ACLS Sznajd model

This time we consider another generalization of the Sznajd model, which also includes four opinions. But now they are no longer one-dimensional! In this Authoritarian-conservative-libertarian-socialist (ACLS) Sznajd model opinions are two-dimensional. Namely, there are two binary dimensions: attitudes towards political and personal freedoms are considered at the same time [1].

The model

The ACLS model is based on the idea that political spectrum could be not one-dimensional (the left-right divide), but in fact have two intertwined dimensions: economic (we will use \( E_k \) to denote economic dimension) and social (personal; we will use \( S_k \) to denote social dimension) freedoms. This idea is heavily inspired by the Political Compass.

Based on their binary (restrict or endorse) economic and personal freedom attitudes agents can be split into four types:

• Authoritarian agents restrict both economic and personal freedoms. In our notation such agents have \( E_k = -1 \) and \( S_k = -1 \). These agents are marked using red-ish color in the app.
• Conservative agents endorse economic freedoms, but restrict personal freedoms. In our notation such agents have \( E_k = -1 \) and \( S_k = 1 \). These agents are marked using blue-ish color in the app.
• Libertarian agents endorse both economic and personal freedoms. In our notation such agents have \( E_k = 1 \) and \( S_k = 1 \). These agents are marked using purple-ish color in the app.
• Socialist agents restrict economic freedoms, but endorse personal freedoms. In our notation such agents have \( E_k = -1 \) and \( S_k = 1 \). These agents are marked using green-ish color in the app.

As the dimensions under consideration are notably different, most people perceive economic freedoms to be less intimate than personal freedoms, different assumptions are made about the opinion dynamics on these dimensions. Namely, economic discussions occur through Sznajd model with "social validation" rule and without "discord" rule, while discussions about personal freedoms occur via Glauber dynamics at zero temperature.

To be a bit more clear about the algorithm let us outline it for you. In this outline, when using mathematical expressions we will assume that agents are distributed in the one-dimensional space (line), though in the app below they are distributed in the two-dimensional space (grid).

First choose two random neighboring agents. If these two agents agree about personal freedom (if \( S_{i} = S_{i+1} \) in one-dimensional case), they transmit their economic attitudes to their other neighbors (\( E_{i-1} = E_{i} \) and \( E_{i+1} = E_{i+2} \)). Otherwise the transmission of economic attitudes occurs with probability \( p_t \) (here t stands for tolerance).

Next choose a random agent and his two neighbors. For the sake of simplicity we choose two neighbors on the opposite sides of the agent. If the neighbors agree on economic issues (if \(E_{i-1} = E_{i+1} \)), then transmission of social attitudes occurs. Otherwise the transmission of social attitudes occurs with probability \( p_t \).

Transmission of social attitudes occurs via Glauber dynamics at zero temperature. Namely, if the neighbors agree about the personal freedom (if \( S_{i-1}=S_{i+1} \)), then the selected agent adopts their views on personal freedom (\(S_i = S_{i-1} \)). Otherwise agent flips his views on personal freedom with probability \( 1/2 \).

HTML 5 app

Explore the formation of consensus in this case. Note that the agents reach agreement only on economic dimension, while personal (social) dimension stays noisy. These is because of the different mechanisms used for each dimension.

This app has two additional parameters \( p_e \) and \( p_s \). Both of these set the probabilities for each agent to endorse economic and social (personal) freedoms during the initialization.

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

• K. Sznajd-Weron, J. Sznajd. Who is left, who is right? Physica A 351: 593-604 (2005). doi: 10.1016/j.physa.2004.12.038.