On the second Monday of May this year, James Willstrop, a 24-year-old British National Champion from Leeds, played David Palmer, an Australian from New South Wales, in the finals of the British Open. The match was the longest of the tournament, but it was the fifth game that will cause this match to be referred to as one of the most dramatic finals in the 80-year history of the event.
Willstrop, the winner of the fourth game, continued his momentum to steadily build a 9-6 lead in the fifth. Between points, each player showed signs of exhaustion, Palmer putting his hands on his knees and Willstrop repeatedly leaning against the sidewall. During play, though, neither player slowed down, and the points became increasingly exciting as seeming put-aways were somehow scraped up by each player—sometimes multiple times in one rally.
Palmer won three points in a row to even the score at 9-9.
At this point, the match had seen both large and small changes in momentum, but the seesaw swung back and forth even more quickly in the next series of points. With the 11-point scoring system, the winner of a game needs to win by two points, and Willstrop had the first opportunity to close out the match at 10-9. Palmer fought off the match point, only to see Willstrop win the next point to gain a second match point at 11-10. But then Palmer did something extraordinary: He won three long points in a row to take his fourth title.
The rapid fire changes in emotion and match balls had a lot to do with the 11-point scoring system: When the score reached 9-all, every point thereafter either was a match point or an opportunity for a match point. The excitement of the sold-out crowd at the Echo Arena in Liverpool was palpable as they alternated from gasps of amazement to shrieks of anguish and delight.
The only thing missing was a decision that has been one of the cornerstones of tight squash matches until the 11-point system: What to choose when your opponent catches you at the end of a game.
For anyone who is not a PSA pro, that decision is one of the most interesting parts of squash. In softball for the masses, the nine-point scoring system is used where a player has to have served and won the rally in order to score a point, and at 8-8, the player who got to eight first has to decide whether the game will be played to nine (set one) or to ten (set two).
When I was a graduate student at business school, I had started to necessarily curtail my pro squash career, but it still was on my mind all the time. When I took a surprisingly complicated course in statistics, I thought my professor might be able to help me answer a question I had wondered about since I was first introduced to squash: Is there a correct answer as to what to call?
My professor, Dev Joneja, found the problem quite intriguing, and it turned out both that its analysis required a sophisticated mathematical technique called a Markov Chain and that the problem was solvable1. I couldn’t believe it.
Because the end of a game in squash has an infinite number of potential sequences, there is more than one possible ‘reality’ of how the game might evolve. This is called a stochastic process in statistics, and the Markov Chain is used for stochastic processes where what happens next is statistically independent of what happened before. In other words, one could theoretically describe the present score with full information and thus capture all the information that could influence the future score.
It turns out, though, that all you really need to know is your chance of winning a particular point against your opponent. If it is less than 38%, you should call set one; if it is greater than that, you should call set two. That’s it. The secret is revealed.
Hold on. If my chance is less than 38% of winning any particular point, how is it that I got to eight first?
Well, even though flipping a coin has a 50-50 chance of coming up heads, there still are coin flipping sequences with 10, 20 or more heads in a row. In other words, you might have just beaten the odds in getting to eight first.
This, of course, is the trick: Who wants to think that they got there through chance? Even when playing someone who is much better than you, you might think that you are there because you are playing better than your opponent on that day. And maybe you are. In any case, it almost certainly behooves you to think positively.
Then there are all the other factors that consistently change during a match: Level of fatigue, level of anxiety, the referee’s calls. How, then, can you predict your chance of winning a point? The best way is to play the person over and over again and see how many rallies you take versus your opponent. The more you play, the more you will know, and the more you know, the better you will know your odds.
So, if this year’s British Open had been played to nine instead of eleven, what should Willstrop have done? Theoretically, it could be figured out. It is just statistic probability.
When Palmer reached match ball at 12-11, he noticed that his racquet was cracked and walked off court to replace it. When play resumed, his winning shot on his first match point was a mishit off the frame of his new racquet. The ball hit the sidewall nick and rolled out giving him his fourth Open title.
What are the odds?
(Footnotes)
1 In fact, my professor co-wrote a paper based on my question. See Broadie, M., Joneja, D. (1993) “An Application of Markov Chain Analysis to the Game of Squash”, Decision Sciences, Vol. 24, No. 5, 1023–1035.
