Antifragility
Just as a package sent by mail can bear a stamp "fragile",
"breakable" or "handle with care", consider the exact opposite: a package that
has stamped on it "please mishandle" or "please handle carelessly". The contents
of such package are not just unbreakable, impervious to shocks, but have
something more than that , as they tend to benefit from shocks. This is beyond
robustness.
So let us coin the appellation "antifragile" for anything that,
on average, (i.e., in expectation) benefits from variability. Alas, I found no
simple, noncompound word in any of the main language families that expresses the
point of such fragility in reverse. To see how alien the concept to our minds,
ask around what's the antonym of fragile. The likely answer will be: robust,
unbreakable, solid, wellbuilt, resilient, strong, somethingproof (say
waterproof, windproof, rustproof), etc. Wrong — and it is not just individuals,
but branches of knowledge that are confused by it; this is a mistake made in
every dictionary. Ask the same person the opposite of destruction, they
will answer construction or creation. And ask for the opposite
of concavity, they will answer convexity.
A verbal definition of convexity is: benefits more than it
loses from variations; concavity is its opposite. This is key: when I tried
to give a mathematical expression of fragility (using sums of pathdependent
payoffs), I found that "fragile" could be described in terms of concavity to
a source of variation (random or nonrandom), over a certain range of
variations. So the opposite of that is convexity — tout simplement.
A grandmother's health is fragile, hence concave, with respect
to variations in temperature, if you find it preferable to make her spend two
hours in 70? F instead of an hour at 0? F and another at 140? F for the exact
70? F on average. (A concave function of a combination f(½ x1+½ x2) is higher
than the combination ½ f(x1)+ ½ f(x2).
Further, one could be fragile to certain events but not others:
A portfolio can be slightly concave to a small fall in the market but not to
extremely large deviations (Black Swans).
Evolution is convex (up to a point) with respect to variations
since the DNA benefits from disparity among the offspring. Organisms benefit, up
to a point, from a spate of stressors. Trial and error is convex since errors
cost little, gains can be large.
Now consider the Triad in the Table. Its elements are those for
which I was able to find general concavities and convexities and catalogue
accordingly.
The Triad
FRAGILE

ROBUST

ANTI
FRAGILE 

Mythology — Greek

Sword of
Damocles, Rock of Tantalus 
Phoenix

Hydra

Biological & Economic Systems

Efficiency

Redundancy

Degeneracy (functional redundancy, in the EdelmanGaly sense)

Science/Technology

Directed Research

Opportunistic research

Stochastic Tinkering (convex bricolage)

Human Body

Mollification, atrophy, "aging", sarcopenia

Recovery

Hypertrophy,
Hormesis, Mithridatism 
Political Systems

NationState;
Centralized 
Statelings, vassals under a large empire

CityState; Decentralized

Income

Companies

Income of Executives (bonuses)
 
Civilization

Postagricultural
Modern urban 
Ancient settlements

Nomadic and huntergatherer tribes

Decision Making

Modelbased probabilistic
decision making 
Heuristicbased decision making

Convex heuristics

Knowledge

Explicit

Tacit

Tacit with convexity

Epistemology

TrueFalse

SuckerNonsucker
 
Ways of Thinking

Modernity

Medieval Europe

Ancient Mediterranean

Errors

Hates mistakes

Mistakes are just information

Loves mistakes

Learning

Classroom

Real life, pathemata mathemata

Real life and library

Medicine

Additive treatment (give medication)

Subtractive treatment (remove items from consumption, say carbs,
etc.)
 
Finance

Short Optionality

Long Optionality
 
Decision Making

Acts of commission

Acts of omission ("missed opportunity")
 
Literature

EReader

Book

Oral Tradition

Business

Industry

Small Business

Artisan

Finance

Debt

Equity

Venture Capital

Finance

Public Debt

Private debt with no bailout


General

Large

Small but specialized

Small but not specialized

General

Monomodal payoff

Barbell polarized payoff
 
Finance

Banks, Hedge funds managed by economists

Hedge Funds (some)

Hedge Funds
(some) 
Business

Agency Problem

Principal Operated
 
Reputation (profession)

Academic, Corporate executive, Pope, Bishop, Politician

Postal employee, Truck driver, train conductor

Artist, Writer

Reputation (class)

Middle Class

Minimum wage persons

Bohemian,
aristocracy, old money 
The larger the corporation, the more concave to some squeezes
(although on the surface companies they claim to benefit from economies of
scale, the record shows mortality from disproportionate fragility to Black Swan
events). Same with government projects: big government induces fragilities. So
does overspecialization (think of the Irish potato famine). In general most
topdown systems become fragile (as can be shown with a simple test of concavity
to variations).
Worst of all, an optimized system becomes quickly concave to
variations, by construction: think of the effect of absence of redundancies and
spare parts. So about everything behind the mathematical economics revolution
can be shown to fragilize.
Further we can look at the unknown, just like model error, in
terms of antifragility (that is, payoff): is what you are missing from a model,
or what you don't know in real life, going to help you more than hurt you? In
other words are you antifragile to such uncertainty (physical or epistemic)? Is
the utility of your payoff convex or concave? Pascal was first to express
decisions in terms of these convex payoffs. And economics theories produce
models that fragilize (except rare exceptions), which explains why using their
models is vastly worse than doing nothing. For instance, financial models based
on "risk measurements" of rare events are a joke. The smaller the probability,
the more convex it becomes to computational error (and the more concave the
payoff): an 25% error in the estimation of the standard deviation for a Gaussian
can increase the expected shortfall from remote events by a billion (sic) times!
(Missing this simple point has destroyed the banking system).
II
Jensen's Inequality as the Hidden Engine of
History
Now the central point. By a simple mathematical property, one
can show why, under a model of uncertainty, items on the right column will be
likely to benefit in the long run, and thrive, more than shown on the surface,
and items on the left are doomed to perish. Over the past decade managers of
companies earned in, the aggregate, trillions while retirees lost trillions (the
fact that executives get the upside not the downside gives them a convex payoff
"free option"). And aggressive tinkering fares vastly better than directed
research. How?
Jensen's inequality says the following: for a convex payoff,
the expectation of an average will be higher than the average of
expectations. For a concave one, the opposite (grandmother's health is
worse if on average the temperature is 70 than in an average temperature of
70).
Squaring is a convex function. Take a die (six sides) and
consider a payoff equal to the number it lands on. You expect 3½. The square of
the expected payoff will be 12¼ (square 3½). Now assume we get the square of the
numbers on the die, 15.1666, so, the average of a square payoff is higher than
the square of the average payoff.
The implications can be striking as this second order effect
explains so much of hidden things in history. In expectation, anything that
loves Black Swans will be present in the future. Anything that fears it will be
eventually gone — to the extent of its concavity.
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