The Black Swan: The Impact of the Highly Improbable

The object is not in fact a lens cap. These two photos illustrate scale Invariance: the terrain is fractal. Compare it to man-made objects such as a car or a house. Source: Professor Stephen W. Wheatcraft, University of Nevada Reno.

In the rest of this chapter I will explain how I can endorse Mandelbrotian fractals as a representation of much of randomness without necessarily accepting their precise use. Fractals should be the default, the approximation, the framework. They do not solve the Black Swan problem and do not turn all Black Swans into predictable events, but they significantly mitigate the Black Swan problem by making such large events conceivable. (It makes them gray. Why gray? Because only the Gaussian give you certainties. More on that, later.)

THE LOGIC OF FRACTAL RANDOMNESS (WITH A WARNING) [55]

I have shown in the wealth lists in Chapter 15 the logic of a fractal distribution: if wealth doubles from 1 million to 2 million, the incidence of people with at least that much money is cut in four, which is an exponent of two. If the exponent were one, then the incidence of that wealth or more would be cut in two. The exponent is called the “power” (which is why some people use the term power law). Let us call the number of occurrences higher than a certain level an “exceedance”—an exceedance of two million is the number of persons with wealth more than two million. One main property of these fractals (or another way to express their main property, scalability) is that the ratio of two exceedances[56] is going to be the ratio of the two numbers to the negative power of the power exponent.

FIGURE 13: THE PURE FRACTAL STATISTICAL MOUNTAIN

The degree of inequality will be the same in all sixteen subsections of the graph. In the Gaussian world, disparities in wealth (or any other quantity) decrease when you look at the upper end—so billionaires should be more equal in relation to one another than millionaires are, and millionaires more equal in relation to one another than the middle class. This lack of equality at all wealth levels, In a nutshell, is statistical self-similarity.

TABLE 2: ASSUMED EXPONENTS FOR VARIOUS PHENOMENA[57]

Phenomenon	Assumed Exponent (vague approximation)
Frequency of use of words	1.2
Number of hits on websites	1.4
Number of books sold in the U.S.	1.5
Telephone calls received	1.22
Magnitude of earthquakes	2.8
Diameter of moon craters	2.14
Intensity of solar flares	0.8
Intensity of wars	0.8
Net worth of Americans	1.1
Number of persons per family name	1
Population of U.S. cities	1.3
Markets moves	3 (or lower)
Company size	1.5
People killed in terrorists attacks	2 (but possibly much lower exponent)

Let us illustrate this. Say that you “think” that only 96 books a year will sell more than 250,000 copies (which is what happened last year), and that you “think” that the exponent is around 1.5. You can extrapolate to estimate that around 34 books will sell more than 500,000 copies—simply 96 times (500,000/250,000)^-1.5. We can continue, and note that around 8 books should sell more than a million copies, here 96 times (l,000,000/250,000)^-1.5.

Let me show the different measured exponents for a variety of phenomena.

Let me tell you upfront that these exponents mean very little in terms of numerical precision. We will see why in a minute, but just note for now that we do not observe these parameters; we simply guess them, or infer them for statistical information, which makes it hard at times to know the true parameters—if it in fact exists. Let us first examine the practical consequences of an exponent.

TABLE 3: THE MEANING OF THE EXPONENT