WHERE IS THE GRAY SWAN?

I have written this entire book about the Black Swan. This is not because I am in love with the Black Swan; as a humanist, I hate it. I hate most of the unfairness and damage it causes. Thus I would like to eliminate many Black Swans, or at least to mitigate their effects and be protected from them. Fractal randomness is a way to reduce these surprises, to make some of the swans appear possible, so to speak, to make us aware of their consequences, to make them gray. But fractal randomness does not yield precise answers. The benefits are as follows. If you know that the stock market can crash, as it did in 1987, then such an event is not a Black Swan. The crash of 1987 is not an outlier if you use a fractal with an exponent of three. If you know that biotech companies can deliver a megablockbuster drug, bigger than all we’ve had so far, then it won’t be a Black Swan, and you will not be surprised, should that drug appear.

Thus Mandelbrot’s fractals allow us to account for a few Black Swans, but not all. I said earlier that some Black Swans arise because we ignore sources of randomness. Others arise when we overestimate the fractal exponent. A gray swan concerns modelable extreme events, a black swan is about unknown unknowns.

I sat down and discussed this with the great man, and it became, as usual, a linguistic game. In Chapter 9 I presented the distinction economists make between Knightian uncertainty (incomputable) and Knightian risk (computable); this distinction cannot be so original an idea to be absent in our vocabulary, and so we looked for it in French. Mandelbrot mentioned one of his friends and prototypical heroes, the aristocratic mathematician Marcel-Paul Schützenberger, a fine erudite who (like this author) was easily bored and could not work on problems beyond their point of diminishing returns. Schützenberger insisted on the clear-cut distinction in the French language between hasard and fortuit. Hasard, from the Arabic az-zahr, implies, like alea, dice—tractable randomness; fortuit is my Black Swan—the purely accidental and unforeseen. We went to the Petit Robert dictionary; the distinction effectively exists there. Fortuit seems to correspond to my epistemic opacity, l’imprévu et non quantifiable; hasard to the more ludic type of uncertainty that was proposed by the Chevalier de Méré in the early gambling literature. Remarkably, the Arabs may have introduced another word to the business of uncertainty: rizk, meaning property.

I repeat: Mandelbrot deals with gray swans; I deal with the Black Swan. So Mandelbrot domesticated many of my Black Swans, but not all of them, not completely. But he shows us a glimmer of hope with his method, a way to start thinking about the problems of uncertainty. You are indeed much safer if you know where the wild animals are.

* The nontechnical reader can skip from here until the end of the chapter.

* By using symmetry we could also examine the incidences below the number.

* Clearly, you do not observe 100 percent in a finite sample.

* The defense mechanism when you question their work is to say that they “do science, not philosophy” and to berate my approach of worrying about model errors. This is a common trap: people think that science is about formulating predictions (even when wrong). To me, science is about how not to be a sucker.

Chapter Seventeen

LOCKE’S MADMEN, OR BELL CURVES IN THE WRONG PLACES *