gaussian and paretian. gaussian – heights of individuals tallest man (robert pershing wadlow) 272...
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Gaussian and Paretian
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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
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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
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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
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Wealthiest person (John D Rockfeller) ~ 189.6 billion
Poorest (?) ~1000
Ratio =189.000.000
Paretian: wealth
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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
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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
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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
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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
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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)
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We live immersed in an universe surrounded by power laws inside and
outside us
Why does it matter?
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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?
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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
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1st case
Financial markets and extreme events
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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
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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”
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2nd case
Risk, Hollywood and skewed industry
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Budget, revenue and profit in a typical year
Budget, Revenue, & Profit in the US Movie Industry in 1999 (Longstaff et al. 2004).
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Extreme events: outliers support the industry
‘The Blair Witch Project,’ – Cost = $60,000 – Revenue = 140 million
‘Waterworld,’ – Cost = $175 – Revenue = $88 million
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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”
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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
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3rd case
Managing niches: Chris Anderson’s Long Tail
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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
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Markets without ends
Anderson (2006) The Long Tail
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Anderson (2006) The Long Tail
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Anderson (2006) The Long Tail
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The death of the 80/20 rule of profit
Anderson (2006) The Long Tail
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Fractal markets
Anderson (2006) The Long Tail
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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
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Latent markets
Anderson (2006) The Long Tail
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Aggregators
Rhapsody & iTunes
Netflix
eBay
Advertising
Music
Movies
Physical goods and merchants
Aggregate
the long tail of
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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
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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
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The danger of averages
Thank you
Any questions?