Sports, a new arena for scaling

A recent paper in the open access New Journal of Physics, might help you outsmart your bookie. At least if your bookie covers tennis.

In this country, we like to think of all men being equal. But when it comes to things we can measure, like height, we know all men are not equal. If you make a histogram of heights for men, you get the famous “bell curve.” There exists some average height that corresponds to the peak of the curve, in the neighborhood of 6 feet. Most men are a bit shorter or taller than average, but even extreme heights are still within 2 feet of average. It’s exceedingly unlikely to find someone 10 feet tall. So, by this measure, all men are equal-ish.

But all men are not equal in wealth. The Pareto principle, worth another post in itself, says that 20 percent of the people have 80 percent of the wealth. Further, with wealth, there is no upper bound on income. As a result, the histogram of wealth looks vastly different from the histogram of height. It is a “power law” distribution. There is no average amount of wealth. It is also called a scale-free distribution: if you zoom in on any area of the graph, it will look exactly the same. And extreme wealth, unlike 10 feet tall men, becomes a real possibility.

In the article mentioned above, the authors extend the wealth analogy to sports. It’s a natural fit here: individuals are not equal, they have rank. Anyone who watches or plays sports knows there is no average. One can be an Olympic-level sprinter, but every so often a Usain Bolt will come around to humble you. The authors of the article look at the distribution of ranks (or prizemoney, where appropriate) for different sports and indeed find a power law relationship.  (There is an exponential tail, but that is a mathematical detail to be discussed among the truly interested.)

Actually, the running analogy is a little off, because presumably there is some biological limit to sprinting speed, though apparently we aren’t there yet.

Switching focus, tennis is a good example of a sport where we don’t seem to be up against any biological limits, and there is no upper limit on skill, the crucial piece of the game. All pro players are good, but Serena Williams or Rafael Nadal will crush most of them.

In the paper, the authors further go down the rabbit hole with tennis, as it is a sport with an extremely detailed record of head-to-head meetings. Based on the previous analysis of the rank distributions, they are able to come up with an equation (that fits the data) predicting the likelihood of a win. Based solely on the difference in rank of two players, they know the exact odds.

Call your bookie.