taleb thales’ secret, or the intelligence of antifragility

A Simple Rule, Perhaps Too Simple 59

10/19/11 © Copyright 2011 by N. N. Taleb. This draft version cannot be disseminated or quoted.

Chapter 4. Thales’ Secret, or The Intelligence of Antifragility

I landed early (once) — A simple heuristic to get an inheritance — Where we discuss the idea of doing instead place of walking the Great Walk — Where the philosophers' stone was staring at us — ideas matter less than fragility

Now, the central chapter. A bit technical, but central. Or perhaps not so

technical after all, but still very central.

A SIMPLE RULE, PERHAPS TOO SIMPLE

A story, present in the rabbinical literature (Midrash Tehillim), probably

originating in 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.

This is a potent illustration of how fragility stems from nonlinear

effects. Let us leave side the idea of circumventing rules and other lessons

one might derive from it, and focus on the very simple point, in fact, that

defines fragility:

For the fragile shocks bring higher harm as their intensity increases (up to the

point of breaking).

I’ve used the intuition to show why large corporations hurt more (when

they fall) than small ones and why speed is not a good thing —whether in

traffic or in business. Your car is fragile. If you drive it into the wall at fifty

miles per hour, it would cause more damage than if you drove it into the

same wall ten times at five mph. The harm at fifty miles per hour is more

than ten times the harm at five mph.

Other examples. Drinking seven bottles of wine in one sitting, then

water 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.

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 freefall. Or 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 (2 and a half centimeters).

Figure 7- 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 inwards, with a steeper and steeper vertical slope).

Let me explain the central argument —why is fragility necessarily in the

nonlinear and not in the linear? Just as with the stone hurting more than the

pebbles, 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 be already 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. The fragility



that comes from linearity is immediately visible, so we rule it out because the

object would be already broken and the person already dead. 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 hits of large

size (or high speed) are rarer than ones of small size (and slow speed).

Let me rephrase it again in connection with Black Swans and extreme

events. There are a lot more ordinary events than extreme events, say, in

ordinary life, a million more hits of (to take an arbitrary measure), say, one

hundredth of a pound per square inch than hits of a hundred pounds per

square inch, or, in the financial markets, there are at least ten thousand time

more events of .1% than events of 10%. So 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. There are close to eight thousand micro-earthquakes

daily on planet earth, that is, those below 2 on the Richter scale —about

three million a year. That is These are totally harmless, and, with three

million per year, you would need them to be so. But shocks of 6 and higher

make the newspapers.

Let me try again and re-express my previous rule.

For the fragile, the cumulative effect of small shocks is smaller than the single effect of a large shock.

This leaves me with the definition that the fragile is hurt a lot more by

extreme events. Finito —and there is no other definition.

One more illustration. Consider that objects handled by humans, say a

coffee cup or a cell phone gets cumulative impacts equivalent to, say, tons

per square inch over the years, but break at the slightest fall.

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 —what is known heuristically by weightlifters. 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 mean strengthening of the body, muscle growth and beach-friendly

looks). 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 weightlifters call “failure”.

We will have more illustrations of these two simple points; they are

quite central as they allow us to immediately compare objects and classify

them in the Triad of Chapter 1.

For now, note the reach of this simple curve: if affects about anything

in sight, even medical error or size of government —anything that touches

uncertainty. And, of course, innovation.

When to Smile and When to Frown

Linearity comes into two kinds. Concave (curves inward) or its opposite

convex (curves outwards).

Figures 5 and 6 show the following simplifications of nonlinearity: the

convex and the concave.

Figure 8 The different types of nonlinearities. The convex (left) and the concave (right). The convex curves outward, the concave inward.



Figure 9 A better way to understand convexity and concavity. What curves inwards looks like a smile —and what curves outwards 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

Traffic in Heathrow

Another example of the convexity effect. Traffic is highly nonlinear. When I

take the day flight from New York to London, and I leave my residence

around 5 AM (yes, I know), it takes me around 26 minutes to reach the

British Air terminal at JFK airport. At that time, New York is empty. When I

leave my place at 6 AM for the later flight, there is almost no difference in

travel time although traffic is a bit denser. You can add more and more cars

on the highway, with no or minimal impact concerning time spent in traffic.

Then, mystery, you increase the number of cars by 10% and the travel

time jumps up by 50% (I am using approximate numbers). Look at the

convexity effect at work: the average number of cars does not matter at all

for traffic speed. If you have 90,000 cars for one hour, then 110,000 cars for

another hour, traffic would be much, much slower than if you had 100,000

cars for two hours. But the hitch: travel time is a negative, so I count it as a

cost, like an expense, and a rise is a bad thing.

So travel cost is fragile to the volatility of the number of cars on the

highway; it does not depend so much on their average number. Every

additional car increases travel time more than the previous one.

This is a hint to a central problem of the world today, that of the

misunderstanding of nonlinear response by those involved in creating

“efficiencies” and “optimization” of systems. For instance, European airports

and railroads are stretched, seeming overly efficient. They operate at close to

maximal capacity, with minimal redundancies and idle capacity, hence

acceptable costs; but a small additional congestion, say 5% more planes in

the sky owing to a tiny backlog can set chaos in airports and cause scenes of

unhappy travelers camping on floors, with for sole solace some bearded

fellow playing French folk songs on his guitar.

We will see applications of the point across economic domains: central

banks can print money; they print and print with no effect (and claim the

“safety” of such measure) then, "unexpectedly", the printing causes a jump in

inflation. Chapter x uses the simple idea to show how many economic results

are completely cancelled by convexity effects. The tools (and culture) of

policy makers are based on the overly linear, ignoring these hidden effects.

I have rapidly put a (very hypothetical) graph of the response in Figure

6. Note for now the curved shape of the graph. It curves inward.

Figure 10- The graph (vertical) shows how the author’s travel time (and travel costs) to JFK depend, beyond a certain point, nonlinearly on the number of cars. We show travel costs as curving inward —concave, not a good thing.

The Scaling Property

I will keep pounding the reader with illustrations and explanations from

several angles. Another intuitive way to look at convexity effects is in

considering the scaling property. If you double the exposure, do you more



than double the harm? If so, then this is a situation of fragility. Otherwise,

you are antifragile.

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 that clearly when put in graphical form.

Figure x illustrates the effect of harm and the unexpected. The more

concave an exposure, the more harm from the unexpected, and

disproportionately so.

Figure 11- Two exposures, one linear, one nonlinear (with negative convexity). An unexpected event affects the nonlinear disproportionately more. The larger the event, the larger the difference.

How to Exit a Movie Theatre

Another example. Imagine how people exit a movie theatre. Someone shouts

“fire”, and you have a dozen persons squashed to death. So we have a

fragility of the theater to size, stemming from the fact that every additional

person exiting brings more and more trauma (a negative convexity effect). A

thousand people exiting (or trying to exit) in one minute is not the same as

the same number exiting in half an hour. Someone unfamiliar with the

business who naively optimizes the size of the place (as Heathrow airport, for

example) might miss the idea that smooth functioning at regular times is

different from the rough functioning at times of stress.

This is the problem of the “squeeze” (largely ignored): when people

have no choice but to do something, and do it right away, regardless of the

cost. They are squeezed. It so happens that modern economic optimized life

causes us to build larger and larger theaters, but with the exact same door.

They do not make this mistake too often while building cinemas and movie

theaters, but we tend to do that in other domains, with, for instance, natural

resources, or as I have just mentioned, central bank policies that ignore

nonlinear responses*.

A “Balanced Meal”

Another example of missing the hidden dimension, that is, variability. We

are currently told by the Soviet-Harvard U.S. health authorities to eat set

quantities of nutrients (total calories, protein, vitamins, etc.) every day, in

some recommended amounts (say the optimal amounts of each, the

equivalent to the optimal condition of seventy degrees in the story of the

grandmother). Aside from the lack of empirical rigor in the way these

recommendations are currently derived (as we will see in Chapter x), there is

another sloppiness in the edict: an insistence in the discourse on the

regularity. Those recommending the nutritional policies fail to understand

that “steadily” getting your calories and nutrients throughout the day, with

“balanced” composition and metronomic regularity does not necessarily

carry the same effect as having them unevenly or randomly distributed, say

by having a lot of proteins one day, fasting completely another, feasting the

third, etc.

For a long time, nobody even bothered to try to figure out whether

variability in distribution mattered just as much as long term composition. I

will go deeper into the issue in medical discussion in Part III but it turned

out that the effect of variability in food sources and the nonlinearity in the

physiological response is central to biological systems. Consuming no

* The other problem is that of the misunderstanding of the nonlinearity of natural

resources, or anything particularly scarce and vital. Economists have the so-called law

of scarcity, by which things increase in value according to their demand —but they

ignore the consequences of nonlinearities. My former thesis director Hélyette Geman

and I are currently proposing a “law of convexity” that makes commodities,

particularly vital ones, even more dear than previously thought.

When The Average is Irrelevant 63


protein at all on Monday, and catching up on Wednesday seemingly causes a

different —better— physiological response, possibly because the deprivation,

as a stressor, activates some pathways that facilitate the subsequent

absorption of the nutrients (or something similar). And, until a few recent

(and disconnected) empirical studies, this convexity effect has been totally

missed by science —though not by religions, ancestral heuristics, and

traditions. And if scientists get some convexity effects (doctors understand

here and there nonlinearities in dose-response), the notion of convexity

effects itself, though, appears completely missed from the language and

methods, particularly biology, economics and other complex systems where

it belongs the most. No wonder there is no word for antifragility in

vocabularies.

Why Planes Don't Arrive Early

Travelers (typically) do not like uncertainty. This is a straight application of

convexity effects: fragility dislikes variability and volatility.

To see how these convexity effects play a role with any estimation and

model error, consider the following. I've taken the very same Paris-New York

flight most of my life. The flight takes about 8 hours, the equivalent of a

French novel plus a brief polite chat with a neighbor and a meal with

Bordeaux wine. I recall many instances in which I arrived early, about twenty

minutes, no more. But there have been instances in which I got there more

than 2 and 3 hours late, and, in at least one instance, it has taken me more

than two days to reach my destination.

Because travel time cannot be really negative, uncertainty tends to

cause delays, making arrival time increase, almost never decrease. Or it

makes arrival time just decrease by minutes, and increase by hours, an

obvious asymmetry. Anything unexpected, any shock, any volatility is likely

to extend the total flying time. As we will see it is very similar (though in

opposite effect) to a financial option: you cannot lose more than the

premium you spend for it and, because of that, any volatility benefits it.

This also explains the irreversibility of time, in a way, if you consider

the passage of time as an increase in disorder.

WHEN THE AVERAGE IS IRRELEVANT

We just saw with the number of cars on the highway that in the presence of

nonlinearities, the variability (or volatility) matters as much as, and

sometimes even a lot more than, the average. This is crucial, as there is a

mathematical property called “Jensen’s inequality” that shows that in

systems with positive convexity, the average underestimates the long term

benefits, and in those with negative convexity (that is, concave) ones, it

overestimates it. Let us start with the case of a grandmother’s thermal

happiness.

How to Lose a Grandmother

You are just being informed that your grandmother will spend the next two

hours at the very desirable temperature of seventy degrees Fahrenheit

(about twenty one degrees Celsius). Excellent you should think, since seventy

degrees is the optimal temperature for grandmothers. Since you went to

Business School, you are a “big picture” type of person and are satisfied with

the summary information.

But there is a second piece of data. Your grandmother, it turns out, will

spend the first hour at zero degrees Fahrenheit (around minus eighteen

Celsius), and the second hour at one hundred and forty degrees (around 60º

C), for an average of the very desirable Mediterranean-style seventy degrees

(21º C). So it looks like you will most certainly end up with no grandmother,

a funeral and, possibly, an inheritance.

Clearly, temperature changes become more and more harmful as they

deviate from seventy degrees. As you see, the second piece of information,

the variability, turned out to be more important than the first. The notion of

average here is of no significance when one is fragile to variations —the

dispersion in possible thermal outcomes here matters much more than the

average. Your grandmother is fragile to variations of temperature, to the

volatility of the weather. Let us call that second piece of information the

second order effect, or, more precisely the convexity effect.

Here, consider that, as much as a good simplification the notion of

average can be, it can also be a Procrustean bed. The information that the

average temperature is seventy degrees Fahrenheit does not simplify the

situation for your grandmother. It is an information squeezed into a

Procrustean bed —and these are necessarily committed by scientific

When The Average is Irrelevant 64


modelers since a model is, by its very nature, a simplification. You just don't

want the simplification to distort the situation to the point of being harmful.

Do not cross a river if it is four feet deep. We will see why numerical

summaries bring sucker problems.

Figure 4 shows the fragility of the health of the grandmother to

variations. If I plot health in the vertical axis, and temperature on the

horizontal one, I see a shape that curves inward —a “concave” shape, or

negative convexity effect.

If the grandmother’s response was “linear” (no curve, a straight line),

then the harm of temperature below seventy degrees would be offset with

benefits of temperature above it.

Figure 12 Fragility: Health as a function of temperature curves inward. A combination of 0 and 140 degrees (F) is worse for your grandmother's health than just 70 degrees. In fact almost any combination averaging 70 degrees is worse than just 70 degrees*. The graph shows concavity or negative convexity effects —curves inward.

Take this for now as we rapidly move to the more general attributes; in

the case of the grandmother’s health response to temperature:

a) there is nonlinearity (the response is not a straight line, not “linear”),

* I am simplifying a bit. There may be a few degrees variations around 70 for

which the grandmother might be better off than just 70, but I skip this nuance here. In

fact younger humans are antifragile to thermal variations, up to a point, benefiting

from some variability, then lose such antifragility with age (or disuse, as I suspect that

thermal comfort ages people and makes them fragile).

b) it curves inward, too much so.

c) the more nonlinear the response, the less relevant the average, and

the more relevant the stability around such average.

The Average is For Nerds

And, what is crucial, the absence of relevance of the average in many

domains has an epistemological dimension, linked to the idea that has

haunted me since childhood that the simplification of nerds tends to fragilize

or misses something in its reduction—which is another way to show why

fragilistas are more likely to be nerds.

The key, as we saw in Chapter x is the table of equivalence is that what

benefits from disorder benefits from the unknown.

And, a hint of what is to come in the discussion on rationalism,

remarkably, as I said earlier, those we deem intelligent, as they tend to

succeed in classes (particularly mathematics) and do well on SAT-style

exams, then make it to, say, MIT, in other words, the nerds, are even more

vulnerable to this mental distortion, and cause horrendous harm. Why?

Since the very definition of intelligence we use is grounded in their ability to

focus, hence contract and simplify, deal with, say, the average instead of a

richer set, and become blind to these small nuances. And the core of things

—of life— can reside in these nuances.

I said that reduction, compression of information in cases distorts. But

then “opinion” will also be irrelevant. Just like saying “seventy degrees” is

meaningless in the case of the grandmother, many other similar statements

such as “True/False” or “on balance” can be equally flawed. But there is

more: the convex will outperform, but we discuss a bit later with Thales’s

insight.

Walk, Don’t Run

Another illustration, this time a situation that benefits from variation—

positive convexity effects. Take two brothers, Castor and Polydeuces, who

need to travel a mile. Castor walks the mile at a leisurely pace and arrives at

destination in twenty minutes. Polydeuces spends fourteen minutes playing

with his handheld device getting updates on the gossip, then runs the same

mile in six minutes, to arrive at the same time as Castor.

Antifragility and Positive Convexity Effects 65


So both persons have covered the exact same distance, in exactly the

same time —same average. Castor who walked all the way presumably will

not get the same health benefits and gains in strength as Polydeuces who

sprinted. Health benefits are convex to speed (up to a point, of course).

The very idea of exercise is to gain from antifragility to workout

stressors —as we saw, lifting weight, exerting exercise are just exploitations

of convexity effects.

ANTIFRAGILITY AND POSITIVE CONVEXITY EFFECTS

I am here equating and mapping everything we call fragility and antifragility

into beneficial and harmful convexity effects. It makes matters simple to

explain and tie together scientifically and provides a universal tool to detect

fragilities, a simple test, and one that will prove useful throughout the book.

The idea of nonlinear effects had been occupying my mind since the days

when I was a student, and I chose to build a professional career around it,

but I did not realize that there could be a systematic way to see clearly

through the connections, sort of a general theory until I (accidentally) coined

the term antifragility. When the universality of the application of idea

became obvious —for instance it is completely behind Black Swan events

(sensitivity to large deviations comes from harmful convexity effects)— I

wrote it down on paper in very technical language and submitted it to a

technical journal in June 2011. My idea is not just that convexity effects are

fragility (and antifragility), but that we could identify them with very simple

method that measure convexity, which makes life much, much simpler. It

means that there is a very simple mathematical property behind things,

behind the reason things survive and flourish.

So we have with convexity effects the hidden expression of antifragility

—and a way to show how things have managed to survive and flourish

against that inexorable debunker of fragility, time, time that smartest of all.

And a formal, precise, and mathematical way to express the difference

between items in the triad, the central classification we made in Table 1.

The thought of applying the exact same test as the grandmother’s

temperature to everything that matters came to me one day as I was looking

at a porcelain cup. It dawned on me —the idea I expressed at the beginning

of the chapter—that the cup was breakable because, craving a certain form of

stability, it does not like disparity of outcomes —or, in other words, as we

said, higher intensity brings more harm (up to a point). To repeat the logic,

and present another view of negative convexity effects as not liking variation

(for a given average), if I drop the cup from a height of five inches once and

one tenth of an inch ninety nine times, if will break. But the average is about

one tenth of an inch. So the cup does not depend on the average height of the

drop, but on the variation around such average. I realized that everything

fragile has to have such property, the convexity effect, and that was it.

This intuition came to me as a trader as I specialized in anything

nonlinear. I may be a little ahead here, as options are coming later, so what

is to note is that I saw that a move of twenty percent in the markets was a

hundred times better for me than a move of four percent. That was the entire

foundation of my work.

We said that if the porcelain cup was linearly harmed, it would be

affected every time you put your finger on it. But there is a difference

between the mechanical and the organic. A cup may be harmed by long term

use, owing to what is called “material fatigue”, and never strengthens

organically from use, but humans and other biological entities not only do

not suffer material fatigue but , actually, as we saw in Chapter 1, age faster

when they are not used.*

The Effect on the Unseen

Let me stop and summarize the point of this (very central) l chapter. Behind

fragility and antifragility there is nonlinearity (convexity or concavity). The

method of ferreting out hidden convexity effects can be generalized; it is not

much more complicated than that but obvious points tend to disappear from

learned minds.

A more important discussion in this chapter is the extension to the

problems of knowledge: since the convexity effect is unseen, not visible, it

carries epistemological considerations. Those who do not see it or take it into

account will have a sucker problem. And boy, quite a sucker problem.

Further, convexity is a replacement for intelligence.

* The type of accumulated harm called “material fatigue” causing sudden

(discontinuous) break is different from progressive disintegration as they play a

different role in overall breakage.

Thales of Miletus 66


Let us now forget about the grandmother, traffic, etc. for a few pages,

and go to the source of this convexity effect: both the notion of optionality

and the very beginning of the idea of philosophy.

The Secret Language of Convexity

This second order effect, the source of both fragility and antifragility, is the

secret of life, behind evolution, almost everything —though hidden in the

background. It is a dimension that few manage to see consciously, let alone

notice; as we saw in Chapter 1, this is similar to cultural without biological

colorblindness, causing us to be less aware of it culturally than practically.

Imagine the following Procrustean bed situation: living in a world in

three dimensions (our world, in 3-D), but seeing things in only two

dimensions (2-D). The missing dimension (variability) will be a problem,

but (I assume) not too big a deal if you are an ant and live in a perfectly flat

surface with no consequential indentations. If you happen to be a bird,

however, three dimensions would matter a lot more for you. In the case of

the grandmother, the missing dimension is linked to the variability around

the average. The first dimension is the desired average temperature (you are

interested in seventy degrees); the second one is the effect of the variability

of the temperature on her health.

Where is this second order effect ignored? The rest of the book will

present longer discussions; let me say for now that it is ignored almost

everywhere where it matters: medicine, business, economics, government

policies, risk studies, Harvard-Soviet inspired policies, computation of

governmental deficits, projection of cash flows by firms, etc. Anything

modern and man-made will be affected. The convexity effect is sometimes

identified and studied locally for a special situation, but never systematically

or globally. Scholars who get it in one domain over a problem miss it in

another.

And remarkably, this effect seems to provide a secret language of

antifragility, the language of nature, as nature is master at these convexity

effects.

THALES OF MILETUS

An anecdote appears in Aristotle's Politics concerning the pre-Socratic

philosopher and mathematician Thales of Miletusxxx. This story, barely

covering half a page, is at the center of both this entire idea of antifragility

and its denigration. And the remarkable aspect of this story is that Aristotle,

arguably the most influential thinker of all times, got the central point of his

own anecdote exactly backwards. So did his followers, particularly after the

enlightenment and the scientific revolution. I am not saying that to denigrate

Aristotle, but to assert the main idea of this book: intelligence makes you

discount antifragility and ignore convexity effects.

Thales was a Greek-speaking Ionian of Phoenician stock philosopher

from the coastal town of Miletus in Asia Minor, and like some philosophers,

enjoyed what he was doing. Miletus was a trading post and had the

mercantile spirit usually attributed to Phoenician settlements. But Thales, as

a philosopher, was characteristically poor. So he got tired of his buddies with

more transactional lives telling him that "those who can, do, and others

philosophize". He set to prove that he could both “do” and philosophize, and

that he chose to philosophize out of love and respect for the occupation, not

because he had no other option. So he performed the following prowess: he

put a down payment on the seasonal use of every olive press in the vicinity of

Miletus and Chios which he got at low rent. The harvest turned out to be

extremely bountiful and there was demand for olive presses, so he let the

owners of olive presses on his own terms, realizing large sums of money.

What he collected was large, perhaps not enough to become massively

wealthy, but enough to make the point that he could talk the talk and was

truly above, not below, wealth. This in my vernacular I've called "f*** you

money" —a monetary sum large enough to get most of the advantages of

wealth (the most important one being independence and the ability to

occupy your mind with matters that interest you) but not its side effects of

filth, conversations with the name-dropping class, chronic stresses

associated with a large estate and the multiplicity of servants, and hidden

punishment from material benefits. Worse, imagine the greatest punishment

for a philosopher: having to attend a black-tie charity event and being forced

to listen to polite exposition of the details of the marble-rich house

renovation —beyond a certain level of wealth and independence, people tend

to be less and less personable and their conversation less and less

interesting.

Strong and Weak Antifragility 67


The story has many morals, all of which permeate this book. But the

central one is related to the following account by Aristotlexxxi: "But from his knowledge of astronomy he had observed while it was still winter that there was going to be a large crop of olives..." So for Aristotle, clearly, the

stated reason was Thales' superior knowledge. Knowledge.

If we look at it with the eyes of antifragility, the story is altogether

different. It is, rather, the skilled expression and exploitation of ignorance,

not knowledge. Thales put himself in a position to take advantage of his lack

of knowledge —and the secret property of convexity effects. The key to the

message of this book is that he did not need to understand too much the

messages from the stars.

Simply, he had, an option, “the right but not the obligation”, which he

bought cheap: there was no need to be right on average —so long as you pay

a low price that allows you to have greater upside than downside. His payoff

was so large that it could have afforded him to be wrong very, very often and

still make a bundle in the long run. Explained in another way: he had a

positive exposure to Black Swans, convex and antifragile to variations —

recall graph 3 on page x.

This is the center of my ideas about knowledge, as we shall see in

chapter x as Fat Tony rules —and my association of antifragility in exposure

and the problems of knowledge. We just don't need to know what's going on

when we buy cheaply —when we have positive convexity effects. But this

property goes beyond buying cheaply: we do not need to understand things

when we have some edge. And, I repeat, the edge is in the larger payoff when

you are right

How is there convexity when one has limited downside? Look at Figure

x. The vertical axis has the profits, the horizontal axis the rent. I use in this

example, for currency of Asia Minor, the stater, a variant of the Phoenician

Thekel, sometimes spelled Shekel, which means “weight” in Semitic

languages. Figure x [7] the asymmetry as in this situation, the payoff is larger

one way (if you are right, you “earn big time”) than another (if you are

wrong, you “lose small”).

Figure 13- Thales' antifragility- He pays little to get a huge potential. We can see positive convexity effects in his payoff as his payoff curves outward (think “smile” as in Figure x), particularly on the left side of the graph. Note that trial-and-error tends to have the same payoff of limited downside.

All the reader needs to note from the picture is the asymmetry I

mentioned earlier. It is convex, owing to its shape that curves outward. The

opposite to the health of the grandmother in Figure x. And the exact same

shape (though in reverse) as what we saw in the graph of traffic to JFK in

Figure x.

STRONG AND WEAK ANTIFRAGILITY

Economics is a number... the sky is the limit. When we work with an

unbounded variable...

Figure 14- Strong Antifragility (Extremistan)

Options 68


Figure 15 – Weak Antifragility (Mediocristan), with bounded maximum

Other Morals

There are other morals related to the theme of this book, antifragility, and

the ethical considerations developed in Chapter x. Financial independence

makes you robust; it allows you to make the right choices without

modernity’s middle class disease of fitting ethics to profession. And a thinker

who does not have a patron does not have the fragilities, the fear of offending

the hand that feeds him. Further, it allows freedom from the university

system and avoids the prostitution of academia (academia is to knowledge

what prostitute is to love).

But there is the notion of internal motivations: plenty of people are

poor against their initial wish and only become robust by spinning a story

that makes them lose their fragility to income and develop the illusion of

being poor by choice, as if they had the option. But they don’t really have the

option —they constructed it. The essayist Michel de Montaigne sees the

Thales episode as a story of avoidance of sour grapes: you need to know

whether you do not like the pursuit of money and wealth because you

genuinely do not like it, or because you are rationalizing your inability to be

successful at it with the argument that the grapes you cannot reach are sour.

Are you fooling yourself? So the episode enlightened Thales about his own

choices in life –how genuine his pursuit of philosophy was. He had other options. And, it is worth repeating, options, any options, by allowing you

more upside than downside, have positive convexity effects, hence harbor

antifragility*.

*I suppose that the main benefit of being rich (over being just independent) is to

be able to despise rich people (a good concentration of whom you find in glitzy ski

resorts) without any "sour grapes" (i.e., when someone convinces himself that he is not

A second moral is that there are two varieties of people: those who

write books or those who write checks for other people to write books,

create, innovate, or pursue knowledge and claim some credit for having

facilitated it. Rich people can enhance their obituary or buy some form of

minor immortality by having a building, a hallway or, perhaps, a small

staircase named after them. They can also buy some derivative social

prominence in ceremonies for patrons of the arts. The recipients would look

at them with a mixture of scorn or resentment, sometimes with a modicum

of gratitude (the snobbish conductor von Karajan called the Carnegie Hall

philanthropists the “fur coat set” buying their way into creativity). Thales,

however, by funding his own philosophy, became his own Maecenas, perhaps

the highest rank one can attain: that of being both independent and

intellectually productive. He now had even more options.

OPTIONS

We can formulate this rule about asymmetry: If you make more when you are right than you are hurt when you are equally wrong then you have positive convexity effects and you will benefit, in the long run, from volatility (and the reverse). This asymmetry equates to my rule of

accelerated benefits.

An option (as opposed to an obligation) has an asymmetry: because you

have the upside (you have the option of taking the good and neglecting the

bad), with little downside (no obligations). You will be antifragile when you

have it unless, of course, you pay too much for it. The next few examples will

attempt to makes matters clearer. The next few vignettes will present the

notion of options situations similar to those of Thales.

Saturday Evening In London

A first example of what an option is. It is Saturday afternoon in London. I am

coping with a major source of stress: where to go tonight. I am fond of the

brand of the unexpected one finds in parties (going to parties, we will see, is

an option, perhaps the best advice for someone who wants to benefit from

uncertainty with low downside). My fear of eating alone in a restaurant while

interested wealth in order to feel good for being wealthy). It is even sweeter when they

don't know that you are richer than they are.

The Thalesian and The Aristotelian 69


re-reading the same chapter of Seneca's Letters that I have been carrying for

a decade to read whenever I am eating alone; my fear was alleviated by a

telephone call. Someone, not a close friend, upon hearing that I was in town,

invited me to a dinner gathering in Kensington, but somehow did not ask

me to commit, with the "drop by if you want". Going to the party is better

than eating alone with Seneca's Letters, but these are not very interesting

people (many are involved in the City and people employed in financial

institutions are rarely interesting and even more rarely likeable) and I know

I can do better, but I am not certain to be able to do so. So I can call around:

if I can do better than the Kensington party, with, say, a dinner that groups

any of my friends J.G, B.A, and W.G., or similarly charming and erudite

people with, I would go there. Otherwise I would take a black taxi to

Kensington. I have an option, not an obligation. It came at no cost since I did

not even solicit it. So I have a small, nay nonexistent downside, a big upside.

Your Rent

Second example: consider living arrangements. Assume I am the official

tenant of a rent-controlled apartment in New York City. I have the option of

staying in it as long as I wish, and no obligation to do so. Should I decide to

move to Ulan Bator, Mongolia, and start a new life there, I can simply notify

the landlord a certain number of days in advance, and thank you good bye.

Otherwise, the landlord is obligated to let me live there somewhat

permanently, at a predictable rent. Should rents in town increase

enormously, and real estate experience a bubble-like explosion, I am largely

protected. On the other hand, should rents collapse, I can easily switch

apartments and reduce my monthly payments --or even buy a new

apartment and get a mortgage with lower monthly payments.

So consider the asymmetry. I benefit from lower rents, but am not hurt

from higher ones. How? Because here again, I have an option, not an

obligation. In a way uncertainty increases the worth of such privilege. Should

I have a high uncertainty about future outcomes, with possible huge

decreases in real estate value, or huge possible increases in them, my option

would become more valuable. The more uncertainty, the more valuable the

option.

Books, Again

Let us assume that there is for your reputation the equivalent to an Amazon

review (ranking between one and five stars, one star meaning horrible and

five stars meaning, literally, stellar), freely posted in the public domain by

people you may or may not know. You are fragile —in reputation— if you

prefer to have 100 pct four stars rather than 80% five stars and 20% one-

star. Just like the grandmother, you do not want dispersion as you feel severe

harm from the one-star comments. On the other hand you are robust, or

antifragile when you do not care about the bad reviews and, like an option,

focus on the good ones hence love dispersion.

Let me repeat: you care more about the average than about the

volatility (or dispersion) around the average. Fragile is when you don’t like

dispersion, antifragile when you prefer it.

Further, when it comes to real books or ideas, the convexity effects is

much more accentuated. As I wrote in the last chapter, an author or artist or

even a philosopher is much better off when a very small segment of the

readers people like his work, mildly, on average, rather than have a very

small number of fanatics and a large majority of indifferent or haters. This

should be clear now in light of optionality —this we call a remote option.

Because, as in Thales’ option, all that matters is the upside, and the most

fanatic fans and supporters is what counts. The more uncertainty, the more

upside. Those who do not buy your book or your work do not have a negative

contribution beyond not buying your book. Further, it helps when the

supporters are both enthusiastic and influential. Wittgenstein, for instance,

was largely considered as a lunatic (he almost had no publications to his

name), but had very few main fans creating a cult, and some, like Bertrand

Russell and J.M. Keynes, were massively influential.

THE THALESIAN AND THE ARISTOTELIAN

The Thalesian focuses on the payoff, the consequence of the actions (hence

includes convexity effects). The Aristotelian focuses on being right and

wrong—raw logic. They intersect less often than you think.

For Fat Tony, the distinction maps into sucker-nonsucker. Things are

always simpler with Fat Tony.

In real life, exposure is more important than knowledge; decision-

effects supersede logic. Textbook “knowledge” misses a dimension, the

The Thalesian and The Aristotelian 70


hidden convexity effect —just like the notion of average. The need to focus on

the payoff from your actions rather than the structure of the world (or

understanding the “True” and the “False”) has been largely missed in

intellectual history. Horribly missed. The payoff, what happens to you (benefits or harm from it), is always the most important thing, not the event itself. In other words, let me repeat it, that we need to be smarter in doing than knowing, better in acting, than understanding —wiser, less error

prone, and, of course, better for long term survival of our species. But the

giant of rationalism, the Medieval philosopher Averroes (Ibn Rushd)

considered Aristotle the supreme expression of human intellect — precisely

because the latter represented “rationalism”, the pure reliance on reasoning,

that thing we call reasoning, in comprehending things. And visibly this is

the very reason Aristotle missed the point, because he overestimated the

reach of human reasoning. Thales' success was automatically imparted to

knowledge about the stars, or about the future coming from the stars, not

from the nature of the bet.xxxii

So Aristotle overvalued knowledge and discounted the action, or

decision, taken off that knowledge.

My point is that True and False (hence what we call “belief”) play a

poor, secondary, role in human decisions; it is the payoff from the True and

the False that dominates —and it is almost always asymmetric, with one

consequence much bigger than the other, i.e. harboring positive and negative

convexity effects. Let me explain. We check people for weapons before they

board the plane. Do we believe that they are terrorists, True or False? False,

as they are not likely to be terrorists (a small probability). But we check them

nevertheless because we are interested in the payoff, and the consequence, or

payoff of the True (that they turn out to be terrorists) is too large and the

costs of checking are too low. Do you think the nuclear reactor is likely to

explode in the next year? False. Yet you want to behave as if it were True and

spend millions on additional safety. A third example: Do you think that this

random medicine will harm you? False. Do you ingest these pills? No, no,

no.*

* Philosophers are split into two categories, those who believe in absolute truth

and falsehoods, usually analytical tradition, and those who believe in the relativity of

truth and falsehood, found in many traditions including the school called the

pragmatists (as well as those called continental and postmodern philosophers). My

message is entirely within the analytical tradition (if not even more extreme); it is not

If you sat with a pencil and jotted down all the decisions you’ve taken in

the past week, or, if you can, over your lifetime, you will realize that almost

all of them have asymmetric payoff, with one side carrying a larger

consequence than the other. There is a positive or negative convexity effect

somewhere since what has big downside, small upside is concave and the

reverse.

So we go by the True- False distinction only in situations of symmetry,

say flipping a coin, where the gains of one outcome are equal to the loss of

the other, and these seem to only exist in logical textbooks. Now I am not

getting into post-modern denial of True or False, to the contrary; I am just

saying that it is insufficient a representation, like using a two dimensional

drawing for something in three dimensions.

That payoffs are more important than events is intuitively grasped in

some applications. But all of the examples above (terrorist, nuclear plant,

medication), representing a small probability of a very large adverse

outcome, are easy to understand as we humans are well wired for risk

aversion (when I talk about Black Swans most people tend to immediately

imagine negative things). These examples correspond to situations of

fragility, the ones I showed in column one of Table 1 and in Figure 2. And we

still have enough instinct (unless we studied economics) to understand that

mitigating fragility is more important than knowledge. Now, the Thales

story represents the exact mirror image of these unlikely favorable outcomes,

of a small probability of a large and unbounded favorable outcome (to

Thales). It is antifragility, and this has not penetrated our philosophical

consciousness†.

Confidence levels. Let me rephrase the idea of the irrelevance of

True/False in decision-making in the real world, particularly when

probabilities are involved. Scientists have something called “confidence

level”; a result obtained with a 95% confidence level means that there is no

more than 5% probability of the result being wrong. The idea of course is

inapplicable for the same reason of size of effects, and extreme events. If I

tell you that some result is true with 95% confidence level, you would be

that True and False are irrelevant, but rather that the distinction is insufficient for

decision-making.

† Thales’ bet is not at all similar to a lottery ticket (which is a human artificial

contraption), mostly because the outcome is both unknown and unbounded, i.e., we do

not know the upper limit.

Bachelier, Jensen and Friends 71


quite satisfied. But what if I told you that the plane was safe with 95%

confidence level? Even 99% confidence level would not do, as a 1%

probability of a crash would be quite a bit alarming (today commercial

planes operate with less than 1 in 100,000 probabilities of crashing and the

ratio is improving). So the probability (hence True/False) does not work in

the real world; it is the payoff that matters.

How to be Stupid

Accordingly, you don't have a need for much of what is commonly called

intelligence, knowledge, insights, skills, and these complicated things that

take place in the brain cells. For you don't have to be right that often. All you

need is wisdom to not do foolish things to hurt yourself (some acts of

omission) and know if an outcome if fine (after its occurrence), not before.

Otherwise, if convexity effects work against you, sorry, but you are doomed,

no matter how intelligent you are and how many PhDs from Harvard are on

your staff—for there may be a small thing that will escape you and hurt you

very badly. The hair holding the sword of Damocles will eventually break, in

time, with certainty.

I learned about convexity effects in class at the Wharton School, in the

lecture on financial options that determined my career, and immediately

realized that the professor did not understand it himself —he understood it

in spots, but not everywhere. It hides where we don't want it to hide. I will

repeat that options benefit from variability and convexity effects.

BACHELIER, JENSEN AND FRIENDS

The discovery of convexity effects (though not the connection to fragility)

was made by several men, one of whom got a bit of mishandling by history*.

On March 29, 1900, a student at the Sorbonne who worked as a

stockbroker in order to support himself, Louis Bachelier, defended a doctoral

thesis in mathematics. The idea was about how to value financial options,

these asymmetric contracts that give the right but not the obligation to buy a

stock at a specified price (also called contingent claims). Now finance being

* A technical point. The following researchers discovered first order convexity

effects; fragility and antifragility are second (and higher) order convexity effects. See

Appendix X.

an uninteresting (and largely despicable) subject, there is no need to focus

on the subject and figure out what a financial option is beyond that it is “a

right not an obligation” (as with the dinner in London), that one can buy for

small and that can carry an occasional (and rare) payoff, large upside and no

or small downside (for the owner). In short, the option has convexity effects†.

Bachelier’s doctoral thesis was poorly received by the head of the

committee, no less a person than the great mathematician and scientific big-

picture thinker Henri Poincaré. So Bachelier received the grade

euphemistically called "honorable", not the "très honorable" that was

necessary to get a real academic position. His work was said to lack in rigor but there was also this unattractiveness of the financial topic for the

committee: finance was never seen in France as particularly respectable

intellectually. Bachelier never managed to have a decent academic career as

he was plagued with the stigma, along with an additional black ball when, in

his fifties, he was about to get his first real position of professor. Many

people later rediscovered his results in the pricing of derivatives, and

something I find scandalous, two men, Robert Merton and Myron

Scholesxxxiii, received the Bank of Sweden Prize in Economic Sciences (called

the "Nobel" in economics) as the Swedish academy, rather poor in

knowledge of the history of ideas, had the illusion that they discovered his

equation. Furthermore, Robert C. Merton, while trying to pass for “option

guru” had spent his career developing models in finance that increase risks

precisely that they missed convexity effects.

In addition, in that very same doctoral thesis, Louis Bachelier observed

properties of randomness that were rediscovered (and publicized) by

Einstein five years later. More depressingly the man who disparaged him,

quite unfairly, Henri Poincaré, has been my intellectual hero most of my

conscious life and that of most people who like the notions complexity.

Another giant who missed the point of antifragility –there will be many

more.

Note that Bachelier was an option trader (who disliked his career) —

and this book, which has almost nothing to do with finance, will be driven by

similar intuitions by yours truly, a former option trader (who disliked his

career). Now that I got the sad story of Bachelier off my chest, let's forget

about economics for a few chapters.

† Further, my experience shows that the only interesting options are the “hidden”

ones, those that are not recognized as financial options.

Why Some Errors Go One Direction 72


Bachelier was using a result he took for granted, but that was

formalized (in much simpler form) five years later, completely

independently. The more formal presentation of that result took place as

follows, five years later. On 17 January 1905, one Johann Ludwig Jensen, an

obscure mathematician, by day Danish employee of a telephone company,

exposed in a presentation some derivation showing the convexity effect on

the long term average, showing in the grandmother’s story that the average

health of the grandmother across temperatures is worse, “unequal” to her

health for a single average temperature.xxxiv Jensen, in fact, was generalizing

an earlier, almost a century old result by the French mathematician

Augustin-Louis Cauchy. Neither he nor the audience realized how

fundamental this result was for about everything —no more than the

audience in the thesis committee of Monsieur Bachelier, knew the import of

that poor man's work. The paper by the Dane was published the following

year in French in the Swedish mathematical journal Acta Mathematica with

the eloquent title “On convex funtions and their inequality for average

values” ("sur les fonctions convexes et les inégalités entre les valeurs

moyennes") —and went unnoticed for a long time, a very long time, as, 105

years later I hardly found anyone who took its consequences to their

conclusion. I built on his result to extend it from the effect of convexity on

the “average” to “dispersion”, hence fragility and antifragility.

The plot is even thicker. It was recently discovered that a French

intellectual and stockbroker Jules Regnault in Calcul des chances et philosophie de la bourse , published in 1863, also discovered a milder form

of these effects —but unlike Bachelier he used his knowledge to make f***

you money over a few years and become catalogued a rentier.

WHY SOME ERRORS GO ONE DIRECTION

We saw that errors and uncertainty tend to make planes land later, not

earlier. Same with traffic, disturbances tend to increase travel time from

Kensington to Picadilly circus, never shorten it.

The errors are one-sided: that’s a negative convexity effect (the

opposite of, say, a Thales bet or an option-style position in which errors tend

to be positive). So this typically causing both underestimation of randomness

and underestimation of harm owing to the fact that one is more exposed to

harm than benefits; as we call it he is “short an option”.

Example of such situations: Predictions of projects, wars, deficits...

{discussion on the graphs}

Projects and prediction

Just as when you add uncertainty to a flight, the planes tend to land later,

not earlier, projects tend to cost more (and take longer). This applies to

many, in fact, almost all projects. In The Black Swan I showed that the

underestimation of the random structure of the world (Mediocristan as

opposed to Extemistan) caused such problems —these unexpected Black

Swan events tend to hit by lengthening, not shortening project time. Black

Swan blindness was the source.

The puzzle was of course that many large-scale projects one and a half

centuries ago were completed on time; many of the tall buildings and

monuments we see today were completed within, and often ahead of

schedule. These include not just the Empire State Building (still standing in

New York), but such items such as the Crystal Palace erected during the

Great Exhibition of 1851, the hallmark of Victorian reign, based on the

inventive ideas of a gardener. The Palace, which housed the exhibition, went

from its organization to the grand opening in just nine months. The building

took the form of a massive glass house, 1848 feet long by 454 feet wide and

was constructed from cast iron-frame components and glass made almost

exclusively in Birmingham and Smethwick.

The obvious is usually missed here: the Crystal Palace project did not

use computers, and the parts were built not far away from the source, with a

small number of entities as part of the food chain. Further, there were,

thankfully, no business schools at the time to teach something called “project

management” and increase overconfidence. And there were no consulting

firms and the agency problem was weak. In other words, it was a much more

linear economy —less complex— than today.

I had been telling anyone who would listen to me that Black Swan

effects had to be increasing, necessarily as a result of complexity,

interdependence between parts, globalization, and the beastly thing called

“efficiency” that make people now sail too close to the wind. Add to that

consultants and business schools. One problem somewhere can halt the

entire project —so the projects tend to get as weak as their weakest link in

the chain (an acute negative convexity effect). The world is getting less and

Why is The Large Fragile? 73


less predictable, and we rely more and more on technologies that have errors

and interactions that are harder to estimate, let alone predict.

And the information economy is the culprit. My colleague Bent

Flyvbjerg showed that the problem of cost overruns and delays is much more

acute in the presence of Information Technologies (IT), as computer projects

cause a large share of these costs overruns and it is better to focus on these

principally (Black Swan Risks are often solved with small rules, not

complicated ones)xxxv. But even outside of these IT-heavy projects, we tend to

have very severe delays. And, of course there is the fallacy of prediction:

these are underestimated, and chronically so.

But the logic is simple: negative convexity effects are the cause. There is

an asymmetry in the way errors hit you. Decision scientists and business

psychologists have theorized something called the “planning fallacy”, in

which they try to explain the fact that projects take longer rarely shorter with

recourse to psychological factors, which play a role but less than Black Swan

effects. Decision scientists ground it in human errors, not in exposure to

Extremistan, with in this case exposure to negative Black Swans rather than

positive Black Swans. But no psychologist realized that, at the core, it is not

the psychological problem, but part of the nonlinear structure of the project.

Just as time cannot be negative, a three month project cannot be completed

in zero or negative time. So errors add to the right end, not the left end of it.

If uncertainty were linear we would observe some projects completed early

(just as we would arrive sometimes early, sometimes late). But this is not the

case.

Wars, Deficits, and Bonds

The second war was estimated to last only a few month; by the time it was

over it got France and Britain heavily in debt, at least ten times what they

thought their financial costs would be, aside from all the destruction. The

same of course for the second war caused the U.K. to become heavily

indebted, mostly to the United States.

In the United States the prime example remains the Iraq war, expected

by George W. Bush and his friends to cost thirty to sixty billions, and so far

can be at more than two trillion. Complexity, once again.

But wars are only illustrative of the way governments underestimate

convexity effects and why they should not be trusted with finances.

Governments do not need wars to run deficits: the underestimation is

chronic for the very same reason ninety-eight percent of modern projects

have overruns.

WHY IS THE LARGE FRAGILE?

We can apply the idea of convexity effects here —in fat the exact same graph

as the one about travel time — on the idea of size. When one is large, one

becomes vulnerable to these errors going in one direction.

To see how size becomes a handicap, consider the reasons one should

not own an elephant as a pet, regardless of what emotional attachment you

may have with an animal of such size. Say you can afford an elephant as part

of your household budget and have one delivered to your backyard. Should

there be a water shortage, you would have to pay a higher and higher price

for each additional gallon of water. That’s fragility, right there, a negative

convexity effect coming from getting too big. The unexpected cost, in

percentage of the total, would be monstrous. Owning, say, a cat or a dog

would not bring about such high unexpected additional costs over the

regular at times of squeeze —adjusting of for the size of each animal.

In spite of what is studied in Business School, size hurts you at times of

stress; it is not good for fragility.

Some economists have been wondering why mergers of corporations do

not appear to play out. The combined unit is now much larger, hence more

powerful and more “efficient”. But numbers show no gain, at best —that was

already true in 1978, as people then voiced the “hubris hypothesis” finding it

irrational for companies to engage in mergers given the poor record of the

idea. And recent data, more than three decades later, confirm the behavior.

There appear to be something with size that is harmful to corporations.

Well, like the elephant as pet, squeezes are much, much more

expensive (relative to size) for large corporations. The gains from size are

visible but the risks are hidden, and some concealed risks seems to bring

frailties into the companies —hence the system.

Let us look at a case study. On January 21, 2008, the Parisian bank

Société Generale rushed to sell in the market close to seventy billion dollars

of stocks, a very large amount for any single “fire sale”. Markets were not

very active (called “thin”) as it was Martin Luther King day in the United

States and markets worldwide dropped precipitously, close to ten percent,

costing the company close to six billion dollars in losses from their fire sale.

74


For they had, over the weekend, uncovered a fraud. Jerome Kerviel, a rogue

back office employee, was playing with humongous sums in the market and

hiding these exposure from the main computer system. They were squeezed

and had no choice but to sell, immediately sell these stocks they didn’t know

they owned.

Now to see the effect of fragility from size, look at Figure x showing

losses as a function of quantity sold. A fire sale of 70 billion leads to a loss of

6 billion. But a fire sale of 5 or 10 billion has no loss at all, as markets would

absorb the quantities without panic. So this tells us that if, instead of having

one very large bank, with Monsieur Kerviel as a rogue trader, we had ten

smaller banks, each with a proportional Monsieur Mini-Kerviel, and each

had his rogue trading independently and at random times, the total losses for

the ten banks would be nothing.

Figure 16

We will use the argument again when discussing corporate size during

my apology of artisanal economies.

WHY IS THE “EFFICIENT” NOT EFFICIENT?

HORMESIS

Figure 17- Hormesis for an organism: we can see a stage of benefits as the dose increase (initially convex) slowing down into a phase of harm as we increase the dose (initially concave), then things flattening out at the level of maximum harm (beyond a certain point, the organism is dead so there is such a thing as a bounded and known worst case scenario in biology) – Note that (medical papers and textbooks make the mistake of having concave curve at the early stages, which would be mathematically impossible).

A Second Graphical Interlude 75


Figure 18- Wrong graph showing initial convexity

A SECOND GRAPHICAL INTERLUDE*

Let us look at the point graphically†, but in a different manner, this time

looking at probabilities.

The Horizontal line presents outcomes, the vertical one their

probability (i.e., their frequency). This is a different representation —this

time, probabilistic — of the outcomes. Before that we saw functions and

variable, with nonlinear responses between one and the other. And before

that we saw what is called the time series: what happens over a certain

period, with the passage of time

Absence of convexity effects are shown to the first graph, Figure x —the

symmetric case, as the potential gain is somewhat equal to potential harm.

Figure 19- Case 1, the Symmetric. Injecting uncertainty in the system makes us move from one bell-shape —the first, with narrow possible spate of outcomes—to the second, a lower peak but more spread out. So it causes an

* The intelligent reader innocent of social science and economics can most

certainly skip these graphs as there is nothing for him to unlearn.

† Technical note: when a payoff has negative skewness, increases in dispersion (or

variance, volatility) lead to a degradation of the expectation, making it more negative.

Hence underestimation of uncertainty implies underestimation of the expected

variable. I show in the appendix (as well as in Taleb, 2011) how a concave payoff from

a symmetric random variable is itself a payoff with negative skewness and the

consequences for fragility, underestimation of the mean, and increases in risk.



increase of both positive and negative surprises, both positive and negative Black Swans*.

Negative convexity effects lead to Figure x; there is a possibility of a

severe unfavorable outcome (left), much more than a hugely favorable one,

as the left side is thicker than the right one.

Figure 20- Case 2, Negative Convexity Effects, Limited gains, larger losses. Fragile, prone to negative asymmetries, negative convexity effects (for example, projects). Increasing uncertainty in the system causes an augmentation of mostly (sometimes only) negative outcomes, just negative Black Swans.

* Technical Comment: Note that I am not using in these example the classical bell-

shaped Gaussian, rather, distributions with power-law “fat tails”.

Figure 21- Case 3, Positive Convexity Effects, with Limited losses, unlimited benefits. Antifragile, prone to positive asymmetries, positive convexity effects. Increasing randomness and uncertainty in the system raise the probability of very favorable outcomes, and accordingly expands the expected payoff. Note that it is the EXACT opposite of figure x {previous}, which means that discovery is, mathematically, exactly like an anti-airplane delay.

Let us apply this analysis to how planners make the mistakes we discussed

earlier, and why deficits tend to do worse than planned:



Figure 22- The Gap between predictions and reality: probability distribution of outcomes from costs of projects in the minds of planners (above) and in reality (below). In the top graph they assume that the costs will be both low and quite certain. The lower graph show outcomes to be both worse and more spread out, particularly with higher possibility of unfavorable outcomes. These undesirable events are on the left and we can see a left “tail” forming). This misunderstanding of the effect of uncertainty applies to government deficits, plans that have IT component, travel time (to a lesser degree) and many more.

The greater dispersion shows underestimation of uncertainty. The

worse average outcome shows underestimation of the expected outcome.

Innovation, on the other hand, have exact opposite properties of the

graph in Figure x: errors tend to cause more benefits than harm. When you

inject more uncertainty in the system, it improves.

Next: Discovery as an Anti-deficit

This chapter got deeper into the plumbing behind antifragility, evolution,

and survivorship. We used the grandmother story to present convexity

effects —sort of, the grammar of antifragility —and the story of Thales to

present this notion of “convexity”, “optionality”, something that tends to

benefit from variation and lessens our dependence on knowledge; how

convexity can supersede understanding and how Aristotle —and about most

of traditional Greek and Levantine philosophy missed the point. The

opposite situation is that of the grandmother suffering from thermal

variations.

Next, let us discuss discovery —and how it is grounded in antifragility,

hardly anything else. Just think of an airplane ride, or the costs and duration

of a project, as represented in Figure x: almost every bit of uncertainty tends

to increase your flying time, your project costs, and worsen your situation.

With discovery (and antifragile situations) the reverse holds as about every

bit of uncertainty improves your situation. Let us see how.

taleb thales’ secret, or the intelligence of antifragility

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