17 S M A L L I S B E A U T I F U L : R I S K ,S C A L E A N D C O N C E N T R AT I O N

Chapter Summary 17: We extract the effect of size on the degradationof the expectation of a random variable, from nonlinear response. Themethod is general and allows to show the "small is beautiful" or "decen-tralized is effective" or "a diverse ecology is safer" effect from a responseto a stochastic stressor and prove stochastic diseconomies of scale andconcentration (with as example the Irish potato famine and GMOs). Weapply the methodology to environmental harm using standard sigmoiddose-response to show the need to split sources of pollution across inde-pendent(nonsynergetic) pollutants.

��.� ������������ : ��� ����� �� �����Diseconomies and Harm of scale Where is small beautiful and how can wedetect, even extract its effect from nonlinear response? 1 Does getting largermakes an entity more vulnerable to errors? Does polluting or subjecting theenvironment with a large quantity cause disproportional "unseen" stochasticeffects? We will consider different types of dose-response or harm-responseunder different classes of probability distributions.

The situations convered include:

1. Size of items falling on your head (a large stone vs small pebbles).

2. Losses under strain.

3. Size of animals (The concavity stemming from size can be directly de-rived from the difference between allometic and isometric growth, as ani-mals scale in a specific manner as they grow, an idea initially detected byHaldane,[33] (on the "cube law"(TK)).

4. Quantity in a short squeeze

5. The effect of crop diversity

6. Large vs small structures (say the National Health Service vs local entities)

7. Centralized government vs municipalities

1 The slogan "small is beautiful" originates with the works of Leonard Kohr [42] and his studentSchumacher who thus titled his influential book.

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Figure 17.1: The Tower ofBabel Effect: Nonlinear re-sponse to height, as tallertowers are disproportion-ately more vulnerable to,say, earthquakes, winds,or a collision. This illus-trates the case of truncatedharm (limited losses).Forsome structures with un-bounded harm the effect iseven stronger.

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8. Large projects such as the concentration of health care in the U.K.

9. Stochastic environmental harm: when, say, polluting with K units is morethan twice as harmful than polluting with K/2 units.

��.�.� First Example: The Kerviel Rogue Trader Affair

The problem is summarized in Antifragile [78] as follows:

On January 21, 2008, the Parisian bank Societé Générale rushed tosell in the market close to seventy billion dollars worth of stocks, avery large amount for any single "fire sale." Markets were not veryactive (called "thin"), as it was Martin Luther King Day in the UnitedStates, and markets worldwide dropped precipitously, close to 10percent, costing the company close to six billion dollars in lossesjust from their fire sale. The entire point of the squeeze is that theycouldn’t wait, and they had no option but to turn a sale into a firesale. For they had, over the weekend, uncovered a fraud. JeromeKerviel, a rogue back office employee, was playing with humongoussums in the market and hiding these exposures from the main com-puter system. They had no choice but to sell, immediately, thesestocks they didn’t know they owned. Now, to see the effect offragility from size (or concentration), consider losses as a functionof quantity sold. A fire sale of $70 billion worth of stocks leads to aloss of $6 billion. But a fire sale a tenth of the size,$7 billion would re-sult in no loss at all, as markets would absorb the quantities withoutpanic, maybe without even noticing. So this tells us that if, instead ofhaving one very large bank, with Monsieur Kerviel as a rogue trader,we had ten smaller units, each with a proportional Monsieur Micro-Kerviel, and each conducted his rogue trading independently and atrandom times, the total losses for the ten banks would be close tonothing.

��.�.� Second Example: The Irish Potato Famine with a warning on GMOs

The same argument and derivations apply to concentration. Consider thetragedy of the Irish potato famine.

In the 19th Century, Ireland experienced a violent potato famine coming fromconcentration and lack of diversity. They concentrated their crops with the"lumper" potato variety. "Since potatoes can be propagated vegetatively, all ofthese lumpers were clones, genetically identical to one another."2

Now the case of genetically modified organism (GMOs) is rich in fragilities (andconfusion about the "natural"): the fact that an error can spread beyond localspots bringing fat-tailedness, a direct result ofthe multiplication of large scaleerrors. But the mathematical framework here allows us to gauge its effect from

2 the source is evolution.berkeley.edu/evolibrary but looking for author’s name.

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Figure 17.2: Integrating theevolutionary explanation ofthe Irish potato famine intoour fragility framework, courtesyhttp://evolution.berkeley.edu/evolibrary.

loss of local diversity. The greater problem with GMOs is the risk of ecocide,examined in Chapter x.

��.�.� Only Iatrogenics of Scale and Concentration

Note that, in this discussion, we only consider the harm, not the benefits of con-centration under nonlinear (concave) response. Economies of scale (or savingsfrom concentration and lack of diversity) are similar to short volatility exposures,with seen immediate benefits and unseen deferred losses.

The rest of the discussion is as follows. We will proceed, via convex trans-formation to show the effect of nonlinearity on the expectation. We start withopen-ended harm, a monotone concave response, where regardless of proba-bility distribution (satisfying some criteria), we can extract the harm from thesecond derivative of the exposure. Then we look at more natural settings repre-sented by the "sigmoid" S-curve (or inverted S-curve) which offers more complexnonlinearities and spans a broader class of phenomena.

Unimodality as a general assumption Let the variable x, representing thestochastic stressor, follow a certain class of continuous probability distributions(unimodal), with the density p(x) satisfying: p(x) � p(x + e) for all e > 0, andx > x⇤ and p(x) � p(x� e) for all x < x⇤ with {x⇤ : p(x⇤) = maxx p(x)}. The den-sity p(x) is Lipschitz. This condition will be maintained throughout the entireexercise.

��.� ��������� ��������� �������In this section, we assume an unbounded harm function, where harm is a mono-tone (but nonlinear) function in C2, with negative second derivative for all val-ues of x in R+; so let h(x), R+ ! R� be the harm function. Let B be the size ofthe total unit subjected to stochastic stressor x, with q(B) = B + h(x).

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Stressor

Damage !or Cost"

Figure 17.3: Simple Harm Func-tions, monotone: k = 1, b =3/2, 2, 3.

We can prove by the inequalities from concave transformations that, the expecta-tion of the large units is lower or equal to that of the sum of the parts. Becauseof the monotonocity and concavity of h(x),

h

N

Âi=1

wi x

!

N

Âi=1

h(wi x), (17.1)

for all x in its domain (R+), where wi are nonnegative normalized weights, thatis, ÂN

i=1 wi = 1 and 0 wi 1.

And taking expectations on both sides, E(q(B)) E⇣

ÂNi=1 q(wi B)

⌘

: the meanof a large unit under stochastic stressors degrades compared to a series of smallones.

��.�.� Application

Let h(x) be the simplified harm function of the form

h(x) ⌘ �k xb, (17.2)

k 2 (0, •) , b 2 [0, •).

Table 22: Applications with unbounded convexity effectsEnvironment Research h(x)LiquidationCosts

Toth etal.,[82],Bouchaudet al. [9]

�kx32

Bridges Flyvbjerg et al[31]

�x( log(x)+7.110 )

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Example 1: One-Tailed Standard Pareto Distribution Let the probability dis-tribution of x (the harm) be a simple Pareto (which matters little for the exercise,as any one-tailed distribution does the job). The density:

pa,L(x) = a La x�a�1 for x � L (17.3)

The distribution of the response to the stressor will have the distribution g =(p � h)(x).

Given that k the stressor is strictly positive, h(x) will be in the negative domain.Consider a second change of variable, dividing x in N equal fragments, so thatthe unit becomes x = x/N, N 2 N�1:

ga,L,N(x) = �

a

aN�a

⇣

� x

k

⌘�a/b

b x

, (17.4)

for x �k⇣

LN

⌘

b

and with a > 1 + b. The expectation for a section x/N, Mb

(N):

Mb

(N) =Z � kLb

N

�•x g

a,L,N(x) dx = �a k Lb Na

⇣

1b

�1⌘

�1

a � b

(17.5)

which leads to a simple ratio of the mean of the total losses (or damage) com-pared to a k number of its N fragments, allowing us to extract the "convexityeffect" or the degradation of the mean coming from size (or concentration):

k Mb

(kN)M

b

(N)= k

a

⇣

1b

�1⌘

(17.6)

With b = 1, the convexity effect =1. With b = 3/2 (what we observe in orderflowand many other domains related to planning, Bouchaud et al., 2012, Flyvbjerget al, 2012), the convexity effect is shown in Figure 23.

2 4 6 8 10N

0.2

0.4

0.6

0.8

1.0

Expected total loss for N units

Convexity Effects

Table 23: The mean harm in total as a result of concentration. Degradation of the meanfor N=1 compared to a large N, with b = 3/2

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Unseen Harm The skewness of ga,L,N(x) shows effectively how losses have

properties that hide the mean in "small" samples (that is, large but insufficientnumber of observations), since, owing to skewness, the observed mean loss withtend to be lower than the true value. As with the classical Black Swan exposures,benefits are obvious and harm hidden.

��.� � ������ ����� : ��� ����������� �������Now the biological and physical domains (say animals, structures) do not incurunlimited harm, when taken as single units. The losses terminate somewhere:what is broken is broken. From the generalized sigmoid function of [? ], whereSM(x) = ÂM

k=1ak

1+exp(bk(ck�x)) , a sum of single sigmoids. We assume as a specialsimplified case M = 1 and a1 = �1 so we focus on a single stressor or sourceof harm S(x), R+ ! [�1, 0] where x is a positive variable to simplify and theresponse a negative one. S(0) = 0, so S(.) has the following form:

S(x) =�1

1 + e b (c�x) +1

1 + eb c (17.7)

The second term is there to ensure that S(0) = 0. Figure 24 shows the differentcalibrations of b (c sets a displacement to the right).

2 4 6 8 10Harm

!1.0

!0.8

!0.6

!0.4

!0.2

Response

Table 24: Consider the object broken at �1 and in perfect condition at 0[backgroundcolor=lightgray] The sigmoid, S(x) in C• is a class of generalizedfunction (Sobolev, Schwartz [69]); it represents literally any object that has pro-gressive positive or negative saturation; it is smooth and has derivatives of allorder: simply anything bounded on the left and on the right has to necessarilyhave to have the sigmoid convex-concave (or mixed series of convex-concave)shape.

The idea is to measure the effect of the distribution, as in 3.14. Recall that theprobability distribution p(x) is Lipshitz and unimodal.

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Convex Response

Higher scale

(dispersion or

variance)

Harm

Response

The second derivative S00(x) = b2eb(c+x)(ebx�ebc)(ebc+ebx)3 . Setting the point where S00(x)

becomes 0, at x = c, we get the following: S(x) is concave in the interval x 2 [0, c)and convex in the interval x 2 (c, •).

The result is mixed and depends necessarily on the parametrization of thesigmoids. We can thus break the probability distributions into two sections, the"concave" and "convex" parts: E = E� + E+. Taking x = x/N, as we did earlier,

E� = NZ c

0S(x) p(x) dx ,

andE+ = N

Z •

cS(x) p(x) dx

The convexity of S(.) is symmetric around c,

S00(x)|x=c�u= �2b2 sinh4✓

b u2

◆

csch3(b u)

S00(x)|x=c+u= 2b2 sinh4✓

bu2

◆

csch3(b u)

We can therefore prove that the effect of the expectation for changes in Ndepends exactly on whether the mass to the left of a is greater than the mass tothe right. Accordingly, if

R a0 p(x) dx >

R •a p(x) dx, the effect of the concentration

ratio will be positive, and negative otherwise.

��.�.� Application

Example of a simple distribution: Exponential Using the same notations as17.2.1, we look for the mean of the total (but without extracting the probabilitydistribution of the transformed variable, as it is harder with a sigmoid). Assumex follows a standard exponential distribution with parameter l, p(x) ⌘ lel(�x)

Ml

(N) = E (S(x)) =Z •

0lel(�x)

✓

� 1

eb(c� xN ) + 1

+1

ebc + 1

◆

dx (17.8)

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Ml

(N) =1

ebc + 1� 2F1

✓

1,Nl

b;

Nl

b+ 1;�ebc

◆

where the Hypergeometric function 2F1(a, b; c; z) = •k=0

akbkzk

k!ck.

The ratio k Ml

(kN)M

l

(N) doesn’t admit a reversal owing to the shape, as we can seein 17.3.1 but we can see that high variance reduces the effect of the concentration.

However high variance increases the probability of breakage.

Λ " 0

Different values of Λ # (0,1]

2 4 6 8 10Κ

0.2

0.4

0.6

0.8

1.0

ΚMΛ !Κ"

MΛ !1"

Example of a more complicated distribution: Pareto type IV Quasiconcavebut neither convex nor concave PDF: The second derivative of the PDF forthe Exponential doesn’t change sign, ∂

2

∂x2 (l exp(�lx)) = l

3el(�x), so the dis-tribution retains a convex shape. Further, it is not possible to move its meanbeyond the point c where the sigmoid switches in the sign of the nonlinearity.So we elect a broader one, the Pareto Distibution of Type IV, which is extremelyflexible because, unlike the simply convex shape (it has a skewed "bell" shape,mixed convex-concave-convex shape) and accommodates tail exponents, hencehas power law properties for large deviations. It is quasiconcave but neitherconvex nor concave. A probability measure (hence PDF) p : D ! [0, 1] is quasi-concave in domain D if for all x, y 2 D and w 2 [0, 1] we have:

p(wx + (1 � w)y) � min (p(x), p(y)).

Where x is the same harm as in Equation 17.7:

pa,g,µ,k(x) =

ak�1/g(x � µ)1g

�1✓

⇣

kx�µ

⌘�1/g

+ 1◆�a�1

g

(17.9)

for x � µ and 0 elsewhere.

The Four figures in 17.3.1 shows the different effects of the parameters on thedistribution.

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2 4 6 8 10

x

!1.0

!0.8

!0.6

!0.4

!0.2

PDF

2 4 6 8 10 12 14

x

0.05

0.10

0.15

0.20

0.25

0.30

0.35

PDF

2 4 6 8 10 12 14

x

0.1

0.2

0.3

0.4

0.5

0.6

0.7

PDF

2 4 6 8 10 12 14

x

0.1

0.2

0.3

0.4

0.5

PDF

The mean harm function, Ma,g,µ,k(N) becomes:

Ma,g,µ,k(N) =

ak�1/g

g

Z •

0(x � µ)

1g

�1✓

1ebc + 1

� 1

eb(c� xN ) + 1

◆

✓

kx � µ

◆�1/g

+ 1

!�a�1

dx (17.10)

M(.) needs to be evaluated numerically. Our concern is the "pathology"where the mixed convexities of the sigmoid and the probability distributionsproduce locally opposite results than 17.3.1 on the ratio

kMa,g,µ,k(N)

Ma,g,µ,k(N) . We produce

perturbations around zones where µ has maximal effects, as in 17.6. Howeveras shown in Figure 17.4, the total expected harm is quite large under theseconditions, and damage will be done regardless of the effect of scale.

��.�.� Conclusion

This completes the math showing extracting the "small is beautiful" effect, aswell as the effect of dose on harm in natural and biological settings where theSigmoid is in use. More verbal discussions are in Antifragile.

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S''(x)=0

1 2 3 4 5 6Μ

"0.5

"0.4

"0.3

"0.2

"0.1

Harm for N#1

Figure 17.4: Harm increases as themean of the probability distributionshifts to the right, to become maxi-mal at c, the point where the sig-moid function S(.) switches fromconcave to convex.

2 4 6 8 10Κ

0.5

1.0

1.5

ΚMk,Α,Γ,Μ !Κ"

Mk,Α,Γ,Μ !1"

Figure 17.5: Different values of µ:we see the pathology where 2 M(2)is higher than M(1), for a value ofµ = 4 to the right of the point c.

1 2 3 4

Μ

0.3

0.4

0.5

0.6

M !2"

M !1"

Figure 17.6: The effect of µ on theloss from scale.

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Acknowledgments

Yaneer Bar-Yam, Jim Gatheral (naming such nonlinear fragility the "Tower ofBabel effect"), Igor Bukanov, Edi Pigoni, Charles Tapiero.

294