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.

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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.

NY = LA, more or less?

This was an exercise (at least the first few paragraphs were) for the Santa Fe Workshop I attended in May.

Geoffrey West wants to let us in on a little-known tidbit. New York and Los Angeles are basically the same city. You probably think that’s preposterous. New York is the land of the subway, L.A. is the land of the freeway. New York has city lights in the nights, L.A. has sunshine-drenched days. They’re different.

West audaciously gives us an additional morsel. Their sameness can be mathematically proven. Now you’re tempted to use this paper to line the birdcage. How could something as human as a city, in all its qualitative complexity, be quantitatively characterized?

Inhabiting the halls of the prestigious Santa Fe Institute, Dr. West is truly 70 years young. He’ll elegantly answer a simple question in no less than five minutes, never pausing for a breath, his natural British charm turned up all the way. He started out as a particle physicist, but for the most recent act of his long career, decided to take on a surprising challenge. West wanted to know if we could use principles of physics to study cities. In other words, blurring our eyes to the cultural specifics, can cities be described by an equation?

West believes they can. West argues that the math describing cities shows remarkable parallels to the math describing biological organisms, a framework he also helped to develop. And while they are similar, they are also distinct in an important and tantalizing way. West hopes this picture will one day help us plan better cities, prevent crime, and achieve sustainability.

The mathematical framework to describe both systems is called scaling.
Scaling, as a scientific term, has a precise and technical meaning, though it’s quite understandable. Imagine you have a stack of $1 bills. You don’t know how much it is worth. But you know that if you have a stack twice as high, it will be worth twice as much. This is a simple example of scaling, usually called “proportional” or “linear” scaling.

There are other kinds of scaling as well. A 5 pound chicken has a daily (resting, not active) calorie requirement of about 100 calories per day. You might expect a 50 pound bulldog then to require 1000 calories per day. In actuality, the bulldog only requires about 550 calories per day. Similarly, the proverbial 500 pound gorilla will require about 3000 calories, not the 10,000 you might expect based on the chicken’s numbers. It’s not the simple linear scaling from the stack of bills, but is quite predictable. (You can take the ratio of 3000 to 550 and see it’s about the same as 550 to 100). Looking into the numbers more deeply, notice that the chicken requires about 20 calories per pound, while gorilla only requires about 6 calories per pound. The gorilla is more efficient than the chicken! West calls this an “economy of scale” in that bigger is better, using resources more conservatively. Countless biological systems follow a similar trend.

Interestingly, cities exhibit this same economy of scale. This “sublinear” scaling manifests itself in both actual jungles and urban jungles. The caveat is this only works for the impersonal elements of a city, for example the number of gas stations or square miles of pavement. As the population of a city increases, the number of gas stations also increases, but at a slower rate. In parallel to the gorilla, you need less resources per person as a city grows. Cities, compared to rural areas, are more efficient. “Cities are green,” as West likes to say.

However, when we consider things dominated by human interactions, we get “superlinear” behavior, which is the tantalizing difference between cities and nature. Back to the money stack. Imagine your stack doubles in size, but the value has actually increased by more that, someone put a few twenties in there. Examples of data exhibiting this kind of scaling include positive things like patent production per capita and average personal wealth but also include bad things like the murder rate. So cities are more productive, in good and bad ways. West remains optimistic though, “while cities are the source of many problems, they are ultimately the source of the solutions, too.”

To illustrate this superlinear scaling further we return to our cities. The New York and L.A. metro areas have roughly the same population. For the sake of argument, we’ll round both to 15 million people. The scaling argument then predicts both will have about the same per capita income. The data support this to within a few percent. By the same argument, Atlanta, 1/3 the size, will have a per capita income about ¼ as high (math details eliminated). Again, this is true to within a few percent! West and his colleagues have extended this analysis to over 300 US metropolitan areas: they all follow this same trend. The trend holds for all of the 4 variables they measured, so if you tell me the size of your city, the trend can tell you (to within a few percent), the gross production, the number of patents produced per year, the crime rate, and the per capita income.

While the data basically follow this trend, it’s a slightly fuzzy swarm of points, due to the few percent deviations mentioned before. For gross product, New York follows the trend line almost exactly. The other 3 variables also show little deviation. So strangely, New York is an “average” city. L.A. underperforms slightly for its size, having 5% less gross product than expected. West thinks these deviations, which are surprisingly constant over many decades, give a hint at the true characters of cities.

All of this analysis has revealed 5 “families” of cities that depart from the (scaling) norm in similar ways. West suggests that perhaps this taxonomy will help us in the future: similar cities may need similar solutions. Let’s return to the beginning of this article. According to West’s research, a slightly revised, more accurate statement might be: New York is just a swollen version of San Francisco, Seattle, or surprisingly, Indianapolis.