Deffuant et al. bounded confidence model

All of the opinion dynamics models we have considered so far had discrete opinions. However it would be rather natural to think about opinions as being continuous. Opinions become discrete only due to the way we observe them, namely ballots in the elections and questionnaires in the polls can have only discrete options (even in case you can write in your own preference). Also discrete opinions are easier to analyze, only then one can talk about the majority or compare their popularity.

Nevertheless there are few interesting models with continuous opinions. Usually these models are based on the concept of bounded confidence. Meaning that people tend to listen to other people who have a relatively similar opinion to theirs. Here in this post we will discuss one of these bounded confidence models proposed by Deffuant et al. in [1].

The model

In the Deffuant model we consider a population \( N \) agents. These agents might be put on a graph (interacting only with their immediate neighbors), but let us consider only the simplest case when the agents are on the complete graph (agents are able to interact with all other agents). Each of these agents is initially assigned opinion \( x_i \), which is randomly chosen from the interval \( [0,1] \).

Afterwards at each time tick two random agents are selected. If their opinions are not too different, \( | x_i - x_j | < \epsilon \), they interact in the following manner:

\begin{equation} x_i(t+1) = x_i(t) - \mu ( x_i(t) - x_j(t)) , \end{equation}

\begin{equation} x_j(t+1) = x_j(t) + \mu ( x_i(t) - x_j(t)) . \end{equation}

In the above \( \epsilon \) could be interpreted as tolerance to differing opinion, while \( \mu \) is the adjustment rate. Obviously \( \epsilon \) lies in the interval \( [0,1] \), while \( \mu \) in \( [0,0.5] \). In the extreme case if \( \mu=0.5 \), then the agents assume the same opinion, which is average of their opinions prior to their interaction.

Not much is known analytically about the dynamics of the Deffuant model, but many things are known from the numerical simulations. From the definition of the model it is pretty clear that opinions will cluster (they will not become exactly the same, but will grow extremely similar) and the number of clusters will be dependent on \( \epsilon \). Actually it is known that roughly the dependence is \( n_c \sim \frac{1}{2 \epsilon} \).

HTML 5 app

Feel free to explore the dynamics of the Deffuant model using the app below. Just note that the app might be somewhat memory hungry. This is mainly because of the top figure, which shows opinion trajectories of all agents (100 of them are used in the app). The bottom figure shows current distribution of the opinions.

Acknowledgment. This post was written while reviewing literature relevant to the planned activities in postdoctoral fellowship ''Physical modeling of order-book and opinion dynamics'' (09.3.3-LMT-K-712-02-0026) project. The fellowship is funded by the European Social Fund under the No 09.3.3-LMT-K-712 ''Development of Competences of Scientists, other Researchers and Students through Practical Research Activities'' measure.

References

• G. Deffuant, D. Neau, F. Amblard, G. Weisbuch. Mixing beliefs among interacting agents. Advances in Complex Systems 3: 87-98 (2000). doi: 10.1142/S0219525900000078.