The purpose of this post is to present a possible research topic for a mathematics or computer science student.

For the last several years I’ve been an avid squash player. For those of you who don’t know, squash is played in a court like a racquetball court (although smaller). The racket is long and thin. And the ball is small, soft, and barely bounces. It is a wonderfully fast game with a good mixture of power and fineness. (My favorite squash-related quote is from Ian McEwan’s book Saturday: “There’s no such thing as a gentle game of squash.”)

At the end of 2008 the World Squash Federation (WSF) voted to abandon the current scoring method and adopted a new one, starting April 1, 2009.

Traditional scoring (the old scoring method)
1. The server chooses which side to serve from (left or right).
2. After each successful rally, the server switches sides.
3. Only the server can win a point.
4. If the server loses a rally, the other player becomes the server.
5. The first player to 9 wins the game, unless the score becomes 8-8.
6. If the score is 8-8, the receiver decides whether to play to 9 or to 10.
7. A match is 3 games out of 5.

Point-a-Rally (PAR) scoring (the new scoring method)
1. The server chooses which side to serve from (left or right).
2. After each successful rally, the server switches sides.
3. The winner of each rally earns a point
4. If the server loses a rally, the other player becomes the server.
5. The first player to 11 wins the game, unless the score becomes 10-10 in which case a player must win by two points.
6. A match is 3 games out of 5.

I’ve been playing PAR scoring for the last few weeks and I like it a lot, although it is very different. Traditional scoring is like climbing a very steep hill, PAR scoring is like rafting down a fast river. One nice thing about PAR scoring is that weaker players tend to have more respetable-looking scores. While the outcome of a traditional match between two unevenly matched players may be 9-0, 9-2, 9-1, a match under PAR scoring may be 11-4, 11-5, 11-3, which doesn’t feel like as much as a blow-out.

One of my colleagues tells me that volleyball made a similar transition several years ago (I don’t know the details). He said that a computer science student in our department analyzed the affect on the game and the most notable difference was that the standard deviation of the game length was much smaller under PAR scoring. Very interesting.

I think it would be a great project for a student to do something similar for traditional vs. PAR scoring in squash.

Here’s how one might set this up. Assume that the probability that player 1 wins when serving from his strong side is p₁ and from his weaker side p₂ (p₁p₂). Similarly, assume that the probability that player 2 wins when serving from his strong side is q₁ and from his weaker side q₂ (q₁q₂).

Choose fixed values of p₁, p₂, q₁, and q₂. Use a computer to run many games under both traditional and PAR scoring rules and see what happens. Determine how often the “better” player wins, compute margins of victory, measure game lengths, standard deviations, etc. Then repeat for different values of p₁, p₂, q₁, and q₂. [The more I think about it, taking p₁=p₂ and q₁=q₂ would be sufficient for understanding the differences between the scoring methods.]

I wouldn’t be surprised if the WSF already performed analysis like this. But if so, I don’t know what the findings were.