Andy Murray and the Longest Break-Per-Match Streaks

Among Andy Murray‘s many accomplishments in 2016, he achieved an impressive–though obscure–feat. In each one of his 87 matches, he broke serve at least once. He has broken at least once per match since failing to do so against Roger Federer in the 2015 Cincinnati semifinals, for an active streak of 107 matches.

Where does that place him among the greats of men’s tennis? Just how unusual is it to break serve in every match for an entire season? As is the case with too many tennis statistics, we don’t know. Someone finds an impressive-sounding stat, and that’s the end of the story. We can’t always fix that, but in this case, we can add some context to Murray’s accomplishment.

Full break-per-match seasons

I’ve collected break stats for matches back to 1991, though we need to keep in mind that there are some mistakes in the 1990s data. Further, Davis Cup presents a problem, as it is excluded entirely. Sometimes we can tell from the scoreline that a player broke serve–as with all of Murray’s Davis Cup matches this year–but often we cannot. I’ll have more to say about that in specific cases below.

Since 1991, there have been at least 14, and perhaps as many as 20 instances in which a player broke serve in every match of a season. (Minimum 40 tour-level matches, and I’ve excluded retirements when calculating both minimums and the streaks themselves.) I say “instances” because several players–Andre Agassi, Lleyton Hewitt, Rafael Nadal, and Nikolay Davydenko–pulled it off more than once. Hewitt’s 2001 season had the most matches–95–of any of them, followed by Murray’s 2016 and Nadal’s 2005, at 87 each.

Here is the complete list:

Player                  Season  Matches  (Unsure)  
Andy Murray               2016       87         0  
Juan Monaco               2014       41         0  
Novak Djokovic            2013       83         0  
Rafael Nadal              2010       79         0  
Nikolay Davydenko         2008       73         0  
Nikolay Davydenko         2007       82         0  
Lleyton Hewitt            2006       46         0  
Rafael Nadal              2005       87         0  
David Nalbandian          2005       63         0  
Andre Agassi              2003       55         0  
Lleyton Hewitt            2001       95         0  
Lleyton Hewitt            2000       76         1  
Hernan Gumy               1997       53         1  
Alex Corretja             1997       67         0  
Andre Agassi              1995       81         0  
Magnus Gustafsson         1994       40         0  
Carlos Costa              1992       60         0  
Guillermo Perez Roldan    1991       40         2  
Ivan Lendl                1991       72         0  
Boris Becker              1991       61         2

(The “Unsure” column indicates how many matches are missing stats and may not have included a break of serve.)

Several more players came close. Federer broke serve in all but one match in three separate seasons. Agassi, Novak Djokovic, David Ferrer, and Thomas Muster all did so twice.

We shouldn’t be surprised that so many players–especially the greats–have broken so often. It’s very rare to win a match without breaking serve: Of the 2,570 ATP tour-level matches from this season for which I have match stats, the winner broke serve in all but 30 of them. Even losers break serve in more than two out of every three matches: In 2016, the loser broke serve in 1,843 of the 2,570 matches, 72% of the time.

Still, there are enough dominant servers on tour that it is difficult to last an entire season without being shut out of the break column. In 1995, Muster broke serve in 99 matches, but failed to do so when he drew the big-serving (and completely unheralded) qualifier TJ Middleton on the carpet in St. Petersburg. Murray’s current streak is all the more impressive because, in his 107 matches, he has faced Milos Raonic six times, John Isner four times, Kevin Anderson and Nick Kyrgios twice each, and Ivo Karlovic once. Given the chance, he probably would’ve broken TJ Middleton as well.

Break-per-match streaks

For Murray to surpass the longest streaks in this category, it will take several more months of high-quality returning. As we saw above, Davydenko and Hewitt may have gone two full years breaking serve in every match they played. In both cases, the lack of ITF data makes their records unclear, but regardless of those details, Davydenko has set an extremely high bar.

Here are all the break-per-match streaks of 100 or more matches since 2000:

Player             Start   End  Streak  Possible  
Nikolay Davydenko   2006  2009     159       182  
Rafael Nadal        2004  2006     156            
Rafael Nadal        2009  2011     146            
Andre Agassi        2002  2004     143            
Novak Djokovic      2012  2014     127            
Lleyton Hewitt      1999  2002     124       230  
Andy Murray         2015  2016     107         ∞  
David Nalbandian    2004  2006     104

This season, Murray didn’t play his 53rd match until August at the Olympics; he’ll need to break serve at least once in that many matches to reach the top of this list.

The exact length of Davydenko’s streak hinges on his 2008 Davis Cup semifinal match against Juan Martin del Potro, which he lost in straight sets. If he broke serve in that match, his streak stretched into early 2009, spanning 182 matches.

(Edit: Thanks to Andrew Moss, we now know that Davydenko did break serve in that match, according to this contemporaneous report.)

Hewitt’s best streak is even more unclear. I don’t have break stats for his 6-3 6-3 loss to Max Mirnyi at the 2000 Olympics. If he didn’t break Mirnyi–a definite possibility, given The Beast’s serving prowess–the streak is “only” 124 matches. If he did, the streak is at least 187, and the exact length depends on more unknowns, including both of his singles matches in the 1999 Davis Cup final against France.

(Edit #2! Thanks to Carl, we know that Hewitt broke Mirnyi, so his streak is at least 187 matches. The next issue is his last match of the 1999 season, a dead rubber against Sebastian Grosjean in that year’s Davis Cup final. Hewitt lost 6-4 6-3, but Grosjean was hardly an overpowering server. Hewitt lost his previous Davis Cup match in straight sets as well, a live rubber against Cedric Pioline, and a match report establishes that Hewitt broke serve. If he broke Grosjean, the streak stretches back to April 1999, and numbers the full 230 matches.)

In any case, Murray has already earned himself a place among the greatest returners in modern tennis. In 2017, we’ll see just how far he can climb this list.

Why Novak Djokovic is Still Number One

Two weeks ago, Andy Murray took over the ATP #1 ranking from Novak Djokovic. Yesterday, he defeated Djokovic in their first meeting since June, securing his place at the top of the year-end ranking table. Murray has been outstanding in the second half of this season, winning all but three of his matches since the Roland Garros final, and he capped the year in style, beating four top-five players to claim the title at the World Tour Finals.

Despite all that, Murray is not the best player in the world. That title still belongs to Djokovic. Since June, Murray has closed the gap, establishing himself as part of what we might call the “Big Two,” but he hasn’t quite ousted his rival. There’s no question that over this period, Murray has played better–that sort of thing is occasionally debatable, but this season it’s just historical fact–but identifying the best player implies something more predictive, and it’s much more difficult to determine by simply looking over a list of recent results.

The ATP rankings generally do a good job of telling us which players are better than others. But the official system has two major problems: It ignores opponent quality, and it artificially limits its scope to the last 52 weeks. Pundits and fans tend to have different problems: They often give too much credit to opponent quality (“He beat Djokovic, so now he’s number one!”) and exhibit an even more extreme recency bias (“He’s looked unbeatable this week!”).

Two systems that avoid these issues–Elo and Jrank–both place Djokovic comfortably ahead of Murray. These algorithms handle the details of recent matches and opponent quality differently from each other, but what they share in common is more important: They consider opponent quality and they don’t use an arbitrary time cutoff like the ATP ranking system does.

Here’s how the three methods would forecast a Djokovic-Murray match, were it held today:

  • ATP: Murray favored, 51.6% chance of winning
  • Elo: Djokovic favored, 61.6% chance of winning
  • Jrank: Djokovic favored, 57.0% chance of winning

Betting markets favored Djokovic by a margin of slightly more than 60/40 yesterday, though bettors probably gave him some of that edge because they thought Murray would be fatigued after his marathon match on Saturday.

As I wrote last week, Elo doesn’t deny that Murray has had a tremendous half-season. Instead, it gives him less credit than the official algorithm does for victories over lesser opponents (such as John Isner in the Paris Masters final), and it recognizes that he started his current run of form at an enormous disadvantage. With his title in London, Murray reached a new peak Elo rating, but it still isn’t enough to overtake Djokovic.

Even though Elo still prefers Novak by a healthy margin, it reflects how much the situation at the top of the ranking list has changed. At the beginning of 2016, Elo gave Djokovic a 76.5% chance of winning a head-to-head against Murray, and that probability rose as high as 81% in April. It fell below 70% after the Olympics, and the gap is now the smallest it has been since February 2011.

Last week illustrates how difficult it will be for Murray take over the #1 Elo ranking place. The pre-tournament Elo difference of 91 points between the two players has shrunk by only 8%, to 84 points. Murray’s win yesterday was worth a bit more than a measly seven points, but Djokovic had several opportunities to nudge his rating upwards in his first four matches, as well. Despite some of Novak’s head-scratching losses this fall, he still wins most of his matches–some of them against very good players–slowing the decline of his Elo rating.

Of course, Elo is just a measuring stick–like any ranking system, it doesn’t tell us what’s really happening on court. It’s possible that Murray has made a significant (and semi-permanent) leap forward or that Djokovic has taken a major step back. On the other hand, streaks happen even without such leaps, and they always end. The smart money is usually on small, gradual changes to the status quo, and Elo gives us a way to measure those changes.

For Elo to rate Murray ahead of Djokovic, it will probably require several more months of these gradual changes. The only faster alternative is for Djokovic to start losing more matches to the likes of Jiri Vesely and Sam Querrey. When faced with dramatic evidence, Elo makes more dramatic changes. While Djokovic has occasionally provided that evidence this season, he has usually offered enough proof–like four wins at the World Tour Finals–to comfortably maintain his position at the top.

Factchecking the History of the ATP Number One With Elo

As I wrote at The Economist this week, Andy Murray might sit atop the ATP rankings, but he probably isn’t the best player in tennis right now. That honor still belongs to Novak Djokovic, who comes in higher on the Elo ranking list, which uses an algorithm that is more predictive of match outcomes than the ATP table.

This isn’t the first time Elo has disagreed with the official rankings over the name at the top. Of the 26 men to have reached the ATP number one ranking, only 18 also became number one on the Elo list. A 19th player, Guillermo Coria, was briefly Elo #1 despite never achieving the same feat on the ATP rankings.

Four of the remaining eight players–Murray, Patrick Rafter, Marcelo Rios, and John Newcombe–climbed as high as #2 in the Elo rankings, while the last four–Thomas Muster, Carlos Moya, Marat Safin, and Yevgeny Kafelnikov–only got as high as #3. Moya and Kafelnikov are extreme cases of the rankings mismatch, as neither player spent even a single full season inside the Elo top five.

By any measure, though, Murray has spent a lot of time close to the top spot. What makes his current ascent to the #1 spot so odd is that in the past, Elo thought he was much closer. Despite his outstanding play over the last several months, there is still a 100-point Elo gap between him and Djokovic. That’s a lot of space: Most of the field at the WTA Finals in Singapore this year was within a little more than a 100-point range.

January 2010 was the Brit’s best shot. At the end of 2009, Murray, Djokovic, and Roger Federer were tightly packed at the top of the Elo leaderboard. In December, Murray was #3, but he trailed Fed–and the #1 position–by only 25 points. In January, Novak took over the top spot, and Murray closed to within 16 points–a small enough margin that one big upset could make the difference. Altogether, Murray has spent 63 weeks within 100 points of the Elo top spot, none of those since August 2013.

For most of the intervening three-plus years, Djokovic has been steadily setting himself apart from the pack. He reached his career Elo peak in April of this season, opening up a lead of almost 200 points over Federer, who was then #2, and 250 points over Murray. Since Roland Garros, Murray has closed the gap somewhat, but his lack of opportunities against highly-rated players has slowed his climb.

If Murray defeats Djokovic in the final this week in London, it will make the debate more interesting, not to mention secure the year-end ATP #1 ranking for the Brit. But it won’t affect the Elo standings. When two players have such lengthy track records, one match doesn’t come close to eliminating a 100-point gap. Novak will end the season as Elo #1, and he is well-positioned to maintain that position well into 2017.

Dominic Thiem and the Best Deciding-Sets Seasons in ATP History

Yesterday at the ATP World Tour Finals, Dominic Thiem won a three-set match against Gael Monfils, his 22nd deciding-set victory of 2016. Despite losing to Novak Djokovic in three sets on Sunday, Thiem is enjoying one of the best deciding-set seasons in ATP history.

The loss to Djokovic was only Thiem’s third in 25 deciding sets this year. He began the season with 14 consecutive deciding-set wins, including back-to-back third-set tiebreaks in Buenos Aires against Rafael Nadal and Nicolas Almagro. He strung together another seven straight between May and September, including a grass-court upset of Roger Federer in Stuttgart.

Among players who contested at least 20 deciding sets in a season, Thiem’s winning percentage of 88% is the fifth-best record in the ATP’s modern era. Not every player reaches the 20-decider threshold–some, like Djokovic, avoid it by winning most of their matches in straight sets–but it’s no statistical oddity. There have been nearly 1,000 player-seasons with at least 20 deciders since the 1970s, including Andy Murray’s 17-6 record in 2016.

Outstanding single-season deciding-set records don’t guarantee long-term success. Thiem appears on this list amid a mix of famous and lesser-known names, from Federer to Onny Parun:

Player           Year  Deciders  Wins  Win Perc  
Mario Ancic      2006        24    22     91.7%  
Ilie Nastase     1971        23    21     91.3%  
Tom Okker        1974        20    18     90.0%  
Roger Federer    2006        20    18     90.0%  
Dominic Thiem    2016        25    22     88.0%  
Kei Nishikori    2014        24    21     87.5%  
Stan Smith       1972        22    19     86.4%  
Joakim Nystrom   1984        22    19     86.4%  
Guillermo Vilas  1977        29    25     86.2%  
Onny Parun       1975        34    29     85.3%

Parun’s 1975 season is particularly notable, as no other player has won so many deciding sets in a single year. In 1996, Yevgeny Kafelnikov came close, winning 28. One gets the idea he was trying: He played 105 matches that year, 40 of which went the distance. In more recent years, big names have played more limited schedules, and Thiem is the only active player to win at least 22 deciding sets in a single season. Dmitry Tursunov gave it a shot in 2006, playing 37 deciders, but he won only 20.

Like so many tennis stats, this one can be fluky. For every Kei Nishikori–who has won an incredible 77% of deciding sets at tour level, including some record-setting streaks--there is a Grigor Dimitrov, who won 18 of 22 deciding sets in 2014, then barely broke even the following year, claiming only 11 of 21. Of the 27 players who have posted a 20-decider, 80% winning percentage season, not a single one managed an 80% winning percentage the following year.

For all of his talents, Thiem probably won’t follow in Nishikori’s footsteps. The Austrian won only half of his 40 deciding sets before this season. But a more modest record in these matches is hardly insurmountable. In 1996, Pete Sampras put together his best deciding-sets record, winning 83% of his 24 deciders. The following year, his record fell to a pedestrian 56%, which didn’t keep him from winning two Grand Slams and finishing the season at the top of the rankings.

If Thiem is to continue climbing the rankings, he’ll be better off taking Djokovic’s path, winning most of his deciding sets, but playing them much less frequently. In the last decade, Novak has played 20 deciding sets in a season only three times, and he has only gone the distance 10 times in 2016. Even Nishikori would have to agree: Djokovic’s method is working just fine.

Forecasting the 2016 ATP World Tour Finals

Andy Murray is the #1 seed this week in London, but as I wrote for The Economist, Novak Djokovic likely remains the best player in the world. According to my Elo ratings, he would have a 63% chance of winning a head-to-head match between the two. And with the added benefit of an easier round-robin draw, the math heavily favors Djokovic to win the tournament.

Here are the results of a Monte Carlo simulation of the draw:

Player        SF      F      W  
Djokovic   95.3%  73.9%  54.6%  
Murray     86.3%  58.3%  29.7%  
Nishikori  60.4%  24.9%   7.8%  
Raonic     50.9%  16.3%   3.3%  
Wawrinka   29.4%   7.8%   1.6%  
Monfils    33.2%   8.7%   1.4%  
Cilic      23.9%   5.8%   1.1%  
Thiem      20.7%   4.1%   0.5%

I don’t think I’ve ever seen a player favored so heavily to progress out of the group stage. Murray’s 86% chance of doing so is quite high in itself; Novak’s 95% is otherworldly. His head-to-heads against the other players in his group are backed up by major differences in Elo points–Dominic Thiem is a lowly 15th on the Elo list, given only a 7.4% chance of beating the Serb.

If Milos Raonic is unable to compete, Djokovic’s chances climb even higher. Here are the probabilities if David Goffin takes Raonic’s place in the bracket:

Player        SF      F      W  
Djokovic   96.8%  75.2%  55.4%  
Murray     86.2%  60.7%  30.6%  
Nishikori  60.7%  26.3%   8.1%  
Monfils    47.7%  12.4%   1.8%  
Wawrinka   29.3%   8.5%   1.7%  
Cilic      23.8%   6.2%   1.1%  
Thiem      29.5%   5.8%   0.7%  
Goffin     26.0%   4.9%   0.5%

The luck of the draw was on Novak’s side. I ran another simulation with Djokovic and Murray swapping groups. Here, Djokovic is still heavily favored to win the tournament, but Murray’s semifinal chances get a sizable boost:

Player        SF      F      W  
Djokovic   92.8%  75.1%  54.9%  
Murray     90.9%  58.1%  29.8%  
Nishikori  58.4%  26.9%   7.5%  
Raonic     52.3%  14.3%   3.3%  
Wawrinka   26.9%   8.4%   1.6%  
Monfils    35.3%   7.5%   1.4%  
Cilic      21.9%   6.2%   1.0%  
Thiem      21.6%   3.4%   0.5%

Elo rates Djokovic so highly that he is favored no matter what the draw. But the draw certainly helped.

Doubles!

I’ve finally put together a sufficient doubles dataset to generate Elo ratings and tournament forecasts for ATP doubles. While I’m not quite ready to go into detail, I can say that, by using the Elo algorithm and rating players individually, the resulting forecasts outperform the ATP rankings about as much as singles Elo ratings do.

Here is the forecast for the doubles event at the World Tour Finals:

Team               SF      F      W  
Herbert/Mahut   76.4%  49.5%  32.1%  
Bryan/Bryan     68.7%  36.8%  19.9%  
Kontinen/Peers  55.7%  29.1%  13.8%  
Dodig/Melo      58.4%  28.1%  13.2%  
Murray/Soares   48.3%  20.8%   8.6%  
Lopez/Lopez     37.7%  16.4%   6.2%  
Klaasen/Ram     30.2%  11.9%   4.0%  
Huey/Mirnyi     24.6%   7.3%   2.2%

This distribution is more like what round-robin forecasts usually look like, without a massive gap between the top of the field and the rest. Pierre-Hugues Herbert and Nicolas Mahut are the top rated team, followed closely by Bob Bryan and Mike Bryan. Max Mirnyi was, at his peak, one of the highest Elo-rated doubles players, but his pairing with Treat Huey is the weakest of the bunch.

The men’s doubles bracket has some legendary names, along with some players–like Herbert and Henri Kontinen–who may develop into all-time greats, but it has no competitors who loom over the rest of the field like Murray and Djokovic do in singles.

How To Keep Round Robin Matches Interesting, Part Two

Earlier this week, I published a deep dive into the possible outcomes of four-player round robin groups and offered an ideal schedule that would minimize the likelihood of dead rubbers on the final day. I’ve since heard from a few readers who pointed out two things:

  1. You might do better if you determined the schedule for day two after getting the results of the first two matches.
  2. Major tournaments such as the ATP and WTA Tour Finals already do this, pairing the winners of the first two matches and the losers of the first two matches on day two.

This is an appealing idea. You’re guaranteed to end the second day with one undefeated (2-0) player, two competitors at 1-1, and the last at 0-2. The two participants at 1-1 have everything to play for, and depending on day three’s schedule and tiebreak factors, the 0-2 player could still be in the running as well.

Best of all, you avoid the nightmare scenario of two undefeated players and two eliminated players, in which the final two matches are nearly meaningless.

However, this “contingent schedule” approach isn’t perfect.

Surprise, surprise

We learned in my last post that, if we set the entire schedule before play begins, the likelihood of a dead rubber on the final day is 17%, and if we choose the optimal schedule, leaving #4 vs #1 and #3 vs #2 for the final day, we can drop those chances as low as 10.7%.

(These were based on a range of player skill levels equivalent to 200 points on the Elo scale. The bigger the range of player skills–for instance, the ATP finals is likely to have a group with a range well over 300–the more dramatic the differences in these numbers.)

In addition, we discovered that “dead/seed” matches–those in which one player is already eliminated and the other can only affect their semifinal seeding–are even more common. When the schedule is chosen in advance, the probability of a dead rubber or a “dead/seed” match is always near 40%.

If the day two schedule is determined by day one outcomes, the overall likelihood of these “mostly meaningless” (dead or “dead/seed”) matches drops to about 30%. That’s a major step in the right direction.

Yet there is a drawback: The chances of a dead rubber increase! With the contingent day two schedule, there is a roughly 20% chance of a completely meaningless match on day three.

Our intuition should bear this out. After day two, we are guaranteed one 2-0 player and one 0-2 player. It is somewhat likely that these two have faced each other already, but there still remains a reasonable chance they will play on day three. If they do, the 0-2 player is already eliminated–there will be two 2-1 players at the end of day three. The 2-0 player has clinched a place in the semifinals, so the most that could be at stake is a semifinal seeding.

In other words, if the “winner versus winner” schedule results in a 2-0 vs 0-2 matchup on day three, the odds are that it’s meaningless. And this schedule often does just that.

The ideal contingent schedule

If the goal is to avoid dead rubbers at all costs, the contingent schedule is not for you. You can do a better job by properly arranging the schedule in advance. However, a reasonable person might prefer the contingent schedule because it completely avoids the risk of the low-probability “nightmare scenario” that I described above, of two mostly meaningless day three matches.

Within the contingent schedule, there’s still room for optimization. If the day one slate consists of matches setting #1 against #3 and #2 against #4 (sorted by ranking), the probability of a meaningless match on day three is about average. If day one features #1 vs #2 and #3 vs #4, the odds are even higher: about a 21% chance of a dead rubber and another 11% chance of a “dead/seed” match.

That leaves us with the optimal day one schedule of #1 vs #4 and #2 vs #3. It lowers the probability of a dead rubber to 19% and the chances of a “dead/seed” match to 9.7%. Neither number represents a big difference, but given all the eyes on every match at major year-end events, it seems foolish not to make a small change in order to maximize the probability that both day three matches will matter.

How To Keep Round Robin Matches Interesting

Round robins–such as the formats used by the ATP and WTA Tour Finals–have a lot going for them. Fans are guaranteed at least three matches for every player, and competitors can recover from one (or even two) bad outings. Best of all, when compared to a knockout-style draw, it’s twice as much tennis.

On the other hand, round robins have one major drawback: They can result in meaningless matches. It’s fairly common that, after two matches, a player is guaranteed a spot in the semifinals (sometimes even a specific seed) or eliminated from contention altogether. At a high-profile event such as the Tour Finals, with sky-high ticket prices, do we really want to run the risk of dead rubbers?

I don’t claim to have the answer to that question. However, we can take a closer look at the round robin format to answer several relevant questions. What is the probability that the final day of a four-player group will include at least one dead rubber? What about the final match? And most importantly, before the event begins, can we set the schedule in such a way to minimize the likelihood of dead rubbers?

The range of possibilities

As a first step, let’s determine all of the possible outcomes of the first four matches in a four-player round robin group. For convenience, I’ll refer to the players as A, B, C, and D. The first day features two matches, A vs B and C vs D. The second day is A vs C and B vs D, leaving us with a final day of A vs D and B vs C.

Each match has four possible outcomes: the first player wins in two sets, the first player wins in three, the second wins in two, or the second wins in three. (Sets won are important because they are used as a tiebreaker when, for instance, three players win two matches apiece.) Thus, there are 4 x 4 x 4 x 4 = 256 possible arrangements of the group standings entering the final day of round robin play.

Of those 256 permutations, 32 of them (12.5%) include one dead rubber on the final day. In those cases, the other match is played only to decide semifinal seeding between the players who will advance. Another 32 of the 256 permutations involve one “almost-dead” match, between a player who has been eliminated and a player who is competing only to determine semifinal seeding.

In other words, one out of every four possible outcomes of the first two days results in a day three match that is either entirely or mostly meaningless. Later on, we’ll dig into the probability that these outcomes occur, which depends on the relative skill levels of the four players in the group.

Before we do that, let’s take a little detour to define our terms. Because of the importance of semifinal seeding, some dead rubbers are less dead than others. Further, it is frequently the case that one player in a match still has a shot at the semifinals and the other doesn’t. Altogether, from “live” to “dead,” there are six gradations:

  1. live/live — both players are competing to determine whether they survive
  2. live/seed — one player could advance or not; the other will advance, and is playing to try to earn the #1 group seed
  3. live/dead — one player is trying to survive; the other is eliminated
  4. seed/seed — both players will advance; the winner gets the #1 group seed
  5. seed/dead — one player is in the running for the #1 seed; the other is eliminated
  6. dead/dead — both players are eliminated

All else equal, the higher a match lies on that scale, the more engaging its implications for the tournament. For the remainder of this article, I’ll refer only to the “dead/dead” category as “dead rubbers,” though I will occasionally discuss the likelihood of “dead/seed” matches as well. I’ll assume that the #1 seed is always more desirable than #2 and ignore the fascinating but far-too-complex ramifications of situations in which a player might prefer the #2 spot.

The sixth match

As we’ve seen, there are many sequences of wins and losses that result in a dead rubber on day three. Once the fifth match is played, it is even more likely that the seedings have been determined, making the sixth match meaningless.

After five matches, there are 1,024 possible group standings. (256 permutations after the first four matches, multiplied by the four possible outcomes of the fifth match.) Of those, 145 (14.1%) result in a dead sixth rubber, and another 120 (11.7%) give us a “dead/seed” sixth match.

We haven’t yet determined how likely it is that we’ll arrive at the specific standings that result in dead sixth rubbers. So far, the important point is that dead rubbers on day three aren’t just flukes. In a four-player round robin, they are always a real possibility, and if there is way to minimize their likelihood, we should jump at the chance.

Real scenarios, really dead rubbers

To figure out the likelihood of dead rubbers in practical situations, like the ATP and WTA Tour Finals, I used a hypothetical group of four players with Elo ratings spread over a 200-point range.

Why 200? This year’s Singapore field was very tightly packed, within a little bit more than 100 points, implying that the best player, Angelique Kerber, had about a 65% chance of beating the weakest, Svetlana Kuznetsova. By contrast, the ATP finalists in London are likely to be spread out over a 400-point range, giving the strongest competitor, Novak Djokovic, at least a 90% edge over the weakest.

I’ve given our hypothetical best player a rating of 2200, followed by a field of one player at 2130, one at 2060, and one at 2000. Thus, our favorite has a 60% chance of beating the #2 seed, a 69% chance of defeating the #3 seed, and a 76% chance of besting the #4 seed.

For any random arrangement of the schedule, after the first two days of play, this group has a 17% chance of giving us a dead rubber on day three, plus a 23% chance of a “dead/seed” match on day three.

After the fifth match is contested, there is a 16% chance of that the sixth match is meaningless, with an additional 12% chance that the sixth match falls into the “dead/seed” category.

The wider the range of skill levels, the higher the probability of dead rubbers. This is intuitive: The bigger the range between the top and bottom, the more likely that the best player will win their first two matches–and the more likely they will be straight-setters. Similarly, the chances are higher that the weakest player will lose theirs. The higher the probability that players go into day three with 2-0 or 0-2 records, the less likely that day three matches have an impact on the outcome of the group.

How to schedule a round robin group

A 17% chance of a dead rubber on day three is rather sad. But there is a bright spot in my analysis: By rearranging the schedule, you can raise that probability as high as 24.7% … or drop it as low as 10.7%.

Remember that our schedule looks like this:

Day one: A vs B, C vs D

Day two: A vs C, B vs D

Day three: A vs D, B vs C

We get the lowest possible chance of a day three dead rubber if we put the players on the schedule in order from weakest to strongest: A is #4, B is #3, and so on:

Day one: #4 vs #3, #2 vs #1

Day two: #4 vs #2, #3 vs #1

Day three: #4 vs #1, #3 vs #2

There is a small drawback to our optimal arrangement: It increases the odds of a “dead/seed” match. It turns out that you can only optimize so much: No matter what the arrangement of the competitors, the probability of a “dead/dead” or “dead/seed” match on day three stays about the same, between 39.7% and 41.7%. While neither type of match is desirable, we’re stuck with a certain likelihood of one or the other, and it seems safe to assume that a “dead/seed” rubber is better than a totally meaningless one.

Given how much is at stake, I hope that tournament organizers heed this advice and schedule round robin groups in order to minimize the chances of dead rubbers. The math gets a bit hairy, but the conclusions are straightforward and dramatic enough to make it clear that scheduling can make a difference. Over the course of the season, almost every tennis match matters–it would be nice if every match at the Tour Finals did, too.

(I wrote more about this, which you can read here.)