Enhanced or distorted wisdom of crowds? An agent-based model of opinion formation under social influence
PP. Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence
Submitted for publication
Enhanced or distorted wisdom of crowds?An agent-based model ofopinion formation under social influence
Pavlin Mavrodiev, Frank Schweitzer
Chair of Systems Design, ETH Zurich, Switzerland
Abstract
We propose an agent-based model of collective opinion formation to study the wisdom ofcrowds under social influence. The opinion of an agent is a continuous positive value, denotingits subjective answer to a factual question. The wisdom of crowds states that the average ofall opinions is close to the truth, i.e. the correct answer. But if agents have the chance toadjust their opinion in response to the opinions of others, this effect can be destroyed. Ourmodel investigates this scenario by evaluating two competing effects: (i) agents tend to keeptheir own opinion (individual conviction β ), (ii) they tend to adjust their opinion if they haveinformation about the opinions of others (social influence α ). For the latter, two differentregimes (full information vs. aggregated information) are compared. Our simulations showthat social influence only in rare cases enhances the wisdom of crowds. Most often, we findthat agents converge to a collective opinion that is even farther away from the true answer.So, under social influence the wisdom of crowds can be systematically wrong. Problems of collective decisions are tackled in different scientific disciplines, from biology to so-ciology, from computer science to management science, from robotics to statistical physics. Butthe overarching research interest is the same: Collective decision processes should, in the bestcase, ensure an unanimous outcome, often denoted as consensus [5]. Taking a social perspective,individuals likely have deviating opinions regarding a particular issue. But thanks to social in-teractions during the process of collective opinion formation, individuals are able to adapt theiropinions such that eventually a majority converges to the same opinion. This ideal picture is dis-torted in reality in several ways. To put forward the problem addressed in this paper: convergingto the same opinion does not imply converging to the right opinion. Collective decision processescan easily converge to a consensus that is wrong in an objective sense, because of social influenceamong individuals.The term social influence encompasses several aspects that are underestimated, or not yet un-derstood regarding their consequences for opinion dynamics [10]. For example, what is the role1/18 a r X i v : . [ phy s i c s . s o c - ph ] A ug . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication of social pressure, external influences, or affective involvement in collective decisions? Of partic-ular importance is the question how individuals respond to the information they obtain aboutthe opinions of others. This question will be further addressed in our paper. For a systematicinvestigation, we utilize agent-based models that allow to vary such responses and to study theirimpact on the collective opinion.Many agent-based models of opinion dynamics use a binary characterization of opinions, e.g. { , } or {− , + } . A particular class of stochastic models, the so-called “voter models” [6, 8], studyhow agents respond to the frequencies of these opinions in a predefined neighborhood [2, 9]. In thelinear voter model, the majority rule assumes that the opinion of the (local) majority is adoptedwith a probability directly proportional to the frequency of a given opinion. It can be shownanalytically that such assumptions usually result in consensus. The question thus is how longit takes before this outcome is reached [38]. Interestingly, a slow-down of the opinion dynamicson the agent level, i.e. a certain reluctance to change the opinion, can lead to a speed up of theopinion dynamics on the systemic level: consensus is reached faster [37]. Also, the topology ofthe social network that facilitates agent interactions has an important influence on the consensusformation [26], as well as possible nonlinear responses to the frequency of opinions (e.g. minorityvoting, voting against the trend, etc.) [35]. Eventually, such models can be extended to includeassumptions of social impact theory [17, 27], such as different weights of opinions, persuasionfrom agents with different opinions and support from agents with the same opinion [15].The advantage of this model class is its analytical tractability, in addition to computer simula-tions, its disadvantage is in the many simplifications made [33]. More realistic models are basedon a continuous characterization of opinions, e.g. as positive real numbers in a given interval, [ , ] . The most prominent class of such models is the “bounded confidence model” that assumesagents only interact if the difference in their opinions is less than a certain threshold ε [7, 14].In this case, agents converge toward the mean of their opinions because of their interactions.The possibility to reach consensus then depends on the value of ε [18]. To foster consensus,assumptions about the topology of social interactions (e.g. group interactions) [25], hierarchicaldecisions [28], the influence of in-groups (agents known from previous interactions) [13] can betaken into account.Still, this model class is rather abstract, focusing on effects on the conceptual level. Towardsmore realistic models, the next step is to consider multi-dimensional opinions [30, 31]. Usually anindividual’s opinions on different subjects are not independent, but show correlations such that,out of a multi-dimensional opinion space, characteristic dimensions, e.g. left-right , conservative-liberal emerge [30]. In addition to the simple attractive force assumed in the bounded confidencemodel (convergence toward the mean), now we have to consider also repulsive forces: agentsincrease the distance of their opinions as a result of their interaction. In political space, thisoften leads to polarized opinions [3, 21]. Consensus in such scenarios is a rare exception, studies2/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication rather focus on the possibility to obtain qualified majorities [1, 4, 12]. Such models, while morecomplex, already allow to capture relations in real-world surveys. They can be further enhancedby including the affective response of individuals and emotional influences [30, 36], in additionto social ones.In conclusion, to formally model collective decision processes, we can build on a number of agent-based approaches at different levels of complexity. In this paper, we will use the second modelclass described above, which uses a continuous opinion representation and a mutual convergenceof individual opinions, to describe a particular phenomenon of collective decisions, namely the wisdom of crowds (WoC). It applies to scenarios with factual questions where a true opinion T exists, even if it is not known to individuals [11]. Our running example: “How long is the borderbetween Switzerland and Italy in kilometers?” [19] has a true answer (734 km), while individualsusually only have raw estimates. The WoC effect then states that the average over all individualopinions is remarkably close to the true opinion. So, it denotes a purely statistical effect but,interestingly, there is a lot of empirical evidence for it [16, 20, 29]. This is quantified by the small collective error, E , which is the squared difference between the average and the true opinion.According to Surowiecki [39], four criteria are required to form a wise crowd : diversity of opinions,independence of opinions, some expertise, and the ability to aggregate individual opinions intocollective opinions. “Expertise” only refers to the basic ability to give a meaningful answer to thequestion, e.g. the border of Switzerland is neither 0.1 kilometers, nor 3 million miles. The abilityto aggregate implies that someone, e.g. the social planner, indeed has access to all individualopinions, to calculate the aggregate. The other two criteria are more important to us. Thediversity of opinions is large only if individuals have independent opinions. But what happensif individuals obtain information about the opinions of others and then have the opportunity torevise their own opinion? This question was investigated in a large empirical study [19], whereindividuals had to answer the same question consecutively a number of times, while receivingdifferent information from others. In the so-called aggregated-information regime, they obtainedat each time step information about the aggregated opinion of others. In the full-informationregime, they instead obtained at each time step information about each individual opinion.These two information regimes allowed them to revise their own opinion accordingly.The study had two important findings: (i) In the presence of information about the opinions ofothers, the group diversity of opinions drastically decreased. That means, individuals tried toconverge with their opinions, i.e. subjectively they got the impression to reach consensus. (ii)Despite this convergence, the collective error, which measures the distance to the truth, did not decrease, but increased. That means, the group collectively converged to an opinion which wasobjectively the wrong one. This scenario, which was obtained quite frequently, is very dangerousbecause the group, converging to a common opinion, was collectively convinced that the wrong3/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication opinion was the right one. This effect was stronger in the full-information regime compared tothe aggregated-information regime.To answer the question, under what conditions this will happen, we need an agent-based modelthat reflects these two different information regimes and further allows to vary the strength ofthe social influence of other opinions, in comparison to the own conviction. The aim of our paperis to provide such a model, as an extension of the bounded confidence model. In particular,we want to model the full-information regime, which has shown the strongest effects, while theaggregated-information regime is used as a reference case.The rest of the paper is organised as follows. Section 2 introduces our agent-based model of opin-ion dynamics and relates it to macroscopic indicators to quantify the WoC effect. Section 3 thenpresents the results of extensive agent-based simulations with respect to the three macroscopicmeasures of the wisdom of crowds: the collective error, the group diversity of opinions, and theWoC indicator. Finally, the main conclusions are summarised in Section 4.
With the term “micro dynamics” we refer to the dynamics of the system elements, i.e. theagents. We build on the framework of
Brownian agents [32], which considers that agents havea continuous internal degree of freedom . In our case, this is the opinion x i ( t ) , different for eachagent i = , ..., N . We assume that the values of x i can be mapped to non-zero positive realnumbers, i.e. x i ( t ) > , but are not bound to a defined interval. This reflects the experimentalsituations described before [19].These opinions can change over time because of influences from other agents or simply becauseagents change their mind. x i ( ) denotes the initial value. We propose the following generaldynamics [34]: dx i ( t ) dt = − βx i ( t ) + N ∑ j F ij ( t ) + S i ( t ) (1)This dynamics resembles the Langevin equation to describe Brownian motion, therefore we callthese Brownian agents. The term − βx i is a damping term , i.e. it describes, in the absence ofother influences, a relaxation process toward zero at a time scale β . The term S i ( t ) is an additivestochastic force that describes random influences on the dynamics of opinions. We assume thatthese fluctuations are centered around the initial opinion x i ( ) , to reflect the fact that individualconvictions about certain subjects are likely to be long-lived. Hence, ⟨ S i ( t )⟩ ∝ x i ( ) up to a4/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication constant. We find it more convenient to rewrite the stochastic part as: S i ( t ) = βx i ( ) + Aξ i ( t ) ; ⟨ ξ i ( t ) = ⟩ ; ⟨ ξ i ( t ) ξ i ( t ′ )⟩ = δ ( t − t ′ ) (2) ξ i ( t ) is Gaussian white noise, i.e. it is not correlated in time and zero on average, A denotes thestrength of the stochastic force and is set equal to all agents.The term F ij ( t ) eventually describes how the change of opinion of agent i is influenced by theopinion of other agents j . To better understand its impact, let us first assume F ij ( t ) ≡ , i.eagents do not have any information about the opinions of others. We call this the no-informationregime . It implies that agents update their opinions stochastically without considering any otherinformation. With Eqs. (1), (2) the opinion dynamics for this case simply reads: dx i ( t ) dt = β [ x i ( ) − x i ( t )] + Aξ i ( t ) (3)This stochastic equation denotes a standard Ornstein-Uhlenbeck process which also has an ana-lytic solution [22]. In this case the time average of the individual opinion, x i ( t ) , equals x i ( ) forlarge t with decreasing variance.At difference with this trivial case, in this paper we discuss the case that agent i has full in-formation about the opinions of others and can take these into account in weighted manner, toupdate her own opinion. This is reflected in the following assumption for F ij ( t ) : F ij ( t ) = w ij [ x j ( t ) − x i ( t )] (4)That means the social influence from the opinion of other agents increases with the differencebetween opinions. While this sounds like a simplified assumption, it has been empirically justifiedin [24], therefore we use it here.The coupling variable w ij is chosen to be inversely proportional to the difference in opinions: w ij = N i + exp {∣ x j − x i ∣ / α } ; N − i = N ∑ k = + exp {∣ x k − x i ∣ / α } (5)with a normalization constant N i , such that ∑ Nj = w ij = . The parameter α acts as a measureof the strength of the social influence in the population. Small values for α indicate that agentsare less susceptible to others’ opinions. Conversely, large values for α imply that even largedifferences in opinons between agents do not matter much, therefore agents can resist strongersocial influence. In the limit case that ∣ x j − x i ∣ can be ignored, which is discussed further below,we see from Eqn. (5) that for α → also w ij → , which makes sense. Further, for the relevantrange of values . ≤ α ≤ . , w ij ∝ α . 5/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication
In summary, the stochastic dynamics for the opinion of agent i in the full information regimebecomes: dx i ( t ) dt = N N ∑ j = N i + exp {∣ x j − x i ∣ / α } [ x j ( t ) − x i ( t )] + β [ x i ( ) − x i ( t )] + Aξ i ( t ) (6)To make the impact of individual conviction, represented by the second term including β , com-parable to the impact of social influence, represented by the first term including α , we chooseparameter values ≤ α ≤ , and ≤ β ≤ . The above dynamics does not easily lend itself toanalytical treatment, hence we will simulate it in Section 3. But to interpret the results, we firstneed to think about some comparison. We want to compare our results to a reference case, in which agents have information only aboutthe average opinion of all other agents. We call this the aggregated-information regime [22, 23].In this case the coupling variable w ij is effectively a constant , equal for all agents: w ij ≡ α . Then,we can express the influence of other opinions as: N N ∑ j = F ij ( t ) = α [⟨ x ( t )⟩ − x i ( t )] ; ⟨ x ( t )⟩ = N N ∑ j = x j ( t ) (7)where ⟨ x ( t )⟩ is denoted as the mean opinion in the following. Eqn. (7) results in the stochasticdynamics: dx i ( t ) dt = α [⟨ x ( t )⟩ − x i ( t )] + β [ x i ( ) − x i ( t )] + Aξ i ( t ) (8)We note that Eq. (8) bears similarities with the bounded confidence model [14, 18, 34], in whichagents update their opinion as follows: dx i ( t ) dt = N ∑ j w b ij [ x j ( t ) − x i ( t )] ; w b ij = α Θ [ z ij ( t )] ; z ij ( t ) = ε − ∣ x i ( t ) − x j ( t )∣ (9)The coupling variable w bij reflects that agents only influence each other if the absolute differencesin their opinions is smaller than a confidence interval ε . Θ [ x ] denotes the Heavyside function,i.e. Θ [ x ] = only for x ≥ and Θ [ x ] = otherwise. This means that agents consider the averageopinions of others, but take this average only over those opinion not to far away from the ownopinion.Compared to this case, our aggregated regime considers the information of all other agents on agiven agent and therefore is similar to a mean-field limit. Additionally, the influence of the initial6/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication opinion and stochastic effects are taken into account.
We remind that the wisdom of crowds (WoC) is expected to work if the diversity of individualopinions is large , while the deviation of the average opinion from the true value T is small .Therefore, in line with previous studies [19], we will use the collective error , E , the group diversity , D , and the wisdom of crowds indicator , W , as macroscopic measures to evaluate these conditions.In order to define these measures, we need to keep in mind that the distribution of opinions in theconsidered experimental setup is very broad, and can be proxied by a log-normal distribution.Therefore, the average opinion is not well represented by the arithmetic mean, ⟨ x ⟩ = ∑ i x i / N ,but by the geometric mean, which is equivalent to the arithmetic mean of the log values, ⟨ ln x ⟩ ,[19]. Further, because opinions change over time according to Eqn. (6), we are only interested inthe long-term (LT) values of these macroscopic measures.The long-term collective error E LT shall be defined as the squared deviation of the average opinionfrom the true value, T : E LT = [ ln T − ⟨ ln x LT ⟩] , (10)The long-term group diversity D LT is given by the variance of the distribution of opinions: D LT = Var [ ln x LT ] = N N ∑ i = [ ln x i LT − ⟨ ln x LT ⟩] = ⟨[ ln x LT ] ⟩ − ⟨ ln x LT ⟩ (11)The WoC indicator W is measured by the deviation of the truth from the long-term medianopinion , ˆ x LT . This means that for N / of all opinions x i LT ≤ ˆ x LT holds, and x i LT ≥ ˆ x LT for theremaining N / . In plain words, the value of W indicates how central the position of true value T is within the distribution of opinions. W achieves a maximum value if the truth can be bracketedby the two most central opinions.More formally, if X N is the set of all N opinions, we have the set of ordered opinions, { ¯ x i ∣ ¯ x i ∈ X N , ¯ x i ≤ ¯ x j , ∀ i < j } . The indicator W is defined as max { i ∣ ¯ x i ≤ T ≤ ¯ x N − i + } . It reaches amaximum of [ N / ] when the truth is either the most central opinion or is bracketed by thetwo most central opinions, and a minimum of 0 when T ∉ ( ¯ x , ¯ x N ) . To illustrate this, let usassume we have a set of 100 ordered opinions such that { , , , ..., , } . The two most centralopinions in this set are 49 and 50, or 50 and 51, respectively. If the true value would be 50, it isjust bracketed by the two most central opinions. This would give us W = = N / . If the truevalue would be 70, it is no longer bracketed by the two most central opinions. Instead it is 20positions away from the central opinions, hence W = .7/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication
In order to perform agent-based simulations using Eqs. (6), (8) we need to specify the initialdistribution of opinions, from which N values x i ( ) are sampled. We take as an input twodifferent log-normal distributions P ( ) ( x ) , P ( ) ( x ) with two different mean values µ ( ) ln x =-2.9 and µ ( ) ln x =-3.0, but the same variance σ x =0.72. The histograms of the N = opinions sampledfrom these two initial distributions are plotted in Figure 1.Further, we have considered three different true values, ln T =-2.00, ln T =-2.90, ln T =-3.12. Withthese values and the parameters of the initial distributions, we can determine the initial collectiveerrors E( ) from Eqn. (10) and the initial group diversity D( ) from Eqn. (11). With these Estimate F r equen cy -5 -4 -3 -2 -1 0 Estimate-6 -5 -4 -3 -2 -1
Figure 1: Histograms of initial opinions, x i ( ) , sampled from two different log-normal distribu-tions. The blue lines indicate the different mean values µ ( ) ln x =-2.9 (left) and µ ( ) ln x =-3.0 (right),the variance σ x =0.72 is the same. Note the logarithmic values of the x -axis.specifications, we run N = agent-based simulations in parallel. The dynamics are solvedusing the th order Runge-Kutta method for the full-information regime, Eqn. (6), and the Eulermethod for the aggregated-information regime, Eqn. (8). We used a constant time step ∆ t = . ,and a final time t = , to obtain the long-term values. The noise intensity was chosen as arather small value, A = − . We present all our results as heat maps of the relevant quantities, E LT , D LT or W LT , dependenton the two model parameters, social influence, α , and individual conviction, β , for which we haveperformed a thorough parameter sweep. The right column always refers to the full-information8/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication regime, the focus of our paper. The left column shows the matching results for the referencecase, the aggregated-information regime, which has been discussed in [23] regarding agent-basedsimulations and in [22] regarding analytic solutions for the macroscopic measures.Figure 2 shows the results for E LT for three different initial conditions (a)-(c). The color codeindicates the value of E LT : (red) for high values, which is bad, and (blue) for low values, which isgood. We note that for each plot the color code represents different values. A black line indicatesparameter combinations ( α, β ) for which E LT = E( ) . Mostly, these lines are not noticeablebecause they coincide with α = .In Figure 2(a) the initial condition is chosen such that the average initial opinion is far away from the true value, i.e. the initial collective error is high . Then, we find that an increase of socialinfluence α considerably decreases the collective error, which means it improves the wisdom ofcrowds, whereas individual conviction has little impact on the outcome. This positive finding ismuch stronger in the full-information regime, in which agents have access to the opinions of allother agents. It decreases if they have access only to the aggregated opinion.The situation inverts if instead the initial condition is chosen such that the average initial opinionis close to the true value, i.e. the initial collective error is low . Then an increase in social influence α can worsen the outcome, leading the average opinion farther away from the true value, asshown in Figure 2(a). This is the most dangerous case: agents collectively converge to a commonopinion, but this, from an objective perspective, is the wrong one. This effect, again, is muchstronger in the full-information regime. In the aggregated-information regime, we could stillidentify parameter ranges with low α , where the wisdom of crowds is not much distorted. Butmore information for the agents destroys this possibility. Individual conviction, as the secondinfluential parameter, impacts the results mainly for the aggregated-information regime, but isless noticeable in the full-information regime.This leads us to the question why there is this monotonous deterioration of the wisdom of crowdswith increasing social influence. A theoretical investigation of the average opinion ⟨ ln x ( t )⟩ in theaggregated-information regime [22] tells us that it can only increase over time, d ⟨ ln x ( t )⟩ / dt > .Therefore, if initially ⟨ ln x ( )⟩ < ln T , there is a chance that ⟨ ln x ( t )⟩ → ln T over time, if α is large enough and β is not too strong. However, if initially ⟨ ln x ( )⟩ > ln T , the interactiondynamics of the agents can only lead their average opinion further away from the true value.This is the case shown in Figure 2(b), and it becomes worse both if social influence increases orif full information about the opinion of others is provided.Consequently, we should also expect situations without a monotonous deterioration. This isshown in Figure 2(c), which illustrates a non-monotonous dependence of the collective error onthe social influence. Here, the initial collective error is also low, as in (b), but now the initialdistribution of opinions is such that ⟨ ln x ( )⟩ < ln T . Hence, we find that for low values of thesocial influence the collective opinion indeed converges to the true value (indicated by deep blue).9/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication . . . I nd i v i dua l C on v i c t i on (a) . . . I nd i v i dua l C on v i c t i on . . . (b) . . . . . . Social Influence I nd i v i dua l C on v i c t i on (c) . . . Social Influence I nd i v i dua l C on v i c t i on Figure 2: Agent-based simulations of the long-term collective error E LT (color coded) depen-dent on the values of social influence α ( x -axis) and individual conviction β ( y -axis). (left) Aggregated-information regime, Eqn. (8), (right) full-information regime, Eqn. (6). Param-eters: N =100, t =3000, A = − , ∆ t =0.01. Different initial conditions: (a) E (0)=0.80, ln T =-2.00, ⟨ ln x ( )⟩ =-2.9, (b) E (0)=0.02, ln T =-3.12, ⟨ ln x ( )⟩ =-3.0, (c) E (0)=0.01, ln T =-2.90, ⟨ ln x ( )⟩ =-3.0. Black contour lines indicate regions in the parameter space where E LT = E( ) .Note that these are vertical lines at α =0 in all plots, in (c) there is an additional line.But the parameter range is rather small, and much smaller than in the aggregated-informationregime. Hence, we can conclude that only low social influence can improve the wisdom of crowds,10/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication whereas large social influence most likely deteriorates it.The impact of the second model parameter, individual conviction β , becomes more visible forthe chosen initial conditions (c). Taking a fixed social influence, e.g. α =0.5, we find that indi-vidual conviction can improve the collective error, because it acts as a kind of reluctance against converging to the (wrong) collective opinion.We also point to the visible black line which indicates the conditions under which the collectiveerror does not change, i.e. is equal to the initial value. In the full-information regime, this isnot a straight line as found for the reference case. That means, non-linear effects become muchstronger if agents have access to all other opinions. The second macroscopic measure for the wisdom of crowds is shown in Figure 3, again in com-parison to the reference case. Note that the long-term group diversity D LT does not depend onthe initial conditions, but only on ( α, β ) and ⟨ δ ( )⟩ . Therefore the plot is the same for all initialconditions shown in Figure 2. The color scale is chosen such that a large group diversity, whichis good, is indicated by red , whereas a small group diversity is shown in blue . We find that in thefull-information regime the group diversity is drastically reduced by means of social influence.Its high initial value cannot be maintained, not even for the case of a high initial conviction β .This becomes a problem for those parameter ranges, where a low group diversity is combined witha high collective error. In this case, the agent population collectively converges to the (objectively) wrong opinion. We see that this is most likely the case for the full-information regime, whereasin the reference case the parameter range for acceptable values of the group diversity is muchlarger. Finally we discuss the results for the long-term WoC indicator W LT , which measures the distancebetween the truth and the median of the opinion distribution. Ideally the true value should be“central” with respect to the opinion distribution, which means for the given configurations, itshould have values around [ N / ] =50. The color code is chosen such that (red) indicates goodvalues, (blue) a bad outcome.Figure 5 presents the results for the full-information regime and the reference case, for twodifferent initial conditions, which are also sketched in Figure 4. Initial condition (a), which is thesame as in Figure 2(b), has a small initial collective error, but the true value ln T is smaller thanthe initial collective opinion ⟨ ln x ( )⟩ (which is not the median, but the mean). This implies, in11/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication . . . Social Influence . . . Social Influence
Figure 3: Agent-based simulations of the long-term group diversity D LT (color coded) depen-dent on the values of social influence α ( x -axis) and individual conviction β ( y -axis). (left) Aggregated-information regime, Eqn. (8), (right) full-information regime, Eqn. (6). Parame-ters: N =100, t =3000, A = − , ∆ t =0.01. Initial condition: D (0)=0.72. ln T ln x (a) ln xln T (b)Figure 4: Sketch of the initial opinion distribution (blue solid line) and the median (red solid line).Two different initial conditions (also used in Figure 5): (a) W( ) =43, E( ) =0.02, ln T =-3.12. (b) W( ) =46, E( ) =0.01, ln T = − . . Both: ⟨ ln x ( )⟩ =-3.0, D( ) =0.72. The position of thefinal opinion distribution (blue dashed line) and the median (red dashed line) is determined bythe parameters ( α, β ) . Despite the more favorable initial condition (b), a stronger social influence α leads to a worse outcome.accordance with the discussion of Figure 2(b) that the collective opinion cannot converge to thetrue value. Further, ln T is not very central with respect to the initial distribution of opinions.That means the initial WoC indicator is much lower than [ N / ] =50, which corresponds to themedian. So, in summary, condition (a) reflects rather bad initial condition.Initial condition (b), on the other hand, which is the same as in Figure 2(c), is comparably bettersuited for reaching the true value. It also has a small initial collective error, but the true value ln T is larger than the initial collective opinion ⟨ ln x ( )⟩ , which means the collective opinion12/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication can possibly converge to the true value. Further, ln T is more central with respect to the initialdistribution of opinions. That means the initial WoC indicator is much higher. . . . Social Influence I nd i v i dua l C on v i c t i on (a) . . . Social Influence I nd i v i dua l C on v i c t i on . . . Social Influence I nd i v i dua l C on v i c t i on (b) . . . Social Influence I nd i v i dua l C on v i c t i on Figure 5: Agent-based simulations of the long-term wisdom-of-crowds indicator W LT (colorcoded) dependent on the values of social influence α ( x -axis) and individual conviction β ( y -axis). (left) Aggregated-information regime, Eqn. (8), (right) full-information regime,Eqn. (6). Parameters: N =100, t =3000, A = − , ∆ t =0.01. Different initial conditions: (a) W( ) =43, E( ) =0.02. (b) W( ) =46, E( ) =0.01. Both: ⟨ ln x ( )⟩ =-3.0, D( ) =0.72.Still, as we illustrate in Figures 4, 5, the more favorable initial condition (b) does not automat-ically lead to a better wisdom of crowds. This is due to the opinion dynamics , which is rathercontrolled by the parameters ( α, β ) . Because of the interplay between social influence α andindividual conviction β , agents change their opinions such that (i) the final opinion distributionbecomes more narrow, and (ii) the mean ⟨ ln x LT ⟩ shifts toward higher values. How far it shiftsto the right depends on these two social parameters.As Figure 4 indicates, even the better initial condition can lead to the worse outcome. This hasa systematic and a parameter dependent cause. As for the systematic one, because the collectiveopinion shifts toward higher values, also the range of estimates needed to bracket the true value13/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication will increase. This explains why the WoC indicator most often ends up with lower than the initialvalues.The parameter dependence displayed in Figure 5 shows that in most cases an increasing socialinfluence α leads to a decreasing WoC indicator (blue region). This effect again is strongerin the full-information regime. But an increasing individual conviction is able to mitigate thedeterioration.For the initial condition (b), we observe a non-monotonous dependence of W LT on the strengthof the social influence, very similar to the dependence shown in Figure 2(c) for the collectiveerror. In this case, rather small values of α enhance the chance to converge to the true value,which is shown by a high WoC indicator. Higher values of β are able to strengthen this effect,enlarging the favorable red region in Figure 5(b), even in the full-information regime.So, we can conclude that for both the collective error, Figure 2, and the WoC indicator, Figure 5,we find the same parameter regions where social influence can be beneficial for the wisdom ofcrowds. Provided suitable initial conditions, the collective opinion can even converge to the truevalue, indicated by the minimum collective error, E LT = and the maximum WoC indicator, W LT = [ N / ] . Despite the loss of group diversity, shown by low values of D LT in Figure 3, thecrowd becomes “wiser” for moderate values of the social influence. It is non-trivial and interestingto see that all observed effects are much stronger in the full-information regime, in comparison tothe reference case. Because the opinion dynamics is quite sensitive to the available information,the parameter ranges for a favorable outcome considerably decrease in the presence of strongeragent interactions, i.e. in the full-information regime. The wisdom of crowds refers to cases in which a larger number of individuals has an opinionabout a question for which a true answer exists, although this is not public knowledge. Then, itwas observed already by Galton in 1907 [11] that the average opinion is remarkably close to thetrue opinion. This works best if the diversity of individual opinions is large and all opinions areindependent. In most real-world scenarios, however, individuals become aware of the opinions ofothers and then tend to adjust their own opinion based on this information. To model such anopinion dynamics is the aim of the current paper.We have proposed an agent-based model that reflects two competing influences: (i) the individualconviction about the own initial opinion (parameter β ), (ii) the influence of the opinions of others(parameter α ). The latter reflects a mutual social influence via the exchange of information.Two different regimes are considered: a full-information regime in which agents get to know theopinion of each other agent, and an aggregated-information regime in which agents only get to14/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication know the average opinion. The model assumes that agents have a stronger incentive to adjusttheir opinion if the difference to other opinions, or to the average opinion, is larger. This isreasonable: if someone thinks the length of the border between Switzerland and Italy is 500 kmand then sees many opinions with values between 2000 and 3000 km, there is a stronger “force” toadapt to the majority (even that this is wrong: the true value is 734 km). This has been confirmedby experiments, already [19, 24]. Hence, social influence eventually leads to a decreasing groupdiversity of opinions. This is perceived as converging to some sort of consensus, which likely givesmore confidence in the outcome of the collective decision process.Unfortunately, as the same experiments have also shown, because of this convergence crowdseasily convince themselves that their collective opinion is the right one, even if it is objectivelythe wrong one. Thus, the main objective of our paper is to understand under which conditionsthis distorted wisdom of crowds could emerge. Our agent-based model plays an valuable rolein systematically evaluating these conditions (a) regarding the parameters involved, and (b)regarding the impact of the initial conditions.Our results demonstrate that the role of social influence cannot be reduced to simplistic messages.Instead, they reflect that collective opinion dynamics, as most social processes, are more complex.Specifically, we do find scenarios in which social influence is beneficial for the wisdom of crowds.If the average initial opinion is far away from the true opinion, i.e. the initial collective erroris large , social influence can help to reduce the collective error. But in cases where the initialcollective error is already small, social influence mostly distorts the wisdom of crowds, leadingto an increase of the collective error. There is only one situation, where the wisdom of crowdscan still be improved, namely if the initial average opinion is below the true opinion and thesocial influence is very moderate. The dependency of the WoC effect on the initial conditions israther critical because, under practical circumstances, these initial conditions are unknown. Thismeans, we cannot have a-priori knowledge whether social influence will enhance or distort theWoC.Our insights help to explain the “range reduction effect” found empirically [19]. Because of theadjustment of individual opinions, the opinion distribution, in particular its mean and its medianshifts over time such that the true opinion is displaced to peripheral regions of the opiniondistribution, while the collective opinion becomes narrowly centered around a wrong value. Thisgenerates a dangerous situation because relying on the average opinion in this case would give thewrong information. Our investigations show the range of parameters ( α, β ) where social influencegenerates such outcome, effectively distorting the wisdom of crowds.Comparing the full-information regime with our reference case, the aggregated-informationregime, we note that all effects are much stronger if the information about the opinions ofall agents becomes available. Individual conviction could counterbalance the social influence, butplays a lesser role. This is also understandable: in the full-information regime, someone not only15/18 . Mavrodiev, F. Schweitzer:Enhanced or distorted wisdom of crowds?An agent-based model of opinion formation under social influence Submitted for publication realizes the differences in opinions, in comparison to the own one, but also how many otherindividuals deviate from it. This is hidden in the aggregated-information regime, where some-one only knows the difference to the average opinion, hence its impact may be smaller from apsychological perspective.One could rightly argue against the latter point, because the wisdom of crowds only works withrespect to the average opinion which should be close to the true opinion. So, someone with abasic understanding of the wisdom of crowds should find the average opinion much more reliable,and influential, than knowing all other opinions. This argument ignores one main point: thewisdom of crowds only works if we have a larger number of independent opinions. Only then, theaverage opinion may generate the better signal. Once individuals get to know other opinions, theiropinions are no longer adjusted independently, but in response to the social influence generated.Hence, even a small social influence has the potential to distort the wisdom of crowds, as it wasfound in experiments and confirmed by our agent-based simulations.
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