Author: *Andrew*

This is part 1 of a two stage post, with some introduction to some ideas in this post, then some of my conclusions in part 2 (following in a couple of days). Warning: there's a bit of "probability theory heavy lifting" in this post, but I hope you'll stay with me along the way...

Alternative universes are something of a flavor of the month with the release of *"The Golden Compass"* - though, as a fan, I'd recommend you read the book rather than see the movie, which is a botch on the order of, oh, Ana Ivanovic at the RG 2007 Womens Final. Attractive prospect, but two hours of your life you'd never get back. But I digress...

(Actually, before I leave the topic of *"The Golden Compass"* - can you imagine how competitive sports like, say, the NFL would play out with a bunch of *daemons* on the field? And for those of you whose eyes glaze over at the thought of math, probabilities and statistics, here's a question: what would Nadal's *daemon* be? Or Federer's? Or your own? The math geeks can keep reading...)

I'm going to describe the 2007 ATP season for two players - Roger Federer and Rafael Nadal - in a novel fashion. Instead of simply recapping the season that was - tournament by tournament, match by match, I'll talk about seasons that might have been.

As you know, I never make a firm prediction about the outcome of a match, preferring instead to assign a probability to one player being the victor. So, for example, in September of this year for the US Open 2007 Mens Final, I gave Roger Federer a 75% chance of beating Novak Djokovic: in other words, were Federer and Djokovic to be cast into some kind of eternal Groundhog Day, I'd expect the Serb to win about once every four times they played.

Interestingly, you can find another (betting) site on the web called TennisInsight which does the same for all the major matches. I've never looked at the site before making my assessments for Picks Games, but have found (in the past) that their view and mine haven't been totally out of line.

Obviously, no tennis match is ever played over again under the same conditions, but we do get players competing against each other under similar conditions - over (say) a clay court season or a hard court season. So, for example, we can think of Nadal vs Federer on clay, of which there have been seven matches so far, and see those as a set of repeated tests (albeit with a very small sample size).

Suppose we assign Nadal (hypothetically) a 65% chance of beating Federer in any given encounter on clay: what are the possible outcomes if they meet three times in a season (eg Monte Carlo, Rome, Roland Garros)? There are four possible outcomes: Nadal 3-0, Nadal 2-1, Federer 2-1 and Federer 3-0. You can construct a probability tree for the various outcomes, with these results:

Nadal Wins Probability

3 27%

2 44%

1 24%

0 4%

In a tennis "Groundhog Day" with both players playing each other a million times, you'd expect Nadal to win about 650 000 times - the chance that Federer would come out ahead is trillions to one against. But in a short season, where the players play each other infrequently, odd things can happen. Guillermo Canas and David Nalbandian can take out Federer in back to back matches, for example.

In the table above, note that it's more likely (72% - 28%) that Nadal would end a "season clay series" ahead of Federer, but about 1 time in 4 Federer, assumed to be the weaker clay court player, would come out ahead.

Now, an interesting generalization can be made; instead of looking at just three matches, we can look at an entire tennis season. In other words, we can assign probabilities to Federer and Nadal for each of the matches that they play in each of the tournaments that they play, and then see how likely certain outcomes are. For example: how likely is it that Nadal might lead Federer in ATP Ranking Points gained over the course of the season? How many times might we expect the two players to meet in a final (and who might be expected to win the more matches?) And is that fabled Holy Grail, a calendar year Grand Slam, a feasible proposition?

So I built an Excel spreadsheet, and began to assign some probabilities to the likelihood of surviving certain rounds, and match outcomes between the two players:

Here are some examples for how the table works. Say you're thinking about the Rome Masters tournament, which isn't a Grand Slam event. Federer is assumed to have a 95% chance of winning R32/R16, then a 91% chance of winning the QF/SF/F matches, unless he comes up against Nadal in the final, in which case his probability of winning the final drops to 35%. And he is likely to see the Spaniard across the net, since Nadal's chances of surviving each match in Rome are 97% for each of the first two matches and 95% in the QFs and SFs.

Phew.

Another example: in my model, assuming the input probabilities, Federer has a 98% chance of making it through the first round of the Australian Open, Nadal an 86% probability. But to get to the QF, the two men both have to survive the first four rounds. The probability that Federer will be successful is (98%*98%*98%*98%) or 92%: pretty near a lock. Nadal is more likely than not to make the QF, but 86%^4 (86% to the power of 4) is "only" 55% - so his odds of making the QF are (using this model) about 5/9.

Then I set up all the tournaments each man played in, from Chennai in January to Paris Bercy in November. I (figuratively) had my computer flip coins tens of thousands of times, determining the outcome of hundreds of thousands of matches that might have been. I've made some simplifying assumptions (for example, that the two players only meet in a final).

In technical terms, I ran the model 50 000 times, each time coming out with different outcomes based on the luck of the draw. In one run, Nadal holds the Melbourne trophy aloft, with Federer eliminated in the SF (shades of 2005). In another, Nadal falls at the QF stage, while Federer takes his third AO title (as actually happened). In a third, the two men contest an epic final, with Nadal recording his first victory in a non clay GS. Each of these individual outcomes is possible - but the number of times each occurs, given a large number of computer runs, converges on the expected probability.

And that's just one tournament. The good news (for nerds) is that today's laptops can simulate a whole season in under a minute - 38 seconds on my Dell laptop.

So using this model, I can find out the probability that Nadal wins multiple majors, or Federer none. However, Garbage In, Garbage Out, as they say. Is there any reason to believe the model outputs?

At the end of the (real) Paris Bercy tournament, Federer had **6530** ATP Ranking points for 2007, and Nadal had **5535**. My computer model, run 50 000 times, came up with an average of **6536** points for Federer, and **5537** points for Nadal. Pretty close.

If you were wondering how I came up with the probabilities in the table - a bit of tinkering went on. I would estimate some probabilities, run the model, then re-estimate to get the model output closer to the actual result. Given the number of inputs, these aren't the only probabilities that would work, but I submit that they both give a calibrated answer, and satisfy our intuitive, gut sense of the strengths of the two men.

But now we're in a position to start exploring these alternative universes. In how many did Nadal stand before the Shanghai crowd and hold aloft the ATP Race trophy, having surpassed Federer in the rankings during the 2007 season? In how many parallel universes does Roger Federer raise the US Open trophy aloft, the first man to complete a Calendar Year Grand Slam since Rod Laver?

And in these universes (and ours), how often should we expect Federer and Nadal to play one another in any year? And given their prowess on the different surfaces, who might we expect to come out ahead?

You can use the comments to this post to explore these ideas. Do the assigned probabilities match up with your own sense of the strengths of the two players? And what probability would you assign to a Federer calendar year Grand Slam, or a Nadal triumph at Wimbledon or Flushing Meadows?

In part 2, I'll tell you what I found.

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