2011 : WHAT SCIENTIFIC CONCEPT WOULD IMPROVE EVERYBODY'S COGNITIVE TOOLKIT?

I

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, well-built, resilient, strong, something-proof (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 path-dependent 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

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 top-down 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.