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)

HARRY M. MARKOWITZ, Nobel Laureate in Economics “Levy, Solomon and Levy's Microscopic Simulation of Financial Markets points us towards the future of financial economics ”

Stock market shock explained Physicists model recent trading frenzy. Market 'spikes' are seen by traders as freak events. Physicists expect them

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)

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 Rigorously that DE is ALWAYS wrong in dim >0 ! b. = ( a -  )  b –  b 2

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 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

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 = !!!

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 Global analysis 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

MOVIE

Spatial Correlation of Number of enterprizes per capita One can see the forming of a spatially correlated patches The risk of being unfair, the unstable fate of globalization. Louzoun. Y. Mazurski. D., Goldberg. J. Solomon. S. Artificial Life 4(9) 357 (2003)

Growth Rate Spatial Correlations Significant only during first 4-5 years THEN: uniform country growth rate (by diffusion ) [ THEOREM ] Co-Evolutionist Stochastic Dynamics: Emergence of Power- Laws in stochastic Lotka-Volterra-Eigen- Schuster Systems S. Solomon, P. Richmond, O. Biham and O. Malcai, (2003)

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 : Stable power laws in variable economies; Lotka-Volterra implies Pareto-Zipf S. Solomon and P. Richmond Eur. Phys. J. B 27, (2002)

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/  NOT disputed by Yakovenko etc 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