simulating the coordination of individual economic decisions

Physica A 287 (2000) 613–630www.elsevier.com/locate/physa

Simulating the coordination of individualeconomic decisions

Andrzej Nowaka;b;∗, Marek Ku�sa;c , Jakub Urbaniaka , Tomasz ZaryckiaaInstitute for Social Studies, Warsaw University, ul. Stawki 5=7, 00-183 Warsaw, Poland

bAdvanced School of Social Psychology, Warsaw, PolandcCenter for Theoretical Physics, Polish Academy of Sciences, Warsaw, Poland

Received 26 June 2000; received in revised form 10 July 2000

Abstract

The model of dynamic social in uence is used to describe the coordination of individualeconomic decisions. Computer simulations of the model show that the social and economictransitions occur as growing clusters of “new” in the sea of old. The model formulated at theindividual level may be used to derive another one concerning the aggregate level. The aggregatelevel model was used to simulate spatio-temporal dynamics of the number of privately ownedenterprises in Poland during the transition from centrally governed to the market economy.Analysis revealed the similarity between the model predictions and economic data. c© 2000Elsevier Science B.V. All rights reserved.

1. Introduction

The pervasive feature of di�erent aspects of economic activity is clustering in space.Innovations spread as expanding clusters, farming practices tend to be similar in neigh-boring regions, rent agreements in farming tend to be similar in farms located nearby,and industries tend to be located in the vicinity of other industries, to name just a fewexamples. The theories constructed to account for clustering in space have been formu-lated on the aggregate level, and considered economic factors. Many decisions whicha�ect the economic processes, however, are made by individuals. Well tested and wellunderstood, psychological laws describe the e�ect of social context on the individual

∗ Correspondence address: Institute for Social Studies, Warsaw University, ul. Stawki 5=7, 00-183 Warsaw,Poland.

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(00)00397 -6

614 A. Nowak et al. / Physica A 287 (2000) 613–630

decision making. Computer simulations have shown that the clustering of opinions isa pervasive feature of a wide variety of models, where individual opinions are in u-enced by other individuals. In the present paper, we investigate the relation betweenthe individual and aggregate level models of economic development. We argue that theinclusion of the individual level theory allows one to predict, new properties of eco-nomic growth. We compare the outcomes of computer simulations to the economic dataconcerning Poland in the period of transition between the centrally governed economyand the market economy.

Many economic factors were indicated by di�erent location theories as factorsresponsible for spatial clustering. Since the cost of transportation [1] adds to the costof production, it pays to produce close to the market and in vicinity of other �rms[2]. The cost of transportation also a�ects consumers, who tend to buy at the clos-est location. If the local demand exceeds a threshold, an entrepreneur will o�er thegood. Local demand leads to the emergence of central places [3] of economic activity.Assumptions of higher-order function in the system lead to the emergence of a hi-erarchy of urban centers, some more central than others [3]. Other economic factorsconsidered by modern location theories include: labor costs and labor quality, �scalregulations, institutional infrastructure, quality of life in the region, etc.

Dynamic interaction of innovative enterprises concentrated in a particular place, usu-ally an urban center are responsible for locally concentrated upstream (positive), anddownstream (negative) “propulsive e�ects” a�ecting their environments. As Perroux[4] observed, describing a theory of growth poles, “growth does not appear every-where at the same time; it appears at points or poles of growth with varying intensity;it spreads along the various channels and with di�ering overall e�ects on the wholeeconomy”. An important role is played by the frozen accidents. Myrdal [5] suggestedthat quite often the development of particular regions had their origins in the his-torical events of often accidental character, which started the process of cumulativecausation. The growth may spread to neighboring regions. The places with no suchcritical moments in their history, or with negative critical events, follow a trajectoryof cumulative underdevelopment, which in a backwash e�ect may negatively in u-ence neighboring regions. Similar processes are discussed in Friedmann’s [6] theoryof polarized development with its small “centers of change” or Pred’s [7] model ofregional growth, in which a crucial role is played by the replacement of importedgoods by local products. Clustering of economic enterprises may be attributed to theinteraction of increasing returns and transaction costs across the space [8,9]. Otherrecently considered factors promoting clustering include local embeddedness, in u-ence of local infrastructure, institutional, social and cultural practices, transfer andexchange of formal knowledge and information in clusters, to name just a few (e.g.,Ref. [10]). In most general terms Arthur [11] argued, that any factors, by which en-terprises are more likely to be created in the vicinity of other enterprises would leadto clustering.

All the theories described above are formulated on the aggregate level and con-sider economic factors as causes of clustering. In the present paper, we argue that

A. Nowak et al. / Physica A 287 (2000) 613–630 615

in addition to the economic factors an important role in clustering is played by thepsychological factors, speci�cally social in uence. In this view, economic growth sharesmany important features with processes of social change, which occur in groups andsocieties. The central notion is that the rapid social changes occur in a manner thatis remarkably similar to the phase transitions as described in physics. Metaphorically,“islands of new” form in the “sea of old” in a manner similar to the formation of gasbubbles in a liquid that is nearing the boiling point. As the transition progresses, thoseislands or clusters grow and become connected, and begin to encircle the remainingislands of old. During social transitions, then, two distinct realities co-exist-the realityof the old and the reality of the new. This type of social change thus is not only rapid,but also abrupt and nonlinear. To explore this perspective, cellular automata models ofchange are employed. These models are developed in the context of social in uence.Basic properties of the change process, and the factors that shape these processes, areinvestigated. This approach is then generalized to political and economic transitionsoccurring at the societal level. The validity of the model is tested against the empiricaldata collected during the social transitions that occurred in Poland in the late 1980sand early 1990s.

2. The cellular automata model of social in uence

Social in uence is one of the most pervasive forces that operate in groups and so-cieties. Although social in uence can take a variety of distinct forms, the key featurefor the present purposes is that the attitudes, beliefs and decisions of a single indi-vidual are to a large degree dependent on the attitudes, beliefs and decisions of otherindividuals with whom he or she interacts. Social impact theory [12], has been builtas an empirical generalization of this interdependency. According to this theory, theamount of impact other people have on an individual’s attitudes, beliefs and decisionscan be characterized in terms of three variables: the number of people in uencing orbeing in uenced, the respective strength of these people, and their immediacy to oneanother. The function linking the magnitude of impact on these three factors is quitegeneral. Whether the nature of the group in uence concerns conformity, interest incurrent events, forming impression of other individuals, stage fright, or the likelihoodof signing a petition, the in uence of a group grows as a power function of the numberof people involved, usually with an exponent of approximately 5. This means that thejoint e�ects of a group exerting in uence grows as a square root of the number of peo-ple in the group. In uence also grows in proportion to the strength of the individualsexerting the in uence. Strength represents the potential for in uence, and refers bothto relatively stable individual characteristics (e.g. social status or persuasion skills) andtopic-relevant variables (e.g. motivation to persuade others). Finally, in uence dependson proximity and appears to decrease as a square of the distance. There is evidence, forexample, that the probability that two people will discuss matters of mutual importancedecreases as a square of the distance between their physical locations [13]. According


to the theory of social impact, the joint e�ect of these three factors represents a mul-tiplicative function of strength, immediacy, and number.

In this section, we shall describe a particular model of social change, based on thetheory of social impact, in which the opposing opinions in uence the members via pairinteraction. We have used the model previously to describe the evolution of individualattitudes in a society [14]. In the original model, individuals were assumed to haveone of two opinions on an issue (e.g. either for or against a particular referendum),but in later versions this restriction was relaxed, so that many positions were possibleon an issue [21]. This model assumes that each individual can be characterized by hisor her opinion on a topic, persuasive strength, and position in a social space.

In our simulations a social group is assumed to consist of a set of individuals.Each individual is portrayed as a cell on a two-dimensional grid. Each individualis assumed to have an opinion on a particular issue. In the simplest case, it maybe one of two possible “for” or “against” opinions, or a preference for one of twoalternatives, such as choosing between two candidates in elections. In other cases,there may be more possible attitudes or opinions. It is obvious that in all real socialgroups individuals di�er in their respective strength, that is, in their abilities to changeor support each other’s opinions. Each individual is also characterized by strength,which is assigned in the beginning of the simulations and stays the same during thesimulations. Individual di�erences in strength are very important for the behavior ofthe models. People interact most often and are mostly in uenced by those who areclose to them, such as family members, friends, and co-workers. People are also muchmore likely to interact with neighbors, that is, those who live close to them in physicalspace [13,15–18]. The distance between the two individuals in social space is inverselyrelated to their immediacy vis a vis one another.

To model social interactions, we assume that individuals communicate with others toassess the popularity of each of the possible opinions. Opinions of others located closeto the subject and of those who are most in uential are the most highly weighted. Anindividual’s own opinion is also taken into account in this scenario. In the course ofthe simulation, individuals adopt the opinions that they �nd prevailing in the process ofinteracting with others. This simple model of social interactions is not only intuitive butalso agrees with a number of empirical studies, the results of which were incorporatedin the theory of social impact [12]. Our models can incorporate these features ofsocial interaction and lead to similar conclusions [14,19–21]. Closely related modelswere considered also by Kacperski and Ho lyst [22,23]. Other approach, based on theBrownian motion description was proposed in Refs. [24,25].

In the simplest case where each individual chooses one of two opposite opinions weencode the attitude of the ith person in the two-valued variable �i taking values ±1. Thetotal impact of the individual experiences from his social environment consists of twoparts – the persuasive impact of those who hold the opposite opinion and a supportiveone of those who share the same opinion. Both impacts are functions of (social) dis-tances dij between the individuals i and j and supportive or persuasive strengths of theindividuals (denoted by sj and pj, respectively and distributed statistically according


to some – in principle measurable by polls – probability distribution). Obviously,the two impacts have opposite e�ects so the total impact exerted is the di�erenceof the two:

Ii = Ip

(∑j

t(pj)g(dij)

(1 − �i�j))

− Is(∑

j

t′(sj)g(dij)

(1 + �i�j)

): (1)

Two functions Ip and Is can be speci�ed further in concrete models. The same appliesto (seemingly redundant) function g, which can be incorporated in the de�nition of thesocial distance dij. It is, however, an experimental result, that in fact, the importantcomponent of the social distance is the geometric distance in the physical space. Hence,as a simpli�cation, we shall think of dij as the (mean) geometric distance betweenthe individuals i and j and g as a model-dependent function. The function t and t′ arealso model dependent and can be incorporated into the de�nition of the persuasive andsupportive strengths, but we keep them for further convenience.

The total impact experienced by the individual i at time t in uences his attitude atthe next instant t+ 1 (the discrete-time dynamics is quite natural if we think of eventssuch as elections, etc., or if we probe the attitudes in consecutive weeks, month oryears via opinion polls, etc.). The opinion is changed if the impact of the opposingindividuals prevails:

�i(t + 1) = −sign(�iIi) : (2)

Obviously, realistic models should incorporate various non-deterministic componentsof the opinion change. The simplest way of taking into account the randomness of theprocess consists of introducing noise to the dynamics (2),

�i(t + 1) = −sign(�iIi + hi) : (3)

In the simplest case, we assume that hi are random variables that are statisticallyindependent for di�erent individuals and di�erent time instants.

In order to perform numerical simulations of the model, we have to specify thefunctions Ip, Is, g, t and t′. In many psychological experiments [12], it was estab-lished that, assuming approximately equal strengths of the individuals and approxi-mately the same distances the total impact scales as square root of the number ofpersons. On the other hand, the in uence exerted by a particular individual diminisheswith the square of his distance from the person in uenced by him. Assuming constantthat the persuasive and supportive strengths are bounded from above, we also demandthat the total impact remains �nite even if the individuals are spread with constant den-sity over the in�nite two-dimensional surface. The above restrictions leave still someroom for various forms of Ip, Is, and g. In our simulations, we chose the simplest formful�lling all mentioned demands, i.e.,

Ip =

√√√√∑j

p2j

d4ij

(1 − �i�j); Is =

√√√√∑j

s2jd4ij

(1 + �i�j) : (4)


Fig. 1. Results of computer simulations of social in uence. Distribution of attitudes in groups before (a) andafter (b) exchange of opinions. Each box corresponds to an individual. Two di�erent shades of grey representtwo attitudes, heights of boxes correspond to individual strengths. The persuasive and supportive strengths ofa particular individual were assumed equal: pi = si , with two possible values distributed randomly in space.

The singularity of the model at dij = 0 is avoided by restricting the sums to i 6= j andintroducing, instead, a term proportional to the opinion held by the ith individual, withthe proportionality coe�cient measuring “self-supportiveness”. Observe also that in thecase of a discrete two-dimensional grid, the distances dij are bounded from below bythe distance between the closest neighbors.

We started the simulations from a random distribution of opinions. This may beinterpreted as representing the situation where initially each individual comes to his orher opinion unaware of the opinions of others. This opinion may be the result of anumber of factors not accounted for in our model, such as vested interests, previousexperiences, or simple reasoning about the issue.

Each box in Fig. 1a corresponds to an individual. The color (light vs. dark grey) ofthe box denotes the individual’s opinion and the height of the box corresponds to theindividual’s strength.

In Fig. 1a the majority of the population choose the “no” option represented bythe light color. The minority of the individuals, represented by the dark color, are for


the “yes” choice. The height of the bars indicates the strengths of the individuals.The strength is also distributed randomly among the individuals and in this model itdoes not change in the course of the simulation.

As individuals interact in the course of the simulation, those who �nd the oppositeopinion prevailing, change opinions. Finally, after some simulation steps an equilibriumis reached in which no one changes opinion (see Fig. 1b). Comparison of Figs. 1aand b shows two important di�erences. First of all, opinions are no longer randomlydistributed. Note that the minority opinion survives by the formation of clusters oflike-minded people and that these clusters are usually formed around strong individuals.

Clustering is one of the most ubiquitous features of social life. It is di�cult, in fact,to �nd a social phenomenon that does not re ect some degree of clustering. Clusteringis visible in the spread of accents, fashions, beliefs, and political preferences. Thereis evidence, for example, that attitudes tend to cluster in residential neighborhoods[26]. Clustering has also been observed for farming techniques, political beliefs, socialmovements, religions, fashions, and a host of other phenomena. In the simulation model,clustering is due to the local nature of the in uence processes.

The second phenomenon visible in the comparison of Fig. 1a and b is that thenumber of people holding the minority view has declined. Such a phenomenon is oftenreferred to as polarization of opinions, and has been demonstrated in a number ofpsychological experiments [27,28].

Computer simulations [29], as well as analytical considerations [30] have identi-�ed three features that are especially important for the emergence of polarization andclustering: the existence of individual di�erences in strength, non-linearity in attitudechange, and the local nature of in uence dictated by the geometry of social space. In-dividual di�erences, �rst of all, are important because strong individuals (e.g., leaders)are necessary for the survival of minority clusters. This conclusion is consistent withvarious �ndings in sociology and anthropology that stress the crucial role of leadersin resisting in uence within the minority cultures. This is because the strength of aleader’s in uence can outweigh the in uence of outgroup (i.e., majority) individuals.

The second important feature is non-linearity in attitude change. As long as indi-vidual changes occur incrementally in proportion to the strength of social in uence,the members of a group, in the absence of external in uences, will invariably movetoward uniformity in their opinions [31]. Under these conditions, minority clusterscannot survive. The present model, however, assumes a threshold function, such thatin uences below a certain magnitude of strength have no impact and above this thresh-old changes converge on the opposing opinion. Computer simulations [20] have shownthat the non-linearity in attitude change is indeed critical to the survival of minorityclusters. Such a rule implies that attitudes are distributed in a bimodal fashion, in con-trast to the normal distribution generated by linear change rule. Latan�e and Nowak [15]have shown that for issues low in importance, a normal distribution is typically ob-served for unimportant issues, with intermediate values of the attitude emerging as themost common in the group. For important issues, however, attitudes display a bimodaldistribution, with individuals occupying extreme positions on the issue. This suggests


that minorities have a greater chance of survival if the issue in question is personallyimportant to them.

The third crucial feature of the model concerns locality of interactions [19]. Mostof the simulations portray social space as a two-dimensional matrix of n rows andn columns – a reasonable assumption in view of the role of physical proximity instructuring social interactions. Consider a group in which there are virtually no locale�ects, so that each individual interacts equally with all members of the group. In sucha group, no minority member is shielded from majority in uence, although minorityopinion may nonetheless survive for a relatively long time if high value is placed onone’s own opinion and individual di�erences are strongly pronounced [30]. Becausethe social space lacks structure, however, minority opinions cannot cluster. There is nolocality as well when the interaction patterns in a group are random. When this is thecase, minority opinion rapidly decays and the group converges on the majority position[30].

In particular, three variables play a critical role in group-level social in uence pro-cesses [30,29]. One of these, referred to as noise, represents the variety of in uencesexternal to the group that impact on group members such as personal experiences,communication from people outside the group, and selective exposure to media. Inthe simulations, the value of noise is added as a random number to the social in u-ence experienced by each person. When noise is present, attitudes do not stabilize onabsolute equilibria, because change in the direction opposite to that of social in uenceis always possible. Well-de�ned clusters may be formed, however, and these may existfor a very long time. With low values of noise, this picture does not change much,even if from time to time some of the weaker minority members change their opinion,since the stronger group members can restore their initial attitude. When random in u-ences are strong, however, social in uence within the group plays a correspondinglyweaker role, so that the clusters may lose their stability. Because the dynamics heremay re ect long periods of relative stability intermixed with rapid decay, this scenariois referred to as staircase dynamics [30].

The second factor, self-in uence, represents the weight an individual attaches to hisor her own opinion relative to the opinions of others. In everyday terms, this variablere ects such psychological states as self-con�dence, belief certainty, and strength ofconviction. Hence, the stronger the self-in uence, the greater the resistance to socialin uence. With low values of self-in uence, individuals may switch their opinionsseveral times during the course of a simulation. Decreasing values of self-in uencealso tend to destabilize clusters and can ultimately promote uni�cation based on themajority opinion. When self-in uence is high by relative to the combined in uence ofothers, however, there are no dynamics in the absence of noise.

The third factor, bias, can qualitatively change dynamics of opinions. This factorre ects unequal a priori attractiveness of the various attitude positions. As long as theattitude positions in a group do not di�er substantially in their relative desirability,the nature of the dynamics are dictated by the level of noise and the magnitude ofself-in uence. If a minority opinion is more desirable than the majority opinion, there


is strong potential for social change. During a societal transition, there is typically amarked shift in public opinion. An opinion that had been held by a minority of citizenssuddenly becomes prevalent in the society. As demonstrated in the simulation model,minority opinion can survive if it can create coherent clusters. However, for a societyto undergo transition, the minority opinion has to do more than surviving-it in orderto supplant the majority opinion. In the real world, usually some opinions are moreattractive than some others. Some opinions are more compatible with the society’svalue system, more advantageous in some way, or simply more prominent because ofmass media in uence or other external factors. We can represent the joint e�ect ofall such factors in the simulation model by introducing “bias” into the rule describingchanges in opinions. This is done by adding a constant to the impact to favor oneof the positions, which acts in addition to the e�ects of social interaction. If externalsources (i.e., bias) assume a very high value, they can even overwhelm the e�ects ofsocial interactions. In practice, however, the e�ects of social interaction and bias areboth likely to be observed.

Nowak et al. [21] tested these ideas in simulations based on cellular automata modelsof social in uence, similar to those described above. The simulations started from avery low proportion of minority position (e.g. 10%). The minority opinion would notbe able to survive in this con�guration without the presence of bias, because its lowfrequency in the population makes it hard for its advocates to �nd like-minded peoplewith whom they can cluster. Because of bias, however, the minority opinion is ableto grow. It grows by forming clusters around the initial seeds of the new opinion.When the clusters of the initial minority become fully connected, the initial majority isreduced to islands. Finally, a new equilibrium is reached, although clusters of the oldopinion still exist, well entrenched in the sea of new. Those clusters are composed ofindividuals, who can best withstand the pressure of the new majority, i.e., the strongest,and are connected to other strong members of own group. It is therefore very easy,for the old to regain its popularity once the bias is withdrawn.

The simulations and analytical considerations, based on the theory of social impact,conducted on the individual level in general show that there are two di�erent waysin which changes occur in any system. They may occur gradually, when each of theagents undergoes a gradual change, when the change rules are linear. As discussedabove, such scenario is likely to be observed when the issues are perceived to be notvery important. Or rapid, abrupt and dramatic changes may occur often in a mannerlike so-called phase transitions in physics. The latter changes occur usually in a highlynon-uniform manner; “bubbles of new” appear in the sea of “old”; they grow andconnect together, but under some circumstances they decay. Clustering is the necessary�rst step towards the “new”. The success of the transition depends on how e�ectivelyclusters of minority can grow and connect to each other. This scenario is likely tohappen, when the change rules are highly non-linear (e.g. a threshold function) whichare likely to describe attitude changes in important issues.

So far, we have concentrated on the models that describe attitude change. It is naturalto ask whether these models can be related to actual social transitions in a more general


sense. First of all, it is important to stress that change of attitudes itself is a necessarycondition for social transition to occur. People in an open society should be willingto change. A market economy cannot develop without positive attitudes toward privateenterprise, privatization, investments in the stock market, and so forth. Second, thetransition requires the spread of new ideas, innovations, and knowledge. One simplyshould know what to do in order to join the “new”; that is, how to start a business, howand where to invest, etc. Interviews with businessman in Poland [32], have shown thatthey have relied mostly on the personal knowledge from their friends and acquaintanceswhen opening their �rst �rm, in a period of transition form centrally governed to marketeconomy. There is a considerable evidence suggesting that the spread of innovationsoccurs by growing clusters and “bubbles”. We think that our models describe very wellsuch processes and enable an understanding of their dynamics. In general, our modelsdescribe the transition process in which the probability that one of the agents adopts anew position is increased if this agent is surrounded by those who already have doneit. Note that in this general sense our models describe various economic processes.

3. Aggregate model

Almost all social and economic data allow for characterization of aggregates, ratherthan individuals. Available data concern administrative units, territorial units, organi-zations, etc. The models can therefore be tested against the available empirical dataon the aggregate level. Below we discuss, how the model developed on the basis ofknowledge of rules of the individual change (1)–(3) may be adapted to the aggre-gate level. This model instead of individual attitudes, opinions and decisions describesdynamics of aggregate level entities such as changes in economic activity in variousgeographical or administrative regions.

Almost a de�nitional measure of transition from state governed, to market economyis the number of privately owned enterprises. For any spatial consideration, however,a much better measure is provided by the number of privately owned enterprises percapita, what corresponds to the probability, that a given individual does own a privateenterprise. The second measure avoids the problem inherent in the �rst measure, i.e.,higher concentration of enterprises in population centers, even if the likelihood that anindividual owns a �rm does not depend on spatial location.

It is obvious also that the values of global indices (e.g. the number of enterprises percapita in a given region), characterizing the dynamics on the chosen level, result fromthe summation of “microscopic” variables, like particular decisions of the individualsin the region. It is, thus, reasonable to consider the situation in which the individualsare grouped into natural units (administrative or geographic regions).

It is tempting thus to build a model of the kind considered above for the case ofindividual opinion changes. The role played before by the individuals is now attributedto the aggregates (administrative or geographic regions), the individual opinions aresubstituted by some quantitative characteristics of the aggregates (e.g. the number of


enterprises in the considered region) and, obviously, contrary to the situations consid-ered previously, they can take more than two values. The in uence of the neighborhoodis taken into account by the distance-dependent impact of the neighbouring units (wherethe distance is measured, e.g. between the centers of the regions). Such a model indeeddescribes qualitatively the phenomena observed in the development of privately ownedenterprises in Poland [33]. Nevertheless, we �nd the proposed approach not quite sat-isfactory, since, as indicated above, the aggregate indices result from the summationof the individual (“microscopic”) variables, which, notwithstanding the fact that theyare usually not known in the investigations concentrated on the aggregate level are thereal causes of the globally observed changes. The model presented below starts fromthis lowest level.

Let us assume, that for all individuals in a given region the mutual distance betweenany pair of them is the same (which should be a fair approximation in the case of, e.g.small administrative regions). On the other hand, the distance between two individualsfrom di�erent regions depends on the physical distance between the regions (theircenters). It is to expect that in such a hierarchical model we can translate the dynamicsof the individual decisions (2) into dynamics of global variables characterizing theunits. If, in the �rst approximation, we neglect the interaction between the individualsfrom di�erent regions (which, due to the distance dependence, are weaker than theinteractions within the given region), the dynamics in one region, when the number ofinhabitants is large, is well approximated in terms of the mean �eld theory [30]. Thewhole region can be characterized by a single variable, the weighted majority–minoritydi�erence m(t) de�ned as

m=1

N (s+ p)

∑j

(sj + pj)�j ; (5)

where s and p are the mean values of the supportive and persuasive strengths, and Nis the number of individuals in the region. When averaged over the distributions of piand si and the initial distribution of attitudes, 〈m(t)〉 undergoes time evolution in termsof a map

〈m(t + 1)〉 = f(〈m(t)〉) ; (6)

where the function f depends on the distributions of pi and si and the initial distri-bution of opinions.

In order to specialize the abstract model described above to the speci�c situationwe want to describe, let us make some simplifying assumptions partly supported bythe analysis of the available data. First, let us assume that, in fact, the both (supportiveand persuasive) strengths do not depend on the individual (si = s, pi = p). Moreover,people are in uenced only by positive examples: by the number of others in their vicin-ity, who own private enterprises. Individuals are not signi�cantly in uenced by otherswho do not own enterprises, i.e., p = 0. There are two possible explanations of thisasymmetry. First of all, it is possible, that in uence processes become asymmetricalwhen strong personal interests are engaged. It is clearly more advantageous to own


an enterprise, than not to own it. In such situations individuals may pay dispropor-tional attention to positive examples, while neglecting negative examples. The secondexplanation deals directly with the nature of the social impact function. As discussedabove, social in uence scales as a square root of the number of individuals exertingin uence. Since a small proportion of individuals own enterprises, a small di�erence intheir number changes signi�cantly the magnitude of the in uence. The same absolutenumber, however, may change to a negligibly small degree the in uence of the major-ity. If the second explanation is correct, than not only economic behaviors, and otherwhere there are strong vested interests would behave asymmetrically, but one couldexpect asymmetry in all the cases when minority is very small.

Let us now assume that the variable �j takes the values of 1 and 0 (which di�ersfrom the situations considered above by a linear rescaling) and interpret the value 1as the attitude toward establishing (or continuing to run) an enterprise and 0 as theopposite decision of the jth individual. It is now clear that the quantity m given byEq. (5) measures the total number of enterprises in the region (and can be transformedto the number of enterprises per capita, or the density by scaling by an appropriateextensive quantity – the total number of inhabitants or the area).

When the interactions between the individuals from di�erent regions are included, itis to be expected that the evolution of variables characterizing the regions as a wholelike 〈m〉 above, is caused by the “intrinsic” dynamics (6) and (distance dependent)interactions between the regions stemming from the averaged interactions between in-dividuals from di�erent regions, thus the change of the total number of enterprises inthe region � is given as

m�(t + 1) − m�(t) = A0� + A1�m�(t) +∑�6=�

B��m�g(d��)

; (7)

where the �rst two terms on the right-hand side correspond to the linearized versionof Eq. (6), and A0�, A1�, and B�� are coe�cients to be determined empirically.

Several numerical experiments with the available data established that we can dis-pense with a lot of �-dependence on the right-hand side of Eq. (7) if we chooseappropriately scaled variables to work with. In the following we report the results ofnumerical simulations of changes of number of privately owned enterprises per capitain Poland. The dynamics is governed by the following discrete-time map:

li(t + 1) = li(t) + B0 + B1ki(t) +∑p

bpNp

∑rp6dij6rp+1

kj(t) ; (8)

where li is the number of privately owned enterprises in the ith territorial unit, ki is thedensity of privately owned enterprises (the number of enterprises per unit area), Np –the number of neighboring territorial units located at distance larger than rp and smallerthan rp+1 from the given one. In our simulation the territorial units were counties (thesmallest administrative units in Poland of diameter of the order of 10 km), and theconsecutive radii rp=p×10 km. The coe�cients B0, B1, and bp are determined empir-ically, as unstandardised regression coe�cients by the analysis of multiple regression,


Table 1Values of the standardised coe�cients in multiple regression for number of enterprises per capita and numberof enterprises per area as predictors and di�erence in the number of enterprises per capita in 91–90 as thedependent variable

Distance Ownin uence 0–10 11–20 21–30 31–40

Number of enterprises per capita 0.02 0.01 0.03 0.04 0.0Number of enterprises per area 0.55 0.13 0.18 0.12 0.02

where the di�erence in the number of enterprises in speci�c territorial unit, in twoconsecutive years is the dependent variable, and statistics (e.g. number of enterprisesper area) in all territorial units located at a given distance (e.g. between 10 and 20km) constitute independent variables.

Analysis and simulation described further were performed by Urbaniak [33]. We usedin our analysis and simulations the number of privately owned enterprises registeredin REGON database, for years 1989–1992, i.e., for years of transition from centrallygoverned to the market economy. Regression analysis revealed, that the changes inthe number of enterprises per capita can be predicted much better from the numberof enterprises per area in the previous year, then from the number of enterprises percapita. The relevant standardized coe�cients for the di�erence between 90 and 91, i.e.,the year when the Polish economy was restructured are given in Table 1.

These results indicate that the in uence in economic matters, i.e., whether to open aprivate �rm is not symmetrical. The decision to open a new enterprise is in uenced bythe total, weighted by distance, number of enterprises located within a certain radius,which in the case of Polish transition seems to be in the order of 40 km.

Before the economic transition 90–98, the in uence of number of enterprises percapita was not signi�cant, and the only signi�cant predictor was own in uence (au-toregression term) which had a negative value of −0:48, indicating that the moreenterprises were in the given county, the more would cease to exist. For the di�erence92–91 both there was a strong e�ect of the own in uence of measures both per capita(0.49) and per area (0.55), and signi�cant in uence of the number per area of closelylocated counties (up to 30 km).

Computer simulations were used to compare the model predictions with empiricaldata, collected during the Polish transition. The data from the previous year were usedas an input to the simulation program. Empirically established coe�cients for the 91–90di�erence were used as model parameters.

Figs. 2–8 show the comparison, between the empirical data and model predictions.In 1989, there is no visible clustering of privately owned enterprises. As we can see,there are marked di�erences between the data and the model predictions for 90. Themodel predicts clustering while the only local e�ects visible in the data are the back-wash e�ects in regions with relatively small number of enterprises.This is the resultone could expect, since the model was built to predict the changes in the number ofenterprises in market economy and in 90 the economic reform was only introduced by


Fig. 2. Number of privately owned enterprises per capita in 1989; empirical data.


Fig. 4. Number of privately owned enterprises per capita in 1990; results of computer simulations.



Balcerowicz, while the economy was still mainly centrally governed. In 91–90 thereis a remarkable similarity between the reality and the model prediction. Model predic-tions match economic reality also in 92–91. The respective correlations between thepredicted and observed values for all counties are: for 90–89 correlation r = 0:34, for91–90 r = 0:71, and for 92–91 r = 0:80. As discussed above, model predictions arequite accurate, with the exception of the �rst year, the year before economic reformwas introduced.

4. Conclusions

Individual decisions underlie many economic processes. The laws of social in uencequite precisely describe, how other people in uence individual opinions, attitudes, anddecisions, and may be used to construct computer model of social change on the indi-vidual level. Robust features of this model’s dynamics include polarization (reductionof minority) and clustering. In the presence of bias, which breaks the symmetry be-tween di�erent attitudes, opinions and decisions, allows the minority to grow, pavingthe way for social change. Social change occurs as growing clusters, or bubbles of“new” in the sea of “old”. This scenario gives practical suggestions as how to facil-itate social changes. First, one should establish clusters of the new. In order to dothis, one should identify the social groups that are best prepared to help the transitionoccur. It is essential to assist those groups in creating a cluster, by changing their localenvironment and providing some aid. This aid might be discontinued as soon as thecluster reaches a con�guration in which it can survive as a minority.

The signi�cance of even seemingly small events, such as opening an e�cient foreignfactory, may go well beyond its economic role. It may serve as a seed of the “new”,and provide and anchor for the growth of a whole cluster, spreading new attitudes to-wards work, new organizational ideas, or new standards of quality. Second expansionof existing clusters is a high priority. This is done most e�ciently by a�ecting the


immediate neighborhood of the clusters, and may take a form of aid, or even an infor-mation campaign. The goal is to expand the borders of the clusters of “new”. Finally,development of connections between the di�erent clusters may stabilize clusters, andgive them the advantage over initially more numerous “old”. Locally such connectionsmay take a form of common activities, formation of coalitions, etc. On a larger scale,development of communication, makes it easier for isolated clusters to �nd others likethem.

It is possible to formulate on the aggregate level a theory that would parallel thedynamic theory of social impact formed on the individual level. This theory describesthe dynamics of the proportion of individuals who have a speci�c property (e.g. aspeci�c attitude) as the dynamic variable in a dynamical system model in discrete time.Let us consider changes in a single unit (composed of a number of individuals locatedclose together). We can assume that the distances within the unit are too small to besigni�cant, and all the individuals within a unit may be treated as located approximatelyat the same distance. Mean �eld theory may be used to derive aggregate level propertiesof an ensemble of individuals. The dynamics of each unit depend additionally on thein uence from other units located in their vicinity. The strength of couplings decreaseswith the physical distance, and may be determined empirically for each particular case.Computer simulations produce results similar to the model. Combination of individualand aggregate model could in the future allow one to test individual models on theaggregate level and to aggregate level theories on the individual level. Hopefully, thisapproach may pave the way, for future integration, in coherent models, of factorscoming from both individual and group level. On the theoretical level this approach mayfacilitate introduction of known psychological factors and mechanisms into economictheories. After all, all decisions are made by humans, even if they use rational modelsas a guide.

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