# A. Ishii trust and suspicion model with three agents

Recently one of the well-known researchers in the field has proposed an interesting opinion dynamics model, which aims to explain polarization and group formation in human societies . His model is based on two antipodal concepts in human interactions: trust and suspicion.

In this post we discuss how the model is extended to three agent case (which will eventually lead us to many agent case) and its relationship with well-known Hegselmann-Krause model, which we have discussed in an earlier post.

So now Type I update rule is given by:

\begin{equation} O_{i} (t+1) = O_{i} (t) + \sum_{j=1}^{3} D_{ij} O_{j} (t) \Delta t. \end{equation}

While Type II update rule is given by:

\begin{equation} O_{i} (t+1) = O_{i} (t) + \sum_{j=1}^{3} D_{ij} \Phi(|O_{i} (t)-O_{j} (t)|) ( O_{j} (t) - O_i(t) ) \Delta t. \end{equation}

As we generalize, note that we require that $$D_{ii}$$ is zero. Meaning that agent does not have any impact on his own opinion. This might not be as important for Type II model (as $$\Delta O = 0$$), but it might play important role in Type I model.

With three agents there are variety of different scenarios to explore. For example, what happens if two agents mildly distrust each other, but both strongly trust the other one? In such case that agent could mediate or enforce his own opinion in Type II model. In Type I model the third agent would bring about a temporary consensus, but strong divergence of opinion would eventually follow.

What would happen if all agents would have asymmetric relationships? There are also numerous other scenario to be tested results of which would depend not only on $$\mathbf{D}$$, but also on the initial conditions. Feel free to try variety of scenarios using the app below. 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

• A. Ishii. Opinion Dynamics Theory Considering Trust and Suspicion in Human Relations. In: Group Decision and Negotiation: Behavior, Models, and Support, Lecture Notes in Business Information Processing, 351: 193-204. Springer, 2019. doi: 10.1007/978-3-030-21711-2_15.