Interacting Agents and Continuous Opinions Dynamics

Gérard Weisbuch$^{*}$
Guillaume Deffuant$^{**}$, Frederic Amblard$^{**}$
and Jean Pierre Nadal$^{*}$
$^{*}$Laboratoire de Physique Statistique1
de l'Ecole Normale Supérieure,
24 rue Lhomond, F-75231 Paris Cedex 5, France.
$^{**}$Laboratoire d'Ingénierie pour les Systèmes Complexes (LISC)
Cemagref - Grpt de Clermont-Ferrand
24 Av. des Landais - BP50085
F-63172 Aubière Cedex (FRANCE)
email:weisbuch@lps.ens.fr

Abstract:

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 towards an average opinion, whereas low thresholds result in several opinion clusters. The model is further generalised to network interactions, threshold heterogeneity, adaptive thresholds and binary strings of opinions.

Introduction

Many models about opinion dynamics (Fölmer 1974, Arthur 1994, Orléan 1995, Latané and Nowak 1997, Weisbuch and Boudjema 1999), are based on binary opinions which social actors update as a result of social influence, often according to some version of a majority rule. Binary opinion dynamics have been well studied, such as the herd behaviour described by economists (Fölmer 1974, Arthur 1994, Orléan 1995). One expect that in most cases the attractor of the dynamics will display uniformity of opinions, either 0 or 1, when interactions occur across the whole population. Clusters of opposite opinions 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 influence, is introduced, (Latané and Nowak 1997, Weisbuch and Boudjema 1999).

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 might be related to the kind of information used by agents to validate their own choice:

On the empirical side, there exist well documented studies about social norms concerning sharing between partners. Henrich et al. (2001) compared through experiments shares accepted in the ultimatum game and showed that people agree upon what a "fair" share should be, which can of course vary across different cultures. Young and Burke (2000) report empirical data about crop sharing contracts, whereby a landlord leases his farm to a tenant laborer in return for a fixed share of the crops. In Illinois as well as in India, crop sharing distributions are strongly peaked upon "simple values" such as 1/2-1/2 or 1/3-2/3. The clustering of opinions about "fair shares" is the kind of stylised fact that our model tries to reproduce. More generally, we expect such opinion 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. The present paper was motivated by changes towards environmental-friendly practices in agriculture under the influence of the new Common Agricultural Policy in Europe.

Modeling of continuous opinions dynamics was earlier started by applied mathematicians and focused on the conditions under which a panel of experts would reach a consensus, (Stone 1961, Chatterjee and Seneta 1977, Cohen et al.1986, Laslier 1989, Krause 2000, Hegselmann and Krause 2002, further referred to as the "consensus" literature).

The purpose of this paper is to present results concerning 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 discussion might be for instance lack of understanding, conflicting interest, or social pressure. The threshold would then correspond to some openness character. Another interpretation is that the threshold corresponds to uncertainty: the agents have some initial views with some degree of uncertainty and would not care about other views outside their uncertainty range.

Many variants of the basic idea can be proposed and the paper is organised as follows:

A previous paper (Deffuant et al.2000) reported more complete results on mixing across a social network and on binary strings of opinions.

The basic case: Complete Mixing and one fixed threshold

Let us consider a population of $N$ agents $i$ with continuous opinion $x_i$. We start from an initial distribution of opinions, most often taken uniform on [0,1] in the computer simulations. At each time step any two randomly chosen agents meet: they re-adjust 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$$\vert x-x'\vert<d$ opinions are adjusted according to:
$\displaystyle x$ $\textstyle =$ $\displaystyle x + \mu \cdot (x'-x)$ (1)
$\displaystyle x'$ $\textstyle =$ $\displaystyle x' + \mu \cdot (x-x')$ (2)

where $\mu$ 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 mode2.

The evolution of opinions may be mathematically predicted in the limiting case of small values of $d$ (Neau 2000)3. Time variations of opinions' density $\frac{d \rho(x)}{dt}$ obey the following dynamics:

\begin{displaymath}\frac{d \rho(x)}{dt}= - \frac{d^3}{2}\cdot \mu \cdot (1-\mu) \cdot\frac{\partial ^2(\rho ^2)}{{\partial x}^2}\end{displaymath}


This implies that starting from an initial distribution of opinions in the population, any local higher opinion density is amplified. Peaks of opinions increase and valleys are depleted until very narrow peaks remain among a desert of intermediate opinions.

For finite thresholds, computer simulations show that the distribution of opinions evolves at large times towards clusters of homogeneous opinions.

\begin{figure}\centerline {\epsfxsize=90mm\epsfbox{convergence.eps}}\end{figure}
Figure: Time chart of opinions ($d=0.5 \quad \mu=0.5 \quad N=2000$). One time unit corresponds to sampling a pair of agents.
\begin{figure}\centerline {\epsfxsize=90mm\epsfbox{2pics.eps}}\end{figure}
Figure 2: Time chart of opinions for a lower threshold $d=0.2$ ($ \mu=0.5, \quad N=1000$) .
Obtaining clusters of different opinions does not surprise an observer of human societies, but this result was not a priori obvious since we started from an initial configuration where transitivity of opinion propagation was possible 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 don't 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 34).

\begin{figure}\centerline {\epsfxsize=90mm\epsfbox{phases2.eps}}\end{figure}
Figure 3: Statistics of the number of opinion clusters as a function of $d$ on the x axis for 250 samples ($ \mu=0.5, \quad N=1000$).

Social Networks

The literature on social influence and social choice also considers the case when interactions occur along social connections between agents (Fölmer 1974) rather than randomly across the whole population. Apart from the similarity condition, 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 un-significant subjects could be influenced by complete strangers, we expect important decisions to be influenced by advice taken either from professionals (doctors, for instance) 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 simplification and carried out our simulations on square lattices: square lattices are simple, allow easy visualisation of opinion configurations and contain many short loops, a property that they share with real social networks.

We then started from a 2 dimensional network of connected agents on a square grid. Any agent can only interact with his four connected neighbours (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 non-local 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 majority of agents who have reached consensus around $x=0.5$ while 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.

\begin{figure}\centerline {\epsfxsize=60mm\epsfbox{run33a.ps}}\end{figure}
Figure: Display of final opinions of agents connected on a square lattice of size 29x29 ($d=0.3 \quad \mu=0.3$ 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''.
Interesting differences are noticeable for the smaller values of $d<0.3$ as observed in figure 5.
\begin{figure}\centerline {\epsfxsize=60mm\epsfbox{run15.3}}\end{figure}
Figure: Display of final opinions of agents connected on a square lattice of size 29x29 ($d=0.15 \quad \mu=0.3$ after 100 000 iterations) . Color code: purple 0.14, light blue 0.42, red 0.81 to 0.87. Note the presence of smaller clusters with similar but not identical opinions.
For connectivity 4 on a square lattice, only cluster percolates (Stauffer and Aharony 1994) across the lattice. All agents of the percolating 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 homogenisation occurred independently among the different clusters: the resulting opinion depends on fluctuations of initial opinions and histories from one cluster to the other. The same increase in fluctuations compared to the full mixing case is observed from sample to sample with the same parameter values.

The qualitative results obtained with 2D lattices should be observed with any connectivity, either periodic, random, or small world.

The above results where obtained when all agents have the same invariant threshold. The purpose of the following sections is to check the general character of our conclusions:


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.
\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{plo3mai.eps}}\end{figure}
Figure 6: Time chart of opinions ($ N=200$). Red '+' represent narrow minded opinions (192 agents with threshold 0.2), green 'x' represent open minded opinions (8 agents with threshold 0.4).
One observes that in the long run convergence of opinions into one single cluster is achieved due to the presence of the few "open minded" agents (the single cluster convergence time is 12000, corresponding to 60 iterations per agent on average, for the parameters of figure 6). But in the short run, a metastable situation with two large opinion clusters close to opinions 0.35 and 0.75 is observed due to narrow minded agents, with open minded agents opinions fluctuating around 0.5 due to interactions 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 towards the average until the difference in opinions between the two clusters falls below the low threshold: at this point the two clusters collapse.

This behaviour is generic for any mixtures of thresholds. At any time scale, the number of clusters obeys a "generalized 1/2d rule":

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 restrict the validity of this prediction when threshold dynamics are themselves taken into account.

Threshold Dynamics

Let us interprete 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 uncertainty. If we interprete 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 $\alpha $ and the other agent's opinion $x'(t-1)$ by $1-\alpha$, with $0<\alpha<1$.$\alpha $ can be re-written $\alpha=1-\frac{1}{n}$ 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 interpretation, updates of both opinion $x$ and variance $v$ should be written:

$\displaystyle x(t)$ $\textstyle =$ $\displaystyle \alpha \cdot x(t-1) + (1-\alpha) \cdot x'(t-1)$ (3)
$\displaystyle v(t)$ $\textstyle =$ $\displaystyle \alpha \cdot v(t-1) + \alpha (1-\alpha) \cdot [x(t-1) - x'(t-1)]^2$ (4)


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 $\sigma(t)$ according to:

\begin{displaymath}d(t)= \nu \sigma(t) ,\end{displaymath} (5)
where $\nu$ is a constant parameter often taken equal to 1 in the simulations.

When an agent equally values collected opinions independently of how old they are, he should also update $\alpha $ connected to $n-1$ the number of previously collected opinions:

\begin{displaymath}\alpha(t) = \frac{n(t)-1}{n(t)} \alpha(t-1)\end{displaymath} (6)
This expression is also used in the literature about "consensus" building to describe "hardening" of agents opinions as in Chatterjee and Seneta (1977) and Cohen et al. in 1986.

Another possible updating choice is to maintain $\alpha $ constant which corresponds to taking a moving average on opinions and giving more importance to the $n$ later collected 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 attractors, provided that one starts from an initial number of supposed trials $n(0)$ corresponding to the same $\alpha $.

Scaling

Constant $\alpha $

In the case of constant $\alpha $, a simplified computation valid in the limit of small $\nu$ predicts an exponential decay of thresholds. Neglecting the second term in the dynamics of variance 5 :
 
 
\begin{displaymath}v(t) = \alpha \cdot v(t-1)\end{displaymath} (7)
gives :
\begin{displaymath}v(t) = \alpha^t \cdot v_0 \\ .\end{displaymath} (8)
Writing $\alpha $ as $1-\frac{1}{n}$ and approximating it as exp$(\frac{-1}{n})$ for large $n$, we see that the variance decays exponentially with a characteristic time of $n$ and that the thresholds vary as:
\begin{displaymath}d(t) \simeq d_0 \cdot exp [- \frac{t}{2n}]\end{displaymath} (9)
A parallel estimation for the dynamics of convergence of opinions towards some average opinion $x_{\infty}$ (corresponding to the attractor) can be made by replacing $x'(t-1)$ by $x_{\infty}$ in equation (3) describing the dynamics of $x(t)$. After subtracting $x_{\infty}$ to both members, the deviation of opinions from their attractor can be written as:
\begin{displaymath}x(t)-x_{\infty} = \alpha \cdot (x(t-1)-x_{\infty})\end{displaymath} (10)
Equation 10 shows that opinions also converge exponentially towards the attractor with the same time constant as variance.

Varying $\alpha $

Equivalent computations were also made for the case when $\alpha $ varies as $1- \frac{1}{n(t)}$. For instance, the dynamics of variance is described by the following set of equations:
\begin{displaymath}v(t) = (1- \frac{1}{n(t)}) \cdot v(t-1)\end{displaymath} (11)
\begin{displaymath}v(t) = v_{0} \prod_{n'=n_0}^{n_0+t} (1- \frac{1}{n'})\end{displaymath} (12)
and with some approximation:
\begin{displaymath}v(t) \simeq v_0 \frac{n_0}{n_0+t}\end{displaymath} (13)
Thresholds then vary as the inverse square root of the number of interactions.

The equivalent computation for the evolution of opinion deviation from the attractor also gives an hyperbolic decay:

\begin{displaymath}x(t)-x_0 \simeq (x(0)-x_{\infty}) \frac{ n_0}{n_0+t}\end{displaymath} (14)
The above expressions allow us to predict average trends for the dynamics of thresholds and opinions.

Simulation Results

When comparing to constant threshold dynamics, decreasing thresholds results in a larger variety of final opinions. For initial thresholds values which would have ended in opinion consensus, one observes a number of final clusters which decreases with $\alpha $ (and thus with $n$). Smaller values of $\alpha $ correspond to a fast decrease of the thresholds, which prevents the aggregation of all opinions into large clusters.

Observing the chart of final opinions versus initial opinions on figure 7, one sees that most opinions converge towards two clusters (at $x=0.60$ and $x=0.42$) which are closer than those one could obtain with constant thresholds (typically around $x=0.66$ and $x=0.33$): initial convergence gathered opinions which would had aggregate at the initial threshold values, but which later segregate due to the decrease in thresholds. Many outliers are apparent on the plot.

\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{zerin1.eps}}\end{figure}
Figure: Each point on this chart represents the final opinion of one agent versus its initial opinion (for constant $\alpha=0.7 \quad \nu=1.0 \quad N=1000$, initial threshold 0.5). 

Large $\alpha $ and $n$

Large values of $\alpha $, close to one, e.g. $n>7$, correspond to averaging on many interactions. The interpretation of large $\alpha $ 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 $\alpha $, the observed dynamics is not very different from what we obtained with constant thresholds.

The exponential decay of thresholds predicted by equation 9 is verified on figure 8 plotted for the same parameters values. The observed decay constant on figure 8 is 1.7, slightly less than 2, the theoretical prediction based on equation 9 which neglects the possible increase of variance due to other opinions.

\begin{figure}\centerline {\epsfxsize=100mm\epsfbox{plog312aug.eps}}\end{figure}
Figure: Exponential decay of summed variances (for constant $\alpha=0.9 \quad d(0)= 0.4 \quad \nu=1.0 \quad N=1000$, initial threshold 0.4).
For $\alpha $ varying according to equation 13, the variance dynamics is hyperbolic as observed on the log-log plot of figure 9. The observed slope on figure 9 is not far from the predicted value, -1.0.

In both cases the scaling of variance and thresholds is verified on more than one decade, but deviations are observed:

\begin{figure}\centerline {\epsfxsize=100mm\epsfbox{plvaral.eps}}\end{figure}
Figure 9: Power-law decay of summed variances for varying $\alpha $ ( log-log plot, initial $\alpha=0.7 \quad d(0)= 0.4 \quad \nu=0.5 \quad N=200$).

Small $\alpha $ and $n$

\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{plosum.eps}}\end{figure}
Figure 10: Variation of the dispersion index $y$ with $n$, the initial "subjective" number of collected opinions ( $\alpha=1-1/n, \quad d(0)= 0.5 \quad \nu=1.0 \quad N=1000$). The initial threshold value of 0.5 if kept constant would yield consensus with only one cluster.
A more complicated dynamics is observed for lower values of $n$ and $\alpha $. Apart from the expected main clusters, one also observes large and small clusters plus isolated individuals (outliers).

For $d(0)=0.5$ (which would yield consensus with only one cluster if kept constant) and $\alpha=0.5$ (corresponding to$n=2$, i.e. agents giving equal weight to their own opinion and to the external opinion), more than ten clusters unequal in size are observed plus isolated outliers. One way to characterise the dispersion of opinions with varying $\alpha $ is to compute $y$ the relative value of the squared cluster sizes with respect to the squared number of opinions.
 
 

\begin{displaymath}y = \frac{\sum_{i=1}^n s_i^2}{(\sum_{i=1}^n s_i)^2}\end{displaymath} (15)
For $m$ clusters of equal size, one would have $y=1/m$. The smaller $y$, the more important is the dispersion in opinions. Figure 10 shows the increase of the dispersion index $y$ with $n$ ($n-1$ is the initial "subjective" weight of agent's own opinion).

As previously noticed on figure 7, some outliers do not aggregate in the large clusters. The origin of these isolated agents is due to randomness of:

\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{ploev31aug.eps}}\end{figure}
Figure: Time chart of opinions and thresholds (for constant $\alpha=0.7 \quad d(0)= 0.4 \quad \nu=0.5 \quad N=1000$). Red '+' represent opinions and green 'x' represent thresholds.
The time pattern of thresholds appearing as green bands on figure 11 give us some insight on these effects. One band corresponds to a given number of opinion exchanges experienced by the agents: the upper band corresponds to the variance after one exchange, the second upper to two exchanges and so on. The lower bound of a band corresponds to the result of interactions between very close opinions when the second term in equation 4 is negligible. The vertical width of a band is due to this second term, which relative importance to the first can be estimated from the figure: it is roughly 10 perc. (for $\nu=0.5$). The horizontal width of a band corresponds to the fact that different agents are experiencing the same number of updating at different times: rough evaluations made on figure 11 show that most agents have their first exchange between time 0 and 4000, and their fifth exchange between 1000 and 12 000.

When the decrease of threshold and the clustering of opinions is fast, those agents which 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 important when convergence is fast, i.e. when $n$ and $\alpha $ 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! For $\alpha=0.7$$d(0)= 0.4$$\nu=0.5$ and $N=1000$, the parameters of figure 11, we found that mainstream agents in the two attractors account for 43 and 42 perc. of the population while 15 perc. are in the minority peaks.

The results of the dynamics are even more dispersed for lower values of $\alpha $. In this regime, corresponding to "insecure agents" who don't value their own opinion more than those of other agents, we observe more clusters which importance and localisation 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 this regime resembles quenching to a frozen configuration, thus maintaining many "defects" (e.g. here the outliers), while the opposite regime it resembles annealing (with suppression of defects).

In fact, one way to evaluate the influence of the two effects, randomness of the time at which agents are sampled from randomness of pairing, is to compare the standard random updating iteration that we have used to parallel updating: in a parallel updating, a random pair matching of all agents is first realised and all pairs are then updated simultaneously. Parallel updating then suppresses randomness of updating time: only randomness of pairing remains.

We found by comparing the two algorithms, random and parallel, for the same set of parameters, that parallel updating only slightly reduces the number of outliers. We can then conclude that most of the observed disorder results from the randomness of pairing.

Distribution and Dynamics of thresholds

Finally, an obvious set of simulations to perform is to have a distribution of thresholds and to let these thresholds evolve according to one of the two rules, $\alpha $ constant or decreasing, according to equation 5.
\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{plogtvar.eps}}\end{figure}
Figure: Time chart of opinions for a distribution of varying thresholds (constant $\alpha=0.7 \quad \nu=1 \quad N=200$). The initial thresholds are uniformly distributed on $[0,0.4]$.
We used a uniform distribution of thresholds on $[0, d_{max}]$ and observed, not surprisingly since some thresholds are close to zero even at initial times, that clusters and outliers are in larger number than for single initial thresholds. Figure 12 displays the time evolution of opinions for a constant value of $\alpha=0.7$$\nu=1$ and $ N=200$. The initial thresholds were uniformly distributed on $[0,0.63]$. Clusters correspond to agents with larger thresholds, outliers to thresholds close to 0.

Convergence times differ according to the size of the clusters: agents in large clusters have more occasions to update their opinion in proportion to the number of agents in the same cluster. Small clusters then need longer times to get stabilised.

These results generalise and summarise our previous findings in sections 4, 5.1 and 5.2.


Vector opinions

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 characterised by a vector of $m$ binary 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 between the two vectors). Here, we only treat the case of complete mixing; any pair of agents might interact and adjust opinions according to how many opinions they share.6 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 adjustment are as follows: when opinions on a subject differ, one agent (randomly selected from the pair) is convinced by the other agent with probability $\mu$. Obviously this model has connections with population genetics in the presence of sexual recombination when reproduction only occurs if genome distance is smaller than a given threshold. Such a dynamics results in the emergence of species (see Higgs and Derrida 1991). 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.

Results

We observed once again that $\mu$ and $N$ only modify convergence times towards equilibrium; the most influential factors are threshold $d$ and $m$ the number of subjects under discussion. Most simulations were done for $m=13$. For $N=1000$, convergence times are of the order of 10 million pair iterations. For $m=13$: The same kind of results are obtained with larger values of $m$: two regimes, uniformity of opinions for larger $d$ values and extreme diversity for smaller $d$ values, are separated by one $d_c$ value for which a small number of clusters is observed (e.g for $m=21$,$d_c=5$$d_c$ seems to scale in proportion with $m$ ).

Figure 13 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 $d$.

\begin{figure}\centerline {\epsfxsize=120mm\epsfbox{afflog.eps}}\end{figure}
Figure 13: Log-log plot of average populations of clusters of opinions arranged by decreasing order for $N=1000$ agents ($\mu =1$).

Conclusion

The main lesson from this set of simulations is that opinion exchanges restricted by a proximity threshold result into clustering of opinions. 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.

Continuous opinions and binary strings share some similarity: clustering with a number of clusters decreasing with $d$. But continuous opinions display a regular decrease of cluster number with $d$ while binary strings display a phase transition from consensus to a large multiplicity of clusters.

When one introduces dynamics on thresholds on the basis that agents interprete opinion exchange as sampling a distribution of opinions:

One can of course question the genericity of the results that were obtained with such simple models. In fact the main result, namely clustering, would not be canceled but rather re-inforced by two most direct generalisations of the model: We can then conclude that clustering into different opinions groups is the rule as soon as openness is limited.
 
 

Acknowledgments:

We thank David Neau and the members of the IMAGES FAIR project, Edmund Chattoe, Nils Ferrand and Nigel Gilbert for helpful discussions. GW 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. Thanks to Rainer Hegselmann for pointing us the references to the "consensus" literature. This study has been carried out with financial support from the Commission of the European Communities, Agriculture and Fisheries (FAIR) Specific RTD program, CT96-2092, "Improving Agri-Environmental Policies : A Simulation Approach to the Role of the Cognitive Properties 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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Footnotes

... Statistique1
Laboratoire associé au CNRS (URA 1306), à l'ENS et aux Universités Paris 6 et Paris 7
... mode2
The "consensus" literature most often uses parallel iteration mode when they suppose that agents average at each time step the opinions of their neighbourhood. Their implicit rationale for parallel iteration is that they model successive meetings among experts.
... 2000)3
The other extreme is the absence of any threshold which yields consensus at infinite time as earlier studied in Stone 1961 and others.
... 34
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 outcome of dynamics such as the 1/2d rule can be expressed as true with a probability close to one in the limit of large $N$; but one can often generate a deterministic sequence of updates which would contradict the "most likely" prediction.
... variance5
In fact adding the second term would compensate the decay in variance due to the multiplication by $\alpha $ in the limit of large $\nu$; for finite $\nu$, partial compensation depends on the form of the distribution of opinions, but anyway, variance decays exponentially with a smaller rate than when $\nu$ is close to 0
... share.6
The bit string model shares some resemblance with Axelrod's model of disseminating culture (Axelrod 1997) based on adjustment of cultures as sets of vectors of integer variables characterising agents on a square lattice.

Gerard Weisbuch

2001-11-26