Local Fragility and the Nonlinearity Principle

Local Fragility and Nonlinearity

Several equations are missing and some exponents are flattened in these source notes, so the technical derivation is incomplete. In the formulation from Taleb and Douady (2012), the local fragility of a random variable Xλ depending on parameter λ, at stress level K and semi-deviation level s–(λ) with pdf f λ is its K-left-tailed semi-vega sensitivity (“vega” being sensitivity to some measure of volatility), V(X, fλ, K, s–) to s–, the mean absolute semi-deviation below Ω, here (images omitted). The inherited fragility of Y with respect to X at stress level L = φ(K) and left-semi-deviation level s–(λ) of X is the partial derivative (image omitted). Note that the stress level and the pdf are defined for the variable Y, but the parameter used for differentiation is the left-semi-absolute deviation of X. For antifragility, the flip above Ω, in addition to robustness below the same stress level K. The transfer theorems relate the fragility of Y to the second derivative of the transformation φ, evaluated at K and show the effect of convex (concave or mixed nonlinear) transformations on the tails via the transfer function HK. For the antifragile, use s+, the integral above K.

Fragility is not psychological: We start from the definition of fragility as tail vega sensitivity and end up with nonlinearity as a necessary attribute of the source of such fragility in the inherited case—a cause of the disease rather than the disease itself. However, there is a long literature by economists and decision scientists embedding risk into psychological preferences—historically, risk has been described as derived from risk aversion as a result of the structure of choices under uncertainty with a concavity of the muddled concept of “utility” of payoff; see Pratt (1964), Arrow (1965), Rothschild and Stiglitz (1970, 1971). But this “utility” business never led anywhere except the circularity, expressed by Machina and Rothschild (2008), “risk is what risk-averters hate.” Indeed limiting risk to aversion to concavity of choices is a quite unhappy result.

The porcelain cup and its concavity: Clearly, a coffee cup, a house, or a bridge doesn’t have psychological preferences, subjective utility, etc. Yet each can be fragile to concentrated stress. Taking z as a stress level and Π(z) as a negative payoff representing harm, it suffices to see that, with n>1, Π(n z) < n Π(z) for all 0*< n z<Z**, where Z* is the level (not necessarily specified) at which the item is broken. This inequality describes harm whose magnitude grows more than proportionally with stress. A concave negative payoff, with a negative second derivative over the relevant range, models that response. For a cup, n separate small stresses of intensity Z can cause less damage than a single stress of intensity n Z. In this model the negative payoff is concave before breakage. This physical response does not require the cup to have subjective preferences.

Scaling in a positive way, convexity of cities: Bettencourt and West (2010, 2011), West (2011). Cities are 3-D items like animals, and these beneficial nonlinearities correspond to efficiencies. But consider traffic!

“More Is Different”: Anderson (1972).

Comparative fragility of animals: Diamond (1988).

Flyvbjerg and colleagues on delays: Flyvbjerg (2009), Flyvbjerg and Buzier (2011).

Small Is Beautiful, the romantic views: Dahl and Tufte (1973), Schumacher (1973) for the soundbite. Kohr (1957) for the first manifesto against the size of the governing unit.

Size of government: I can’t find people thinking in terms of convexity effects, not even libertarians—take Kahn (2011).

Small states do better: A long research tradition on governance of city-states. It looks like what we interpret as political systems might come from size. Evidence in Easterly and Kraay (2000).

The age of increasing fragility: Zajdenwebber, see the discussion in The Black Swan. Numbers recalculated in The Economist, “Counting the Cost of Calamities,” Jan. 14, 2012.

Convexity effect on mean: Jensen (1906), Van Zwet (1966). While Jensen deals with monotone functions, Van Zwet deals with concave-convex and other mixtures—but these remain simple nonlinearities. Taleb and Douady (2012) applies it to all forms of local nonlinearities.

Empirical record of bigger: Mergers and hubris hypothesis: in Roll (1986); since then Cartwright and Schoenberg (2006).

Debt in ancient history: Babylonian jubilees, Hudson et al. (2002). Athens, Harrison (1998), Finley (1953). History of debt, Barty-King (1997), Muldrew (1993), Glaeser (2001). The latter has an anarchist view. He actually believes that debt precedes barter exchange.

Food networks: Dunne et al. (2002), Perchey and Dunne (2012), Valdovinos and Ramos-Jiliberto (2010). Fragility and resources, Nasr (2008, 2009).

Fannie Mae: They were concave across all meaningful variables. Some probability-and-nonlinearity-challenged fellow in the Obama commission investigating the cause of the crisis spread the rumor that I only detected interest rate risk of Fannie Mae: not true.

Costs of execution: “Price impact,” that is, execution costs, increase with size; they tend to follow the square root—meaning total execution cost is convex and grows at exponent 3/2, while the corresponding negative payoff is concave. But the problem is that for large deviations, such as the Société Générale case, it is a lot worse; transaction costs accelerate, in a less and less precise manner—all these papers on price impact by the new research tradition are meaningless when you need them. Remarkably, Bent Flyvbjerg found a similar effect, with somewhat weaker curvature, for bridges and tunnels with proportional costs growing at 10 Log[x] of size.

Small Is Beautiful, a technical approach: To explain how city-states, small firms, etc. are more robust to harmful events, take X, a random variable for the “unintended exposure,” the source of uncertainty (for Soc Gen it was the position that it did not see, for a corporation it might be an emergency need to some inventory, etc.). Assume the size of this unintended harm is proportional to the size of the unit—for smaller entities engage in smaller transactions than larger ones. We use for probability distribution the variable of all unintended exposures ∑Xi where Xi are independent scaled copies of the exposure, written Xi = X/N. This scaling notation does not mean that the copies are the same random draw. With k the tail amplitude and α the tail exponent, π(k, α, X) = α k α x-1-α. The N-convoluted Pareto distribution for the unintended total position N ∑ Xi: π(k/N, α, X)N where N is the number of convolutions for the distribution. The mean of the distribution, invariant with respect to N, is α k/α−1).

Losses from squeezes and overruns: for the loss function, take C[X]= -b Xβ, where costs of harm is a concave function of X. Note that for small deviations, β = 3/2 in the microstructure and execution literature.

Resulting probability distribution of harm: As we are interested in the distribution of y, we make a transformation of stochastic variable. The harm y=C[X] has for distribution: π[C-1[x]]/C’[C-1[x]]. Consider that it follows a Pareto distribution with tail amplitude kβ and tail exponent α/β, (image omitted) which has for mean (image omitted). Now the sum: for the convoluted sum of N entities, the asymptotic distribution becomes: (image omitted) with mean (owing to additivity) as a function of the variables which include N: (image omitted). If we check the ratio of expected losses in the tails for N=1 to N=10 at different values of the ratio of β over α, the ratio of the expectation for 1 unit over 10 units (image omitted) reveals the “small is beautiful” effect across different levels of concavity.