Scalable Distributions (Fat-Tailed)

Once Again, a Happy Solution

First, in assuming a scalable distribution, I accept that an arbitrarily large number is possible. In other words, inequalities should not stop above some known maximum bound.

Say that the book The Da Vinci Code sold around 60 million copies. (The Bible sold about a billion copies but let’s ignore it and limit our analysis to lay books written by individual authors.) Although we have never known a lay book to sell 200 million copies, we can consider that the possibility is not zero. It’s small, but it’s not zero. For every three Da Vinci Code–style bestsellers, there might be one superbestseller, and though one has not happened so far, we cannot rule it out. And for every fifteen Da Vinci Codes there will be one superbestseller selling, say, 500 million copies.

Apply the same logic to wealth. Say the richest person on earth is worth $50 billion. There is a nonnegligible probability that next year someone with $100 billion or more will pop out of nowhere. For every three people with more than $50 billion, there could be one with $100 billion or more. There is a much smaller probability of there being someone with more than $200 billion—one third of the previous probability, but nevertheless not zero. There is even a minute, but not zero probability of there being someone worth more than $500 billion.

This tells me the following: I can make inferences about things that I do not see in my data, but these things should still belong to the realm of possibilities. There is an invisible bestseller out there, one that is absent from the past data but that you need to account for. Recall my point in Chapter 13: it makes investment in a book or a drug better than statistics on past data might suggest. But it can make stock market losses worse than what the past shows.

Wars are fractal in nature. A war that kills more people than the devastating Second World War is possible—not likely, but not a zero probability, although such a war has never happened in the past.

Second, I will introduce an illustration from nature that will help to make the point about precision. A mountain is somewhat similar to a stone: it has an affinity with a stone, a family resemblance, but it is not identical. The word to describe such resemblances is self-affine, not the precise self-similar, but Mandelbrot had trouble communicating the notion of affinity, and the term self-similar spread with its connotation of precise resemblance rather than family resemblance. As with the mountain and the stone, the distribution of wealth above $1 billion is not exactly the same as that below $1 billion, but the two distributions have “affinity.”

Third, I said earlier that there have been plenty of papers in the world of econophysics (the application of statistical physics to social and economic phenomena) aiming at such calibration, at pulling numbers from the world of phenomena. Many try to be predictive. Alas, we are not able to predict “transitions” into crises or contagions. My friend Didier Sornette attempts to build predictive models, which I love, except that I can not use them to make predictions—but please don’t tell him; he might stop building them. That I can’t use them as he intends does not invalidate his work, it just makes the interpretations require broad-minded thinking, unlike models in conventional economics that are fundamentally flawed. We may be able to do well with some of Sornette’s phenomena, but not all.