# A. Ishii trust and suspicion model

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 [1]. His model is based on two antipodal concepts in human interactions: trust and suspicion.

In this post we will introduce the trust and suspicion model for the case with two agents.

So let us assume that there two agents. Each of them has certain opinion \( O_i \) on certain subject. These two agents need to define their outlook towards each other, \( D_{ij} \) and \( D_{ji} \). These two can take any real value, with positive values standing for trusting relationship and negative values standing for suspicion.

At each discrete time tick (the original model uses continuous time, but does so in a manner I do not like), we will choose one random agent and update its opinion. There are two different opinion update rules in [1], one is based on the earlier works of the same group (we will refer to this update rule as Type I model), the other is based on bounded confidence models (we will refer to this update rule as Type II model).

Type I update rule is given by:

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

This update rule assumes that agents views push on other agent views towards positive or negative infinities. This somewhat corresponds to the echo chambers, in which opinions get more strength from the like minded. Also opinions get stronger whenever suspicious agent tries to spread opposite views.

Type II update rule is given by:

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

This update rule assumes that agents converge towards shared view point given \( \Phi ( \Delta O ) > 0 \), which we call indifference function. This function implements a bounded confidence rule, with smooth transition between taking opinion into account and ignoring it. There are multiple possible ways to select \( \Phi \) function, but here we will consider sigmoid function used in the original paper [1]:

\begin{equation} \Phi ( \Delta O ) = \frac{1}{1+\exp(\beta [ \Delta O - \varepsilon ])} , \end{equation}

where \( \beta \) describes how sharp the transition is around point the boundary point \( \varepsilon \). If difference in opinions, \( \Delta O \), is small then \( \Phi ( \Delta O ) \approx 1 \). If difference in opinions is large then \( \Phi ( \Delta O ) \) quickly goes to zero. Meaning that only agents with similar enough opinions will influence each other.

Note that in the both update rules \( \Delta t \) is a short time interval. For the sake of simplicity we assume that \( \Delta t = 1 / N \), where \( N \) is the number of agents (in this case it is \( 2 \)).

While this model appears to be simple, it still has quite rich dynamics. For the model with Type I update rule, the following observations should hold:

- If two agents trust each other and their opinions are both positive or negative, then they will run off together towards positive or negative infinity respectively. They will be stopped only by bounded confidence. Note that this stoppage occurs only in our interpretation of the model.
- If two trusting agents have opposite opinions, then they will converge to neutral stance, but one of them will reach zero faster and then they will run off towards positive or negative infinity (depending on who has reached the zero first).
- If two agents do not trust each other, their opinions will eventually diverge.
- If the relationship is not reciprocal, meaning that one trusts the other, but the other distrusts the first, then we will observe chasing behavior with cycles.

For the model with Type II update rule, the following observations should hold:

- If two agents trust each other, then their opinions will converge to some middle ground.
- If two agents distrust each other, then their opinions will diverge until they are different by more than \( \varepsilon \).
- If the relationship between two agents is asymmetric, then we will observe run away chasing behavior.

Check our intuitions and maybe find your own insights using the app below. Also try changing initial opinions of the agents, \( O_1(0) \) and \( O_2(0) \), as they may also play a role in the observed dynamics.

**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.