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; Nobel for Physics 77) (more is more than more) Complex “Macroscopic” properties may be the collective effect of many simple “microscopic” components Phil Anderson “Real world is controlled … –by the exceptional, not the mean; –by the catastrophe, not the steady drip; –by the very rich, not the ‘middle class’. we need to free ourselves from ‘average’ thinking.”

“MORE IS DIFFERENT” Complex Systems Paradigm MICRO - the relevant elementary agents INTER - their basic, simple interactions MACRO - the emerging collective objects Intrinsically (3x) interdisciplinary: -MICRO belongs to one science -MACRO to another science -Mechanisms: a third science Traders, investors transactions herds,crashes,booms Decision making, psychology economics statistical mechanics, physics math, game theory, info

HARRY M. MARKOWITZ, Nobel Laureate in Economics 1990 “Levy, Solomon and Levy's Microscopic Simulation of Financial Markets points us towards the future of financial economics ”

Suppose you are the president of a region, or the president of its industrialists association The new statistics are in: the economy is decaying by 10%. Is it good news or bad news? On top of it, some of the major enterprises (representing 50% of the economy) are decaying by 40%. Is it good news or bad news?

If the average growth rate is -10% and the major enterprises (50% of the economy) are going down by -40%, it means that there are enterprises in the other 50% that are growing by + 20%. If you let them develop, in 4 years: they will grow by (1.20) 4 = double ! (they alone will equal the volume of the total initial economy). From that moment on, the region economy will have a total growth rate of ~ 20%

What would be the worse thing to do? To try to insure a uniform rate of growth by differential taxation and subsidies: Put together the -40% of the losers, with the +20% of the successful and get together a uniform NEGATIVE growth rate -10%: everybody collapses!

These scenarios look oversimplified, unrealistic and unpractical but actually this is somewhat what happened [in both directions] in quite a number of countries around the 1990’s.

I present in the sequel data and theoretical study of Poland's 3000 counties over 15 years following the 1990 liberalization of the economy. The data tells a very detailed story similar to the above but a little bit more sophisticated. To understand it we have to go back in time more then 200 years ago in Holland. (but don't worry, we will soon get back toTorino (Pareto, Volterra) to get more info).

Malthus : autocatalitic proliferation/ returns : B+A  B+B+A death/ consumption B  Ø dw/dt = a  w 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

Verhulst way out of it: B+B  B The LOGISTIC EQUATION dw/dt = a w – c w 2 c=competition / saturation Solution: exponential ==========  saturation at X= a / c w = #B

almost all the social phenomena, except in their relatively brief abnormal times obey the logistic growth. “ Social dynamics and quantifying of social forces ” Elliott W. Montroll US National Academy of Sciences and American Academy of Arts and Sciences '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 …” Nature Robert McCredie, Lord May of Oxford, President of the Royal Society

SAME SYSTEM RealityModels Complex Trivial Adaptive Fixed dynamical law Localized patches Spatial Uniformity Survival Death Discrete Individuals Continuum Density Development Decay 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 Misfit was always assigned to the neglect of specific details.

Logistic Equation usually ignored spatial distribution, Introduce discreteness and randomeness ! w. = ( conditions x birth rate - death  x w + diffusion w - competition w 2 conditions is a function of many spatio-temporal distributed discrete individual contributions rather then totally uniform and static

Phil Anderson “Real world is controlled … –by the exceptional, not the mean; –by the catastrophe, not the steady drip; –by the very rich, not the ‘middle class’ we need to free ourselves from ‘average’ thinking.”

Instead: emergence of singular spatio-temporal localized collective islands with adaptive self-serving behavior => resilience and sustainability even for << 0 ! that the continuum, differential logistic equation prediction: Time Differential Eqations ( continuum << 0 approx ) Multi-Agent  a   prediction Is ALWAYS wrong ! Shnerb, Louzoun, Bettelheim, Solomon,[PNAS (2000)] proved by (FT,RG)

Electronic Journal of Probability Vol. 8 (2003) Paper no. 5, pages 1–51. Branching Random Walk with Catalysts Harry Kesten, Vladas Sidoravicius Shnerb et al. (2000), (2001) studied the following system of interacting particles on Zd: There are two kinds of particles, called A-particles and B-particles. The A-particles perform continuous time simple random walks, independently of each other. The jumprate of each A-particle is DA. The B-particles perform continuous time simple random walks with jumprate DB, but in addition they die at rate and a B-particle at x at time s splits into two particles at x during the next ds time units with a probability NA(x, s)ds+o(ds), where NA(x, s) (NB(x, s)) denotes the number of A-particles (respectively B-particles) at x at time s. Conditionally on the A-system, the jumps, deaths and splittings of different B-particles are independent. Thus the B-particles perform a branching random walk, but with a birth rate of new particles which is proportional to the number of A-particles which coincide with the appropriate B-particles. One starts the process with all the NA(x, 0), x 2 Zd, as independent Poisson variables with mean μA, and the NB(x, 0), x 2 Zd, independent of the A-system, translation invariant and with mean μB. Shnerb et al. (2000) made the interesting discovery that in dimension 1 and 2 the expectation E{NB(x, t)} tends to infinity, no matter what the values of,,DA, DB, μA, μB 2 (0,1) are.

We have only changed the notation slightly and made more explicit assumptions on the initial distributions than Shnerb et al. (2000). Shnerb et al. (2000) indicates that in dimension 1 or 2 the B-particles “survive” for all choices of the parameters,,DA,DB, μA, μB > 0. However, they deal with some form of continuum limit of the system and we found it difficult to interpret what their claim means for the system described in the abstract. For the purpose of this paper we shall say that the B-particles survive if lim sup t!1 P{NB(0, t) > 0} > 0, (1.1) where P is the annealed probability law, i.e., the law governing the combined system of both types of particles. We shall see that in all dimensions there are choices of,,DA,DB, μA, μB > 0 for which the B-particles do not survive in the sense of (1.1). A much weaker sense of survival is that lim sup t!1 ENB(0, t) > 0. (1.2) Our first theorem confirms the discovery of Shnerb et al. (2000) that even more than (1.2) holds in dimension 1 or 2 for all positive parameter values. Note that E denotes expectation with respect to P, so that this theorem deals with the annealed expectation. Theorem 1. If d = 1 or 2, then for all,,DA,DB, μA, μB > 0 ENB(0, t) ! 1 faster than exponentially in t. (1.3) Despite this result, it is not true that (1.1) holds for all,,DA,DB, μA, μB > 0. In fact our principal result is the following theorem, which deals with the quenched expectation (i.e., in a fixed realization of the catalyst system). Here FA := -field generated by {NA(x, s) : x 2 Zd, s 0}.

Movie By Gur Ya’ari

Logistic Diff Eq prediction: Time Differential Equations continuum  a  << 0 approx ) Multi-Agent stochastic  a   prediction w. = a w – c w 2 GDP Poland Nowak, Rakoci, Solomon, Ya’ari

The GDP rate of Poland, Russia and Ukraine (the 1990 levels equals 100 percent) Poland Russia Ukraine

Year% change Slovakia Kazahstan Hungary Belarus

Nowak, Rakoci, Solomon, Ya’ari

d w i / dt = (a i -∑ j r ji )w i + ∑ j r ij w j – ∑ j w i c ik w k d w i / dt = the growth rate of county i a i w i =endogenous proliferation rate in county i ∑ j r ij w j = the growth due to transfer from other counties - ∑ j r ji w i =the capital transfer to other counties ∑ j w i c ik w k = the competition and other interaction factors with other counties and the environment One may represent the dynamics of the counties economies by the following system of coupled differential equations

Predicted Scenario: First the singular educated centers W MAX develop while the others W SLOW decay d w MAX / dt ~ (a MAX - ∑ j r j ;MAX )w MAX >>0 d w SLOW / dt ~ (a SLOW -∑ j r jSLOW )w SLOW <<0 Then, as W MAX >> W SLOW, the transfer becomes relevant and activity spreads from MAX to SLOW and all develop with the same rate a MAX - ∑ j r ji but preserve large inequality d w SLOW / dt ~ r SLOW, MAX w MAX =>w SLOW /w MAX ~ r SLOW MAX / (a MAX -a i )

These predictions are confirmed strongly by the data

Nowak, Rakoci, Solomon, Ya’ari

Case 1: low level of capital redistribution r j, MAX << (a MAX – a j ) -high income inequality w i /w MAX ~ r iMAX /(a MAX -a i ) -outbreaks of instability (e.g. Russia, Ukraine). Case2: high level of central capital redistribution (as in the previous, socialist regime) r j, MAX >> (a MAX – a j ) - slow growth or even regressing economy (Latvia) but quite - uniform wealth in space and time. Case 3 : Poland seems - optimal balance : a j, MAX are large enough to insure adaptability and sustainability over a large number of counties yet the a MAX - ∑ j r jMAX is still large enough to insure overall growth. Couthy Growth= Local Proliferation + transfer from others + saturation d w i / dt = (a i -∑ j r ji )w i + ∑ j r ij w j – ∑ j w i c ik w k

Poland Russia Ukraine Romania Latvia

Instability of over-localized civilizations

Uniform distribution (unefficient but stable (decay)) Very few localized growth centers (occasionally efficient but unequal and unstable) Intermediate Range

Prediction the economic inequality (Pareto exponent) and the economic instability (index anomalous fluctuations exponent) It is also strongly confirmed (the data we had were from western economies) Forbes 400 richest by rank 400  

What next?

PIEMONTE MAP Measure Changes in a i due to Fiat plant closure With Prof Terna’s group Check alternatives