Complexity Emergence in Economics Sorin Solomon, Racah Institute of Physics HUJ Israel Scientific Director of Complex Multi-Agent Systems Division, ISI Turin and of the Lagrange Interdisciplinary Laboratory for Excellence In Complexity Coordinator of EU General Integration Action in Complexity Science Chair of the EU Expert Committee for Complexity Science MORE IS DIFFERENT (Anderson 72) (more is more than more) Complex “Macroscopic” properties may be the collective effect of simple interactions between many elementary “microscopic” components MICRO - Investors, individual capital,shares INTER - sell/buy orders, gain/loss MACRO - social wealth distribution, market price fluctuations (cycles, crashes, booms, stabilization by noise)
SAME SYSTEM RealityModels Complex Trivial Adaptive Fixed dynamical law Localized patches Spatial Uniformity Survival Death Discrete Individuals Continuum Density Development Decay Misfit was always assigned to the neglect of specific details. We show it was rather due to the neglect of the discreteness. Once taken in account => complex adaptive collective objects. emerge even in the worse conditions
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 (assume 0 dim!!!) Simplest Model: A= gain opportunities, B = capital WELL KNOWN Logistic Equation (Malthus, Verhulst, Lotka, Volterra, Eigen) (no diffusion)
Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior => resilience and sustainability even for << 0 ! Diff Eq prediction: Time Differential Equations continuum a << 0 approx ) Multi-Agent stochastic a prediction One Proved that using the DiffEq is ALWAYS wrong in dim >0 ! b. = ( a - ) b – b 2
Microscopic fluctuations Time exponentiationMacroscopic adaptive islands Most singular ( large a= n A ) – rarest – at the beginning (t=0) irrelevant -> in the end (t-> ∞) dominating b (t) = b(0) ∑ prob (n A ) e t ( n A - ) The Importance of Being Discrete; Life Always Wins on the Surface
Discrete A Individuals microscopic noise Autocatalytic B proliferation amplification Collective Macroscopic Objects - Power Laws: - wealth distribution- - Levy, fractal, market fluctuations- -Emergent Properties : Adaptability - Most singular, rarest fluctuations dominate the system dynamics The Importance of Being Discrete; Life Always Wins on the Surface = !!!
Movie By Gur Ya’ari
A one –dimensional simple example continuum prediction A B level a =2/14 = 1 1/2 A level
A B level 2/14 ×× 1 -1/2 = -5/14 A level A one –dimensional simple example continuum prediction a -- ×× t (1-5/14) t continuum prediction
b 4 (t+1) = (1 + 1 × – ) b 4 (t) A A b 13 (t+1) = (1 + 0 × – ) b 13 (t) B level A one –dimensional simple example discrete prediction t (1-5/14) t continuum prediction
AA t (9/14) t Initial exponential decay B level continuum prediction A one –dimensional simple example discrete prediction (3/2) (½)(½) (½)
AA t (9/14) t (3/2) t B level continuum prediction A one –dimensional simple example discrete prediction (3/2) 2 (½)2(½)2 (½) 2
AA t (9/14) t (3/2) t growth B level continuum prediction A one –dimensional simple example discrete prediction (3/2) 3 (½)3(½)3 (½) 3
AA t (9/14) t (3/2) t growth continuum prediction A one –dimensional simple example discrete prediction (3/2) 4 (½)4(½)4 (½) 4
AA t (3/2) t growth (9/14) t continuum prediction A one –dimensional simple example discrete prediction (3/2) 5 (½)5(½)5 (½) 5
and Branching Random Walk Theorems (2002) that : - In all dimensions d: D a > 1-P d always suffices P d = Polya ’ s constant ; P 2 = 1 -On a large enough 2 dimensional surface, the total B population always grows! No matter how fast the death rate , how low the A density, how small the proliferation rate The Importance of Being Discrete; Life Always Wins on the Surface One can prove rigorously by RG
A The Role of DIFFUSION The Emergence of Adaptive B islands Take just one A in all the lattice:
A
A
A B diffusion
A
A Growth stops when A jumps to a neighboring site
AA
A Growth will start on the New A site B population on the old A site will decrease WIDEN THE B ISLAND LINEARLY IN TIME
A
A
A AA
A AA
A AA
A AA B diffusion
A AA
A AA
A AA
A A A A Growth stops when A jumps again (typically after each time interval 1/D A )
A A A AA
A A AAA
A A A Growth stops again when A jumps again (typically after each time interval 1/D A ) A
A A A If the time interval d/D A is small, when A jumps back it still finds B ’ s to proliferate A
TIME S P C A E ln b TIME (A location and b distribution ) The strict adherence of the elementary particles A and B to the basic fundamental laws and the emergence of complex adaptive entities with self-serving behavior do not interfere one with another. Yet they determine one another. Emergent Collective Dynamics: B-islands search, follow, adapt to, and exploit fortuitous fluctuations in A density. Is this a mystery? Not in the AB model where all is on the table ! This is in apparent contradiction to the “fundamental laws” where individual B don’t follow anybody
Polish Economy after Liberalization Data Andrzej Nowak (+group) Kamil Rakocy Gur Ya’ari, SS(+group)
EXAMPLE of Theory Application APPLICATION: Liberalization Experiment Poland Economy after MICRO growth ___________________ => MACRO growth 1990 MACRO decay (90) 1992 MACRO growth (92) 1991 MICRO growth (91) GNP THEOREM (RG, RW) one of the fundamental laws of complexity Diff Eq prediction Complexity prediction Education 88 MACRO decay Maps Andrzej Nowak ’ s group (Warsaw U.), CO 3 collaboration
GNP Complexity prediction Maps Andrzej Nowak ’ s group (Warsaw U.), CO 3 collaboration
fractal space distribution Prediction of campaign success (15/17) Goldenberg Air-view of a sub-urban neighborhood; crosses on the roofs indicate air-conditioner purchase
- Microscopic seeds and Macroscopic Oases MICRO –individual plants INTER –growth, water fixation, MACRO – bushes, vegetation patches N.M. Shnerb, P. Sarah, H. Lavee, and S. Solomon Reactive glass and vegetation patterns Phys. Rev. Lett. 90, (2003) Mediterranean; uniform Semi-arid; patchy Desert; uniform
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 scaling fits 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 EXAMPLES Nr of Species vs individuals size Nr of Species vs number of specimens Nr of Species vs their life time Nr of Languages vs number of speakers Nr of countries vs population / size Nr of towns vs. population Nr of product types vs. number of units sold Nr of treatments vs number of patients treated Nr of patients vs cost of treatment Nr of moon craters vs their size Nr of earthquakes vs their strenth Nr of meteorites vs their size Nr of voids vs their size Nr of galaxies vs their size Nr of rives vs the size of their basin
VERY NON TRIVIAL PREDICTION Relating Market Index Dynamics to Individual Wealth Distribution:
Zipfplot of thewealthsof the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding model results are shown in the inset. Dell Buffet 20 ALLEN GATES WALMART 1/ Individual Wealth Distribution Pareto- ZIPF law
Mantegna and Stanley The distribution of stock index variations for various values of the time interval The probability of the price being the same after as a function of the time interval : P(0, – Market Index Dynamics NOT disputed by the faster then Levy tail decay analysis ! Stock Index Stability in time Probability of “no significant fluctuation”
Zipfplot of thewealthsof the investors in the Forbes 400 of 2003 vs. their ranks. The corresponding model results are shown in the inset. Dell Buffet 20 ALLEN GATES WALMART Stock Index Stability in time Time Interval (s) Probability of “no significant fluctuation” Rank in Forbes 400 list Log INDIVIDUAL WEALTH Theoretical Prediction Forbes 400 richest by rank 400 Confirmed brilliantly Pioneers on a new continent: on physics and economics Sorin Solomon and Moshe Levy Quantitative Finance 3, No 1, C
Conclusions The classical Logistic systems have very non-trivial predictions when treated correctly as made of discrete individuals We have presented 2 applications: –The confirmation of the model prediction in the Poland liberalization experiment –The qualitative and quantitative confirmation of the model predictions in the scaling properties of market economy and other logistic systems)