Antifragile: Things That Gain from Disorder (57 page)
Read Antifragile: Things That Gain from Disorder Online
Authors: Nassim Nicholas Taleb
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”),
(b) it curves inward, too much so, and, finally, (c) the more nonlinear the response, the less relevant the average, and the more relevant the stability around such average.
3
Much of medieval thinking went into finding the philosopher’s stone. It is always good to be reminded that chemistry is the child of alchemy, much of which consisted of looking into the chemical powers of substances. The main efforts went into creating value by transforming metals into gold by the method of
transmutation
. The necessary substance was called the philosopher’s stone—
lapis philosophorum
. Many people fell for it, a list that includes such scholars as Albertus Magnus, Isaac Newton, and Roger Bacon and great thinkers who were not quite scholars, such as Paracelsus.
It is a matter of no small import that the operation of transmutation was called the
Magnus Opus
—the great(est) work. I truly believe that the operation I will discuss—based on some properties of optionality—is about as close as we can get to the philosopher’s stone.
The following note would allow us to understand:
(a) The severity of the problem of conflation (mistaking the price of oil for geopolitics, or mistaking a profitable bet for good forecasting—not convexity of payoff and optionality).
(b) Why anything with optionality has a long-term advantage—and how to measure it.
(c) An additional subtle property called Jensen’s inequality.
Recall from our traffic example in
Chapter 18
that 90,000 cars for an hour, then 110,000 cars for the next one, for an average of 100,000, and traffic will be horrendous. On the other hand, assume we have 100,000 cars for two hours, and traffic will be smooth and time in traffic short.
The number of cars is the
something,
a variable; traffic time is the
function of something
. The behavior of the
function
is such that it is, as
we said, “not the same thing.” We can see here that the
function of something
becomes different from the
something
under nonlinearities.
(a) The more nonlinear, the more the
function of something
divorces itself from the
something
. If traffic were linear, then there would be no difference in traffic time between the two following situations: 90,000, then 110,000 cars on the one hand, or 100,000 cars on the other.
(b) The more volatile the
something
—the more uncertainty—the more the
function
divorces itself from the
something
. Let us consider the average number of cars again. The function (travel time) depends more on the volatility around the average. Things degrade if there is unevenness of distribution. For the same average you prefer to have 100,000 cars for both time periods; 80,000 then 120,000, would be even worse than 90,000 and 110,000.
(c) If the function is convex (antifragile), then the average of the function
of something
is going to be higher than the function of the average
of something
. And the reverse when the function is concave (fragile).
As an example for (c), which is a more complicated version of the bias, assume that the function under question is the squaring function (multiply a number by itself). This is a convex function. Take a conventional die (six sides) and consider a payoff equal to the number it lands on, that is, you get paid a number equivalent to what the die shows—1 if it lands on 1, 2 if it lands on 2, up to 6 if it lands on 6. The square of the expected (average) payoff is then (1+2+3+4+5+6 divided by 6)
2
, equals 3.5
2
, here 12.25. So the
function of the average
equals 12.25.
But the average of the function is as follows. Take the square of every payoff, 1
2
+2
2
+3
2
+4
2
+5
2
+6
2
divided by 6, that is, the average square payoff, and you can see that
the average of the function
equals 15.17.
So, since squaring is a convex function, the average of the square payoff is higher than the square of the average payoff. The difference here between 15.17 and 12.25 is what I call the hidden benefit of antifragility—here, a 24 percent “edge.”
There are two biases: one elementary convexity effect, leading to mistaking the properties of the average of something (here 3.5) and those of a (convex) function of something (here 15.17), and the second, more involved, in mistaking an average of a function for the function of an average, here 15.17 for 12.25. The latter represents optionality.
Someone with a linear payoff needs to be right more than 50 percent of the time. Someone with a convex payoff, much less. The hidden benefit of antifragility is that you can guess worse than random and still end up outperforming. Here lies the power of optionality—your
function of something
is very convex, so you can be wrong and still do fine—the more uncertainty, the better.
This explains my statement that you can be dumb and antifragile and still do very well.
This hidden “convexity bias” comes from a mathematical property called Jensen’s inequality. This is what the common discourse on innovation is missing. If you ignore the convexity bias, you are missing a chunk of what makes the nonlinear world go round. And it is a fact that such an idea is missing from the discourse. Sorry.
4
Let us take the same example as before, using as the function the square root (the exact inverse of squaring, which is concave, but much less concave than the square function is convex).
The square root of the expected (average) payoff is then √(1+2+3+4+5+6 divided by 6), equals √3.5, here 1.87. The
function of the average
equals 1.87.
But the average of the function is as follows. Take the square root of every payoff, (√1+√2+√3+√4+√5+√6), divided by 6, that is, the average square root payoff, and you can see that
the average of the function
equals 1.80.
The difference is called the “negative convexity bias” (or, if you are a stickler, “concavity bias”). The hidden harm of fragility is that you need to be much, much better than random in your prediction and knowing where you are going, just to offset the negative effect.
Let me summarize the argument: if you have favorable asymmetries, or positive convexity, options being a special case, then in the long run you will do reasonably well, outperforming the average in the presence of uncertainty. The more uncertainty, the more role for optionality to kick in, and the more you will outperform. This property is very central to life.
1
The method does not require a good model for risk measurement. Take a ruler. You know it is wrong. It will not be able to measure the height of the child. But it can certainly tell you if he is growing. In fact the error you get about the rate of growth of the child is much, much smaller than the error you would get measuring his height. The same with a scale: no matter how defective, it will almost always be able to tell you if you are gaining weight, so stop blaming it.
Convexity is about acceleration. The remarkable thing about measuring convexity effects to detect model errors is that even if the model used for the computation is wrong, it can tell you if an entity is fragile and by how much it is fragile. As with the defective scale, we are only looking for second-order effects.
2
I am simplifying a bit. There may be a few degrees’ variation around 70 at which the grandmother might be better off than just at 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 losing such antifragility with age (or disuse, as I suspect that thermal comfort ages people and makes them fragile).
3
I remind the reader that this section is technical and can be skipped.
4
The grandmother does better at 70 degrees Fahrenheit than at an average of 70 degrees with one hour at 0, another at 140 degrees. The more dispersion around the average, the more harm for her. Let us see the counterintuitive effect in terms of
x
and function of
x
,
f
(
x
). Let us write the health of the grandmother as
f
(
x
), with
x
the temperature. We have a function of the average temperature,
f
{(0 + 140)/2}, showing the grandmother in excellent shape. But {f(o) + f(140)}/2 leaves us with a dead grandmother at
f
(0) and a dead grandmother at
f
(140), for an “average” of a dead grandmother. We can see an explanation of the statement that the properties of
f
(
x
) and those of
x
become divorced from each other when
f
(
x
) is nonlinear. The average of
f
(
x
) is different from
f
(average of
x
).
R
ecall that we had no name for the color blue but managed rather well without it—we stayed for a long part of our history culturally, not biologically, color blind. And before the composition of
Chapter 1
, we did not have a name for antifragility, yet systems have relied on it effectively in the absence of human intervention. There are many things without words, matters that we know and can act on but cannot describe directly, cannot capture in human language or within the narrow human concepts that are available to us. Almost anything around us of significance is hard to grasp linguistically—and in fact the more powerful, the more incomplete our linguistic grasp.
But if we cannot express what something is exactly, we can say something about what it is not—the indirect rather than the direct expression. The “apophatic” focuses on what cannot be said directly in words, from the Greek
apophasis
(saying no, or mentioning without mentioning). The method began as an avoidance of direct description, leading to a focus on negative description, what is called in Latin
via negativa
, the negative way, after theological traditions, particularly in the Eastern Orthodox Church.
Via negativa
does not try to express what God is—leave that to the primitive brand of contemporary thinkers and philosophasters with scientistic tendencies. It just lists what God is
not
and proceeds by the process of elimination. The idea is mostly associated with the mystical theologian Pseudo-Dionysos the Areopagite. He was some obscure Near Easterner by the name of Dionysos who wrote powerful mystical treatises and was for a long time confused with Dionysos the
Areopagite, a judge in Athens who was converted by the preaching of Paul the Apostle. Hence the qualifier of “Pseudo” added to his name.
Neoplatonists were followers of Plato’s ideas; they focused mainly on Plato’s forms, those abstract objects that had a distinct existence on their own. Pseudo-Dionysos was the disciple of Proclus the Neoplatonist (himself the student of Syrianus, another Syrian Neoplatonist). Proclus was known to repeat the metaphor that statues are carved by subtraction. I have often read a more recent version of the idea, with the following apocryphal pun. Michelangelo was asked by the pope about the secret of his genius, particularly how he carved the statue of David, largely considered the masterpiece of all masterpieces. His answer was: “It’s simple. I just remove everything that is not David.”
The reader might thus recognize the logic behind the barbell. Remember from the logic of the barbell that it is necessary to first remove fragilities.
Recall that the interventionista focuses on positive action—
doing
. Just like positive definitions, we saw that acts of commission are respected and glorified by our primitive minds and lead to, say, naive government interventions that end in disaster, followed by generalized complaints about naive government interventions, as these, it is now accepted, end in disaster, followed by more naive government interventions. Acts of omission,
not
doing something, are not considered acts and do not appear to be part of one’s mission.
Table 3
showed how generalized this effect can be across domains, from medicine to business.