Physica A 287 (2000) 613–630
www.elsevier.com/locate/physa
Simulating the coordination of individual
economic decisions
Andrzej Nowaka; b; ∗ , Marek Kusa; c , Jakub Urbaniaka , Tomasz Zaryckia
a Institute
for Social Studies, Warsaw University, ul. Stawki 5=7, 00-183 Warsaw, Poland
b Advanced School of Social Psychology, Warsaw, Poland
c Center 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 inuence is used to describe the coordination of individual
economic decisions. Computer simulations of the model show that the social and economic
transitions occur as growing clusters of “new” in the sea of old. The model formulated at the
individual level may be used to derive another one concerning the aggregate level. The aggregate
level model was used to simulate spatio-temporal dynamics of the number of privately owned
enterprises in Poland during the transition from centrally governed to the market economy.
c 2000
Analysis revealed the similarity between the model predictions and economic data. Elsevier Science B.V. All rights reserved.
1. Introduction
The pervasive feature of dierent aspects of economic activity is clustering in space.
Innovations spread as expanding clusters, farming practices tend to be similar in neighboring 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 few
examples. The theories constructed to account for clustering in space have been formulated on the aggregate level, and considered economic factors. Many decisions which
aect the economic processes, however, are made by individuals. Well tested and well
understood, psychological laws describe the eect of social context on the individual
∗
Correspondence address: Institute for Social Studies, Warsaw University, ul. Stawki 5=7, 00-183 Warsaw,
Poland.
c 2000 Elsevier Science B.V. All rights reserved.
0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 3 9 7 - 6
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A. Nowak et al. / Physica A 287 (2000) 613–630
decision making. Computer simulations have shown that the clustering of opinions is
a pervasive feature of a wide variety of models, where individual opinions are inuenced by other individuals. In the present paper, we investigate the relation between
the individual and aggregate level models of economic development. We argue that the
inclusion of the individual level theory allows one to predict, new properties of economic growth. We compare the outcomes of computer simulations to the economic data
concerning Poland in the period of transition between the centrally governed economy
and the market economy.
Many economic factors were indicated by dierent location theories as factors
responsible for spatial clustering. Since the cost of transportation [1] adds to the cost
of production, it pays to produce close to the market and in vicinity of other rms
[2]. The cost of transportation also aects consumers, who tend to buy at the closest location. If the local demand exceeds a threshold, an entrepreneur will oer the
good. 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 hierarchy of urban centers, some more central than others [3]. Other economic factors
considered by modern location theories include: labor costs and labor quality, scal
regulations, institutional infrastructure, quality of life in the region, etc.
Dynamic interaction of innovative enterprises concentrated in a particular place, usually an urban center are responsible for locally concentrated upstream (positive), and
downstream (negative) “propulsive eects” aecting their environments. As Perroux
[4] observed, describing a theory of growth poles, “growth does not appear everywhere at the same time; it appears at points or poles of growth with varying intensity;
it spreads along the various channels and with diering overall eects on the whole
economy”. An important role is played by the frozen accidents. Myrdal [5] suggested
that quite often the development of particular regions had their origins in the historical events of often accidental character, which started the process of cumulative
causation. The growth may spread to neighboring regions. The places with no such
critical moments in their history, or with negative critical events, follow a trajectory
of cumulative underdevelopment, which in a backwash eect may negatively inuence neighboring regions. Similar processes are discussed in Friedmann’s [6] theory
of polarized development with its small “centers of change” or Pred’s [7] model of
regional growth, in which a crucial role is played by the replacement of imported
goods by local products. Clustering of economic enterprises may be attributed to the
interaction of increasing returns and transaction costs across the space [8,9]. Other
recently considered factors promoting clustering include local embeddedness, inuence of local infrastructure, institutional, social and cultural practices, transfer and
exchange 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 enterprises are more likely to be created in the vicinity of other enterprises would lead
to clustering.
All the theories described above are formulated on the aggregate level and consider 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 the
psychological factors, specically social inuence. In this view, economic growth shares
many important features with processes of social change, which occur in groups and
societies. The central notion is that the rapid social changes occur in a manner that
is 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 gas
bubbles in a liquid that is nearing the boiling point. As the transition progresses, those
islands or clusters grow and become connected, and begin to encircle the remaining
islands of old. During social transitions, then, two distinct realities co-exist-the reality
of 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 of
change are employed. These models are developed in the context of social inuence.
Basic properties of the change process, and the factors that shape these processes, are
investigated. This approach is then generalized to political and economic transitions
occurring at the societal level. The validity of the model is tested against the empirical
data collected during the social transitions that occurred in Poland in the late 1980s
and early 1990s.
2. The cellular automata model of social inuence
Social inuence is one of the most pervasive forces that operate in groups and societies. Although social inuence can take a variety of distinct forms, the key feature
for the present purposes is that the attitudes, beliefs and decisions of a single individual are to a large degree dependent on the attitudes, beliefs and decisions of other
individuals with whom he or she interacts. Social impact theory [12], has been built
as an empirical generalization of this interdependency. According to this theory, the
amount of impact other people have on an individual’s attitudes, beliefs and decisions
can be characterized in terms of three variables: the number of people inuencing or
being inuenced, the respective strength of these people, and their immediacy to one
another. The function linking the magnitude of impact on these three factors is quite
general. Whether the nature of the group inuence concerns conformity, interest in
current events, forming impression of other individuals, stage fright, or the likelihood
of signing a petition, the inuence of a group grows as a power function of the number
of people involved, usually with an exponent of approximately 5. This means that the
joint eects of a group exerting inuence grows as a square root of the number of people in the group. Inuence also grows in proportion to the strength of the individuals
exerting the inuence. Strength represents the potential for inuence, and refers both
to relatively stable individual characteristics (e.g. social status or persuasion skills) and
topic-relevant variables (e.g. motivation to persuade others). Finally, inuence depends
on proximity and appears to decrease as a square of the distance. There is evidence, for
example, that the probability that two people will discuss matters of mutual importance
decreases as a square of the distance between their physical locations [13]. According
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A. Nowak et al. / Physica A 287 (2000) 613–630
to the theory of social impact, the joint eect of these three factors represents a multiplicative function of strength, immediacy, and number.
In this section, we shall describe a particular model of social change, based on the
theory of social impact, in which the opposing opinions inuence the members via pair
interaction. We have used the model previously to describe the evolution of individual
attitudes in a society [14]. In the original model, individuals were assumed to have
one 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 possible
on an issue [21]. This model assumes that each individual can be characterized by his
or 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 individual
is assumed to have an opinion on a particular issue. In the simplest case, it may
be one of two possible “for” or “against” opinions, or a preference for one of two
alternatives, 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 social
groups individuals dier in their respective strength, that is, in their abilities to change
or 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 the
simulations. Individual dierences in strength are very important for the behavior of
the models. People interact most often and are mostly inuenced by those who are
close to them, such as family members, friends, and co-workers. People are also much
more likely to interact with neighbors, that is, those who live close to them in physical
space [13,15–18]. The distance between the two individuals in social space is inversely
related to their immediacy vis a vis one another.
To model social interactions, we assume that individuals communicate with others to
assess the popularity of each of the possible opinions. Opinions of others located close
to the subject and of those who are most inuential are the most highly weighted. An
individual’s own opinion is also taken into account in this scenario. In the course of
the simulation, individuals adopt the opinions that they nd prevailing in the process of
interacting with others. This simple model of social interactions is not only intuitive but
also agrees with a number of empirical studies, the results of which were incorporated
in the theory of social impact [12]. Our models can incorporate these features of
social interaction and lead to similar conclusions [14,19–21]. Closely related models
were considered also by Kacperski and Ho lyst [22,23]. Other approach, based on the
Brownian motion description was proposed in Refs. [24,25].
In the simplest case where each individual chooses one of two opposite opinions we
encode the attitude of the ith person in the two-valued variable i taking values ±1. The
total impact of the individual experiences from his social environment consists of two
parts – the persuasive impact of those who hold the opposite opinion and a supportive
one of those who share the same opinion. Both impacts are functions of (social) distances dij between the individuals i and j and supportive or persuasive strengths of the
individuals (denoted by sj and pj , respectively and distributed statistically according
A. Nowak et al. / Physica A 287 (2000) 613–630
617
to some – in principle measurable by polls – probability distribution). Obviously,
the two impacts have opposite eects so the total impact exerted is the dierence
of the two:
!
!
X t(pj )
X t 0 (sj )
(1 − i j ) − Is
(1 + i j ) :
(1)
Ii = Ip
g(dij )
g(dij )
j
j
Two functions Ip and Is can be specied further in concrete models. The same applies
to (seemingly redundant) function g, which can be incorporated in the denition of the
social distance dij . It is, however, an experimental result, that in fact, the important
component of the social distance is the geometric distance in the physical space. Hence,
as a simplication, we shall think of dij as the (mean) geometric distance between
the individuals i and j and g as a model-dependent function. The function t and t 0 are
also model dependent and can be incorporated into the denition of the persuasive and
supportive strengths, but we keep them for further convenience.
The total impact experienced by the individual i at time t inuences his attitude at
the next instant t + 1 (the discrete-time dynamics is quite natural if we think of events
such as elections, etc., or if we probe the attitudes in consecutive weeks, month or
years via opinion polls, etc.). The opinion is changed if the impact of the opposing
individuals prevails:
i (t + 1) = −sign(i Ii ) :
(2)
Obviously, realistic models should incorporate various non-deterministic components
of the opinion change. The simplest way of taking into account the randomness of the
process consists of introducing noise to the dynamics (2),
i (t + 1) = −sign(i Ii + hi ) :
(3)
In the simplest case, we assume that hi are random variables that are statistically
independent for dierent individuals and dierent time instants.
In order to perform numerical simulations of the model, we have to specify the
functions Ip , Is , g, t and t 0 . In many psychological experiments [12], it was established that, assuming approximately equal strengths of the individuals and approximately the same distances the total impact scales as square root of the number of
persons. On the other hand, the inuence exerted by a particular individual diminishes
with the square of his distance from the person inuenced by him. Assuming constant
that the persuasive and supportive strengths are bounded from above, we also demand
that the total impact remains nite even if the individuals are spread with constant density over the innite two-dimensional surface. The above restrictions leave still some
room for various forms of Ip , Is , and g. In our simulations, we chose the simplest form
fullling all mentioned demands, i.e.,
v
v
uX 2
uX 2
pj
sj
u
u
t
(1 − i j ); Is = t
(1 + i j ) :
(4)
Ip =
4
4
d
d
ij
ij
j
j
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A. Nowak et al. / Physica A 287 (2000) 613–630
Fig. 1. Results of computer simulations of social inuence. Distribution of attitudes in groups before (a) and
after (b) exchange of opinions. Each box corresponds to an individual. Two dierent shades of grey represent
two attitudes, heights of boxes correspond to individual strengths. The persuasive and supportive strengths of
a 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 and
introducing, instead, a term proportional to the opinion held by the ith individual, with
the proportionality coecient measuring “self-supportiveness”. Observe also that in the
case of a discrete two-dimensional grid, the distances dij are bounded from below by
the distance between the closest neighbors.
We started the simulations from a random distribution of opinions. This may be
interpreted as representing the situation where initially each individual comes to his or
her opinion unaware of the opinions of others. This opinion may be the result of a
number of factors not accounted for in our model, such as vested interests, previous
experiences, or simple reasoning about the issue.
Each box in Fig. 1a corresponds to an individual. The color (light vs. dark grey) of
the box denotes the individual’s opinion and the height of the box corresponds to the
individual’s strength.
In Fig. 1a the majority of the population choose the “no” option represented by
the light color. The minority of the individuals, represented by the dark color, are for
A. Nowak et al. / Physica A 287 (2000) 613–630
619
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 it
does not change in the course of the simulation.
As individuals interact in the course of the simulation, those who nd the opposite
opinion prevailing, change opinions. Finally, after some simulation steps an equilibrium
is reached in which no one changes opinion (see Fig. 1b). Comparison of Figs. 1a
and b shows two important dierences. First of all, opinions are no longer randomly
distributed. Note that the minority opinion survives by the formation of clusters of
like-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 dicult, in fact,
to nd a social phenomenon that does not reect some degree of clustering. Clustering
is visible in the spread of accents, fashions, beliefs, and political preferences. There
is evidence, for example, that attitudes tend to cluster in residential neighborhoods
[26]. Clustering has also been observed for farming techniques, political beliefs, social
movements, religions, fashions, and a host of other phenomena. In the simulation model,
clustering is due to the local nature of the inuence processes.
The second phenomenon visible in the comparison of Fig. 1a and b is that the
number of people holding the minority view has declined. Such a phenomenon is often
referred to as polarization of opinions, and has been demonstrated in a number of
psychological experiments [27,28].
Computer simulations [29], as well as analytical considerations [30] have identied three features that are especially important for the emergence of polarization and
clustering: the existence of individual dierences in strength, non-linearity in attitude
change, and the local nature of inuence dictated by the geometry of social space. Individual dierences, rst of all, are important because strong individuals (e.g., leaders)
are necessary for the survival of minority clusters. This conclusion is consistent with
various ndings in sociology and anthropology that stress the crucial role of leaders
in resisting inuence within the minority cultures. This is because the strength of a
leader’s inuence can outweigh the inuence of outgroup (i.e., majority) individuals.
The second important feature is non-linearity in attitude change. As long as individual changes occur incrementally in proportion to the strength of social inuence,
the members of a group, in the absence of external inuences, will invariably move
toward uniformity in their opinions [31]. Under these conditions, minority clusters
cannot survive. The present model, however, assumes a threshold function, such that
inuences below a certain magnitude of strength have no impact and above this threshold changes converge on the opposing opinion. Computer simulations [20] have shown
that the non-linearity in attitude change is indeed critical to the survival of minority
clusters. Such a rule implies that attitudes are distributed in a bimodal fashion, in contrast to the normal distribution generated by linear change rule. Latane and Nowak [15]
have shown that for issues low in importance, a normal distribution is typically observed for unimportant issues, with intermediate values of the attitude emerging as the
most common in the group. For important issues, however, attitudes display a bimodal
distribution, with individuals occupying extreme positions on the issue. This suggests
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A. Nowak et al. / Physica A 287 (2000) 613–630
that minorities have a greater chance of survival if the issue in question is personally
important to them.
The third crucial feature of the model concerns locality of interactions [19]. Most
of the simulations portray social space as a two-dimensional matrix of n rows and
n columns – a reasonable assumption in view of the role of physical proximity in
structuring social interactions. Consider a group in which there are virtually no local
eects, so that each individual interacts equally with all members of the group. In such
a group, no minority member is shielded from majority inuence, although minority
opinion may nonetheless survive for a relatively long time if high value is placed on
one’s own opinion and individual dierences are strongly pronounced [30]. Because
the social space lacks structure, however, minority opinions cannot cluster. There is no
locality as well when the interaction patterns in a group are random. When this is the
case, 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 inuence processes [30,29]. One of these, referred to as noise, represents the variety of inuences
external to the group that impact on group members such as personal experiences,
communication from people outside the group, and selective exposure to media. In
the simulations, the value of noise is added as a random number to the social inuence experienced by each person. When noise is present, attitudes do not stabilize on
absolute equilibria, because change in the direction opposite to that of social inuence
is always possible. Well-dened clusters may be formed, however, and these may exist
for 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 inuences are strong, however, social inuence within the group plays a correspondingly
weaker role, so that the clusters may lose their stability. Because the dynamics here
may reect long periods of relative stability intermixed with rapid decay, this scenario
is referred to as staircase dynamics [30].
The second factor, self-inuence, represents the weight an individual attaches to his
or her own opinion relative to the opinions of others. In everyday terms, this variable
reects such psychological states as self-condence, belief certainty, and strength of
conviction. Hence, the stronger the self-inuence, the greater the resistance to social
inuence. With low values of self-inuence, individuals may switch their opinions
several times during the course of a simulation. Decreasing values of self-inuence
also tend to destabilize clusters and can ultimately promote unication based on the
majority opinion. When self-inuence is high by relative to the combined inuence of
others, however, there are no dynamics in the absence of noise.
The third factor, bias, can qualitatively change dynamics of opinions. This factor
reects unequal a priori attractiveness of the various attitude positions. As long as the
attitude positions in a group do not dier substantially in their relative desirability,
the nature of the dynamics are dictated by the level of noise and the magnitude of
self-inuence. If a minority opinion is more desirable than the majority opinion, there
A. Nowak et al. / Physica A 287 (2000) 613–630
621
is strong potential for social change. During a societal transition, there is typically a
marked shift in public opinion. An opinion that had been held by a minority of citizens
suddenly becomes prevalent in the society. As demonstrated in the simulation model,
minority opinion can survive if it can create coherent clusters. However, for a society
to undergo transition, the minority opinion has to do more than surviving-it in order
to supplant the majority opinion. In the real world, usually some opinions are more
attractive than some others. Some opinions are more compatible with the society’s
value system, more advantageous in some way, or simply more prominent because of
mass media inuence or other external factors. We can represent the joint eect of
all such factors in the simulation model by introducing “bias” into the rule describing
changes in opinions. This is done by adding a constant to the impact to favor one
of the positions, which acts in addition to the eects of social interaction. If external
sources (i.e., bias) assume a very high value, they can even overwhelm the eects of
social interactions. In practice, however, the eects of social interaction and bias are
both likely to be observed.
Nowak et al. [21] tested these ideas in simulations based on cellular automata models
of social inuence, similar to those described above. The simulations started from a
very low proportion of minority position (e.g. 10%). The minority opinion would not
be able to survive in this conguration without the presence of bias, because its low
frequency in the population makes it hard for its advocates to nd like-minded people
with whom they can cluster. Because of bias, however, the minority opinion is able
to 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 is
reduced to islands. Finally, a new equilibrium is reached, although clusters of the old
opinion still exist, well entrenched in the sea of new. Those clusters are composed of
individuals, 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 dierent ways
in which changes occur in any system. They may occur gradually, when each of the
agents undergoes a gradual change, when the change rules are linear. As discussed
above, such scenario is likely to be observed when the issues are perceived to be not
very important. Or rapid, abrupt and dramatic changes may occur often in a manner
like so-called phase transitions in physics. The latter changes occur usually in a highly
non-uniform manner; “bubbles of new” appear in the sea of “old”; they grow and
connect together, but under some circumstances they decay. Clustering is the necessary
rst step towards the “new”. The success of the transition depends on how eectively
clusters of minority can grow and connect to each other. This scenario is likely to
happen, when the change rules are highly non-linear (e.g. a threshold function) which
are likely to describe attitude changes in important issues.
So far, we have concentrated on the models that describe attitude change. It is natural
to ask whether these models can be related to actual social transitions in a more general
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A. Nowak et al. / Physica A 287 (2000) 613–630
sense. First of all, it is important to stress that change of attitudes itself is a necessary
condition for social transition to occur. People in an open society should be willing
to change. A market economy cannot develop without positive attitudes toward private
enterprise, privatization, investments in the stock market, and so forth. Second, the
transition requires the spread of new ideas, innovations, and knowledge. One simply
should know what to do in order to join the “new”; that is, how to start a business, how
and where to invest, etc. Interviews with businessman in Poland [32], have shown that
they have relied mostly on the personal knowledge from their friends and acquaintances
when opening their rst rm, in a period of transition form centrally governed to market
economy. There is a considerable evidence suggesting that the spread of innovations
occurs by growing clusters and “bubbles”. We think that our models describe very well
such processes and enable an understanding of their dynamics. In general, our models
describe the transition process in which the probability that one of the agents adopts a
new position is increased if this agent is surrounded by those who already have done
it. 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, rather
than individuals. Available data concern administrative units, territorial units, organizations, etc. The models can therefore be tested against the available empirical data
on the aggregate level. Below we discuss, how the model developed on the basis of
knowledge of rules of the individual change (1) – (3) may be adapted to the aggregate level. This model instead of individual attitudes, opinions and decisions describes
dynamics of aggregate level entities such as changes in economic activity in various
geographical or administrative regions.
Almost a denitional measure of transition from state governed, to market economy
is the number of privately owned enterprises. For any spatial consideration, however,
a much better measure is provided by the number of privately owned enterprises per
capita, what corresponds to the probability, that a given individual does own a private
enterprise. 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 an
individual 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 per
capita in a given region), characterizing the dynamics on the chosen level, result from
the summation of “microscopic” variables, like particular decisions of the individuals
in the region. It is, thus, reasonable to consider the situation in which the individuals
are grouped into natural units (administrative or geographic regions).
It is tempting thus to build a model of the kind considered above for the case of
individual opinion changes. The role played before by the individuals is now attributed
to the aggregates (administrative or geographic regions), the individual opinions are
substituted by some quantitative characteristics of the aggregates (e.g. the number of
A. Nowak et al. / Physica A 287 (2000) 613–630
623
enterprises in the considered region) and, obviously, contrary to the situations considered previously, they can take more than two values. The inuence of the neighborhood
is taken into account by the distance-dependent impact of the neighbouring units (where
the distance is measured, e.g. between the centers of the regions). Such a model indeed
describes qualitatively the phenomena observed in the development of privately owned
enterprises in Poland [33]. Nevertheless, we nd the proposed approach not quite satisfactory, since, as indicated above, the aggregate indices result from the summation
of the individual (“microscopic”) variables, which, notwithstanding the fact that they
are usually not known in the investigations concentrated on the aggregate level are the
real causes of the globally observed changes. The model presented below starts from
this lowest level.
Let us assume, that for all individuals in a given region the mutual distance between
any 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 individuals
from dierent regions depends on the physical distance between the regions (their
centers). It is to expect that in such a hierarchical model we can translate the dynamics
of the individual decisions (2) into dynamics of global variables characterizing the
units. If, in the rst approximation, we neglect the interaction between the individuals
from dierent regions (which, due to the distance dependence, are weaker than the
interactions within the given region), the dynamics in one region, when the number of
inhabitants is large, is well approximated in terms of the mean eld theory [30]. The
whole region can be characterized by a single variable, the weighted majority–minority
dierence m(t) dened as
X
1
(sj + pj )j ;
(5)
m=
N (s + p) j
where s and p are the mean values of the supportive and persuasive strengths, and N
is the number of individuals in the region. When averaged over the distributions of pi
and si and the initial distribution of attitudes, hm(t)i undergoes time evolution in terms
of a map
hm(t + 1)i = f(hm(t)i) ;
(6)
where the function f depends on the distributions of pi and si and the initial distribution of opinions.
In order to specialize the abstract model described above to the specic situation
we want to describe, let us make some simplifying assumptions partly supported by
the analysis of the available data. First, let us assume that, in fact, the both (supportive
and persuasive) strengths do not depend on the individual (si = s, pi = p). Moreover,
people are inuenced only by positive examples: by the number of others in their vicinity, who own private enterprises. Individuals are not signicantly inuenced by others
who do not own enterprises, i.e., p = 0. There are two possible explanations of this
asymmetry. First of all, it is possible, that inuence processes become asymmetrical
when strong personal interests are engaged. It is clearly more advantageous to own
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A. Nowak et al. / Physica A 287 (2000) 613–630
an enterprise, than not to own it. In such situations individuals may pay disproportional attention to positive examples, while neglecting negative examples. The second
explanation deals directly with the nature of the social impact function. As discussed
above, social inuence scales as a square root of the number of individuals exerting
inuence. Since a small proportion of individuals own enterprises, a small dierence in
their number changes signicantly the magnitude of the inuence. The same absolute
number, however, may change to a negligibly small degree the inuence of the majority. If the second explanation is correct, than not only economic behaviors, and other
where there are strong vested interests would behave asymmetrically, but one could
expect 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 diers
from the situations considered above by a linear rescaling) and interpret the value 1
as the attitude toward establishing (or continuing to run) an enterprise and 0 as the
opposite decision of the jth individual. It is now clear that the quantity m given by
Eq. (5) measures the total number of enterprises in the region (and can be transformed
to the number of enterprises per capita, or the density by scaling by an appropriate
extensive quantity – the total number of inhabitants or the area).
When the interactions between the individuals from dierent regions are included, it
is to be expected that the evolution of variables characterizing the regions as a whole
like hmi above, is caused by the “intrinsic” dynamics (6) and (distance dependent)
interactions between the regions stemming from the averaged interactions between individuals from dierent regions, thus the change of the total number of enterprises in
the region is given as
X B m
;
(7)
m (t + 1) − m (t) = A0 + A1 m (t) +
g(d )
6=
where the rst two terms on the right-hand side correspond to the linearized version
of Eq. (6), and A0 , A1 , and B are coecients to be determined empirically.
Several numerical experiments with the available data established that we can dispense with a lot of -dependence on the right-hand side of Eq. (7) if we choose
appropriately scaled variables to work with. In the following we report the results of
numerical simulations of changes of number of privately owned enterprises per capita
in Poland. The dynamics is governed by the following discrete-time map:
X bp
X
kj (t) ;
(8)
li (t + 1) = li (t) + B0 + B1 ki (t) +
Np
p
rp 6dij 6rp+1
where li is the number of privately owned enterprises in the ith territorial unit, ki is the
density 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 smaller
than rp+1 from the given one. In our simulation the territorial units were counties (the
smallest administrative units in Poland of diameter of the order of 10 km), and the
consecutive radii rp = p × 10 km. The coecients B0 , B1 , and bp are determined empirically, as unstandardised regression coecients by the analysis of multiple regression,
A. Nowak et al. / Physica A 287 (2000) 613–630
625
Table 1
Values of the standardised coecients in multiple regression for number of enterprises per capita and number
of enterprises per area as predictors and dierence in the number of enterprises per capita in 91–90 as the
dependent variable
Distance
Own
inuence
0 –10
11–20
21–30
31– 40
Number of enterprises per capita
Number of enterprises per area
0.02
0.55
0.01
0.13
0.03
0.18
0.04
0.12
0.0
0.02
where the dierence in the number of enterprises in specic territorial unit, in two
consecutive years is the dependent variable, and statistics (e.g. number of enterprises
per area) in all territorial units located at a given distance (e.g. between 10 and 20
km) constitute independent variables.
Analysis and simulation described further were performed by Urbaniak [33]. We used
in our analysis and simulations the number of privately owned enterprises registered
in REGON database, for years 1989 –1992, i.e., for years of transition from centrally
governed to the market economy. Regression analysis revealed, that the changes in
the number of enterprises per capita can be predicted much better from the number
of enterprises per area in the previous year, then from the number of enterprises per
capita. The relevant standardized coecients for the dierence between 90 and 91, i.e.,
the year when the Polish economy was restructured are given in Table 1.
These results indicate that the inuence in economic matters, i.e., whether to open a
private rm is not symmetrical. The decision to open a new enterprise is inuenced by
the 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 inuence of number of enterprises per
capita was not signicant, and the only signicant predictor was own inuence (autoregression term) which had a negative value of −0:48, indicating that the more
enterprises were in the given county, the more would cease to exist. For the dierence
92–91 both there was a strong eect of the own inuence of measures both per capita
(0.49) and per area (0.55), and signicant inuence of the number per area of closely
located counties (up to 30 km).
Computer simulations were used to compare the model predictions with empirical
data, collected during the Polish transition. The data from the previous year were used
as an input to the simulation program. Empirically established coecients for the 91–90
dierence 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 dierences between the data and the model predictions for 90. The
model predicts clustering while the only local eects visible in the data are the backwash eects in regions with relatively small number of enterprises.This is the result
one could expect, since the model was built to predict the changes in the number of
enterprises in market economy and in 90 the economic reform was only introduced by
626
A. Nowak et al. / Physica A 287 (2000) 613–630
Fig. 2. Number of privately owned enterprises per capita in 1989; empirical data.
Fig. 3. Number of privately owned enterprises per capita in 1990; empirical data.
Fig. 4. Number of privately owned enterprises per capita in 1990; results of computer simulations.
A. Nowak et al. / Physica A 287 (2000) 613–630
Fig. 5. Number of privately owned enterprises per capita in 1991; empirical data.
Fig. 6. Number of privately owned enterprises per capita in 1991; results of computer simulations.
Fig. 7. Number of privately owned enterprises per capita in 1992; empirical data.
627
628
A. Nowak et al. / Physica A 287 (2000) 613–630
Fig. 8. Number of privately owned enterprises per capita in 1992; results of computer simulations.
Balcerowicz, while the economy was still mainly centrally governed. In 91–90 there
is a remarkable similarity between the reality and the model prediction. Model predictions match economic reality also in 92–91. The respective correlations between the
predicted and observed values for all counties are: for 90 –89 correlation r = 0:34, for
91–90 r = 0:71, and for 92–91 r = 0:80. As discussed above, model predictions are
quite accurate, with the exception of the rst year, the year before economic reform
was introduced.
4. Conclusions
Individual decisions underlie many economic processes. The laws of social inuence
quite precisely describe, how other people inuence individual opinions, attitudes, and
decisions, and may be used to construct computer model of social change on the individual level. Robust features of this model’s dynamics include polarization (reduction
of minority) and clustering. In the presence of bias, which breaks the symmetry between dierent attitudes, opinions and decisions, allows the minority to grow, paving
the 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 facilitate social changes. First, one should establish clusters of the new. In order to do
this, one should identify the social groups that are best prepared to help the transition
occur. It is essential to assist those groups in creating a cluster, by changing their local
environment and providing some aid. This aid might be discontinued as soon as the
cluster reaches a conguration in which it can survive as a minority.
The signicance of even seemingly small events, such as opening an ecient foreign
factory, 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 towards work, new organizational ideas, or new standards of quality. Second expansion
of existing clusters is a high priority. This is done most eciently by aecting the
A. Nowak et al. / Physica A 287 (2000) 613–630
629
immediate neighborhood of the clusters, and may take a form of aid, or even an information campaign. The goal is to expand the borders of the clusters of “new”. Finally,
development of connections between the dierent clusters may stabilize clusters, and
give them the advantage over initially more numerous “old”. Locally such connections
may 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 like
them.
It is possible to formulate on the aggregate level a theory that would parallel the
dynamic theory of social impact formed on the individual level. This theory describes
the dynamics of the proportion of individuals who have a specic property (e.g. a
specic 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 located
close together). We can assume that the distances within the unit are too small to be
signicant, and all the individuals within a unit may be treated as located approximately
at the same distance. Mean eld theory may be used to derive aggregate level properties
of an ensemble of individuals. The dynamics of each unit depend additionally on the
inuence from other units located in their vicinity. The strength of couplings decreases
with the physical distance, and may be determined empirically for each particular case.
Computer simulations produce results similar to the model. Combination of individual
and aggregate model could in the future allow one to test individual models on the
aggregate level and to aggregate level theories on the individual level. Hopefully, this
approach may pave the way, for future integration, in coherent models, of factors
coming from both individual and group level. On the theoretical level this approach may
facilitate introduction of known psychological factors and mechanisms into economic
theories. After all, all decisions are made by humans, even if they use rational models
as a guide.
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