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.


I’ve got (almost) rhythm

Music is one of the things thought to make us human, to differentiate us from animals. If you’ve ever heard a piece of electronically generated music, you can tell in an instant. The beat is too regular, too perfect. It’s a little unsettling. Researchers in Germany have been looking at the flip side: what makes music human?

Actually, a lot of electronically produced music does have imperfections baked into it, to make it sound more human. “White noise” is added, random deviations from the beat. One note is a little too soon, the next is really late, and so on.

But, as the research team discovered, the natural human deviations aren’t purely random. They studied the rhythm of human subjects in two tasks: drumming and voice performance.

Both types showed “pink noise” type variations, meaning that if you were a little too early on the first note, you’re more likely to be a little early on the next note as well. Eventually you will “forget” about the first note, and be just as likely to be early or late.  It’s all still pretty random, but less so than white noise.

The drumming was less random than the voice, displaying almost perfect “1/f” behavior. [I won’t delve into the math.]

To further investigate this, the researchers produced two electronic versions of the same music, and added white noise  rhythm deviations to one sample, and “1/f” deviations the other. Subjects preferred the “1/f” version. You can try it out.

It remains unclear why we prefer this sort of noise – why would this have evolved? Why do we care about deviations? Shouldn’t perfect rhythm be preferable?

Another question is the timescale, the average human deviation was on the order of 10 milliseconds. Why is this pleasing?

This “1/f” noise shows up a lot of places. It has actually been shown to exist for two other aspects of music: pitch and volume. Heart beats and neural signals have been shown to display it. It’s present in electronics as “flicker noise” and can be used to describe phenomena in economics and meteorology. It’s unknown whether this is all the reflection of some universal truth, but it’s speculated it may be.