans screaming at me, these two players aren’t exactly equal. In the 102 points the Spanish duo played on Monday night, Ferrer won 38% on return and Nadal won 47%. For an entire five-set match, those rates work out to a 93% chance of Rafa winning. Maybe that’s not quite high enough, but it’s in the ballpark. Using those figures, Ferrer’s chance of hanging on to win the second set drop significantly, to 57.5%. When you’re winning barely half of your service points, your odds of securing a pair of holds are worse than a coin flip. Had Ferrer won the set, it’s more likely that he would’ve needed to either break Rafa again or come through in a tiebreak.

That’s a pretty big difference between our two initial estimates. 85% sounds good enough to qualify for “almost” (though one study quantifies the meaning of “almost” at above 90%), but 57.5% does not.

That doesn’t quite settle it, though. The win probability model takes all notions of streakiness out of the equation.  According to the formula, there’s no patches of good or bad play, no dips in motivation, so extra energy to finish off a set, etc. I’m not convinced any of those exist in any systematic manner, but it’s tough to settle the question either way. Therefore, if we have the ability to use data from real-life matches, we should.

And here, we can. Let’s start with Nadal. Going back to late 2011, I was able to identify 69 sets in which Rafa was returning down a break at 4-3. (There are probably more; my point-by-point dataset isn’t exhaustive, but the missing matches are mostly random, so the 69 should be representative of the last several years.) Of those 69, he came back to win 21, almost exactly 30%.

Ferrer has been more solid than Nadal’s opponents. (It helps that Ferrer only had to face Rafa once, while Nadal’s opponents had face him every time.) I found 122 sets in which Ferrer served at 4-3, leading by a break. He went on to win the set 109 of those times, or about 89%.

The 89% figure is definitely too high for our purposes: Not only was Ferrer a better player, on average, between 2012 and today, than he is now, but he also had the benefit of facing weaker opponents than Nadal in almost all of those 122 sets. 89%–not far from the theoretical 85% we started with–is a grossly optimistic upper limit.

Even if we take the average of Nadal’s and Ferrer’s real-life results–roughly 90% conversions for Ferru and 70% for Rafa’s opponents–80% is still overshooting the mark. As we’ve established, Ferrer’s numbers refer to a stronger version of the Spaniard, while Rafa is still near the level of his last half-decade. Even 80%, then, is overstating the chances that Nadal would’ve lost a set.

That leaves us with a range between 57%, which assumes Nadal would keep winning nearly half of Ferrer’s service points, and 80%, which is based on the experience of both players over the last several years. Ultimately, any final figure comes down to what we think about Ferrer’s level right now–not as good as it was even a couple of years ago, but at the same time, good enough to come within two games of taking a set from the top-ranked player in the world.

It would take a lot more work to come up with a more precise estimate, and even then, we’d still be stuck not only trying to establish Ferrer’s current ability level, but also his ability level in that set. Just as the word “almost” refers to a range of probabilities, I’m happy to call it a day with my own range. Taking all of these calculations together, we might settle on a narrower field of, say, 65-70%, or about two in three. There’s a good chance a healthy Ferrer would have taken that set from his long-time tormentor, but it was far from a sure thing … or even, given the usual meaning of the word, an “almost” sure thing.

Measuring a Season’s Worth of Luck

In Toronto last week, Stefanos Tsitsipas was either very clutch, very lucky, or both. Against Alexander Zverev in Friday’s quarter-final, he won fewer than half of all points, claiming only 56.7% of his service points, compared to Zverev’s 61.2%. The next day, beating Kevin Anderson in the semi-final in a third-set tiebreak, he again failed to win half of total points, holding 69.9% of his service points against Anderson’s 75.5%.

Whether the Greek prospect played his best on the big points or benefited from a hefty dose of fortune, this isn’t sustainable. Running those serve- and return-points-won (SPW and RPW) numbers through my win probability model, we find that–if you take luck and clutch performance out of the mix–Tsitsipas had a 27.8% chance of beating Zverev and a 26.5% chance of beating Anderson. These two contests–perhaps the two days that have defined the youngster’s career up to this point–are the very definition of “lottery matches.” They could’ve gone either way, and over a long enough period of time, they’ll probably even out.

Or will they? Are some players more likely to come out on top in these tight matches? Are they consistently–dare I say it–clutch? Using this relatively simple approach of converting single-match SPW and RPW rates into win probabilities, we can determine which players are winning more or less often than they “should,” and whether it’s a skill that some players consistently display.

Odds in the lottery

Let’s start with some examples. When one player wins more than 55% of points, he is virtually guaranteed to win the match. Even at 53%, his chances are extremely good. Still, a lot of matches–particularly best-of-threes on fast surfaces–end up in the range between 50% and 53%, and that’s what most interesting from this perspective.

Here are Tsitsipas’s last 16 matches, along with his SPW and RPW rates and the implied win probability for each:

```Tournament  Round  Result  Opponent     SPW    RPW  WinProb
Toronto     F      L       Nadal      62.9%  21.1%       3%
Toronto     SF     W       Anderson   69.9%  24.5%      27%
Toronto     QF     W       A Zverev   56.7%  38.8%      28%
Toronto     R16    W       Djokovic   77.2%  32.0%      85%
Toronto     R32    W       Thiem      83.3%  30.2%      93%
Toronto     R64    W       Dzumhur    82.8%  35.0%      98%
Washington  SF     L       A Zverev   54.7%  25.5%       1%
Washington  QF     W       Goffin     71.2%  32.7%      67%
Washington  R16    W       Duckworth  80.0%  37.5%      98%
Washington  R32    W       Donaldson  59.5%  45.5%      74%
Wimbledon   R16    L       Isner      72.5%  18.0%      10%
Wimbledon   R32    W       Fabbiano   64.0%  55.9%     100%
Wimbledon   R64    W       Donaldson  70.1%  40.9%      95%
Wimbledon   R128   W       Barrere    71.5%  39.0%      94%
Halle       R16    L       Kudla      59.7%  28.8%       8%
Halle       R32    W       Pouille    78.3%  42.9%      99%```

More than half of the matches are at least 90% or no more than 10%. But that leaves plenty of room for luck in the remaining matches. Thanks in large part to his last two victories, the win probability numbers add up to only 9.8 wins, compared to his actual record of 12-4. All four losses were rather one-sided, but in addition to the Toronto matches against Zverev and Anderson, his wins against David Goffin in Washington and, to a lesser extent, Novak Djokovic in Toronto, were far from sure things.

In the last two months, Stefanos has indeed been quite clutch, or quite lucky.

Season-wide views

When we expand our perspective to the entire 2018 season, however, the story changes a bit. In 48 tour-level matches through last week’s play (excluding retirements), Tsitsipas has gone 29-19. The same win probability algorithm indicates that he “should” have won 27.4 matches–a difference of 1.6 matches, or about five percent, which is less than the gap we saw in his last 16. In other words, for the first two-thirds of the season, his results were either unlucky or un-clutch, if only slightly. At the very least, the aggregate season numbers are less dramatic than his recent four-event run.

For two-thirds of a season, a five percent gap between actual wins and win-probability “expected” wins isn’t that big. For players with at least 30 completed tour-level matches this season, the magnitude of the clutch/luck effect extends from a 20% bonus (for Pierre Hugues Herbert) to a 20% penalty (for Sam Querrey, which he reduced a bit by beating John Isner in Cincinnati on Monday despite winning less than 49% of total points). Here are the ten extremes at each end, of the 59 ATPers who have reached the threshold so far in 2018:

```Player                 Matches  Wins  Exp Wins  Ratio
Pierre Hugues Herbert       30    16      13.2   1.22
Nikoloz Basilashvili        34    17      14.0   1.21
Frances Tiafoe              39    24      20.0   1.20
Evgeny Donskoy              30    13      10.9   1.19
Grigor Dimitrov             34    20      17.1   1.17
Lucas Pouille               31    16      13.7   1.17
Gael Monfils                34    21      18.3   1.15
Daniil Medvedev             34    18      15.8   1.14
Marco Cecchinato            33    19      16.7   1.14
Maximilian Marterer         32    17      15.2   1.12
…
Leonardo Mayer              37    19      20.1   0.95
Guido Pella                 37    20      21.2   0.95
Marin Cilic                 38    27      28.8   0.94
Novak Djokovic              37    27      29.3   0.92
Marton Fucsovics            30    16      17.5   0.92
Joao Sousa                  36    18      19.8   0.91
Dusan Lajovic               34    17      18.7   0.91
Fernando Verdasco           43    22      24.5   0.90
Mischa Zverev               39    18      20.7   0.87
Sam Querrey                 30    15      18.8   0.80```

A difference of three or four wins, as many of these players display between their actual and expected win totals, is more than enough to affect their standing in the rankings. The degree to which it matters depends enormously on which matches they win or lose, as Tsitsipas’s semi-final defeat of Anderson has a much greater impact on his point total than, say, Querrey’s narrow victory over Isner does for his. But in general, the guys at the top of this list are ones who have seen unexpected ranking boosts this season, while some of the guys at the bottom have gone the other way.

The last full season

Let’s take a look at an entire season’s worth of results. Last year, a few players–minimum 40 completed tour-level matches–managed at least a 20% luck/clutch bonus, but with the surprising exception of Daniil Medvedev, none of them have repeated the feat so far in 2018:

```Player                 Matches  Wins  Exp Wins  Ratio
Donald Young                43    21      16.2   1.30
Fabio Fognini               58    35      28.5   1.23
Jack Sock                   55    36      29.8   1.21
Jiri Vesely                 45    22      19.3   1.14
Daniil Medvedev             43    22      19.7   1.11
John Isner                  57    36      32.3   1.11
Damir Dzumhur               56    33      29.7   1.11
Gilles Muller               48    30      27.1   1.11
Alexander Zverev            74    53      48.1   1.10
Juan Martin del Potro       53    37      33.6   1.10```

A few of these players have had solid seasons, but posting a good luck/clutch number in 2017 is hardly a guaranteed, as the likes of Donald Young, Jack Sock, and Jiri Vesely can attest. Here is the same list, with 2018 luck/clutch ratios shown alongside last year’s figures:

```Player                 2017 Ratio  2018 Ratio
Donald Young                 1.30        0.89  *
Fabio Fognini                1.23         1.1
Jack Sock                    1.21        0.68  *
Jiri Vesely                  1.14        1.08  *
Daniil Medvedev              1.11        1.14
John Isner                   1.11        0.96
Damir Dzumhur                1.11        1.01
Gilles Muller                1.11        0.84  *
Alexander Zverev             1.10        1.06
Juan Martin del Potro        1.10        1.07```

* fewer than 30 completed tour-level matches

The average luck/clutch ratio of these ten players has fallen to a bit below 1.0.

Unsustainable luck

You can probably see where this is going. I generated full-season numbers for each year from 2008 to 2017, and identified those players who appeared in the lists for adjacent pairs of seasons. If luck/clutch ratio is a skill–that is, if it’s more clutch than luck–guys who post good numbers will tend to do so the following year, and those who post lower numbers will be more likely to remain low.

Across 325 pairs of player-seasons, that’s not what happened. There is almost no relationship between one year of luck/clutch ratio and the next. The r^2 value–a measure of correlation–is 0.07, meaning that the year-to-year numbers are close to random.

Across sports, analysts have found plenty of similar results, and they are often quick to pronounce that “clutch doesn’t exist,” which leads to predictable rejoinders from the laity that “of course it does,” and so on. It’s boring, and I’m not particularly interested in that debate. What this specific finding shows is:

This type of luck, defined as winning more matches than implied by a player’s SPW and RPW in each match, is not sustainable.

What Tsitsipas accomplished last weekend in Toronto was “clutch” by almost any definition. What this finding demonstrates is that a few such performances–or even a season’s worth of them–doesn’t make it any more likely that he’ll do the same next year. Or, another possibility is that the players who stick at the top level of professional tennis are all clutch in this sense, so while Tsitsipas might be quite mentally strong in key moments, he’ll often run up against players who have similar mental skills, and he won’t be able to consistently win these close matches.

If Stefanos is able to maintain a ranking in the top 20, which seems plausible, he’ll probably need to win more serve and return points than he has so far. Fortunately for him, he’s still almost eight years younger than his typical peer, so he has plenty of time to improve. The occasional lottery matches that tilt his way will need to be mere bonuses, not the linchpin of his strategy to reach the top.

Simona Halep and Recoveries From Match Point Down

Italian translation at settesei.it

In yesterday’s French Open quarterfinals, Elina Svitolina held a commanding lead over Simona Halep, up a set and 5-1. Depending on what numbers you plug into the formula, Svitolina’s chance of winning the match at that stage was somewhere between 97% and 99%. Halep fought back to 5-5, and in the second-set tiebreak, Svitolina earned a match point at 6-5. Halep recovered again, won the breaker, and then cruised to a 6-0 victory in the third set.

It’s easy to fit a narrative to that sequence of events: After losing two leads, Svitolina was dispirited, and Halep was all but guaranteed a third-set victory. Maybe. It’s impossible to test that sort of thing on the evidence of a single match, but this is hardly the first time a player has failed to convert match point and needed to start fresh in a new set.

Even without a match point saved, the player who wins the second set has a small advantage going into the decider. In the last six-plus years of women’s Slam matches, the player who won the second set went on to win 51.3% of third sets. On the other hand, if the second set was a tiebreak, the winner of the second set won the decider only 43.7% of the time. Though it sounds contradictory at first, consider what we know about such sets. The second-set winner just barely claimed her set (in the tiebreak), while usually, her opponent took the first set more decisively. Momentum helps a little, but it can’t overcome much of a difference in skill level.

Let’s dig into the specific cases of second-set match points saved. Thanks to the data behind IBM’s Pointstream on Grand Slam websites, we have the point-by-point sequence for most Slam singles matches going back to 2011. (The missing matches are usually those on non-Hawkeye courts and a few small courts at Roland Garros.) That’s over 2,600 women’s singles matches. In just over 1,700 of them, one of the two players earned a match point in the second set. Over 97% of the time, that player converted–needing an average of 1.7 match points to do so–and avoiding playing a third set.

That leaves 45 matches in which one player held a match point in the second set, failed to finish the job, and was forced to play a third set. It’s a limited sample, and it doesn’t wholeheartedly support the third-set-collapse narrative suggested above. 60% of the time–27 of the 45 matches–the player who failed to convert match point in the second set, like Svitolina did, went on to lose the third set. The third set was often lopsided: 5 of the 27 were bagels (including yesterday’s match), and the average score was 6-2. None of the third sets went beyond 6-4.

The other 18 matches–the 40% of the time in which the player with the second-set match point bounced back to win the third set–featured rather one-way deciders, as well. In those, the third-set loser managed an average of only 2.3 games, also never doing better than 6-4.

This is a small sample, so it’s unwise to conclude that this 60/40 margin is anything close to an iron law of tennis. That said, it does provide some evidence that players don’t necessarily collapse after failing to convert a straight-sets win at match point. What happened to Svitolina yesterday is far from certain to happen next time.

Smaller Swings In Big Moments

Italian translation at settesei.it

Despite the name, unforced errors aren’t necessarily bad. Sometimes, the right tactic is to play more aggressively, and in order to hit more winners, most players will commit more errors as well. Against some opponents, increasing the unforced error count–as long as there is a parallel improvement in winners or other positive point-ending shots–might be the only way to win.

Last week, I showed that one of the causes of Angelique Kerber’s first-round loss was her disproportionate number of errors in big moments. But as my podcasting partner Carl Bialik pointed out, that isn’t the whole story. If Kerber played more aggressively on the most important points–one possible cause of more errors–it might be the case that her winner rate was higher, as well. Since the 6-2 6-2 scoreline was so heavily tilted against her, it was a safe bet that Kerber recorded more high-leverage errors than winners. Still, Carl makes a valid point, and one worth testing.

To do so, let’s revisit the data: 500 women’s singles matches from the last four majors and the first four rounds of this year’s French Open. By measuring the importance of each point, we can determine the average leverage (LEV) of every point in each match, along with the average leverage of points which ended with a player hitting an unforced error, or a winner. Last week, we found that Kerber’s UEs in her first-round loss had an average LEV of 5.5%, compared to a LEV of 3.8% on all other points. For today’s purposes, let’s use match averages as a reference point: Her average UE LEV of 5.5% also compares unfavorably to the overall match average LEV of 4.1%.

What about winners? Kerber’s 15 winners came on points with an average LEV of 3.9%, below the match average. Case closed: On more important points, Kerber was more likely to commit an error, and less likely to hit a winner.

Across the whole population, players hit more errors and fewer winners in crucial moments, but only slightly. Points ending in errors are about one percent more important than average (percent, not percentage point, so 4.14% instead of 4.1%), and points ending in winners are about two percent less important than average. In bigger moments, players increase their winner rate about 39% of the time, and they improve their W-UE ratio about 45% of the time. Point being, there are tour-wide effects on more important points, but they are quite small.

Of course, Kerber’s first-round upset isn’t indicative of how she has played at Slams in general. In my article last week, I mentioned the four players who did the best job of reducing errors at big moments: Kerber, Agnieszka Radwanska, Timea Bacsinszky, and Kiki Bertens. Kerber and Radwanska both hit fewer winners on big points as well, but Bacsinszky and Bertens manage a perfect combination, hitting slightly more winners as the pressure cranks up. Among players with more than 10 Slam matches since last year’s French, Bacsinszky is the only one to hit winners on more important points than her unforced errors over 75% of the time.

Compared to her peers, Kerber’s big-moment tactics are remarkably passive. The following table shows the 21 women for whom I have data on at least 13 matches. “UE Rt.” (“UE Ratio”) is similar to the metric I used last week, comparing the average importance of points ending in errors to average points; “W Ratio” is the same, but for points ending in winners, and “W+UE Ratio” is–you guessed it–a (weighted) combination of the two. The combined measure serves as an rough approximation of aggression on big points, where ratios below 1 are more passive than the player’s typical tactics and ratios above 1 are more aggressive.

```Player                     M  UE Rt.  W Rt.  W+UE Rt.
Angelique Kerber          20    0.92   0.85      0.88
Alize Cornet              13    0.92   0.87      0.94
Agnieszka Radwanska       17    0.91   0.95      0.95
Simona Halep              19    0.93   0.94      0.95
Samantha Stosur           13    0.95   0.98      0.96
Timea Bacsinszky          14    0.89   1.02      0.97
Elina Svitolina           15    1.02   0.95      0.97
Karolina Pliskova         18    0.97   0.98      0.97
Caroline Wozniacki        14    0.93   1.00      0.97
Johanna Konta             13    1.00   0.97      0.98
Caroline Garcia           14    0.94   1.02      0.98
Svetlana Kuznetsova       17    0.96   0.98      0.99
Garbine Muguruza          20    1.02   0.94      0.99
Venus Williams            25    1.00   0.97      0.99
Elena Vesnina             13    0.96   1.03      0.99
Anastasia Pavlyuchenkova  15    1.03   0.99      0.99
Coco Vandeweghe           13    1.08   0.95      1.01
Madison Keys              13    1.01   1.02      1.01
Serena Williams           27    0.99   1.05      1.02
Carla Suarez Navarro      14    1.00   1.14      1.05
Dominika Cibulkova        14    1.11   1.03      1.07```

Kerber’s combined measure stands out from the pack. Her point-ending shots–both winners and errors, but especially winners–occur disproportionately on less important points, and the overall effect is double that of the next most passive big-moment player, Alize Cornet. Every other player is close enough to neutral that I would hesitate before making any conclusions about their pressure-point tactics.

Even when Kerber wins, she does so with effective defense at key points. In only two of her last 20 matches at majors did her winners occur on particularly important points. (Incidentally, one of those two was last year’s US Open final.) In general, her brand of passivity works–she won 16 of those matches. But defensive play doesn’t leave very much room for error–figuratively or literally. The tactics were familiar and proven, but against Makarova, they were poorly executed.

Angelique Kerber’s Unclutch Unforced Errors

Italian translation at settesei.it

It’s been a rough year for Angelique Kerber. Despite her No. 1 WTA ranking and place at the top of the French Open draw, she lost her opening match on Sunday against the unseeded Ekaterina Makarova. Adding insult to injury, the loss goes down in the record books as a lopsided-looking 6-2 6-2.

Andrea Petkovic chimed in with her diagnosis of Kerber’s woes:

She’s simply playing without confidence right now. It was tight, even though the scoreline was 2 and 2 but everyone who knows a thing about tennis knew that Angie made errors whenever it mattered because she’s playing without any confidence right now – errors she didn’t make last year.

This is one version of a common analysis: A player lost because she crumbled on the big points. While that probably doesn’t cover all of Kerber’s issues on Sunday–Makarova won 72 points to her 55–it is true that big points have a disproportionate effect on the end result. For every player who squanders a dozen break points yet still wins the match, there are others who falter at crucial moments and ultimately lose.

This family of theories–that a player over- or under-performed at big moments–is testable. For instance, I showed last summer that Roger Federer’s Wimbledon loss to Milos Raonic was due in part to his weaker performance on more important points. We can do the same with Kerber’s early exit.

Here’s how it works. Once we calculate each player’s probability of winning the match before each point, we can assign each point a measure of importance–I prefer to call it leverage, or LEV–that quantifies how much the single point could effect the outcome of the match. At 3-0, 40-0, it’s almost zero. At 3-3, 40-AD in the deciding set, it might be over 10%. Across an entire tournament’s worth of matches, the average LEV is around 5% to 6%.

If Petko is right, we’ll find that the average LEV of Kerber’s unforced errors was higher than on other points. (I’ve excluded points that ended with the serve, since neither player had a chance to commit an unforced error.) Sure enough, Kerber’s 13 groundstroke UEs (that is, excluding double faults) had an average LEV of 5.5%, compared to 3.8% on points that ended some other way. Her UE points were 45% more important than non-UE points.

Let’s put that number in perspective. Among the 86 women for whom I have point-by-point UE data for their first-round matches this week*, ten timed their errors even worse than Kerber did. Magdalena Rybarikova was the most extreme: Her eight UEs against Coco Vandeweghe were more than twice as important, on average, as the rest of the points in that match. Seven of the ten women with bad timing lost their matches, and two others–Agnieszka Radwanska and Marketa Vondrousova–committed so few errors (3 and 4, respectively), that it didn’t really matter. Only Dominika Cibulkova, whose 15 errors were about as badly timed as Kerber’s, suffered from unclutch UEs yet managed to advance.

* This data comes from the Roland Garros website. I aggregate it after each major and make it available here.

Another important reference point: Unforced errors are evenly distributed across all leverage levels. Our instincts might tell us otherwise–we might disproportionately recall UEs that came under pressure—-but the numbers don’t bear it out. Thus, Kerber’s badly timed errors are just as badly timed when we compare her to tour average.

They are also poorly timed when compared to her other recent performances at majors. Petkovic implied as much when she said her compatriot was making “errors she didn’t make last year.” Across her 19 matches at the previous four Slams, her UEs occurred on points that were 11% less important than non-UE points. Her errors caused her to lose relatively more important points in only 5 of the 19 matches, and even in those matches, the ratio of UE leverage to non-UE leverage never exceeded 31%, her ratio in Melbourne this year against Tsurenko. That’s still better than her performance on Sunday.

Across so many matches, a difference of 11% is substantial. Of the 30 players with point-by-point UE data for at least eight matches at the previous four majors, only three did a better job timing their unforced errors. Radwanska heads the list, at 16%, followed by Timea Bacsinszky at 14% and Kiki Bertens at 12%. The other 26 players committed their unforced errors at more important moments than Kerber did.

As is so often the case in tennis, it’s difficult to establish if a stat like this is indicative of a longer-trend trend, or if it is mostly noise. We don’t have point-by-point data for most of Kerber’s matches, so we can’t take the obvious next step of checking the rest of her 2017 matches for similarly unclutch performances. Instead, we’ll have to keep tabs on how well she limits UEs at big moments on those occasions where we have the data necessary to do so.

New at Tennis Abstract: Point-by-Point Stats

Yesterday, I announced the new ATP doubles results on Tennis Abstract. Today, I want to show you something else I rolled out over the offseason: sequential point-by-point stats for more than 100,000 matches.

Traditional match stats can do no more than summarize the action. Point-by-point stats are so much more revealing: They show us how matches unfold and allow us to look much deeper into topics such as momentum and situational skill. These are subjects that remain mysteries–or, at the very least, poorly quantified–in tennis.

As an example, let’s take a look at the new data available for one memorable contest, the World Tour Finals semifinal between Andy Murray and Milos Raonic:

The centerpiece of each page is a win probability graph, which shows the odds that one player would win the match after each point. These graphs do not take player skill into account, though they are adjusted for gender and surface. The red line shows one player’s win probability, while the grey line indicates “volatility”–a measure of how much each point matters. You can see exact win probability and volatility numbers by moving your cursor over the graph. Most match graphs aren’t nearly as dramatic as this one; of course, most matches aren’t nearly as dramatic as this one was.

(I’ve written a lot about win probability in the past, and I’ve also published the code I use to calculate in-match win probability.)

Next comes a table with situational serving stats for both players. In the screenshot above, you can see deuce/ad splits; the page continues, with tiebreak-specific totals and tallies for break points, set points, and match points. After that is an exhaustive, point-by-point text recap of the match, which displays the sequence of every point played.

I’ve tried to make these point-by-point match pages as easy to find as possible. Whenever you see a link on a match score, just click that link for the point-by-point page. For instance, here is part of Andy Murray’s page, showing where to click to find the Murray-Raonic example shown above:

As you can see from all the blue scores in this screenshot, most 2016 ATP tour-level matches have point-by-point data available. The same is true for the last few seasons, as well as top-level WTA matches. The lower the level, the fewer matches are available, but you might be surprised by the breadth and depth of the coverage. The site now contains point-by-point data for almost half of 2016 main-draw men’s Futures matches. For instance, here’s the graph for a Futures final last May between Stefanos Tsitsipas and Casper Ruud.

I’ll keep these as up-to-date as I can, but with my current setup, you can expect to wait 1-4 weeks after a match before the point-by-point page becomes available. I’m hoping to further automate the process and shorten the wait over the course of this season.

Enjoy!

The Most Exciting Matches of the 2016 WTA Season

Italian translation at settesei.it

In my most recent piece for The Economist, I used a metric called Excitement Index (EI) to consider the implications of shortening singles matches to a format like the no-ad, super-tiebreak rules used for doubles. In my simulations, the shorter format didn’t fare well: The most gripping contests are often the longest ones, and the full-length third set is frequently the best part.

I used data from ATP tournaments in that piece, and several readers have asked how women’s matches score on the EI scale. Many matches from the 2016 season rate extremely highly, while some players we tend to think of as exciting fail to register among the best by this metric. I’ll share some of the results in a moment.

First, a quick overview of EI. We can calculate the probability that each player will win a match at any point in the contest, and using those numbers, it’s possible to determine the leverage of every point–that is, the difference between a player’s odds if she wins the next point and her odds if she loses it. At 40-0, down a break in the first set, that leverage is very low: less than 2%. In a tight third-set tiebreak, leverage can climb as high as 25%. The average point is around 5% to 6%, and as long as neither player has a substantial lead, points at 30-30 or later are higher.

EI is calculated by averaging the leverage of every point in the match. The more high-leverage points, the higher the EI. To make the results a bit more viewer-friendly, I multiply the average leverage by 1,000, so if the typical point has the potential for a 5% (0.05) swing, the EI is 50. The most boring matches, like Garbine Muguruza‘s 6-1 6-0 dismantling of Ekaterina Makarova in Rome, rate below 25. The most exciting will occasionally top 100, and the average WTA match this year scored a 53.7. By comparison, the average ATP match this year rated at 48.9.

Of course, the number and magnitude of crucial moments isn’t the only thing that can make a tennis match “exciting.” Finals tend to be more gripping than first-round tilts, long rallies and daring net play are more watchable than error-riddled ballbashing, and Fed Cup rubbers feature crowds that can make the warmup feel like a third-set tiebreak. When news outlets make their “Best Matches of 2016” lists, they’ll surely take some of those other factors into account. EI takes a narrower view, and it is able to show us which matches, independent of context, offered the most pressure-packed tennis.

Here are the top ten matches of the 2016 WTA season, ranked by EI:

```Tournament    Match                Score                    EI
Charleston    Lucic/Mladenovic     4-6 6-4 7-6(13)       109.9
Wimbledon     Cibulkova/Radwanska  6-3 5-7 9-7           105.0
Wimbledon     Safarova/Cepelova    4-6 6-1 12-10         101.7
Kuala Lumpur  Nara/Hantuchova      6-4 6-7(4) 7-6(10)    100.2
Brisbane      CSN/Lepchenko        4-6 6-4 7-5            99.0
Quebec City   Vickery/Tig          7-6(5) 6-7(3) 7-6(7)   98.5
Miami         Garcia/Petkovic      7-6(5) 3-6 7-6(2)      98.1
Wimbledon     Vesnina/Makarova     5-7 6-1 9-7            97.2
Beijing       Keys/Kvitova         6-3 6-7(2) 7-6(5)      96.8
Acapulco      Stephens/Cibulkova   6-4 4-6 7-6(5)         96.7
```

Getting to 6-6 in the final set is clearly a good way to appear on this list. The top fifty matches of the season (out of about 2,700) all reached at least 5-5 in the third. The highest-rated clash that didn’t get that far was Angelique Kerber‘s 1-6 7-6(2) 6-4 defeat of Elina Svitolina, with an EI of 88.2. Svitolina’s 4-6 6-3 6-4 victory over Bethanie Mattek Sands in Wuhan, the top match on the list without any sets reaching 5-5, scored an EI of 87.3.

Wimbledon featured an unusual number of very exciting matches this year, especially compared to Roland Garros and the Australian Open, the other tournaments that forgo a tiebreak in the final set. The top-rated French Open contest was the first-rounder between Johanna Larsson and Magda Linette, which scored 95.3 and ranks 13th for the season, while the highest EI among Aussie Open matches is all the way down at 27th on the list, a 92.8 between Monica Puig and Kristyna Pliskova.

Dominika Cibulkova is the only player who appears twice on this list. That doesn’t mean she’s a sure thing for exciting matches: As we’ll see, elite players rarely are. The only year-end top-tenner who ranks among the highest average EIs is Svetlana Kuznetsova, who played as many “very exciting” matches–those rating among the top fifth of matches this season–as any other woman on tour:

```Rank  Player                M  Avg EI  V. Exc  Exc %  Bor %
1     Kristina Mladenovic  60    59.8      19  55.0%  25.0%
2     Christina McHale     46    59.6      16  50.0%  19.6%
3     Heather Watson       35    58.5      12  48.6%  25.7%
4     Jelena Jankovic      43    57.6      12  55.8%  30.2%
5     Svetlana Kuznetsova  64    57.4      21  48.4%  32.8%
6     Venus Williams       38    57.1      10  55.3%  31.6%
7     Yanina Wickmayer     43    56.5      13  46.5%  30.2%
8     Alison Riske         46    56.5      10  45.7%  32.6%
9     Caroline Garcia      62    56.4      18  43.5%  33.9%
10    Irina-Camelia Begu   42    56.4      14  45.2%  40.5%
```

(Minimum 35 tour-level matches (“M” above), excluding retirements. My data is also missing a random handful of matches throughout the season.)

The “V. Exc” column tallies how many top-quintile matches the player took part in. The “Exc %” column shows the percent of matches that rated in the top 40% of all WTA contests, while “Bor %” shows the same for the bottom 40%, the more boring matches. Big servers who reach a disproportionate number of tiebreaks and 7-5 sets do well on this list, though it is far from a perfect correspondence. Tiebreaks can create a lot of big moments, but if there were many love service games en route to 6-6, the overall picture isn’t nearly so exciting.

Unlike Kuznetsova, who played a whopping 32 deciding sets this year, most of the other top women enjoy plenty of blowouts. Muguruza, Simona Halep, and Serena Williams occupy the very last three places on the average-EI ranking, largely because when they win, they do so handily–and they win a lot. The next table shows the WTA year-end top-ten, with their ranking (out of 59) on the average-EI list:

```Rank  Player        WTA#  Matches  Avg EI  V. Exc  Exc %  Bor %
5     Kuznetsova       9       64    57.4      21  48.4%  32.8%
13    Pliskova         6       66    55.6      19  48.5%  39.4%
16    Keys             8       64    55.4      13  40.6%  35.9%
23    Cibulkova        5       68    54.6      21  42.6%  42.6%
28    Kerber           1       77    54.0      12  42.9%  41.6%
tour average                   53.7          40.0%  40.0%
41    Radwanska        3       69    52.5      12  29.0%  44.9%
51    Konta           10       67    51.2      12  34.3%  46.3%
57    Muguruza         7       51    49.9       5  33.3%  43.1%
58    Halep            4       59    49.6       8  30.5%  50.8%
59    Williams         2       44    48.1       3  27.3%  50.0%
```

It’s a good thing that fans love Serena, because her matches rarely provide much in the way of big moments. As low as Williams and Halep rate on this measure, Victoria Azarenka scores even lower. Her Miami fourth-rounder against Muguruza was her only match this season to rank in the “exciting” category, and her average EI was a mere 44.0.

Clearly, EI isn’t much of a method for identifying the best players. Even looking at the lowest-rated competitors by EI would be misleading: In 56th place, right above Muguruza, is the otherwise unheralded Nao Hibino. EI excels as a metric for ferreting out the most riveting individual matches, whether they were broadcast worldwide or ignored entirely. And the next time someone suggests shortening matches, EI is a great tool to highlight just how much excitement would be lost by doing so.

Measuring the Clutchness of Everything

Italian translation at settesei.it

Matches are often won or lost by a player’s performance on “big points.” With a few clutch aces or un-clutch errors, it’s easy to gain a reputation as a mental giant or a choker.

Aside from the traditional break point stats, which have plenty of limitations, we don’t have a good way to measure clutch performance in tennis. There’s a lot more to this issue than counting break points won and lost, and it turns out that a lot of the work necessary to quantify clutchness is already done.

I’ve written many times about win probability in tennis. At any given point score, we can calculate the likelihood that each player will go on to win the match. Back in 2010, I borrowed a page from baseball analysts and introduced the concept of volatility, as well. (Click the link to see a visual representation of both metrics for an entire match.) Volatility, or leverage, measures the importance of each point–the difference in win probability between a player winning it or losing it.

To put it simply, the higher the leverage of a point, the more valuable it is to win. “High leverage point” is just a more technical way of saying “big point.”  To be considered clutch, a player should be winning more high-leverage points than low-leverage points. You don’t have to win a disproportionate number of high-leverage points to be a very good player–Roger Federer’s break point record is proof of that–but high-leverage points are key to being a clutch player.

(I’m not the only person to think about these issues. Stephanie wrote about this topic in December and calculated a full-year clutch metric for the 2015 ATP season.)

To make this more concrete, I calculated win probability and leverage (LEV) for every point in the Wimbledon semifinal between Federer and Milos Raonic. For the first point of the match, LEV = 2.2%. Raonic could boost his match odds to 50.7% by winning it or drop to 48.5% by losing it. The highest leverage in the match was a whopping 32.8%, when Federer (twice) had game point at 1-2 in the fifth set. The lowest leverage of the match was a mere 0.03%, when Raonic served at 40-0, down a break in the third set. The average LEV in the match was 5.7%, a rather high figure befitting such a tight match.

On average, the 166 points that Raonic won were slightly more important, with LEV = 5.85%, than Federer’s 160, at LEV = 5.62%. Without doing a lot more work with match-level leverage figures, I don’t know whether that’s a terribly meaningful difference. What is clear, though, is that certain parts of Federer’s game fell apart when he needed them most.

By Wimbledon’s official count, Federer committed nine unforced errors, not counting his five double faults, which we’ll get to in a minute. (The Match Charting Project log says Fed had 15, but that’s a discussion for another day.) There were 180 points in the match where the return was put in play, with an average LEV = 6.0%. Federer’s unforced errors, by contrast, had an average LEV nearly twice as high, at 11.0%! The typical leverage of Raonic’s unforced errors was a much less noteworthy 6.8%.

Fed’s double fault timing was even worse. Those of us who watched the fourth set don’t need a fancy metric to tell us that, but I’ll do it anyway. His five double faults had an average LEV of 13.7%. Raonic double faulted more than twice as often, but the average LEV of those points, 4.0%, means that his 11 doubles had less of an impact on the outcome of the match than Roger’s five.

Even the famous Federer forehand looks like less of a weapon when we add leverage to the mix. Fed hit 26 forehand winners, in points with average LEV = 5.1%. Raonic’s 23 forehand winners occurred during points with average LEV = 7.0%.

Taking these three stats together, it seems like Federer saved his greatness for the points that didn’t matter as much.

The bigger picture

When we look at a handful of stats from a single match, we’re not improving much on a commentator who vaguely summarizes a performance by saying that a player didn’t win enough of the big points. While it’s nice to attach concrete numbers to these things, the numbers are only worth so much without more context.

In order to gain a more meaningful understanding of this (or any) performance with leverage stats, there are many, many more questions we should be able to answer. Were Federer’s high-leverage performances typical? Does Milos often double fault on less important points? Do higher-leverage points usually result in more returns in play? How much can leverage explain the outcome of very close matches?

These questions (and dozens, if not hundreds more) signal to me that this is a fruitful field for further study. The smaller-scale numbers, like the average leverage of points ending with unforced errors, seem to have particular potential. For instance, it may be that Federer is less likely to go for a big forehand on a high-leverage point.

Despite the dangers of small samples, these metrics allow us to pinpoint what, exactly, players did at more crucial moments. Unlike some of the more simplistic stats that tennis fans are forced to rely on, leverage numbers could help us understand the situational tendencies of every player on tour, leading to a better grasp of each match as it happens.

How Dangerous Is It To Fix a Single Service Game?

Italian translation at settesei.it

Earlier this week, I offered a rough outline of the economics of fixing tennis matches, calculating the expected prize money that players forgo at various levels when they lose on purpose. The vast gulf between prize money, especially at lower-level events, and fixing fees suggests that gamblers must pay high premiums to convince players to do something ethically repugnant and fraught with risk.

So much for match-level fixes. What about single service games? In Ben Rothenberg’s recent report, a shadowy insider offers the following data points:

Buying a service break at a Futures event cost \$300 to \$500, he said. A set was \$1,000 to \$2,000, and a match was \$2,000 to \$3,000.

In other words, a service break is valued at between 10% and 25% the cost of an entire match. The article doesn’t mention service-break prices at higher levels, so we’ll have to use the Futures numbers as our reference point.

Selling a service break might be a way to have your cake and eat it too, taking some cash from gamblers while retaining the chance to advance in the draw and earn ranking points. But it won’t always work out that way.

I ran some simulations to see how much a service break should cost, based on the simplifying assumption that prices correspond to chances of winning and, by extension, forgone prize money. It turns out that the range of 10% to 25% is exactly right.

Let’s start with the simplest scenario: Two equal men with middle-of-the-road serves, which win them 63% of service points. In an honest match, these two would each have a 50% chance of winning. If one of them guarantees a break in his second service game, he is effectively lowering his chances of winning the match to 38.5%. dropping his expected prize money for the tournament by 23%.

If our players have weaker serves, for instance each winning 55% of service points, the fixer’s chances of winning the match fall to about 42%, only a 16% haircut. With stronger serves, using the extreme case of 70% of points going the way of the server, the fixer’s chances drop to 34%, a loss of 32% in his expected prize money.

This last scenario–two equal players with big serves–is the one that confers the most value on a single service break. We can use that 32% sacrifice as an upper bound for the worth of a single fixed break.

Fixed contests have more value to gamblers when the better player is guaranteed to lose, and in those cases, a service break doesn’t have as much impact on the outcome of the match. If the fixer is considerably better than his opponent, he was probably going to break serve a few times more than his opponent would, so losing a single game is less likely to determine the outcome of the match.

Let’s take a few examples:

• If one player wins 64% of service points and other wins 62%, the favorite has a 60% chance of winning. If he fixes one service break, his chances of winning fall to just below 48%, about a 20% drop in expected prize money.
• When one player wins 65% of service points against an opponent winning 61%, his chances in an honest match are 69.3%. Giving up one fixed service break, his odds fall to 57.4%, a sacrifice of roughly 17%.
• A 67% server facing a 60% server has an 80.8% chance of winning. With one fixed service break, that drops to 70.7%, a loss of 12.5%.
• A huge favorite winning 68% of service points against his opponent’s 58% has an 89.5% chance of advancing to the next round. Guarantee a break in one of his service games, and his odds drop to 82%, a loss of 8.4%.

With the exception of very lopsided matches (for which there might not be as many betting markets), we have our lower bound, not far below 10%.

The average Futures first-rounder, if we can generalize from such a mixed bag of matches, is somewhere in the middle of those examples–not an even contest, but without a heavy favorite. So the typical value of a fixed service break is between about 12% and 20% of the value of the match, right in the middle of the range of estimates given by Rothenberg’s source.

Even in this hidden, illegal marketplace, the numbers we’ve seen so far suggest that both gamblers and players act reasonably rationally. Amid a sea of bad news, that’s a good sign for tennis’s governing bodies: It promises that players will respond in a predictable manner to changing incentives. Unfortunately, it remains to be seen whether the incentives will change.