The Black Swan: The Impact of the Highly Improbable

Exponent	Share of the top 1%	Share of the top 20%
1	99.99%[58]	99.99%
1.1	66%	86%
1.2	47%	76%
1.3	34%	69%
1.4	27%	63%
1.5	22%	58%
2	10%	45%
2.5	6%	38%
3	4.6%	34%

Table 2 illustrates the impact of the highly improbable. It shows the contributions of the top 1 percent and 20 percent to the total. The lower the exponent, the higher those contributions. But look how sensitive the process is: between 1.1 and 1.3 you go from 66 percent of the total to 34 percent. Just a 0.2 difference in the exponent changes the result dramatically—and such a difference can come from a simple measurement error. This difference is not trivial: just consider that we have no precise idea what the exponent is because we cannot measure it directly. All we do is estimate from past data or rely on theories that allow for the building of some model that would give us some idea—but these models may have hidden weaknesses that prevent us from blindly applying them to reality.

So keep in mind that the 1.5 exponent is an approximation, that it is hard to compute, that you do not get it from the gods, at least not easily, and that you will have a monstrous sampling error. You will observe that the number of books selling above a million copies is not always going to be 8—It could be as high as 20, or as low as 2.

More significantly, this exponent begins to apply at some number called “crossover,” and addresses numbers larger than this crossover. It may start at 200,000 books, or perhaps only 400,000 books. Likewise, wealth has different properties before, say, $600 million, when inequality grows, than it does below such a number. How do you know where the crossover point is? This is a problem. My colleagues and I worked with around 20 million pieces of financial data. We all had the same data set, yet we never agreed on exactly what the exponent was in our sets. We knew the data revealed a fractal power law, but we learned that one could not produce a precise number. But what we did know—that the distribution is scalable and fractal—was sufficient for us to operate and make decisions.

The Problem of the Upper Bound

Some people have researched and accepted the fractal “up to a point.” They argue that wealth, book sales, and market returns all have a certain level when things stop being fractal. “Truncation” is what they propose. I agree that there is a level where fractality might stop, but where? Saying that there is an upper limit but I don’t know how high it is, and saying there is no limit carry the same consequences in practice. Proposing an upper limit is highly unsafe. You may say, Let us cap wealth at $150 billion in our analyses. Then someone else might say, Why not $151 billion? Or why not $152 billion? We might as well consider that the variable is unlimited.

Beware the Precision

I have learned a few tricks from experience: whichever exponent I try to measure will be likely to be overestimated (recall that a higher exponent implies a smaller role for large deviations)—what you see is likely to be less Black Swannish than what you do not see. I call this the masquerade problem.

Let’s say I generate a process that has an exponent of 1.7. You do not see what is inside the engine, only the data coming out. If I ask you what the exponent is, odds are that you will compute something like 2.4. You would do so even if you had a million data points. The reason is that it takes a long time for some fractal processes to reveal their properties, and you underestimate the severity of the shock.

Sometimes a fractal can make you believe that it is Gaussian, particularly when the cutpoint starts at a high number. With fractal distributions, extreme deviations of that kind are rare enough to smoke you: you don’t recognize the distribution as fractal.

The Water Puddle Revisited

As you have seen, we have trouble knowing the parameters of whichever model we assume runs the world. So with Extremistan, the problem of induction pops up again, this time even more significantly than at any previous time in this book. Simply, if a mechanism is fractal it can deliver large values; therefore the incidence of large deviations is possible, but how possible, how often they should occur, will be hard to know with any precision. This is similar to the water puddle problem: plenty of ice cubes could have generated it. As someone who goes from reality to possible explanatory models, I face a completely different spate of problems from those who do the opposite.