# Repulsion in controversial debate

Controversial debates are often marked by an emergence of polarized state. In this post we will consider yet another variation of the voter model proposed by a group of German researchers [1], which shows that significant repulsion leads to the polarized fifty-fifty state.

## The model

Unlike in the original voter model in this model the agents have three opinions: "yes" (+1), "undecided" (0) and "no" (-1). One of the two interacting agents is the speaker, while the other is the listener. The speaker must have a strong opinion to express it (the agent can't be "undecided"). As the listener hears the argument four different changes may happen:

**If the listener is undecided**, \( S_l = 0 \) is true. Then the listener might become convinced (1) with probability \( p_{con} \) or repelled (2) with probability \( p_{con} \cdot p_{repel} \). Note that these events are mutually exclusive, so we have to have \( p_{con} ( 1+p_{repel} ) \leq 1 \).**If the listener initially agrees with the speaker**, \( S_l = S_s \) is true. Then the listener can start doubting (3) with probability \( p_{con} \cdot p_{repel} \cdot p_{doubt} \).**If the listener initially disagrees with the speaker**, \( S_l = -S_s \) is true. Then the listener can start doubting (4) with probability \( p_{con} \cdot p_{doubt} \).

The following changes are made to the listeners opinions after interaction, if the described events occur:

\begin{equation} S_l = 0 \quad \Rightarrow \quad S_l = S_s , \end{equation}

\begin{equation} S_l = 0 \quad \Rightarrow \quad S_l = -S_s , \end{equation}

\begin{equation} S_l = S_s \quad \Rightarrow \quad S_l = 0 , \end{equation}

\begin{equation} S_l = -S_s \quad \Rightarrow \quad S_l = 0 . \end{equation}

Otherwise the listener's opinion does not change.

In this model we assume that one time tick corresponds to \( N \) conversations. \( N \) times we randomly choose the speaker and the listener, while checking if the describe changes happen. Note that \( N \) is also the number of agents in simulation and thus on average one agent should have a change both to speak and to listen.

Note that only \( p_{doubt} \) and \( p_{repel} \) parameters have influence on the final state. \( p_{con} \) just adjusts the timescale of the dynamics.

## Interactive app

Use the interactive app below to explore the model dynamics. You can change all of the aforementioned parameters of this model except for the number of agents. For simplicity sake we have simply fixed number of agents at \( N = 100 \). The interactive plots fractions of agents who hold positive (blue), negative (red) and neutral (gray) 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

- S. M. Krause, F. Weyhausen-Brinkmann, S. Bornholdt. Repulsion in controversial debate drives public opinion into fifty-fifty stalemate. arXiv: 1909.06483 [physics.soc-ph].