# Quantifying Cakewalks, or The Time Rafa Finally Got Lucky

During this year’s US Open, much has been made of some rather patchy sections of the draw. Many great players are sitting out the tournament with injury, and plenty of others crashed out early. Pablo Carreno Busta reached the quarterfinals by defeating four straight qualifiers, and Rafael Nadal could conceivably win the title without beating a single top-20 player.

None of this is a reflection on the players themselves: They can play only the draw they’re dealt, and we’ll never know how they would’ve handled a more challenging array of opponents. The weakness of the draw, however, could affect how we remember this tournament.  If we are going to let the quality of the field color our memories, we should at least try to put this year’s players in context to see how they compare with majors in the past.

How to measure draw paths

There are lots of ways to quantify draw quality. (There’s an entire category on this blog devoted to it.) Since we’re interested in the specific sets of opponents faced by our remaining contenders, we need a metric that focuses on those. It doesn’t really matter that, say, Nick Kyrgios was in the draw, since none of the semifinalists had to play him.

Instead of draw difficulty, what we’re after is what I’ll call path ease. It’s a straightforward enough concept: How hard is it to beat the specific set of guys that Rafa (for instance) had to play?

To get a number, we’ll need a few things: The surface-weighted Elo ratings of each one of a player’s opponents, along with a sort of “reference Elo” for an average major semifinalist. (Or finalist, or title winner.) To determine the ease of Nadal’s path so far, we don’t want to use Nadal’s Elo. If we did that, the exact same path would look easier or harder depending on the quality of the player who faced it.

(The exact value of the “reference Elo” isn’t that important, but for those of you interested in the numbers: I found the average Elo rating of every slam semifinalist, finalist, and winner back to 1988 on each of the three major surfaces. On hard courts, those numbers are 2145, 2198, and 2233, respectively. When measuring the difficulty of a path to the semifinal round, I used the first of those numbers; for the difficulty of a path to the title, I used the last.)

To measure path ease, then, we answer the question: What are the odds that an average slam semifinalist (for instance) would beat this particular set of players? In Rafa’s case, he has yet to face a player with a weighted-hard-court Elo rating above 1900, and the typical 2145-rated semifinalist would beat those five players 71.5% of the time. That’s a bit easier than Kevin Anderson‘s path the semis, but a bit harder than Carreno Busta’s. Juan Martin del Potro, on the other hand, is in a different world altogether. Here are the path ease numbers for all four semifinalists, showing the likelihood that average contenders in each round would advance, giving the difficulty of the draws each player has faced:

```Semifinalist   Semi Path  Final Path  Title Path
del Potro           9.1%        7.5%       10.0%
Anderson           69.1%       68.9%       47.1%
Carreno Busta      74.3%       71.2%       48.4%```

(We don’t yet know each player’s path to the title, so I averaged the Elos of possible opponents. Anderson and Carreno Busta are very close, so for Rafa and Delpo, their potential final opponent doesn’t make much difference.)

There’s one quirk with this metric that you might have noticed: For Nadal and del Potro, their difficulty of reaching the final is greater than that of winning the title altogether! Obviously that doesn’t make logical sense–the numbers work out that way because of the “reference Elos” I’m using. The average slam winner is better than the average slam finalist, so the table is really saying that it’s easier for the average slam winner to beat Rafa’s seven opponents than it would be for the average slam finalist to get past his first six opponents. This metric works best when comparing title paths to title paths, or semifinal paths to semifinal paths, which is what we’ll do for the rest of this post.

Caveats and quirks aside, it’s striking just how easy three of the semifinal paths have been compared to del Potro’s much more arduous route. Even if we discount the difficulty of beating Roger Federer–Elo thinks he’s the best active player on hard courts but doesn’t know about his health issues–Delpo’s path is wildly different from those of his semifinal and possible final opponents.

Cakewalks in context

Semifinalist path eases of 69% or higher–that is, easier–are extremely rare. In fact, the paths of Anderson, Carreno Busta, and Nadal are all among the ten easiest in the last thirty years! Here are the previous top ten:

```Year  Slam             Semifinalist               Path Ease
1989  Australian Open  Thomas Muster                  84.1%
1989  Australian Open  Miloslav Mecir                 74.2%
1990  Australian Open  Ivan Lendl                     73.8%
2006  Roland Garros    Ivan Ljubicic                  73.7%
1988  Australian Open  Ivan Lendl                     72.2%
1988  Australian Open  Pat Cash                       70.1%
2004  Australian Open  Juan Carlos Ferrero            69.2%
1996  US Open          Michael Chang                  68.8%
1990  Roland Garros    Andres Gomez                   68.4%
1996  Australian Open  Michael Chang                  66.2%```

In the last decade, the easiest path to the semifinal was Stan Wawrinka‘s route to the 2016 French Open final four, which rated 59.8%. As we’ll see further on, Wawrinka’s draw got a lot more difficult after that.

Del Potro’s draw so far isn’t quite as extreme, but it is quite difficult in the historical context. Of the nearly 500 major semifinalists since 1988, all but 15 are easier than his 9.1% path difficulty. Here are the top ten, all of whom faced draws that would have given the average slam semifinalist less than an 8% chance of getting that far:

```Year  Slam             Semifinalist              Path Ease
2009  Roland Garros    Robin Soderling                1.6%
1988  Roland Garros    Jonas Svensson                 1.9%
2017  Wimbledon        Tomas Berdych                  3.7%
1996  Wimbledon        Richard Krajicek               6.4%
2011  Wimbledon        Jo Wilfried Tsonga             6.6%
2012  US Open          Tomas Berdych                  6.8%
2017  Roland Garros    Dominic Thiem                  6.9%
2014  Australian Open  Stan Wawrinka                  7.0%
1989  Roland Garros    Michael Chang                  7.1%
2017  Wimbledon        Sam Querrey                    7.5%```

Previewing the history books

In the long term, we’ll care a lot more about how the 2017 US Open champion won the title than how he made it through the first five rounds. As we saw above, three of the four semifinalists have a path ease of around 50% to win the title–again, meaning that a typical slam winner would have a roughly 50/50 chance of getting past this particular set of seven opponents.

No major winner in recent memory has had it so easy. Nadal’s path would rate first in the last thirty years, while Carreno Busta’s or Anderson’s would rate in the top five. (If it comes to that, their exact numbers will depend on who they face in the final.) Here is the list that those three men have the chance to disrupt:

```Year  Slam             Winner                  Path Ease
2002  Australian Open  Thomas Johansson            48.1%
2001  Australian Open  Andre Agassi                47.6%
1999  Roland Garros    Andre Agassi                45.6%
2000  Wimbledon        Pete Sampras                45.3%
2006  Australian Open  Roger Federer               44.5%
1997  Australian Open  Pete Sampras                44.4%
2003  Australian Open  Andre Agassi                43.9%
1999  US Open          Andre Agassi                41.5%
2002  Wimbledon        Lleyton Hewitt              39.9%
1998  Wimbledon        Pete Sampras                39.1%```

At the 2006 Australian Open, Federer lucked into a path that was nearly as easy as Rafa’s this year. His 2003 Wimbledon title just missed the top ten as well. By comparison, Novak Djokovic has never won a major with a path ease greater than 18.7%–harder than that faced by more than half of major winners.

Nadal has hardly had it easy as he has racked up his 15 grand slams, either. Here are the top ten most difficult title paths:

```Year  Slam             Winner                Path Ease
2014  Australian Open  Stan Wawrinka              2.2%
2015  Roland Garros    Stan Wawrinka              3.1%
2016  Us Open          Stan Wawrinka              3.2%
2013  Roland Garros    Rafael Nadal               4.4%
2014  Roland Garros    Rafael Nadal               4.7%
1989  Roland Garros    Michael Chang              5.0%
2012  Roland Garros    Rafael Nadal               5.2%
2016  Australian Open  Novak Djokovic             5.4%
2009  US Open          J.M. Del Potro             5.9%
1990  Wimbledon        Stefan Edberg              6.2%```

As I hinted in the title of this post, while Nadal got lucky in New York this year, it hasn’t always been that way. He appears three times on this list, facing greater challenges than any major winner other than Wawrinka the giant-killer.

On average, Rafa’s grand slam title paths haven’t been quite as harrowing as Djokovic’s, but compared to most other greats of the last few decades, he has worked hard for his titles. Here are the average path eases of players with at least three majors since 1988:

```Player           Majors        Avg Path Ease
Stan Wawrinka         3                 2.8%
Novak Djokovic       12                11.3%
Stefan Edberg         4                14.6%
Andy Murray           3                18.8%
Boris Becker          4                18.8%
Mats Wilander         3                19.8%
Gustavo Kuerten       3                22.0%
Roger Federer        19                23.5%
Jim Courier           4                26.4%
Pete Sampras         14                28.9%
Andre Agassi          8                32.3%```

If Rafa adds to his grand slam haul this weekend, his average path ease will take a bit of a hit. Still, he’ll only move one place down the list, behind Stefan Edberg. After more than a decade of battling all-time greats in the late rounds of majors, it’s fair to say that Nadal deserved this cakewalk.

Update: This post reads a bit differently than when I first wrote it: I’ve changed the references to “path difficulty” to “path ease” to make it clearer what the metric is showing.

Nadal and Anderson advanced to the final, so we can now determine the exact path ease number for whichever one of them wins the title. Rafa’s exact number remains 51.4%, and should he win, his career average across 16 slams will increase to about 15%. Anderson’s path ease to the title is “only” 41.3%, which would be good for ninth on the list shown above, and just barely second easiest of the last 30 US Opens.