Meet, Discuss, and Segregate!
GE´RARD WEISBUCH,1 GUILLAUME DEFFUANT,2 FRE´DE´RIC AMBLARD,2 JEAN-PIERRE NADAL1
1Laboratoire de Physique Statistique,* de l’Ecole Normale Supe´rieure, 24, rue Lhomond, F-75231 Paris Cedex
5, France; and 2Laboratoire d’Inge´nierie pour les Syste`mes Complexes (LISC), Cemagref, Grpt de
Clermont-Ferrand 24, avenue des Landais, BP50085, F-63172 Aubie`re Cedex, France
Received December 4, 2001; revised March 18, 2002; accepted March 18, 2002
We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random
binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield
convergence of opinions toward an average opinion, whereas low thresholds result in several opinion clusters.
The model is further generalized to network interactions, threshold heterogeneity, adaptive thresholds, and
binary strings of opinions.
2002 Wiley Periodicals, Inc.
Key Words: opinion dynamics, heterogeneous agents, bounded confidence
T he present work was initially motivated by an empiri-cal study about the diffusion of environmentallyfriendly practices (agri-environmental measures)
among European farmers [1]. From 1992 Common Agricul-
tural Policy (CAP) of European Communities was changed
from the postwar policy of financially supporting produc-
tion to a financial support of environmentally friendly prac-
tices such as input (fertilizers and pesticide) reduction, set-
aside, and preservation of biodiversity. The next problem
was the implementation by farmers of the policy defined at
the highest level, the European Commission. Our “IMAGES”
teams then studied the factors and processes, which favor
(or prevent) the adoption by farmers of the new policy. The
empirical data collected through surveys and interviews
with farmers and agricultural counselors are in accordance
with previous literature on the Sociology of Agriculture [2]
and demonstrate that:
● changing practices involves a lot of uncertainties for the
farmer; for instance, fertilizers reduction combined with
best farming practices might imply a complete reorgani-
zation of crop choice and annual rotation.
● To decrease uncertainties, farmers engage in many dis-
cussions with their peers; in addition to the propagation
of information, the interactions between farmers perform
a normative control and a prestige allocation. The nor-
mative control is based on an agreement about what con-
stitutes ‘good’ or ‘bad’ farming practices.
● The specific interviews done in our project showed that
the decision of adopting an agri-environmental measure
is the result of a process, which involves social and infor-
mational influences, sometimes during several years
(for more information about the interviews and data see
Deffuant [3]).
*Laboratoire associe´ au CNRS (URA 1306), a´ l’ENS et aux
Universite´s Paris 6 et Paris 7.
Corresponding author: Ge´rard Weisbuch (e-mail: weisbuch@
lps.ens.fr)
© 2002 Wiley Periodicals, Inc., Vol. 7, No. 3 C O M P L E X I T Y 55

In this context of numerous information exchanges be-
fore any adoption decision, modeling adoption dynamics
through methods inspired from information contagion is
more appropriate than game theory often used in environ-
mental economics. In the game theoretical approach, pay-
offs are directly learned by switching behaviors, but the
long-term costs of switches are underestimated with re-
spect the “inertial” conditions faced by farmers in Europe:
several-year contracts have to be signed with local author-
ity, and they involve major reorganization of the farm (capi-
tal investment, crop rotation, labor).
A first possibility to model the information diffusion pro-
cess is to use binary state models (the state representing
whether the farmer adopted green practices or not). Such
models correspond to the classical threshold models of in-
novation diffusion developed by sociologists [4, 5]. These
models assume that each individual has a threshold, which
is interpreted as his personal interest to change his behav-
ior. Then, the social pressure is represented as a function of
the number of neighbors who already adopted. The deci-
sion of adoption is based on the sum of the threshold and
this social pressure.
Many models about opinion dynamics [6–10] are based
on binary opinions, which social actors update as a result of
social influence, often according to some version of a ma-
jority rule. Binary opinion dynamics have been well studied,
such as the herd behavior described by economists [6,7,11].
One expects that the attractor of the dynamics will display
uniformity of opinions, either 0 or 1, when interactions oc-
cur across the whole population. Clusters of opposite opin-
ions appear when the dynamics occur on a social network
with exchanges restricted to connected agents. Clustering is
reinforced when agent diversity, such as a disparity in in-
fluence, is introduced [10,12].
One issue of interest concerns the importance of the
binary assumption: what would happen if opinion were a
continuous variable such as the worthiness of a choice (a
utility in economics), or some belief about the adjustment
of a control parameter?
The rationale for binary versus continuous opinions is
related to the kind of information used by agents to validate
their own choice:
● the actual choice of the other agents, a situation common
in economic choice of brands: “do as the others do”;
● or the actual opinion of the other agents, about the
“value” of a choice: “establish one’s opinion according to
what the others think or at least according to what they
say.”
More generally, we expect opinion (rather than choice)
dynamics to occur in situation where agents have to make
important choices and care to collect many opinions before
taking any decisions: adopting a technological change
might often be the case. Political elections also belong to the
same category, because of uncertainties concerning new
candidates, new challenges in the future, long electoral
mandate (especially in Europe), etc. European integration is
an obvious example of a quasi-irreversible decision, which
involved many uncertainties and was subject to all sorts of
discussions among citizens.
Modeling of continuous opinions dynamics was earlier
started by applied mathematicians and focused on the con-
ditions under which a panel of experts would reach a con-
sensus [13–18].
The purpose of this article is to present results concern-
ing continuous opinion dynamics subject to the constraint
that convergent opinion adjustment only proceeds when
opinion difference is below a given threshold. The rationale
for the threshold condition is that agents only interact when
their opinions are already close enough; otherwise they do
not even bother to discuss. The reason for refusing discus-
sion might be for instance lack of understanding, conflicting
interest, or social pressure. The threshold would then cor-
respond to some openness character. Another interpreta-
tion is that the threshold corresponds to uncertainty: the
agents have some initial views with some degree of uncer-
tainty and would not care about other views outside their
uncertainty range.
Social Psychology literature discusses social influence
and the conditions under which individual attitudes and
decisions are influenced by others (see e.g., Petty and Ca-
cioppo [19]). Part of this literature concentrates on how ini-
tial attitudes determine the outcome of interactions [20–22].
One can summarize the general outcome of the reported
experiments as an increase of influence when initial posi-
tions are close enough. The threshold condition that we
introduce here is also used in Axelrod’s model of dissemi-
nating culture [23].
Many variants of the basic idea can be proposed, and the
article is organized as follows:
● We first expose the simple case of complete mixing
among agents under a unique and constant threshold
condition.
● We then check the genericity of the results obtained for
the simplest model to other cases such as localized inter-
actions, distribution of thresholds, varying thresholds,
and binary strings of opinions.
A previous publication [24] and a working paper [25]
report more complete results on several aspects.
2. THE BASIC CASE: COMPLETE MIXING AND ONE
FIXED THRESHOLD
Let us consider a population of N agents i with continuous
opinion xi. We start from an initial distribution of opinions,
most often taken uniform on [0,1] in the computer simula-
56 C O M P L E X I T Y © 2002 Wiley Periodicals, Inc.

tions. At each time step any two randomly chosen agents
meet: they readjust their opinion when their difference in
opinion is smaller in magnitude than a threshold d. Suppose
that the two agents have opinion x and x. Iff |x-x| < d
opinions are adjusted according to
x = x +



x − x (1)
x = x +



x − x (2)
where
is the convergence parameter whose values may
range from 0 to 0.5.
In the basic model, the threshold d is taken as constant
in time and across the whole population. Note that we here
apply a complete mixing hypothesis plus a random serial
iteration mode. (The “consensus” literature most often uses
parallel iteration mode when they suppose that agents av-
erage at each time step the opinions of their neighborhood.
Their implicit rationale for parallel iteration is that they
model successive meetings among experts.)
The evolution of opinions may be mathematically pre-
dicted in the limiting case of small values of d [26]. (The
other extreme is the absence of any threshold, which yields
consensus at infinite time as earlier studied in Stone [13]
and others.) For finite thresholds, computer simulations
show that the distribution of opinions evolves at large times
toward clusters of homogeneous opinions.
● For large threshold values (d > 0.3) only one cluster is
observed at the average initial opinion. Figure 1 repre-
sents the time evolution of opinions starting from a uni-
form distribution of opinions.
● For lower threshold values, several clusters can be ob-
served (see Figure 2). Consensus is then not achieved
when thresholds are low enough.
Obtaining clusters of different opinions does not surprise
an observer of human societies, but this result was not a
priori obvious because we started from an initial configu-
ration where transitivity of opinion propagation was pos-
sible through the entire population: any two agents however
different in opinions could have been related through a
chain of agents with closer opinions. The dynamics that we
describe ended up in gathering opinions in clusters on the
one hand, but also in separating the clusters in such a way
that agents in different clusters do not exchange anymore.
The number of clusters varies as the integer part of 1/2d:
this is to be further referred to as the “1/2d rule” (see Figure
3; notice the continuous transitions in the average number
of clusters when d varies. Because of the randomness of the
initial distribution and pair sampling, any prediction on the
FIGURE 3
Statistics of the number of opinion clusters as a function of d on
the x-axis for 250 samples (µ = 0.5, N = 1000).
FIGURE 1
Time chart of opinions (d = 0.5 µ = 0.5 N = 2000). One time unit
corresponds to sampling a pair of agents.
FIGURE 2
Time chart of opinions for a lower threshold d = 0.2 (µ = 0.5, N
= 1000).
© 2002 Wiley Periodicals, Inc. C O M P L E X I T Y 57

outcome of dynamics such as the 1/2d rule can be ex-
pressed as true with a probability close to one in the limit of
large N, but one can often generate a deterministic se-
quence of updates which would contradict the “most likely”
prediction.).
3. SOCIAL NETWORKS
The literature on social influence and social choice also
considers the case when interactions occur along social
connections between agents [6] rather than randomly
across the whole population. Apart from the similarity con-
dition, we now add to our model a condition on proximity,
i.e., agents only interact if they are directly connected
through a pre-existing social relation. Although one might
certainly consider the possibility that opinions on certain
unsignificant subjects could be influenced by complete
strangers, we expect important decisions to be influenced
by advice taken either from professionals (doctors, for in-
stance) or from socially connected persons (e.g., through
family, business, or clubs). Facing the difficulty of inventing
a credible instance of a social network as in the literature on
social binary choice, we here adopted the standard simpli-
fication and carried out our simulations on square lattices:
square lattices are simple, allow easy visualization of opin-
ion configurations, and contain many short loops, a prop-
erty that they share with real social networks.
We then started from a two-dimensional (2D) network of
connected agents on a square grid. Any agent can only in-
teract with his four connected neighbors (N, S, E, and W).
We used the same initial random sampling of opinions from
0 to 1 and the same basic interaction process between
agents as in the previous sections. At each time step a pair
is randomly selected among connected agents, and opinions
are updated according to Equations 1 and 2 provided of
course that their distance is less than d.
The results are not very different from those observed
with nonlocal opinion mixing as described in the previous
section, at least for the larger values of d (d > 0.3, all results
displayed in this section are equilibrium results at large
times).
As seen in Figure 4, the lattice is filled with a large ma-
jority of agents who have reached consensus around x = 0.5,
whereas a few isolated agents have “extremists” opinions
closer to 0 or 1. The importance of extremists is the most
noticeable difference with the full mixing case described in
the previous section.
Interesting differences are noticeable for the smaller val-
ues of d < 0.3 as observed in Figure 5.
For connectivity 4 on a square lattice, only one cluster
percolates [27] across the lattice. All agents of the percolat-
ing cluster share the same opinion as observed on Figure 4,
but for d < 0.3 several opinion clusters are observed and
none percolates across the lattice.
Similar opinions, but not identical, are shared across
several clusters. The differences of opinions between groups
of clusters relate to d, but the actual values inside a group of
clusters fluctuate from cluster to cluster because homogeni-
zation occurred independently among the different clusters:
the resulting opinions depend on fluctuations of initial
opinions and histories from one cluster to the other. The
same increase in fluctuations compared to the full mixing
FIGURE 5
Display of final opinions of agents connected on a square lattice
of size 29 × 29 (d = 0.15 µ = 0.3 after 100.000 iterations). Color
code: purple 0.14, light blue 0.42, red 0.81–0.87. Note the pres-
ence of smaller clusters with similar but not identical opinions.
FIGURE 4
Display of final opinions of agents connected on a square lattice
of size 29 × 29 (d = 0.3 µ = 0.3 after 100,000 iterations). Opinions
between 0 and 1 are coded by gray level (0 is black and 1 is
white). Note the percolation of the large cluster of homogeneous
opinion and the presence of isolated “extremists.”
58 C O M P L E X I T Y © 2002 Wiley Periodicals, Inc.

case is observed from sample to sample with the same pa-
rameter values.
The qualitative results obtained with 2D lattices should
be observed with any connectivity, either periodic, random,
or small world.
The above results were obtained when all agents had the
same invariant threshold. The purpose of the following sec-
tions is to check the general character of our conclusions:
● when one introduces a distribution of thresholds in the
population;
● when the thresholds themselves obey some dynamics.
4. HETEROGENEOUS CONSTANT THRESHOLD
Supposing that all agents use the same threshold to decide
whether to take into account the views of other agents is a
simplifying assumption. When heterogeneity of thresholds
is introduced, some new features appear. To simplify the
matter, let us exemplify the issue in the case of a bimodal
distribution of thresholds, for instance, 8 agents with a large
threshold of 0.4 and 192 with a narrow threshold of 0.2 as in
Figure 6.
One observes that in the long run, convergence of opin-
ions into one single cluster is achieved because of the pres-
ence of the few “open minded” agents (the single cluster
convergence time is 12,000, corresponding to 60 iterations
per agent on average, for the parameters of Figure 6). How-
ever, in the short run, a metastable situation with two large
opinion clusters close to opinions 0.35 and 0.75 is observed
because of narrow minded agents, with open minded
agents opinions fluctuating around 0.5 because of interac-
tions with narrow minded agents belonging to either high or
low opinion cluster. Because of the few exchanges with the
high d agents, low d agents opinions slowly shift toward the
average until the difference in opinions between the two
clusters falls below the low threshold: at this point the two
clusters collapse.
This behavior is generic for any mixtures of thresholds.
At any time scale, the number of clusters obeys a “general-
ized 1/2d rule”:
● on the long run clustering depends on the higher threshold;
● on the short run clustering depends on the lower threshold;
● the transition time between the two dynamics is propor-
tional to the total number of agents and to the ratio of
narrow minded to open minded agents.
In some sense, the existence of a few “open minded”
agents seems sufficient to ensure consensus after a large
enough time for convergence. The next section restricts the
validity of this prediction when threshold dynamics are
themselves taken into account.
5. THRESHOLD DYNAMICS
5.1. The Model
Let us interpret the basic threshold rule in terms of agent’s
uncertainty: agents take into account others’ opinion on the
occasion of interaction because they are not certain about
the worthiness of a choice. They engage in interaction only
with those agents which opinion does not differ too much
from their own opinion in proportion of their own uncer-
tainty. If we interpret the threshold for exchange as the
agent uncertainty, we might suppose with some rationale
that his subjective uncertainty decreases with the number of
opinion exchanges.
Taking opinions from other agents can be interpreted, at
least by the agent himself, as sampling a distribution of
opinions. As a result of this sampling, agents should update
their new opinion by averaging over their previous opinion
and the sampled external opinion and update the variance
of the opinion distribution accordingly.
Within this interpretation, a “rational procedure” (in the
sense of Herbert Simon) for the agent is to simultaneously
update his opinion and his subjective uncertainty. Let us
write opinion updating as weighting one’s previous opinion
x(t  1) by  and the other agent’s opinion x(t  1) by 1 
, with 0 <  < 1.  is a “confidence” parameter weighting
how much the agent trust his own opinion with respect to
those of others.  can be rewritten  = 1  (1/n), where n
expresses a characteristic number of opinions taken into
account in the averaging process. n  1 is then a relative
weight of the agent previous opinion as compared to the
newly sampled opinion weighted 1. Within this interpreta-
tion, updates of both opinion x and variance v should be
written:
x


t =   x
t − 1 +
1 −   x
t − 1 (3)
v


t =   v
t − 1 +  
1 −   x
t − 1 − x
t − 12. (4)
FIGURE 6
Time chart of opinions (N = 200). Red + represent narrow-minded
opinions (192 agents with threshold 0.2), green × represent open-
minded opinions (8 agents with threshold 0.4).
© 2002 Wiley Periodicals, Inc. C O M P L E X I T Y 59

The second equation simply represents the change in
variance when the number of samples increases from n  1
at time t  1 to n at time t. It is directly obtained from
the definition of variance as a weighted sum of squared
deviations.
As previously, updating occurs when the difference in
opinion is lesser than a threshold, but this threshold is now
related to the variance of the distribution of opinions
sampled by the agent. A simple choice is to relate the
threshold to the standard deviation (t) according to:
d


t = 
t, (5)
where  is a constant parameter often taken equal to 1 in the
simulations.
When an agent equally values collected opinions inde-
pendently of how old they are, he should also update his
confidence parameter  in connection with n(t)  1 the
number of previously collected opinions (This expression is
also used in the literature about “consensus” building to
describe “hardening” of agents’ opinions as in Chatterjee
and Seneta [14] and Cohen et al. in 1986 [15].):



t =
n


t − 1
n


t
 


t − 1. (6)
Another possible updating choice is to maintain  con-
stant, which corresponds to taking a moving average on
opinions and giving more importance to the n later col-
lected opinions. Such a “bounded” memory would make
sense in the case when the agent believes that there exists
some slow shift in the distribution of opinions, whatever its
cause, and that older opinions should be discarded.
Both algorithms were tried in the simulations and give
qualitatively similar results in terms of the number of at-
tractors, provided that one starts from an initial number of
supposed trials n(0) corresponding to the same . The only
difference concerns the dynamics of convergence:
● In the case of constant confidence , convergence is ex-
ponential: thresholds, variances, and distances to attrac-
tors decay exponentially versus the number of updates
experienced by the agents.
● In the case of adjustable confidence , convergence is
hyperbolic: variances and distances to attractors decay as
the inverse number of updates (thresholds vary as the
inverse square root of this number).
These scaling are predicted using simple approximations
and verified by simulation.
5.2. SIMULATION RESULTS
When compared with constant threshold dynamics, de-
creasing thresholds results in a larger variety of final opin-
ions. For initial thresholds values, which would have ended
in opinion consensus, one observes a number of final clus-
ters, which decreases with  (and thus with n).
Observing the chart of final opinions versus initial opin-
ions on Figure 7, one sees that most opinions converge to-
ward two clusters (at x = 0.60 and x = 0.42), which are closer
than those one would obtain with constant thresholds (typi-
cally around x = 0.66 and x = 0.33): initial convergence gath-
ered opinions which would have aggregated at the initial
threshold values (0.5), but which later segregated because of
the decrease in thresholds. Many outliers are apparent on
the plot.
Large values of , close to one, e.g., n > 7, correspond to
averaging on many interactions. The interpretation of large
 and n is that the agent has more confidence in his own
opinion than in the opinion of the other agent with whom
he is interacting, in proportion with n  1. For constant
values of , the observed dynamics is not very different from
what we obtained with constant thresholds (Figure 8).
A more complicated dynamics is observed for lower val-
ues of n and  which correspond to a fast decrease of the
thresholds, thus preventing the aggregation of all opinions
into large clusters. Apart from the main clusters, one also
observes smaller clusters plus outliers (already present on
Figure 7).
For d (0) = 0.5 (which would yield consensus with only
one cluster if kept constant) and  = 0.5 (corresponding to n
= 2, i.e., agents giving equal weight to their own opinion and
FIGURE 7
Each point on this chart represents the final opinion of one agent
versus its initial opinion (for constant  = 0.7  = 1.0 N = 1000,
initial threshold 0.5).
60 C O M P L E X I T Y © 2002 Wiley Periodicals, Inc.

to the external opinion), more than 10 clusters unequal in
size are observed plus isolated outliers. One way to charac-
terize the dispersion of opinions with varying  is to com-
pute y the relative value of the squared cluster sizes with
respect to the squared number of opinions.
y =

i=1
n
si
2



i=1
n
si
2
(7)
For m clusters of equal size, one would have y = 1/m. The
smaller y, the more important is the dispersion in opinions.
Figure 8 shows the increase of the dispersion index y with
n (n  1 is the initial “subjective” weight of agent’s own
opinion).
The time pattern of thresholds appearing as green bands
on Figure 9 gives us some insight on these effects. Because
opinion exchange decays variance by an approximately
constant factor close to , each individual green band cor-
responds to a given number of opinion exchanges experi-
enced by the agents: the upper band corresponds to the
variance after one exchange, the second upper to two ex-
changes, and so on. The horizontal width of a band corre-
sponds to the fact that different agents are experiencing the
same number of updates at different times: rough evalua-
tions made on Figure 9 show that most agents have their
first exchange between time 0 and 4000, and their fifth ex-
change between 1000 and 12,000.
When the decrease of threshold and the clustering of
opinions is fast, those agents that are not sampled early
enough and/or not paired with close enough agents can be
left over from the clustering process. When they are
sampled later, they might be too far from the other agents to
get involved into opinion adjustment. The effect gets im-
portant when convergence is fast, i.e., when n and  are
small.
Let us note that these agents in the minority have larger
uncertainty and are more “open to discussion” than those
in the mainstream, in contrast with the common view that
eccentrics are opinionated!
The results of the dynamics are even more dispersed for
lower values of . In this regime, corresponding to “insecure
agents” who do not value their own opinion more than
those of other agents, we observe more clusters which im-
portance and localization depend on the random sampling
of interacting agents and are thus harder to predict than in
the other regime with a small number of big clusters.
Using a physical metaphor, clustering in the small  re-
gime resembles fast quenching to a frozen configuration,
thus maintaining many “defects” (e.g., here the outliers),
whereas in the opposite large  regime it resembles slow
annealing (with suppression of defects).
6. VECTOR OPINIONS
6.1. The Model
Another subject for investigation is vectors of opinions.
Usually people have opinions on different subjects, which
can be represented by vectors of opinions. In accordance
with our previous hypotheses, we suppose that one agent
interacts concerning different subjects with another agent
according to some distance with the other agent’s vector of
opinions. In order to simplify the model, we revert to binary
opinions. An agent is characterized by a vector of m binary
FIGURE 8
Variation of the dispersion index y with n, the initial “subjective”
number of collected opinions ( = 1 − 1/n, d (0) = 0.5  = 1.0 N
= 1000). Small values of y correspond to several attractors, larger
values close to one to a single attractor. The initial threshold value
of 0.5 if kept constant would yield consensus with only one cluster.
FIGURE 9
Time chart of opinions and thresholds (for constant  = 0.7 d (0)
= 0.4  = 0.5 N = 1000). Red + represent opinions and green ×
represent thresholds.
© 2002 Wiley Periodicals, Inc. C O M P L E X I T Y 61

opinions about the complete set of m subjects, shared by all
agents. We use the notion of Hamming distance between
binary opinion vectors (the Hamming distance between two
binary opinion vectors is the number of different bits be-
tween the two vectors). Here, we only treat the case of com-
plete mixing; any pair of agents might interact and adjust
opinions according to how many opinions they share. (The
bit string model shares some resemblance with Axelrod’s
model of disseminating culture [23] based on adjustment of
cultures as sets of vectors of integer variables characterising
agents on a square lattice.) The adjustment process occurs
when agents agree on at least m  d subjects (i.e., they
disagree on d  1 or fewer subjects). The rules for adjust-
ment are as follows: when opinions on a subject differ, one
agent (randomly selected from the pair) is convinced by the
other agent with probability
. Obviously this model has
connections with population genetics in the presence of
sexual recombination when reproduction only occurs if ge-
nome distance is smaller than a given threshold. Such a
dynamics results in the emergence of species (see Higgs and
Derrida [28]). We are again interested in the clustering of
opinion vectors. In fact clusters of opinions here play the
same role as biological species in evolution.
6.2. RESULTS
We observed once again that
and N only modify conver-
gence times toward equilibrium; the most influential factors
are threshold d and m the number of subjects under dis-
cussion. Most simulations were done for m = 13. For N =
1000, convergence times are of the order of 10 million pair
iterations. For m = 13:
● When d > 3, convergence toward a single opinion (con-
sensus) is observed (with the exception of one or two
outliers for the lower values of d).
● For d = 3, one observes from 2 to 7 significant peaks (with
a population larger than 1%) plus some isolated opinions.
● For d = 2 a large number (around 500) of small clusters is
observed.
The same kind of results are obtained with other values
of m: two regimes, uniformity of opinions for larger d values
and extreme diversity for smaller d values, are separated by
one dc value for which a small number of clusters is ob-
served (e.g., for m = 21, dc = 5. dc seems to scale in propor-
tion with m).
Figure 10 represents these populations of the different
clusters at equilibrium (iteration time was 12,000,000) in a
log-log plot according to their rank-order of size. No scaling
law is obvious from these plots, but we observe the strong
qualitative difference in decay rates for various thresholds d.
7. CONCLUSION
The main lesson from this set of simulations is that opinion
exchanges restricted by a small proximity threshold result
into clustering of opinions as opposed to consensus in the
absence of threshold or for large thresholds.
When one gets more specific, some differences appear
between the case of continuous opinions and binary strings:
● Binary strings display an abrupt phase transition from
consensus to a large multiplicity of clusters when d de-
creases.
● By contrast, for continuous opinions, the change in the
number of attractors as a function of threshold is graded.
The 1/2d rule predicts the outcome of the dynamics in
the simplest cases, but it also provides some qualitative
insight for the case of threshold dynamics.
When we introduced dynamics on thresholds on the ba-
sis that agents interpret opinion exchange as sampling a
distribution of opinions, a surprising result was that the
nature of opinion and threshold updating only changes the
scaling law for the time of convergence; rather, the impor-
tant parameter which determines the distribution of attrac-
tors is  the initial self-confidence of the agents.
On the other hand, what we have shown here is that
clustering of opinions does not automatically imply as a
cause conflicting interests or more simply a diversity of in-
terests and opportunities. In other words, clustering of
opinions might result from the history of agents without any
structural diversity.
The formal model was introduced to represent continu-
ous opinion dynamics among groups of farmers facing un-
certainties concerning the economic and/or social value of
a choice; its possible range of application goes far beyond
technological change issues. Obviously what we said about
opinions sometimes applies to beliefs. The binary string dy-
namics, e.g., is an interesting approach to political debates.
FIGURE 10
Log-log plot of average populations of clusters of opinions ar-
ranged by decreasing order for N = 1000 agents (µ = 1).
62 C O M P L E X I T Y © 2002 Wiley Periodicals, Inc.

The continuous opinion dynamics can also bring some in-
sight on the emergence of discrete coding of continuous
variables, e.g., the emergence of lexicon (how big is big?).
ACKNOWLEDGMENTS
We thank David Neau and the members of the IMAGES
FAIR project, Edmund Chattoe, Nils Ferrand, and Nigel Gil-
bert for helpful discussions. G.W. benefited at different
stages in the project from discussions with Sam Bowles,
Winslow Farell, and John Padgett at the Santa Fe Institute,
whom we thank for its hospitality. We also thank Rainer
Hegselmann for pointing us the references to the “consen-
sus” literature. This study has been carried out with finan-
cial support from the Commission of the European Com-
munities, Agriculture and Fisheries (FAIR) Specific RTD pro-
gram, CT96-2092, “Improving Agri-Environmental Policies:
A Simulation Approach to the Role of the Cognitive Prop-
erties of Farmers and Institutions.” It does not necessarily
reflect its views and in no way anticipates the Commission’s
future policy in this area.
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