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 Uni verTHE
A E S T H E T I C S OF RANDOMNESS 2 6 1
FIGURE 11 : Apparently, a lens cap has been dropped on the ground. Now turn the
page.
sity 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.
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
262 THOSE GRAY SWANS OF EXTREMISTAN
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.
too good for them. It was, as the saying goes, margaritas ante porcos,
pearls before swine.
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 peo-
* The nontechnical reader can skip from here until the end of the chapter.
T H E A E S T H E T I C S O F R A N D O M N E S S 263
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.
pie 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.
* By using symmetry we could also examine the incidences below the number.
2 6 4 THOSE GRAY SWANS OF EXTREMISTAN
TABLE 2: ASSUMED EXPONENTS FOR VARIOUS PHENOMENA*
Phenomenon
Frequency of use of words
Number of hits on websites
Number of books sold in the U.S.
Telephone calls received
Magnitude of earthquakes
Diameter of moon craters
Intensity of solar flares
Intensity of wars
Net worth of Americans
Number of persons per family
name
Population of U.S. cities
Market moves
Company size
People killed in terrorist attacks
Assumed Exponent
(vague approximation)
1.2
1.4
1.5
1.22
2.8
2.14
0.8
0.8
1.1
1
1.3
3 (or lower)
1.5
2 (but possibly a much lower
exponent)
* Source: M.E.J. Newman (2005) and the author's own calculations.
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 ( 5 0 0 , 0 0 0 / 2 5 0 , 0 0 0 ) 1 5 . We can continue, and note that
around 8 books should sell more than a million copies, here 96 times
( l , 0 0 0 , 0 0 0 / 2 5 0 , 0 0 0 ) 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
THE A E S T H E T I C S OF RANDOMNESS 2 6 5
TABLE 3: THE MEANING OF THE EXPONENT
Exponent Share of the top 1% Share of the top 20%
1 99.99%* 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%
* Clearly, you do not observe 100 percent In a finite sample.
true parameters—if it in fact exists. Let us first examine the practical consequences
of an exponent.
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
2 6 6 THOSE GRAY SWANS OF EXTREMISTAN
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,
THE A E S T H E T I C S OF RANDOMNESS 2 6 7
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.
I have just read three "popular science" books that summarize the research
in complex systems: Mark Buchanan's Ubiquity, Philip Ball's Critical
Mass, and Paul Ormerod's Why Most Things Fail. These three authors
present the world of social science as full of power laws, a view with
which I most certainly agree. They also claim that there is universality of
many of these phenomena, that there is a wonderful similarity between
various processes in nature and the behavior of social groups, which I
agree with. They back their studies with the various theories on networks
and show the wonderful correspondence between the so-called critical
phenomena in natural science and the self-organization of social groups.
They bring together processes that generate avalanches, social contagions,
and what they call informational cascades, which I agree with.
Universality is one of the reasons physicists find power laws associated
with critical points particularly interesting. There are many situations,
both in dynamical systems theory and statistical mechanics, where many
of the properties of the dynamics around critical points are independent of
the details of the underlying dynamical system. The exponent at the critical
point may be the same for many systems in the same group, even
though many other aspects of the system are different. I almost agree with
this notion of universality. Finally, all three authors encourage us to apply
techniques from statistical physics, avoiding econometrics and Gaussianstyle
nonscalable distributions like the plague, and I couldn't agree more.
But all three authors, by producing, or promoting precision, fall into
