Abstract

Models of opinion formation are used to investigate many collective phenomena. While social influence often constitutes a basic mechanism, its implementation differs between the models. In this paper, we provide a general framework of social influence inspired by the social psychological concept of cognitive dissonance.We only premise that individuals strive to minimize dissonance resulting from different opinions compared to neighbor individuals in a given social network. Within a game theoretic context, we show that our concept of dissonance reduction exhibits the basic properties of a coordination process. We further show that different models of opinion formation can be represented as best response dynamics within our general framework. Thus, we offer a unifying perspective on heterogeneous models and link them to rational choice theory.

Abstract

How do humans respond to indirect social influence when making decisions? We analysed an experiment where subjects had to repeatedly guess the correct answer to factual questions, while having only aggregated information about the answers of others. While the response of humans to aggregated information is a widely observed phenomenon, it has not been investigated quantitatively, in a controlled setting. We found that the adjustment of individual guesses depends linearly on the distance to the mean of all guesses. This is a remarkable, and yet surprisingly simple, statistical regularity. It holds across all questions analysed, even though the correct answers differ in several orders of magnitude. Our finding supports the assumption that individual diversity does not affect the response to indirect social influence. It also complements previous results on the nonlinear response in information-rich scenarios. We argue that the nature of the response to social influence crucially changes with the level of information aggregation. This insight contributes to the empirical foundation of models for collective decisions under social influence.

Abstract

Measuring culture and its dynamics through surveys has important limitations, but the emerging eld of computational social science allows us to overcome them by analyzing large-scale datasets. In this article, we study cultural dynamics through the votes in the Eurovision song contest, which are decided by a crowd-based scheme in which viewers vote through mobile phone messages. Taking into account asymmetries and imperfect perception of culture, we measure cultural relations among European countries in terms of cultural anity. We propose the Friend-or-Foe coecient, a metric to measure voting biases among participants of a Eurovision contest. To validate how this metric represent cultural anity, we designed a model of a random, biased Eurovision contest. Simulations of this model show how our metrics can detect negative anities and serve as an estimator for positive anities. We apply this estimator to the historical set of Eurovision contests from 1975 to 2012, nding patterns of asymmetry and clustering in the resulting networks. Furthermore, we dene a measure of vote polarization that, when applied to empirical data, shows a sharp increase within countries of the EU during 2010 and 2011. As a result, we measure how the recent political decisions of EU states inuence the way their citizens relate to the culture of other EU members, leading to stronger cultural biases in the way they vote in the Eurovision song contest.

Abstract

We study the role of hierarchical structures in a simple model of collective consensus formation based on the bounded confidence model with continuous individual opinions. For the particular variation of this model considered in this paper, we assume that a bias towards an extreme opinion is introduced whenever two individuals interact and form a common decision. As a simple proxy for hierarchical social structures, we introduce a two-step decision making process in which in the second step groups of like-minded individuals are replaced by representatives once they have reached local consensus, and the representatives in turn form a collective decision in a downstream process. We find that the introduction of such a hierarchical decision making structure can improve consensus formation, in the sense that the eventual collective opinion is closer to the true average of individual opinions than without it. In particular, we numerically study how the size of groups of like-minded individuals being represented by delegate individuals affects the impact of the bias on the final population-wide consensus. These results are of interest for the design of organisational policies and the optimisation of hierarchical structures in the context of group decision making.

Abstract

We study the role of hierarchical structures in a simple model of collective consensus formation based on the bounded confidence model with continuous individual opinions. For the particular variation of this model considered in this paper, we assume that a bias towards an extreme opinion is introduced whenever two individuals interact and form a common decision. As a simple proxy for hierarchical social structures, we introduce a two-step decision making process in which in the second step groups of like-minded individuals are replaced by representatives once they have reached local consensus, and the representatives in turn form a collective decision in a downstream process. We find that the introduction of such a hierarchical decision making structure can improve consensus formation, in the sense that the eventual collective opinion is closer to the true average of individual opinions than without it. In particular, we numerically study how the size of groups of like-minded individuals being represented by delegate individuals affects the impact of the bias on the final population-wide consensus. These results are of interest for the design of organisational policies and the optimisation of hierarchical structures in the context of group decision making.

Abstract

Noise-induced volatility refers to a phenomenon of increased level of fluctuations in the collective dynamics of bistable units in the presence of a rapidly varying external signal, and intermediate noise levels. The archetypical signature of this phenomenon is that –beyond the increase in the level of fluctuations– the response of the system becomes uncorrelated with the external driving force, making it different from stochastic resonance. Numerical simulations and an analytical theory of a stochastic dynamical version of the Ising model on regular and random networks demonstrate the ubiquity and robustness of this phenomenon, which is argued to be a possible cause of excess volatility in financial markets, of enhanced effective temperatures in a variety of out-of-equilibriumsystems and of strong selective responses of immune systems of complex biological organisms. Extensive nu-merical simulations are compared with a mean-field theory for different network topologies.

Abstract

Wisdom of crowds refers to the phenomenon that the aggregate prediction or forecast of a group of individuals can be surprisingly more accurate than most individuals in the group, and sometimes - than any of the individuals comprising it. This article models the impact of social influence on the wisdom of crowds. We build a minimalistic representation of individuals as Brownian particles coupled by means of social influence. We demonstrate that the model can reproduce results of a previous empirical study. This allows us to draw more fundamental conclusions about the role of social influence: In particular, we show that the question of whether social influence has a positive or negative net effect on the wisdom of crowds is ill-defined. Instead, it is the starting configuration of the population, in terms of its diversity and accuracy, that directly determines how beneficial social influence actually is. The article further examines the scenarios under which social influence promotes or impairs the wisdom of crowds.

Abstract

Social groups can be remarkably smart and knowledgeable when their averaged judgements are compared with the judgements of individuals. Already Galton [Galton F (1907) Nature 75:7] found evidence that the median estimate of a group can be more accu-rate than estimates of experts. This wisdom of crowd effect was recently supported by examples from stock markets, political elections, and quiz shows [Surowiecki J (2004) The Wisdom of Crowds]. In contrast, we demonstrate by experimental evidence (N = 144) that even mild social influence can undermine the wisdom of crowd effect in simple estimation tasks. In the exper-iment, subjects could reconsider their response to factual ques-tions after having received average or full information of the responses of other subjects. We compare subjects’ convergence of estimates and improvements in accuracy over five consecutive estimation periods with a control condition, in which no informa-tion about others’ responses was provided. Although groups are initially “wise,” knowledge about estimates of others narrows the diversity of opinions to such an extent that it undermines the wisdom of crowd effect in three different ways. The “social influence effect” diminishes the diversity of the crowd without improvements of its collective error. The “range reduction effect” moves the position of the truth to peripheral regions of the range of estimates so that the crowd becomes less reliable in providing expertise for external observers. The “confidence ef-fect” boosts individuals’ confidence after convergence of their estimates despite lack of improved accuracy. Examples of the revealed mechanism range from misled elites to the recent global financial crisis.

Abstract

Analyzing the data of the wisdom of crowd experiment by Lorenz et al. (1), Farrell (2) points out that information exchange improves individual estimates of answers to factual questions. He furthermore suggests that information exchange increases individual rewards and that this comes with an increase in confidence in their estimates. However, it has to be noted that individual rewards were revealed only after subjects specified their confidence. [This also shows that the reference to Danchin et al. (3) is not fitting, because in the foraging example, subjects were constantly aware of the external metric, namely the success of the foraging activities of other individuals.] Moreover, although the original experiment was designed to study social interaction effects, Farrell applies a psychological perspective. In fact, Farrell's commentary (2) focuses on individual-level effects, whereas Lorenz et al. (1) concentrated on group-level effects and show how aggregating micro-outcomes can lead to unexpected macro-level effects.

Abstract

The joint knowledge of many diverse individuals can outperform experts in estimation and decision-making problems. This wisdom of the crowd has been demonstrated in different societal areas such as internet search engines, political elections or stock markets. Recently, psychologists argued that humans may even simulate a diverse society in their own minds by drawing different answers from their brain (Vul & Pashler, 2008). The underlying idea is that individuals can access different knowledge areas in their brain, whose joint evaluation yields better estimates than their separate consideration. This article presents a mathematical treatment of the wisdom of crowds and two potential mechanisms to quantify the wisdom of crowds in one mind. The implications of both methods are analyzed and applied to new experimental data (N = 144), which contain five consecutive estimates from the same individuals. The theoretical and empirical analysis demonstrates limitations of the wisdom of crowds in one mind: Asking oneself several times is on average less powerful than asking only one other individual. This is due to the smaller diversity of estimates of similar individuals and the larger average bias to which they converge. Further, individuals cannot perform independent draws from an ‘‘internal distribution’’. Hence, there may be other mechanisms at work such as talking oneself into believing initial guesses or eliciting progressively wilder ones. ©2010

Abstract

A large extent of undetected norm violations may have positive effects for society. If many norm violations are hidden, society seems to be in good order so that actors are more willing to comply with the norms themselves. In this sense, igno-rance promotes norm compliance.We challenge this view by arguing that in scenar-ios, in which norms are controlled and enforced by third parties who receive rewards for their success, the opposite is true: Ignorance promotes norm violations. The rea-son is that unsuspicious inspectors who believe that little hidden norm violations are committed will spend less effort for detection, some formerly detected norm viola-tions will go undetected, norm targets will be less deterred from the lower detection probability and will commit more norm violations over time. This article develops a respective mathematical model and confirms the above described intuition.

Abstract

Models of continuous opinion dynamics under bounded confidence show a sharp transition between a consensus and a polarization phase at a critical global bound of confidence. In this paper, heterogeneous bounds of confidence are studied. The surprising result is that a society of agents with two different bounds of confidence (open-minded and closed-minded agents) can find consensus even when both bounds of confidence are significantly below the critical bound of confidence of a homogeneous society. The phenomenon is shown by examples of agent-based simulation and by numerical computation of the time evolution of the agents density. The result holds for the bounded confidence model of Deffuant, Weisbuch and others (Weisbuch, G. et al; Meet, discuss, and segregate!, Complexity, 2002, 7, 55-- 63), as well as for the model of Hegselmann and Krause (Hegselmann, R., Krause, U.; Opinion Dynamics and Bounded Confidence, Models, Analysis and Simulation, Journal of Artificial Societies and Social Simulation, 2002, 5, 2). Thus, given an average level of confidence, diversity of bounds of confidence enhances the chances for consensus. The drawback of this enhancement is that opinion dynamics becomes suspect to severe drifts of clusters, where open-minded agents can pull closed-minded agents towards another cluster of closed-minded agents. A final consensus might thus not lie in the center of the opinion interval as it happens for uniform initial opinion distributions under homogeneous bounds of confidence. It can be located at extremal locations. This is demonstrated by example. This also show that the extension to heterogeneous bounds of confidence enriches the complexity of the dynamics tremendously.

Abstract

Assume two different communities each of which maintain their respective opinions mainly because of the weak interaction between them. In such a case, it is an interesting problem to find the necessary strength of inter-community interaction in order for the two communities to reach a consensus. In this paper, the information accumulation system (IAS) model is applied to investigate the problem. With the application of the IAS model, the opinion dynamics of the two-community problem is found to belong to a wider class of two-species problems appearing in population dynamics or in the competition of two languages, for all of which the governing equations can be described in terms of coupled logistic maps. Tipping diffusivity is defined as the maximal inter-community interaction such that the two communities maintain different opinions. For a problem with a simple community structure and homogeneous individuals, the tipping diffusivity is calculated theoretically. As a conclusion of the paper, the convergence of the two communities to the same value is less possible the more overall interaction, intra-community and inter-community, takes place. This implies, for example, that the increase in the interaction between individuals caused by the development of modern communication tools, such as Facebook and Twitter, does not necessarily improve the tendency towards global convergence between different communities. If the number of internal links increases by a factor, the number of inter-community links must be increased by an even higher factor, in order for consensus to be the only stable attractor.

Abstract

A new theorem on conditions for convergence to consensus of a multiagent time-dependent time-discrete dynamical system is presented. The theorem is build up on the notion of averaging maps. We compare this theorem to results by Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005) about set-valued Lyapunov theory and convergence under switching communication topologies. We give examples that point out differences of approaches including examples where Moreau's theorem is not applicable but ours is. Further on, we give examples that demonstrate that the theory of convergence to consensus is still not complete.

Abstract

In nonlinear voter models the transitions between two states depend in a nonlinear manner on the frequencies of these states in the neighborhood. We investigate the role of these nonlinearities on the global outcome of the dynamics for a homogeneous network where each node is connected to $m = 4$ neighbors. The paper unfolds in two directions. We first develop a general stochastic framework for frequency dependent processes from which we derive the macroscopic dynamics for key variables, such as global frequencies and correlations. Explicite expressions for both the mean-field limit and the pair approximation are obtained. We then apply these equations to determine a phase diagram in the parameter space that distinguishes between different dynamic regimes. The pair approximation allows us to identify three regimes for nonlinear voter models: (i) complete invasion, (ii) random coexistence, and-- most interestingly-- (iii) correlated coexistence. These findings are contrasted with predictions from the mean-field phase diagram and are confirmed by extensive computer simulations of the microscopic dynamics.

Abstract

In this paper histograms of user ratings for movies (1,...,10) are analysed. The evolving stabilised shapes of histograms follow the rule that all are either double-or triple-peaked. Moreover, at most one peak can be on the central bins 2,...,9 and the distribution in these bins looks smooth `Gaussian-like’ while changes at the extremes (1 and 10) often look abrupt. It is shown that this is well approximated under the assumption that histograms are confined and discretised probability density functions of L'evy skew $α$-stable distributions. These distributions are the only stable distributions which could emerge due to a generalized central limit theorem from averaging of various independent random variables as which one can see the initial opinions of users. Averaging is also an appropriate assumption about the social process which underlies the process of continuous opinion formation. Surprisingly, not the normal distribution achieves the best fit over histograms observed on the web, but distributions with fat tails which decay as power-laws with exponent –(1 + $α$) . The scale and skewness parameters of the L'evy skew $α$-stable distributions seem to depend on the deviation from an average movie (with mean about 7.6). The histogram of such an average movie has no skewness and is the most narrow one. If a movie deviates from average the distribution gets broader and skew. The skewness pronounces the deviation. This is used to construct a one parameter fit which gives some evidence of universality in processes of continuous opinion dynamics about taste.

Abstract

A local culture denotes a commonly shared behaviour within a cluster of firms. Similar to social norms or conventions, it is an emergent feature resulting from the firms' interaction in an economic network. To model these dynamics, we consider a distributed agent population, representing e.g. firms or individuals. Further, we build on a continuous opinion dynamics model with bounded confidence ($ epsilon$), which assumes that two agents only interact if differences in their behaviour are less than $ epsilon$. Interaction results in more similarity of behaviour, i.e. convergence towards a common mean. This framework is extended by two major concepts: (i) The agent's in-group consisting of acquainted interaction partners is explicitly taken into account. This leads to an effective agent behaviour reflecting that agents try to continue to interact with past partners and thus to keep sufficiently close to them. (ii) The in-group network structure changes over time, as agents can form new links to other agents with sufficiently close effective behaviour or delete links to agents no longer close in behaviour. Thus, our model provides a feedback mechanism between the agents' behaviour and their in-group structure. Studying its consequences by means of agent-based computer simulations, we find that for narrow-minded agents (low $ epsilon$) the additional feedback helps to find consensus more often, whereas for open-minded agents (high $ epsilon$) this does not hold. This counterintuitive result is explained by simulations of the network evolution.

Abstract

We study an opinion formation model that takes into account that individuals have diverse preferences when forming their opinion regarding a particular issue. We show that the system exhibits a phenomenon called "diversity-induced resonance"[Tessone et al. Phys. Rev. Lett. 97, 194101 (2006)], by which an external influence (for example advertising, or fashion trends) is better followed by populations having the right degree of diversity in their preferences, rather than others where the individuals are identical or have too different preferences. We support our findings by numerical simulations of the model and a mean-field type analytical theory.

Abstract

For the voter model, we study the effect of a memory-dependent transition rate. We assume that the transition of a spin into the opposite state decreases with the time it has been in its current state. Counterintuitively, we find that the time to reach a macroscopically ordered state can be accelerated by slowing down the microscopic dynamics in this way. This holds for different network topologies, including fully connected ones. We find that the ordering dynamics is governed by two competing processes which either stabilize the majority or the minority state. If the first one dominates, it accelerates the ordering of the system. The conclusions of this Letter are not restricted to the voter model, but remain valid to many other spin systems as well.

Abstract

We investigate an extension of the voter model in which voters are equipped with an individual inertia to change their opinion. This inertia depends on the persistence time of a voter's current opinion (ageing). We focus on the case of only two different inertia values: zero if a voter just changed towards a new opinion and nu otherwise. We are interested in the average time to reach consensus, i.e. the state in which all voters have adopted the same opinion. Adding inertia to the system means to slow down the dynamics at the voter's level, which should presumably lead to a slower consensus formation. As an unexpected outcome of our inertial voter dynamics, there is a parameter region of nu where an increasing inertia leads to a faster consensus formation. These results rest on the heterogeneity of voters which evolves through the described ageing. In a control setting of homogeneous inertia values, we only find monotonously increasing consensus times. In the paper, we present dynamical equations for the mean-field case which allow for analytical insight into the observed slower-is-faster effect.

Abstract

The opinion dynamics model introduced by Deffuant and Weisbuch as well as the one by Hegselmann and Krause are rather similar. In both models individuals are assumed to have opinions about an issue, they meet and discuss, and they may adapt their opinions towards the other agents` opinions or may ignore each other if their positions are too different. Both models differ with respect to the number of peers they meet at once. Furthermore the model by Deffuant and Weisbuch has a convergence parameter that controls how fast agents adapt their opinions. By defining the reversed parameter as self-support we can extend the applicability of this parameter to scenarios with more than one interaction partner. We investigate the effects of changing the number of peers met at once, which is done for different population sizes, and the effects of changing the self-support. For describing the dynamics we look at different statistics, i.e. number of cluster, number of major clusters, and Gini coefficient.

Abstract

Models of continuous opinion dynamics under bounded confidence have been presented independently by Krause and Hegselmann and by Deffuant et al in 2000. They have raised a fair amount of attention in the communities of social simulation, sociophysics and complexity science. The researchers working on it come from disciplines as physics, mathematics, computer science, social psychology and philosophy. Agents hold continuous opinions which they can gradually adjust if they hear the opinions of others. The idea of bounded confidence is that agents only interact if they are close in opinion to each other. Usually, the models are analyzed with agent-based simulations in a Monte-Carlo style, but they can also be reformulated on the agent's density in the opinion space in a master-equation style. This paper is to present the agent-based and density-based modeling frameworks including the cases of multidimensional opinions and heterogeneous bounds of confidence; second, to give the bifurcation diagrams of cluster configuration in the homogeneous model with uniformly distributed initial opinions; third to review the several extensions and the evolving phenomena which have been studied so far; and fourth to state some basic open questions.

Abstract

The question how social norms can emerge from microscopic interactions between individuals is a key problem in social sciences to explain collective behavior. In this paper we propose an agent-based model to show that randomly distributed social behavior by way of local interaction converges to a state with a multimodal distribution of behavior. This can be interpreted as a coexistence of different social norms, a result that goes beyond previous investigations. The model is discrete in time and space, behavior is characterized in a continuous state space. The adaptation of social behavior by each agent is based on attractive and repulsive forces caused by friendly and adversary relations among agents. The model is analyzed both analytically and by means of spatio-temporal computer simulations. It provides conditions under which we find convergence towards a single norm, coexistence of two opposing norms, and coexistence of a multitude of norms. For the latter case, we also show the evolution of the spatio-temporal distribution of behavior.

Abstract

In this paper, we investigate the so-called ``Sznajd Model'' (SM) in one dimension, which is a simple cellular automata approach to consensus formation among two opposite opinions (described by spin up or down). To elucidate the SM dynamics, we first provide results of computer simulations for the spatio-temporal evolution of the opinion distribution $L(t)$, the evolution of magnetization $m(t)$, the distribution of decision times $P( tau)$ and relaxation times $P( mu)$. In the main part of the paper, it is shown that the SM can be completely reformulated in terms of a linear VM, where the transition rates towards a given opinion are directly proportional to frequency of the respective opinion of the second-nearest neighbors (no matter what the nearest neighbors are). So, the SM dynamics can be reduced to one rule, ``Just follow your second-nearest neighbor''. The equivalence is demonstrated by extensive computer simulations that show the same behavior between SM and VM in terms of $L(t)$, $m(t)$, $P( tau)$, $P( mu)$, and the final attractor statistics. The reformulation of the SM in terms of a VM involves a new parameter $ sigma$, to bias between anti-and ferromagnetic decisions in the case of frustration. We show that $ sigma$ plays a crucial role in explaining the phase transition observed in SM. We further explore the role of synchronous versus asynchronous update rules on the intermediate dynamics and the final attractors. Compared to the original SM, we find three additional attractors, two of them related to an asymmetric coexistence between the opposite opinions.

Abstract

For a binary choice problem, the spatial coordination of decisions in an agent community is investigated both analytically and by means of stochastic computer simulations. The individual decisions are based on different local information generated by the agents with a finite lifetime and disseminated in the system with a finite velocity. We derive critical parameters for the emergence of minorities and majorities of agents making opposite decisions and investigate their spatial organization. We find that dependent on two essential parameters describing the local impact and the spatial dissemination of information, either a definite stable minority/majority relation (single-attractor regime) or a broad range of possible values (multi-attractor regime) occurs. In the latter case, the outcome of the decision process becomes rather diverse and hard to predict, both with respect to the share of the majority and their spatial distribution. We further investigate how a dissemination of information on different time scales affects the outcome of the decision process. We find that a more “efficient” information exchange within a subpopulation provides a suitable way to stabilize their majority status and to reduce “diversity” and uncertainty in the decision process.