1 The other biographer of Socrates, Xenophon, presents a different picture. The Socrates of the Memorabilia is no-nonsense and down to earth; he despises sterile knowledge, and the experts who study matters without practical consequence when so many useful and important things are neglected (instead of looking at stars to understand causes, figure out how you can use them to navigate; use geometry to measure land, but no more).
2 Adam Smith was first and last a moral philosopher. Marx was a philosopher. Kahneman and Simon are psychologist and cognitive scientist, respectively. The exception is, of course, Hayek.
3 The philosopher Rupert Read convinced me that Hayek harbored in fact a strain of naive rationalism, as did Popper, and presents convincing arguments that the two should not be included in the category of antifragile thinkers.
4 The reader might wonder about the connection between education and disorder. Education is teleological and hates disorder. It tends to cater to fragilistas.
BOOK V
The Nonlinear and the Nonlinear1
Time for another autobiographical vignette. As Charles Darwin wrote in a historical section of his On the Origin of Species, presenting a sketch of the progress of opinion: “I hope I may be excused for entering on these personal details, as I give them to show that I have not been hasty in coming to a decision.” For it is not quite true that there is no exact word, concept, and application for antifragility. My colleagues and I had one without knowing it. And I had it for a long, very long time. So I have been thinking about the exact same problem most of my life, partly consciously, partly without being aware of it. Book V explores the journey and the idea that came with it.
ON THE IMPORTANCE OF ATTICS
In the mid-1990s, I quietly deposited my necktie in the trash can at the corner of Forty-fifth Street and Park Avenue in New York. I decided to take a few years off and locked myself in the attic, trying to express what was coming out of my guts, trying to frame what I called “hidden nonlinearities” and their effects.
What I had wasn’t quite an idea, rather, just a method, for the deeper central idea eluded me. But using this method, I produced close to a six-hundred-page-long discussion of managing nonlinear effects, with graphs and tables. Recall from the prologue that “nonlinearity” means that the response is not a straight line. But I was going further and looking at the link with volatility, something that should be clear soon. And I went deep into the volatility of volatility, and such higher-order effects.
The book that came out of this solitary investigation in the attic, finally called Dynamic Hedging, was about the “techniques to manage and handle complicated nonlinear derivative exposures.” It was a technical document that was completely ab ovo (from the egg), and as I was going, I knew in my guts that the point had vastly more import than the limited cases I was using in my profession; I knew that my profession was the perfect platform to start thinking about these issues, but I was too lazy and too conventional to venture beyond. That book remained by far my favorite work (before this one), and I fondly remember the two harsh New York winters in the near-complete silence of the attic, with the luminous effect of the sun shining on the snow warming up both the room and the project. I thought of nothing else for years.
I also learned something quite amusing from the episode. My book was mistakenly submitted to four referees, all four of them academic financial economists instead of “quants” (quantitative analysts who work in finance using mathematical models). The person who made the submissions wasn’t quite aware of the difference. The four academics rejected my book, interestingly, for four sets of completely different reasons, with absolutely no intersection in their arguments. We practitioners and quants aren’t too fazed by remarks on the part of academics—it would be like prostitutes listening to technical commentary by nuns. What struck me was that if I had been wrong, all of them would have provided the exact same reason for rejection. That’s antifragility. Then, of course, as the publisher saw the mistake, the book was submitted to quantitative reviewers, and it saw the light of day.2
The Procrustean bed in life consists precisely in simplifying the nonlinear and making it linear—the simplification that distorts.
Then my interest in the nonlinearity of exposures went away as I began to deal with other matters related to uncertainty, which seemed more intellectual and philosophical to me, like the nature of randomness—rather than how things react to random events. This may also have been due to the fact that I moved and no longer had that attic.
But some events brought me back to a second phase of intense seclusion.
After the crisis of the late 2000s, I went through an episode of hell owing to contact with the press. I was suddenly deintellectualized, corrupted, extracted from my habitat, propelled into being a public commodity. I had not realized that it is hard for members of the media and the public to accept that the job of a scholar is to ignore insignificant current affairs, to write books, not emails, and not to give lectures dancing on a stage; that he has other things to do, like read in bed in the morning, write at a desk in front of a window, take long walks (slowly), drink espressos (mornings), chamomile tea (afternoons), Lebanese wine (evenings), and Muscat wines (after dinner), take more long walks (slowly), argue with friends and family members (but never in the morning), and read (again) in bed before sleeping, not keep rewriting one’s book and ideas for the benefit of strangers and members of the local chapter of Networking International who haven’t read it.
Then I opted out of public life. When I managed to retake control of my schedule and my brain, recovered from the injuries deep into my soul, learned to use email filters and autodelete functions, and restarted my life, Lady Fortuna brought two ideas to me, making me feel stupid—for I realized I had had them inside me all along.
Clearly, the tools of analysis of nonlinear effects are quite universal. The sad part is that until that day in my new-new life of solitary walker cum chamomile drinker, when I looked at a porcelain cup I had not realized that everything nonlinear around me could be subjected to the same techniques of detection as the ones that hit me in my previous episode of seclusion.
What I found is described in the next two chapters.
1 The nontechnical reader can skip Book V without any loss: the definition of antifragility from Seneca’s asymmetry is amply sufficient for a literary read of the rest of the book. This is a more technical rephrasing of it.
2 A similar test: when a collection of people write “There is nothing new here” and each one cites a different originator of the idea, one can safely say there is something effectively new.
CHAPTER 18
On the Difference Between a Large Stone and a Thousand Pebbles
How to punish with a stone—I landed early (once)—Why attics are always useful—On the great benefits of avoiding Heathrow unless you have a guitar
FIGURE 8. The solicitor knocking on doors in concave (left) and convex (right) position. He illustrates the two forms of nonlinearity; if he were “linear” he would be upright, standing straight. This chapter will show—a refinement of Seneca’s asymmetry—how one position (the convex) represents antifragility in all its forms, the other, fragility (the concave), and how we can easily detect and even measure fragility by evaluating how humped (convex) or how slumped (concave) the courtier is standing.
I noticed looking at the porcelain cup that it did not like volatility or variability or action. It just wanted calm and to be left alone in the tranquility of the home study-library. The realization that fragility was simply vulnerability to the volatility of the things that affect it was a huge personal embarrassment for me, since my specialty was the link between volatility and nonlinearity; I know, I know, a very strange specialty. So let us start with the result.
A SIMPLE RULE TO DETECT THE FRAGILE
A story present in the rabbinical literature (Midrash Tehillim), probably originating from earlier Near Eastern lore, says the following. A king, angry at his son, swore that
he would crush him with a large stone. After he calmed down, he realized he was in trouble, as a king who breaks his oath is unfit to rule. His sage advisor came up with a solution. Have the stone cut into very small pebbles, and have the mischievous son pelted with them.
The difference between a thousand pebbles and a large stone of equivalent weight is a potent illustration of how fragility stems from nonlinear effects. Nonlinear? Once again, “nonlinear” means that the response is not straightforward and not a straight line, so if you double, say, the dose, you get a lot more or a lot less than double the effect—if I throw at someone’s head a ten-pound stone, it will cause more than twice the harm of a five-pound stone, more than five times the harm of a one-pound stone, etc. It is simple: if you draw a line on a graph, with harm on the vertical axis and the size of the stone on the horizontal axis, it will be curved, not a straight line. That is a refinement of asymmetry.
Now the very simple point, in fact, that allows for a detection of fragility:
For the fragile, shocks bring higher harm as their intensity increases (up to a certain level).
FIGURE 9. The King and His Son. The harm from the size of the stone as a function of the size of the stone (up to a point). Every additional weight of the stone harms more than the previous one. You see nonlinearity (the harm curves inward, with a steeper and steeper vertical slope).
The example is shown in Figure 9. Let us generalize. Your car is fragile. If you drive it into the wall at 50 miles per hour, it would cause more damage than if you drove it into the same wall ten times at 5 mph. The harm at 50 mph is more than ten times the harm at 5 mph.
Other examples. Drinking seven bottles of wine (Bordeaux) in one sitting, then purified water with lemon twist for the remaining six days is more harmful than drinking one bottle of wine a day for seven days (spread out in two glasses per meal). Every additional glass of wine harms you more than the preceding one, hence your system is fragile to alcoholic consumption. Letting a porcelain cup drop on the floor from a height of one foot (about thirty centimeters) is worse than twelve times the damage from a drop from a height of one inch (two and a half centimeters).
Jumping from a height of thirty feet (ten meters) brings more than ten times the harm of jumping from a height of three feet (one meter)—actually, thirty feet seems to be the cutoff point for death from free fall.
Note that this is a simple expansion of the foundational asymmetry we saw two chapters ago, as we used Seneca’s thinking as a pretext to talk about nonlinearity. Asymmetry is necessarily nonlinearity. More harm than benefits: simply, an increase in intensity brings more harm than a corresponding decrease offers benefits.
Why Is Fragility Nonlinear?
Let me explain the central argument—why fragility is generally in the nonlinear and not in the linear. That was the intuition from the porcelain cup. The answer has to do with the structure of survival probabilities: conditional on something being unharmed (or having survived), then it is more harmed by a single rock than a thousand pebbles, that is, by a single large infrequent event than by the cumulative effect of smaller shocks.
If for a human, jumping one millimeter (an impact of small force) caused an exact linear fraction of the damage of, say, jumping to the ground from thirty feet, then the person would already be dead from cumulative harm. Actually a simple computation shows that he would have expired within hours from touching objects or pacing in his living room, given the multitude of such stressors and their total effect. The fragility that comes from linearity is immediately visible, so we rule it out because the object would be already broken. This leaves us with the following: what is fragile is something that is both unbroken and subjected to nonlinear effects—and extreme, rare events, since impacts of large size (or high speed) are rarer than ones of small size (and slow speed).
Let me rephrase this idea in connection with Black Swans and extreme events. There are a lot more ordinary events than extreme events. In the financial markets, there are at least ten thousand times more events of 0.1 percent magnitude than events of 10 percent magnitude. There are close to eight thousand microearthquakes daily on planet Earth, that is, those below 2 on the Richter scale—about three million a year. These are totally harmless, and, with three million per year, you would need them to be so. But shocks of intensity 6 and higher on the scale make the newspapers. Take objects such as porcelain cups. They get a lot of hits, a million more hits of, say, one hundredth of a pound per square inch (to take an arbitrary measure) than hits of a hundred pounds per square inch. Accordingly, we are necessarily immune to the cumulative effect of small deviations, or shocks of very small magnitude, which implies that these affect us disproportionally less (that is, nonlinearly less) than larger ones.
Let me reexpress my previous rule:
For the fragile, the cumulative effect of small shocks is smaller than the single effect of an equivalent single large shock.
This leaves me with the principle that the fragile is what is hurt a lot more by extreme events than by a succession of intermediate ones. Finito—and there is no other way to be fragile.
Now let us flip the argument and consider the antifragile. Antifragility, too, is grounded in nonlinearties, nonlinear responses.
For the antifragile, shocks bring more benefits (equivalently, less harm) as their intensity increases (up to a point).
A simple case—known heuristically by weight lifters. In the bodyguard-emulating story in Chapter 2, I focused only on the maximum I could do. Lifting one hundred pounds once brings more benefits than lifting fifty pounds twice, and certainly a lot more than lifting one pound a hundred times. Benefits here are in weight-lifter terms: strengthening the body, muscle mass, and bar-fight looks rather than resistance and the ability to run a marathon. The second fifty pounds play a larger role, hence the nonlinear (that is, we will see, convexity) effect. Every additional pound brings more benefits, until one gets close to the limit, what weight lifters call “failure.”1
For now, note the reach of this simple curve: it affects about just anything in sight, even medical error, government size, innovation—anything that touches uncertainty. And it helps put the “plumbing” behind the statements on size and concentration in Book II.
When to Smile and When to Frown
Nonlinearity comes in two kinds: concave (curves inward), as in the case of the king and the stone, or its opposite, convex (curves outward). And of course, mixed, with concave and convex sections.
Figures 10 and 11 show the following simplifications of nonlinearity: the convex and the concave resemble a smile and a frown, respectively.
FIGURE 10. The two types of nonlinearities, the convex (left) and the concave (right). The convex curves outward, the concave inward.
FIGURE 11. Smile! A better way to understand convexity and concavity. What curves outward looks like a smile—what curves inward makes a sad face. The convex (left) is antifragile, the concave (right) is fragile (has negative convexity effects).
I use the term “convexity effect” for both, in order to simplify the vocabulary, saying “positive convexity effects” and “negative convexity effects.”
Why does asymmetry map to convexity or concavity? Simply, if for a given variation you have more upside than downside and you draw the curve, it will be convex; the opposite for the concave. Figure 12 shows the asymmetry reexpressed in terms of nonlinearities. It also shows the magical effect of mathematics that allowed us to treat steak tartare, entrepreneurship, and financial risk in the same breath: the convex graph turns into concave when one simply puts a minus sign in front of it. For instance, Fat Tony had the exact opposite payoff than, say, a bank or financial institution in a certain transaction: he made a buck whenever they lost one, and vice versa. The profits and losses are mirror images of each other at the end of the day, except that one is the minus sign times the other.
Figure 12 also shows why the convex likes volatility. If you earn more than you lose from fluctuations, you want a l
ot of fluctuations.
FIGURE 12. Pain More than Gain, or Gain More than Pain. Assume you start from the “You Are Here” spot. In the first case, should the variable x increase, i.e., move to the right on the horizontal axis, the gains (vertical axis) are larger than the losses encountered by moving left, i.e., an equivalent decrease in the variable x. The graph illustrates how positive asymmetry (first graph) turns into convex (inward) curving and negative asymmetry (second graph) turns into concave (outward) curving. To repeat, for a set deviation in a variable, in equivalent amounts in both directions, the convex gains more than it loses, and the reverse for the concave.
Why Is the Concave Hurt by Black Swan Events?
Now the idea that has inhabited me all my life—I never realized it could show so clearly when put in graphical form. Figure 13 illustrates the effect of harm and the unexpected. The more concave an exposure, the more harm from the unexpected, and disproportionately so. So very large deviations have a disproportionately larger and larger effect.
FIGURE 13. Two exposures, one linear, one nonlinear, with negative convexity—that is, concavity—in the first graph, positive convexity in the second. An unexpected event affects the nonlinear disproportionately more. The larger the event, the larger the difference.
Antifragile: Things That Gain from Disorder Page 31