Year(s) that might have been - Pt 2
******By Andrew, TW Contributing Editor.
In this second post on alternate (ATP) tennis universes for 2007, I want to get to the results of my inquiry; given how well the No. 1 and No. 2 players were playing in 2007, what (if the dice had rolled slightly differently) might have happened?
Would we expect to see more shots of Federer tasting final defeat like the one on the right, taken after his Madrid loss to Nalbandian (found by Rosangel)?
Incidentally, there are many, many wonderful stories and novels on the topic of "what might have been" had history turned out differently. What if the Spanish Armada had successfully conquered England in 1588? What if the South had won the Civil War? What if Japan and Germany had won WWII? In science fiction, this topic is known as "alternate history."
Here are some of the things I found, using the model I discussed in part 1:
* A calendar year Grand Slam for Federer was a one in ten chance. For Nadal, it was a one in one thousand shot.
* Nadal had a one in five chance of eclipsing Federer in the ATP Race.
* On form, each man might have been expected to win four Masters Series titles. In the "real" 2007, Federer "only" won two titles, Nadal three.
* Playing at this level, in the "regular" season we can expect six matches between Federer and Nadal (three on clay,three on other surfaces).
Remember, in this view of the world, the actual history of 2007 is but one among a set of might-have-beens - but not all of these might-have-beens are equally likely. As I wrote in the last post, it's possible that Federer could have gone three for three against Nadal on clay, but much more likely (in my view) that the Mallorcan would sweep on the surface.
The model is "tuned" to produce a set of outcomes which (a) are reasonable given our knowledge of the two men's strengths - so Nadal is stronger on clay, Federer on other surfaces, and (b) in aggregate, on average we end up with the same ATP Ranking Points for each man for one full season as actually happened in 2007.
To get down to cases, let's look at the outcomes of each of the majors. In my model, I basically split the tournaments into two groups, clay and other. For each of the majors, assuming that the two players are seeded first and second, Federer and Nadal would each need to defeat six players to reach the final, then face either their main rival or someone else (player X) in the final. What are the odds of these outcomes? This calculation doesn't require heavy duty computer simulation:
Roland Garros Other Major (AO, Wimbledon, USO)
Federer Final 73% 80%
Win 34% 67%
Nadal Final 80% 31%
Win 58% 13%
F/N Match 59% 25%
Given these odds, the most likely number of Grand Slam final matchups between the two players is one each year (59% of the time at the French Open, 41% of the time at one of the other majors). This happens 43% of the time, in my model. 30% of the time there are two match ups (as happened in 2006 and 2007). There's an 18% chance that no Grand Slam finals will have a Federer/Nadal pairing, and a slightly under 1% chance of all four finals being No. 1 versus No. 2.
The above table also allows us to calculate the odds of the two players making all the finals, or winning all the finals (a Calendar Year Grand Slam). Given that Nadal comes out with slightly less than under a one in three chance of making the final at Melbourne, Wimbledon or Flushing Meadows, you won't be surprised to learn that his chances of making all of them, plus Roland Garros, is about 2%, a pretty long shot. And a calendar year Grand Slam for Nadal, playing as well as he did in 2007, is a pretty remote proposition: 0.1%, or a one in a one thousand shot.
Federer's odds of making each of the Major finals is 80% * 80% * 80% * 73% - about 37%, slightly above one in three. And, as modelled, a calendar year Grand Slam is about a 10% probability - one in ten.
Now, I've used a pretty simple model to carry out these calculations - and one could fairly argue that a Roddick at Wimbledon or Djokovic at Flushing Meadows is better than a one in ten shot against Federer. So have at it - do you think a one in ten bet against a calendar year Grand Slam is too high, too low, or just right?
We have to use a computer simulation - Monte Carlo simulation for the stats geeks out there - for our next trick, which is looking at the No. 1 ranking for the season (pre-Shanghai).
In the "real" 2007, Federer led Nadal going into Shanghai by 995 points - and in my computer simulation, 50% of the time he did better, 50% of the time worse. And in 20% of these parallel universes, Nadal was the guy holding aloft the ATP Race trophy in Shanghai. Playing as well as he did in 2007, Nadal had a one in five chance of being the points leader before the YEC.
When it comes to tournament victories, we can see whether the players' results fell below, met or exceeded expectations. To do this, I'll use the 10%, 50% and 90% outcomes, representing lower than expected, expected and better than expected outcomes:
From the table, Nadal did about as well as expected in terms of tournament wins, with one fewer Masters shield and one more "other" win (Barcelona plus Stuttgart). Federer's baseline performance in Majors is two per year, and in 2007 he won three, but did less well than expected in Masters tournaments (note that Nadal lost two finals, Hamburg and Paris, while Federer lost three - Monte Carlo, Montreal and Madrid).
Finally, how often, at this level of performance, should we expect the two men to meet? Somewhere between 3 and 8 times, with six (three clay, three hard court matches) being most likely. In fact, in 2007 we had four meetings pre Shanghai, and in 2006 the two met five times.
So there you have it. This exercise has been about extrapolating the large scale (a tennis season) from the small (the estimated probability that a player will prevail in a single match).
We can do this on an even smaller scale - using the statistics for the likelihood of a player winning a serve or return point to estimate their chances of winning a match. But that's a story for another day.