Friday, June 25, 2010

"The greatest match ever": some analysis

The Isner–Mahut match at Wimbledon, which finally ended in Isner's favour with the monumental scoreline of 70-68 in the fifth set, seems out of place in the 21st century. It belongs in an era of unlimited FA Cup replays, timeless Tests and fight-to-the-finish boxing. That there is still room for this sort of encounter in today's quick-fix, penalty shoot-out, winner-takes-all world is I think the biggest positive that we can all take from what some have called the greatest ever tennis match. In terms of quality it probably wasn't that great (I only saw the last of its eleven hours, finishing at close on 4am my time last night) but in sheer size it was truly colossal; nothing before has ever come close. At times Isner could hardly move his legs, but somehow he was able to rely on his brutal serving arm. Both players were presented with a special trophy after the match, as was the umpire. You can find a priceless commentary of this match from the Guardian here.

Rule number one of successful blogging is that all content should be of interest to its audience, not just its author. I should warn you now that I may be about to violate this rule. First, here are a few facts and figures surrounding the three-day contest:
  • Both players' ace counts hit triple digits, Isner serving 112 aces to Mahut's 103.

  • The previous longest match lasted just over 6½ hours. Just the fifth set of this match exceeded eight hours.

  • The number of points played in the match nearly reached quadruple figures, Mahut holding up his half of the bargain with 502 points while Isner managed 478. From what I could tell, Mahut made most of the running on the second day's play as those figures would suggest. He unquestionably looked the fitter of the two men, but he just could not find a way to break the Isner serve.

  • Mahut had to win three qualifying matches to make the main draw. In the second of these he was pushed to a 24-22 deciding third set by Britain's Alex Bogdanovic, a marathon in itself. He then came from two sets down to win his last qualifying match in five.

  • Thiemo de Bakker, Isner's second-round opponent, won his opening round match 16-14 in the fifth set.

  • In 2007 Mahut had a match point in his loss to Andy Roddick in the final of Queen's. He had beaten Nadal earlier in the tournament. That would suggest he knows how to play on grass.

  • Also in 2007, the six-foot-nine Isner burst onto the tennis scene by winning five successive matches, all of them in a tie-break in the final set. In last year's US Open he beat Roddick on a tie-break in the fifth and final set. Then at the start of this year in my home town of Auckland, he picked up the title by beating Arnaud Clément in the final on, guess what, a tie-break in the final set.

What I keep hearing is, "we'll never see anything like this ever again". Is that true? Well even though this match has obliterated every record in the book, given infinite time and no changes to the rules of tennis (more on that later), and that Wimbledon doesn't fall into the sea due to climate change, we will see something similar again, be it in our lifetime, our great-grandchildren's lifetime or some time after that. So it's really a question of whether tennis continues to be played as it is now, hundreds of years hence. If it does (and that's a really big if) how long will we have to wait to see a 70 in the games column again?

First it would be nice to know the chances that the Isner–Mahut match-up, in particular, should produce such a marathon. What were the odds that we'd get to 68-all? Even to do that I'll have to make a few assumptions:

  1. Both players have the same chance of winning a point on serve, which I'll call p. As Isner had a far superior ranking (he was the 23rd seed while Mahut was ranked outside the world's top 100) this is a bit unrealistic. But Mahut had proven his ability on grass before and he was coming into the match on the back of considerably more grass-court play than the American. So this assumption, which implies that both players are of equal strength, isn't so far-fetched after all (as the eventual result showed).

  2. Only this match is used to estimate p. Match-ups are so important in tennis. For instance Mahut's chance of winning a point on the Isner serve is surely smaller than against just about any other player, with the possible exception of Ivo Karlovic. Likewise the surface plays a huge part; even different grass courts play differently. So adding any other opponents or surfaces into the mix will only make my estimate less accurate.

  3. The outcome of each point is independent of all other points, so p is constant throughout the match. Whether this is a realistic assumption I honestly don't know. My guess is that there are "momentum" and "crisis aversion" effects that come into play, as well as (in a match as long as this) a serious "fatigue" effect. Also, players may have a better chance when serving or receiving from a particular side of the court (left or right). That's for another investigation.

Of the 980 points played, 759 of them went with serve. This gives p = 0.77449, a figure much higher than the corresponding 0.66469 taken from all the matches in the first two rounds of the men's tournament combined.So what's the probability of the server winning a game?
He can win it to love [probability p^4], to fifteen [4*p^4*(1-p): there are four ways of winning a game for the loss of one point], to thirty [10*p^4*(1-p)^2] or in a deuce situation. The "deuce" case is a bit trickier as it involves an infinite series, but it works out to be:

Summing these four terms, the probability that the server wins the game (I'll call this g) is:
0.35980 + 0.32456 + 0.18298 + 0.09823 = 0.96556. Wow. So given how Isner and Mahut were serving, you could only expect a break of serve a little over one time in thirty. It's also pretty amazing that the most likely outcome of a game in this situation is a hold to love.

The probability that the match goes into a fifth set, given the assumptions, is 3/8. I'm sure this is a slight overestimate due to the effect of momentum (between two evenly-matched players, I'd imagine whoever wins the first set has an above 50% chance of also winning the second). Now what's the chance that a set goes to six games all? The probability of this is:
[g^10 + 25*g^8*(1-g)^2 + 100*g^6*(1-g)^4 + 100*g^4*(1-g)^6 + 25*g^2*(1-g)^8 + (1-g)^10] * [g^2 + (1-g)^2].

Plugging in our value of 0.96556 for g, that probability is 0.67854. So with two players serving as well (and returning as badly?) as Isner and Mahut, you could expect two-thirds of sets to reach 6-6. Contrast this with the tournament as a whole, where using the figure from the first two rounds, you'd expect just 28% of sets to reach 6-6 (and I'm sure on clay it would be even lower than this).

So the probability at the start of the Isner–Mahut match that it would reach 6-6 in the fifth, i.e. the no-tie-break "overtime" stage, is 3/8 * 0.67854 = 0.25445.

We can continue by working out the probability that the fifth set will run deeper than this:

So there you have it. These two would serve up (literally!) a 70-68 (or longer) fifth set 0.357% of the time, or once every 280 times they step onto the court.

I'm sure most people would think it was far more unlikely than that, somewhere in the thousands or even millions-to-one range. And they're probably (partially, at least) right. When I've watched marathon men's matches in the past, at around 10-all in the decider one man starts to cramp, or loses concentration, or serves a couple of double faults, putting a major dent in
his value of p and opening the door for his opponent. Remarkably, in the Isner–Mahut match, that never happened. If anything, both players' p-values seemed to rise as the match wore on, and it took two out-of-the-blue passing shots from Isner to bring it to an end.

So all I've done here is come up with a figure for one match alone, based on some dodgy assumptions. As for calculating a probability based on all the matches that take place in a tournament, you'd need a whole bunch of p-values for all the possible encounters, many of which are clearly not 50-50 match-ups, and I wouldn't know where to start! Here's someone who has started, and more than that.

This match has inevitably generated some discussion about a possible fifth-set tie-break at Wimbledon (and, for that matter, Melbourne and Paris). Currently the US Open is the only Grand Slam to use a tie-break in the final set. Personally I'm a fan of the "no tie-break" rule in the last set, at least not at 6-6. The Wimbledon finals of 2001 and 2008, as well as the Australian Open semi between Federer and Safin in 2005, are examples of classic matches that were taken "up a notch" by the lack of a deciding tie-break. The same can be said of some great women's Grand Slam finals, particularly at the French Open. But I wouldn't be totally averse to a tie-break coming in later, at least in the early rounds. There are three problems I see in a gargantuan battle like this. Most obviously the winner of the match, in this case Isner, is completely shot when he comes to play his next match. As it happened, Isner won just five games against De Bakker, and incredibly did not serve a single ace. Had Mahut won, I think he would have done better. Secondly it becomes almost irrelevant who wins, partly because the winner is unlikely to advance further. Finally so much attention is given to the match as to almost overshadow the rest of the tournament; in twenty years' time people may well remember this match but have no clue as to who won the final. I reckon a tie-break at 12-all would make sense (it's like an extra set) and as for the final, well that should definitely stay no-limit.

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