Physics, Finance and Logistic Systems

Boltzmann, Pareto, Levy and Malthus

Sorin Solomon,

Hebrew University of Jerusalem and ISI Torino

+

d Xi = (randi(t)

Pierre Verhulst

+ ci

x

Boltzmann

( )

P(Xi) ~ Xi –1-

At each time instance

Vilifredo Pareto

(X.,t)) Xi +j aij Xj

+

Pierre Verhulst

x

Boltzmann

( )

P(X ) ~ ( X )–1-

At short times

Paul Levy

P(X=0) ~ ( t )–1/

d Xi = (randi(t) (X.,t)) Xi +j aij Xj+ ci

+

Pierre Verhulst

x

Boltzmann

( )

d Xi = ( randi + ci(X.,t)) Xi +j aij Xj =

XiX

Andrzej Nowak+ +Kamil Rackozi

Gur Ya’ari+SorinSolomon

Poland

Russia

Ukraine

Power Laws (Pareto-

Zipf)

1 Gates, William Henry III 48,000, Microsoft

2 Buffett, Warren Edward 41,000, Berkshire

3 Allen, Paul Gardner 20,000, Microsoft,

4-8Walton 5X18,000, Wal-Mart

9 Dell, Michael 14,200, Dell

10 Ellison, Lawrence Joseph 13,700, Oracle

Gates Buffett AllenWalton Dell EllisonLn 2 Ln 4 Ln 5 Ln 6Ln 3

Ln 90Ln 48

Ln 41Ln 20

Ln 14.2Ln 13.7

~ population growth rate ~ average family size fixed income (+redistribution) / market returns volatility

economic stability;

Wealth Social Distribution

Forbes 400 richest by rank

Dell

Buffet

20ALLEN

GATES

WALMART

Lo

g IN

DIV

IDU

AL

WE

AL

TH

Rank in Forbes 400 list400

Wealth Social Distribution

Individual Wealth Distribution

red circles: Pareto for 400 richest people is USA

The inset:

the average wealth history 88-2003

and model fit

blue squares: simulation results

No. 6 of the Cowles Commission for Research in Economics, 1941.

HAROLD T. DAVIS

No one however, has yet exhibited a stable social order, ancient or modern, which has not followed the Pareto pattern at least approximately. (p. 395)

Snyder [1939]:

Pareto’s curve is destined to take its place as one of the great generalizations of human knowledge

Later

POWER LAWSConnection:

Wealth Inequality Price Instability

~ population growth rate ~ average family size fixed income (+redistribution) / market returns volatility

economic stability;

Wealth Social Distribution

Stock Index Stability in time

Forbes 400 richest by rank

Time Interval (seconds)400

Probability of “No significant fluctuation”

Time Interval

Dell

Buffet

20ALLEN

GATES

WALMART

Lo

g IN

DIV

IDU

AL

WE

AL

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Rank in Forbes 400 list400

Time Interval (s)

P

rob

abil

ity

of

“n

o s

ign

ific

ant

flu

ctu

atio

n” Stock Index

Stability in time

Logistic Equation

Malthus : autocatalitic proliferation/ returns :B+AB+B+Adeath/ consumption B Ødw/dt = aw

a =(#A x birth rate - death rate)

a =(#A x returns rate - consumption /losses rate)

exponential solution: w(t) = w(0)e a t

a < 0

w= #B

a

TIME

birth rate > death rate

birth rate > death rate

Verhulst way out of it: B+B B The LOGISTIC EQUATION

dw/dt = a w – c w2 c=competition / saturation Solution: exponential ==========saturation

w = #B

• A+B-> A+B+B proliferation

• B-> . death

• B+B-> B competition (radius R)

almost all the social phenomena, …. obey the logistic growth. “Social dynamics and quantifying of social forces” E. W. Montroll I would urge that people be introduced to the logistic equation early in their education… Not only in research but also in the everyday world of politics and economics … Lord Robert May

b.

= ( a -)b – b 2 + diffusion

Simplest Model: A= conditions, B = plants/ animals

WELL KNOWN Logistic Equation (Malthus, Verhulst, Lotka, Volterra, Eigen)

Phil Anderson

“Real world is controlled …

– by the exceptional, not the mean;

– by the catastrophe, not the steady drip; – we need to free ourselves

from ‘average’ thinking.”

Logistic Equation usually ignored spatial distribution, Introduce discreteness and randomeness !

w.

= ( conditions x birth rate - deathx w + diffusion w - competition w2

conditions is a function of many spatio-temporal distributed discrete individual contributions rather then totally uniform and static

+

=Multi-Agent stochastic

prediction

even fora

Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior

=> resilience and sustainability

even for <a> << 0!

Multi-Agent Complex Systems Implications: one can prove rigorously that the LE prediction:

Time

Logistic Equations

(continuum <a> << 0 approx)

Multi-Agent stochastic

a

prediction

Is ALWAYS wrong !

Further Rigorous Theoretical Results: Even in non-stationary, arbitrarily varying conditions (corresponding to wars, revolutions, booms, crashes, draughts)

Indeed it is verified:

the list of systems presenting scalingfits empirically well

the list of systems

modeled in the past by logistic equations !

that stable Power Laws

emerge generically from

stochastic logistic systems

The Theorem predicts:

VALIDATION:Scaling systems logistic systems EXAMPLESNr of Species vs individuals sizeNr of Species vs number of specimensNr of Species vs their life timeNr of Languages vs number of speakersNr of countries vs population / sizeNr of towns vs. populationNr of product types vs. number of units soldNr of treatments vs number of patients treatedNr of patients vs cost of treatmentNr of moon craters vs their sizeNr of earthquakes vs their strenthNr of meteorites vs their sizeNr of voids vs their sizeNr of galaxies vs their size Nr of rives vs the size of their basin

Conclusion

The 100 year Pareto puzzle

Is solved

by combining

The 100 year Logistic Equation of Lotka and Volterra

With the 100 year old statistical mechanics of Boltzmann

Market Fluctuations

Scaling

Market Fluctuations in the Lotka-Volterra-Boltzmann model

O

Paul LevyGauss Levy DATA

instead of Gauss

instead of Gauss

Mantegna and Stanley The distribution of stock index variations for various values of

the time interval

The probability

of the price being the same afteras a function of the time interval :

P(0,–

Market Index Dynamics

Stock Index Stability in time

Pro

bab

ilit

y o

f “

no

sig

nif

ican

t

fl

uct

uat

ion

”

The relative probability of the price being the same afteras a

function of the time interval : P(0,–

1

310

1 3 10

Dell

Buffet

20ALLEN

GATES

WALMART

Stock Index Stability in time

Time Interval (s)

P

rob

abil

ity

of

“n

o s

ign

ific

ant

fl

uct

uat

ion

”

Rank in Forbes 400 list

Lo

g IN

DIV

IDU

AL

WE

AL

TH

Theoretical Prediction

Forbes 400 richest by rank

400

Confirmed brilliantlyPioneers on a new continent: on physics and economics Sorin Solomon and Moshe Levy Quantitative Finance 3, No 1, C12 2003