Variable tolerance in Deffuant's bounded confidence model

Let us briefly come back to the Deffuant's model and look at one of its common generalizations. Namely let us allow tolerance to vary. Varying tolerance to different opinions would be a rather natural assumption as it is usually hard to persuade a radical, while it is somewhat easier to persuade a moderate.

So let us say that our tolerance function has the following form:

\begin{equation} \mathrm{tolerance}(x) = \epsilon_{min} + \epsilon \frac{x^\alpha (1-x)^\beta}{x_s^\alpha (1-x_s)^\beta} . \end{equation}

In the above \( x_s = \frac{\alpha}{\alpha+\beta} \) is the point at which tolerance function has largest value. Note that if both powers are zeros, \( \alpha = \beta = 0 \), this tolerance function is flat and the model becomes equivalent to the original Deffuant's model. You can examine how the tolerance function looks like using app below.

Now that the two randomly selected will have differing tolerances we have to redefine our interaction rules from previously. Namely, let us say that agents will always interact, but each of them separately will be affected by the interaction only if the absolute difference between the opinions is less than their tolerance. In other words agent \( i \) is affected by interaction with agent \( j \), if:

\begin{equation} | x_i - x_j | < \mathrm{tolerance}(x_i) . \end{equation}

This means that after the interaction both agent, a single agent or no agents might update their opinions.

Feel free to explore how the tolerance function parameters change the dynamics of the Deffuant's model. One thing to note is that when \( \alpha \neq \beta \), then most "tolerant" agents are not the moderates (\( x \approx 0.5 \)), but ones having their leanings (\( x < 0.5 \) or \( x > 0.5 \)). The side the most "tolerant" agents will spawn less clusters than the opposite side (one with the less "tolerant" agents).

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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.