Fragility Is Nonlinear

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.