The Black Swan

by Nassim Nicholas Taleb

  True, intellectually sophisticated characters were exactly what I looked for in life. My erudite and polymathic father—who, were he still alive, would have only been two weeks older than Benoît M.—liked the company of extremely cultured Jesuit priests. I remember these Jesuit visitors occupying my chair at the dining table. I recall that one had a medical degree and a PhD in physics, yet taught Aramaic to locals in Beirut’s Institute of Eastern Languages. His previous assignment could have been teaching high school physics, and the one before that was perhaps in the medical school. This kind of erudition impressed my father far more than scientific assembly-line work. I may have something in my genes driving me away from bildungsphilisters.

  Although Mandelbrot often expressed amazement at the temperament of high-flying erudites and remarkable but not-so-famous scientists, such as his old friend Carleton Gajdusek, a man who impressed him with his ability to uncover the causes of tropical diseases, he did not seem eager to trumpet his association with those we consider great scientists. It took me a while to discover that he had worked with an impressive list of scientists in seemingly every field, something a name-dropper would have brought up continuously. Although I have been working with him for a few years now, only the other day, as I was chatting with his wife, did I discover that he spent two years as the mathematical collaborator of the psychologist Jean Piaget. Another shock came when I discovered that he had also worked with the great historian Fernand Braudel, but Mandelbrot did not seem to be interested in Braudel. He did not care to discuss John von Neuman with whom he had worked as a postdoctoral fellow. His scale was inverted. I asked him once about Charles Tresser, an unknown physicist I met at a party who wrote papers on chaos theory and supplemented his researcher’s income by making pastry for a shop he ran near New York City. He was emphatic: “un homme extraordinaire,” he called Tresser, and could not stop praising him. But when I asked him about a particular famous hotshot, he replied, “He is the prototypical bon élève, a student with good grades, no depth, and no vision.” That hotshot was a Nobel laureate.

  THE PLATONICITY OF TRIANGLES

  Now, why am I calling this business Mandelbrotian, or fractal, randomness? Every single bit and piece of the puzzle has been previously mentioned by someone else, such as Pareto, Yule, and Zipf, but it was Mandelbrot who a) connected the dots, b) linked randomness to geometry (and a special brand at that), and c) took the subject to its natural conclusion. Indeed many mathematicians are famous today partly because he dug out their works to back up his claims—the strategy I am following here in this book. “I had to invent my predecessors, so people take me seriously,” he once told me, and he used the credibility of big guns as a rhetorical device. One can almost always ferret out predecessors for any thought. You can always find someone who worked on a part of your argument and use his contribution as your backup. The scientific association with a big idea, the “brand name,” goes to the one who connects the dots, not the one who makes a casual observation—even Charles Darwin, who uncultured scientists claim “invented” the survival of the fittest, was not the first to mention it. He wrote in the introduction of The Origin of Species that the facts he presented were not necessarily original; it was the consequences that he thought were “interesting” (as he put it with characteristic Victorian modesty). In the end it is those who derive consequences and seize the importance of the ideas, seeing their real value, who win the day. They are the ones who can talk about the subject.

  So let me describe Mandelbrotian geometry.

  The Geometry of Nature

  Triangles, squares, circles, and the other geometric concepts that made many of us yawn in the classroom may be beautiful and pure notions, but they seem more present in the minds of architects, design artists, modern art buildings, and schoolteachers than in nature itself. That’s fine, except that most of us aren’t aware of this. Mountains are not triangles or pyramids; trees are not circles; straight lines are almost never seen anywhere. Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand.

  I have said that we seem naturally inclined to Platonify, and to think exclusively in terms of studied material: nobody, whether a bricklayer or a natural philosopher, can easily escape the enslavement of such conditioning. Consider that the great Galileo, otherwise a debunker of falsehoods, wrote the following:

  The great book of Nature lies ever open before our eyes and the true philosophy is written in it. … But we cannot read it unless we have first learned the language and the characters in which it is written. … It is written in mathematical language and the characters are triangles, circles and other geometric figures.

  Was Galileo legally blind? Even the great Galileo, with all his alleged independence of mind, was not capable of taking a clean look at Mother Nature. I am confident that he had windows in his house and that he ventured outside from time to time: he should have known that triangles are not easily found in nature. We are so easily brainwashed.

  We are either blind, or illiterate, or both. That nature’s geometry is not Euclid’s was so obvious, and nobody, almost nobody, saw it.

  This (physical) blindness is identical to the ludic fallacy that makes us think casinos represent randomness.

  Fractality

  But first, a description of fractals. Then we will show how they link to what we call power laws, or scalable laws.

  Fractal is a word Mandelbrot coined to describe the geometry of the rough and broken—from the Latin fractus, the origin of fractured. Fractality is the repetition of geometric patterns at different scales, revealing smaller and smaller versions of themselves. Small parts resemble, to some degree, the whole. I will try to show in this chapter how the fractal applies to the brand of uncertainty that should bear Mandelbrot’s name: Mandelbrotian randomness.

  The veins in leaves look like branches; branches look like trees; rocks look like small mountains. There is no qualitative change when an object changes size. If you look at the coast of Britain from an airplane, it resembles what you see when you look at it with a magnifying glass. This character of self-affinity implies that one deceptively short and simple rule of iteration can be used, either by a computer or, more randomly, by Mother Nature, to build shapes of seemingly great complexity. This can come in handy for computer graphics, but, more important, it is how nature works. Mandelbrot designed the mathematical object now known as the Mandelbrot set, the most famous object in the history of mathematics. It became popular with followers of chaos theory because it generates pictures of ever increasing complexity by using a deceptively minuscule recursive rule; recursive means that something can be reapplied to itself infinitely. You can look at the set at smaller and smaller resolutions without ever reaching the limit; you will continue to see recognizable shapes. The shapes are never the same, yet they bear an affinity to one another, a strong family resemblance.

  These objects play a role in aesthetics. Consider the following applications:

  Visual arts: Most computer-generated objects are now based on some version of the Mandelbrotian fractal. We can also see fractals in architecture, paintings, and many works of visual art—of course, not consciously incorporated by the work’s creator.

  Music: Slowly hum the four-note opening of Beethoven’s Fifth Symphony: ta-ta-ta-ta. Then replace each individual note with the same four-note opening, so that you end up with a measure of sixteen notes. You will see (or, rather, hear) that each smaller wave resembles the original larger one. Bach and Mahler, for instance, wrote submovements that resemble the larger movements of which they are a part.

  Poetry: Emily Dickinson’s poetry, for instance, is fractal: the large resembles the small. It has, according to a commentator, “a consciously made assemblage of dictions, metres, rhetorics, gestures, and tones.”

  Fractals initially made Benoît M. a pariah in the mathematical establishment. French mathematicians
were horrified. What? Images? Mon dieu! It was like showing a porno movie to an assembly of devout Eastern Orthodox grandmothers in my ancestral village of Amioun. So Mandelbrot spent time as an intellectual refugee at an IBM research center in upstate New York. It was a f*** you money situation, as IBM let him do whatever he felt like doing.

  But the general public (mostly computer geeks) got the point. Mandelbrot’s book The Fractal Geometry of Nature made a splash when it came out a quarter century ago. It spread through artistic circles and led to studies in aesthetics, architectural design, even large industrial applications. Benoît M. was even offered a position as a professor of medicine! Supposedly the lungs are self-similar. His talks were invaded by all sorts of artists, earning him the nickname the Rock Star of Mathematics. The computer age helped him become one of the most influential mathematicians in history, in terms of the applications of his work, way before his acceptance by the ivory tower. We will see that, in addition to its universality, his work offers an unusual attribute: it is remarkably easy to understand.

  A few words on his biography. Mandelbrot came to France from Warsaw in 1936, at the age of twelve. Owing to the vicissitudes of a clandestine life during Nazi-occupied France, he was spared some of the conventional Gallic education with its uninspiring algebraic drills, becoming largely self-taught. He was later deeply influenced by his uncle Szolem, a prominent member of the French mathematical establishment and holder of a chair at the Collège de France. Benoît M. later settled in the United States, working most of his life as an industrial scientist, with a few transitory and varied academic appointments.

  The computer played two roles in the new science Mandelbrot helped conceive. First, fractal objects, as we have seen, can be generated with a simple rule applied to itself, which makes them ideal for the automatic activity of a computer (or Mother Nature). Second, in the generation of visual intuitions lies a dialectic between the mathematician and the objects generated.

  Now let us see how this takes us to randomness. In fact, it is with probability that Mandelbrot started his career.

  A Visual Approach to Extremistan/Mediocristan

  I am looking at the rug in my study. If I examine it with a microscope, I will see a very rugged terrain. If I look at it with a magnifying glass, the terrain will be smoother but still highly uneven. But when I look at it from a standing position, it appears uniform—it is almost as smooth as a sheet of paper. The rug at eye level corresponds to Mediocristan and the law of large numbers: I am seeing the sum of undulations, and these iron out. This is like Gaussian randomness: the reason my cup of coffee does not jump is that the sum of all of its moving particles becomes smooth. Likewise, you reach certainties by adding up small Gaussian uncertainties: this is the law of large numbers.

  The Gaussian is not self-similar, and that is why my coffee cup does not jump on my desk.

  Now, consider a trip up a mountain. No matter how high you go on the surface of the earth, it will remain jagged. This is even true at a height of 30,000 feet. When you are flying above the Alps, you will still see jagged mountains in place of small stones. So some surfaces are not from Mediocristan, and changing the resolution does not make them much smoother. (Note that this effect only disappears when you go up to more extreme heights. Our planet looks smooth to an observer from space, but this is because it is too small. If it were a bigger planet, then it would have mountains that would dwarf the Himalayas, and it would require observation from a greater distance for it to look smooth. Likewise, if the planet had a larger population, even maintaining the same average wealth, we would be likely to find someone whose net worth would vastly surpass that of Bill Gates.)

  Figures 11 and 12 illustrate the above point: an observer looking at the first picture might think that a lens cap has fallen on the ground.

  Recall our brief discussion of the coast of Britain. If you look at it from an airplane, its contours are not so different from the contours you see on the shore. The change in scaling does not alter the shapes or their degree of smoothness.

  Pearls to Swine

  What does fractal geometry have to do with the distribution of wealth, the size of cities, returns in the financial markets, the number of casualties in war, or the size of planets? Let us connect the dots.

  The key here is that the fractal has numerical or statistical measures that are (somewhat) preserved across scales—the ratio is the same, unlike the Gaussian. Another view of such self-similarity is presented in Figure 13. As we saw in Chapter 15, the superrich are similar to the rich, only richer—wealth is scale independent, or, more precisely, of unknown scale dependence.

  In the 1960s Mandelbrot presented his ideas on the prices of commodities and financial securities to the economics establishment, and the financial economists got all excited. In 1963 the then dean of the University of Chicago Graduate School of Business, George Shultz, offered him a professorship. This is the same George Shultz who later became Ronald Reagan’s secretary of state.

  FIGURE 11: Apparently, a lens cap has been dropped on the ground. Now turn the page.

  Shultz called him one evening to rescind the offer.

  At the time of writing, forty-four years later, nothing has happened in economics and social science statistics—except for some cosmetic fiddling that treats the world as if we were subject only to mild randomness—and yet Nobel medals were being distributed. Some papers were written offering “evidence” that Mandelbrot was wrong by people who do not get the central argument of this book—you can always produce data “corroborating” that the underlying process is Gaussian by finding periods that do not have rare events, just like you can find an afternoon during which no one killed anyone and use it as “evidence” of honest behavior. I will repeat that, because of the asymmetry with induction, just as it is easier to reject innocence than accept it, it is easier to reject a bell curve than accept it; conversely, it is more difficult to reject a fractal than to accept it. Why? Because a single event can destroy the argument that we face a Gaussian bell curve.

  In sum, four decades ago, Mandelbrot gave pearls to economists and résumé-building philistines, which they rejected because the ideas were too good for them. It was, as the saying goes, margaritas ante porcos, pearls before swine.

  FIGURE 12: 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)*

  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* is going to be the ratio of the two numbers to the negative power of the power 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 estimat
e 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 12 books should sell more than a million copies, here 96 times (1,000,000/250,000)−1.5.

  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*

  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

  Market moves 3 (or lower)