Gaussian and Paretian. Gaussian – heights of individuals Tallest man (Robert Pershing Wadlow) 272...

Gaussian and Paretian

Gaussian – heights of individuals

Tallest man (Robert Pershing Wadlow) 272 cm

Shortest man (He Pingping) 74 cm

Ratio = 3.7

Source : Lada Adamic - http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html

Source: Bak (1996) “How Nature Works”

Krugman on the Zipf law:

“we are unused to seeing regularities this exact in

economics – it is so exact that I find it spooky” (1996)

p.40

Largest city (NYC) pop 8 million

Smallest city (Duffield, Virginia) pop. 52

Ratio = 150000

Paretian: city size

Most frequent word* (rank 1: you = 1222421)

Least frequent word (rank 40000: imperious = 6)

Ratio: = 203737

Paretian: word frequency in TV and movie scripts

*

http://en.wiktionary.org/wiki/Wiktionary:Frequency_lists/TV/2006/

1-1000

Wealthiest person (John D Rockfeller) ~ 189.6 billion

Poorest (?) ~1000

Ratio =189.000.000

Paretian: wealth

Scale-free Networks

Nodes: people (Females; Males)

Links: sexual relationships

SEX web scale-free network

(Liljeros et al. Nature 2001)

4781 Swedes; 18-74; 59% response rate.

Nodes: computers, routers Links: physical lines

(Faloutsos, Faloutsos and Faloutsos, 1999)

Routers web

Allometric growth – cube-surface law

3/4 mistery

West, Brown & Enquist (1997). A General Model for the Origin of Allometric

Scaling Laws in Biology

West and Brown. Life's Universal Scaling Laws

Scaling exponents for urban indicators vs. city size

Y β 95% CI Adj-R2 Observations Country–year

Gasoline stations 0.77 0.74, 0.81 0.93 318 U.S. 2001

New patents 1.27 1.25,1.29 0.72 331 U.S. 2001

R&D employment 1.26 1.18,1.43 0.93 295 China 2002

New AIDS cases 1.23 1.18,1.29 0.76 93 U.S. 2002–2003

Bettencourt et Al. 2007. Growth, innovation, scaling, and the pace of life in cities, PNAS , vol. 104 no. 17 , 7301–7306 ¶

Y(t) = Y0

N(t) β

36 Kinds of “Physical” Power Laws Cities Traffic jams Coastlines Brush-fire damage Water levels in the Nile Hurricanes & floods Earthquakes Asteroid hits Sun Spots Galactic structure Sand pile avalanches Brownian motion Music Epidemics/Plagues Genetic circuitry Metabolism of cells Functional networks in brain Tumor growth

Biodiversity Circulation in plants and animals Langton’s Game of Life Fractals Punctuated equilibrium Mass extinctions/explosions Brain functioning Predicting premature births Laser technology evolution Fractures of materials Magnitude estimate of sensorial stimuli Willis’ Law: No. v. size of plant genera Fetal lamb breathing Bronchial structure Frequency of DNA base chemicals Protein-protein interaction networks Heart-beats Yeast

38 Kinds of “Social” Power Laws Structure of the Internet equipment Internet links # hits received from website/day Price movements on exchanges Economic fluctuations “Fordist” power structure/effects Salaries Labor strikes Job vacancies Firm size Growth rates of firms Growth rates of internal structure Supply chains Cotton prices Alliance networks among biotech firms Entrepreneurship/Innovation Director interlock structure Italian Industrial Clusters

Language—word usage Social networks Blockbuster drugs Sexual networks Distribution of wealth Citations Co-authorships Casualties in war Growth rate of countries’ GDP Delinquency rates Movie profits Actor networks Size of villages Distribution of family names Consumer products Copies of books sold Number of telephone calls and emails Deaths of languages Aggressive behavior among children “No learning” agents (Ormerod)

We live immersed in an universe surrounded by power laws inside and

outside us

Why does it matter?

We live immersed in an universe surrounded by power laws inside and

outside us

What happens if we get it wrong and assume a Gaussian world instead of a

Paretian one?

Power law

y = x - = constant

Exponential

y = e – x e = constant

Exponentials vs. Power Laws

Bell Curve

Power Law

Linear axes Log axes

Power Law

Bell Curve

1st case

Financial markets and extreme events

Do It Yourself (Financial DIY)

Download Dow Jones index numbers from:

http://www.dowjones.com Take daily variation: take log of each daily index

number. Subctract log from following day log Assume variations fit Gaussian and calculate sample

variance s2 or

s2 = (xav –xi )2 / (n-1)

Calculate how typical each crash day is:

z = (xi – xav) / s

Using z score calculate probability

Mandelbrot & Hudson 2004

Probability of financial crushes according to standard financial theory (Mandelbrot, 2004)

August 31, 1998 6.8% Wall Street crush 1 in 20 million August 1997 7.7% Dow Jones 1 in 50 billion July 2002 3 step falls in 7 days 1 in 4 trillion

And finally October 19, 1987 29.2% fall 1 in 10-50

“It is a number outside the scale of nature. You could span the powers of ten from the smallest subatomic particle to the breadth of the measurable universe – and still never meet

such a number”

2nd case

Risk, Hollywood and skewed industry

Budget, revenue and profit in a typical year

Budget, Revenue, & Profit in the US Movie Industry in 1999 (Longstaff et al. 2004).

Extreme events: outliers support the industry

‘The Blair Witch Project,’ – Cost = $60,000 – Revenue = 140 million

‘Waterworld,’ – Cost = $175 – Revenue = $88 million

Unrealistic picture of risk

Gaussian (probability) Pareto (prob)

Home Alone 2.97 *10–16 0.83%

Waterworld –3.41 *10–12 0.45%

(De Vany Hollywood Economics 2003, pp. 219, 284)

it masks the importance of the rare events determining the success of the industry

and

creates a false sense of security ‘(Illusion of Control’ fallacy)

Econometric models used for predictions.

Results: “predictions of total grosses for an individual movie can be expected to be off by as much as a multiplicative factor of 100 high or low”.

(Simonoff and Sparrow, 2000)

Consequence: “studio models focus on forecasting expected values and virtually ignore the variance”

Assume movie industry is Paretian

Gaussian (probability) Pareto (prob)

Home Alone 2.97 *10–16 0.83%

Waterworld –3.41 *10–12 0.45%

(De Vany Hollywood Economics 2003, pp. 219, 284)

1. Risk is unbounded: profit and loss are scalable

2. Calculate slope. It gives indication of variability and consequently real risk

3. Scaling property of Paretian distribution allows the statistical calculation of ratios. For instance

Ratio = N10.000

/ N100.000

= N100.000

/ N1.000.000

= exponent

4. Practices based on expected results (flat fee distribution, contracting) are damaging. Adopt contingent reward and contract

approach

5. ‘Star system’ doesn’t seem to work

3rd case

Managing niches: Chris Anderson’s Long Tail

Two tails of a power law

Casti _126

Find gutemberg

Ricther-Gutenberg Law

Earthquake magnitude (mb

) ~ Log E

Nc

(Ear

thqu

akes

/Yea

r)

Extreme events tail

Small events tail

Markets without ends

Anderson (2006) The Long Tail

Anderson (2006) The Long Tail

Anderson (2006) The Long Tail

The death of the 80/20 rule of profit

Anderson (2006) The Long Tail

Fractal markets

Anderson (2006) The Long Tail

Managing tails as if they

were averages

…………………….

1.000.000.000

100.000.000

10.000.000

1.000.000

100.000

10.000

1.000

100

10

1

1 10 100 1.000 10.000 … 10.000.000 …

Product Rank

• Long tail of rare, high-impact (‘hits’) events

• ‘Economy of scarcity’

• Homogeneous markets

• Consumerism

• 80/20 rule-inspired management

• Long tail of small niches

• ‘Economy of abundance’

• Heterogeneous markets

• Producerism

• Diversity

Revenues

2nd power law region

Latent demand space: new long tail1st power law region

Traditional markets: traditional long tail

Cut-off point

Sales predicted by power law

model

Latent markets

Anderson (2006) The Long Tail

Aggregators

Google

Rhapsody & iTunes

Netflix

eBay

Advertising

Music

Movies

Physical goods and merchants

Aggregate

the long tail of

Gladwell

From management of averages to

management of extreme events tail

Illycaffe’

Transforming TIEs into positive

extreme events

Anderson

From minimum common

denominator to niche aggregation

strategy

Power Law

Gaussian Curve

Hollywood

The management of risk from Gaussian

mean to Paretian tail

Gaussian vs Paretian

In a Gaussian world:

Challenge: manage the population

How: reduce population to the representative agent and define variance (of population)

Manage around mean and variance

In a Paretian world:

Challenge: manage the frontier

Identify outliers and manage the tail of the distribution

Manage tail

The danger of averages

Thank you

Any questions?