Methodology: Contagious Beliefs

post by James Stephen Brown (james-brown) · 2024-10-19T03:58:17.966Z · LW · GW · 0 comments

Contents

  Simulating Political Alignment
    The Symmetrical Valence Matrix
    Belief Pairs and Valence Values
    Matrix Structure and Calculations
    Simulation Grid
      Agent Representation
      Interaction Rules
      Belief Adoption or Rejection
    Example
      Possible coalitions and their strengths:
      Outcome:
    Why Coalition Strength ≠ Average Valence
      Calculating average valence:
    Why We Don’t Require an Increase in Coalition Strength?
    Correlation
      Calculation Using Phi Coefficient
      Phi Coefficient Formula
    Visualisation and Analysis
      Grid Display
      Correlation Log
    Methodological Considerations
      Symmetry and Limitations
      Addressing Limitations
    Conclusion
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Simulating Political Alignment

This methodology concerns a simulation tool which has been developed to model how beliefs, that are not directly related, end up correlated in political identities. It models the transmission of beliefs between nodes on a static hexagonal grid, based on a valence matrix. This methodology allows the user to observe the spread of ideas and the emergence of belief clusters, including correlations between ideas that are not explicitly related.

The model is fully editable but is initially pre-populated with a sample set. This sample set does not reflect objective real-world data.

The Symmetrical Valence Matrix

At the heart of the simulation is the Valence Matrix, where the user can define the relationships between pairs of beliefs. The matrix is symmetrical, meaning that the influence between any two beliefs is the same in both directions. This symmetry simplifies the model and reduces redundancy during data entry—allowing the user to focus on the core interactions without an overwhelming number of parameters.

Belief Pairs and Valence Values

Each belief is represented as a binary pair of opposing extremes (e.g., “Pro-Life” vs. “Pro-Choice”). Valence values range from -100 to +100 and represent the degree of alignment between beliefs:

Matrix Structure and Calculations

Accounting for the symmetry and binary nature of beliefs, when executed, the code deconstructs the binary pairs into individual belief valences.

This can be summarised mathematically:

The value entered in the matrix compares the first (left) side of the binary belief with the first (left) side of the other binary belief. This value is the same when applied to the opposite or values on the right of each binary (due to the symmetrical nature of the model). So, if you have the values {Pro-Tax, Anti-Tax} and the values {Left Wing, Right Wing} with a valence value of 75, this is deconstructed by the program into the following individual valences:

This approach ensures consistency and reduces the number of unique valence entries needed. For  beliefs, the triangular matrix has  entries, significantly simplifying the model. For a model with 16 beliefs, this allows for 120 entries (of which many will be left as zero), then each is broken down into 6 individual valences, each representing 2 symmetrical valences. That’s 120 x 6 = 720 valences from the original 120 entries—all of which can have unique correlations when simulated.

Simulation Grid

The simulation plays out on a static two-dimensional hexagonal grid, where each node represents an individual agent holding a set of beliefs. The hex grid structure means each agent has exactly six neighbors. Beliefs are randomly seeded to some agents.

Agent Representation

Interaction Rules

Belief Adoption or Rejection

Example

Suppose we have an agent with pre-existing beliefs A and B, which share a +10 valence, and an incoming belief C has a +20 valence with A but a -30 valence with C:

Possible coalitions and their strengths:

Outcome:

    •    The highest coalition strength is  for coalition.
    •    Belief  is ejected, and the agent adopts beliefs  and .

Why Coalition Strength ≠ Average Valence

The reason we use the sum of the valences to determine coalition strength, rather than the sum divided by the number of beliefs (the average valence), is because averaging creates a non-intuitive result. An incoming belief with a positive correlation with both of an agent’s two pre-existing beliefs should always be accepted. But if we are looking at average valence, then the incoming belief needs to have an equal or greater valence than the pre-existing beliefs have with each other.

For example:

    •    Existing Beliefs: Beliefs  and  with  .
    •    Incoming Belief: , with    and  .

Calculating average valence:

    •    Existing coalition 
    •    Average valence:  
    •    Proposed coalition 
    •    Sum of valences:  
    •    Number of belief pairs: 3
    •    Average valence:   +13.33

In this case, the average valence decreases when adding belief , even though all valences are positive. This would incorrectly suggest that belief  should be rejected. Using the sum of valences avoids this issue and aligns with the logic that positive correlations should lead to belief adoption.

Why We Don’t Require an Increase in Coalition Strength?

Similarly, if we require an increase in coalition strength in order to adopt a belief, then an agent with no beliefs will never adopt a belief. The logic of the model is that beliefs are accepted freely until they conflict with existing beliefs.

Correlation

Calculation Using Phi Coefficient

The simulation tracks and logs correlations between beliefs over time. The Phi coefficient is used to measure the correlation between pairs of beliefs.

Phi Coefficient Formula

For binary variables  and :


Where:

Interpretation

Due to the effects of indirect valences, strong positive or negative correlations can arise between beliefs where there was no assigned valence, or sometimes even in the opposite direction to an assigned valence.

Visualisation and Analysis

Grid Display

Correlation Log

Methodological Considerations

Symmetry and Limitations

  1. Symmetrical Relationships: While symmetry simplifies the model, it limits the ability to represent asymmetric influence, where one belief might influence another differently than vice versa.
  2. Binary Beliefs: Representing beliefs as binary opposites simplifies the interactions but doesn’t capture nuances such as degrees of belief or non-binary positions.
  3. Ignores Social Pressure: While a propensity to increase cognitive coherence and avoid cognitive dissonance is characteristic of belief adoption in human brains, social pressure is also very important and hasn’t been explicitly modelled in this simulation.
  4. Doesn't explicitly address Bias: Bias is also a key factor in belief adoption which isn't explicitly addressed in this model.

Addressing Limitations

  1. Symmetrical Relationships: Using symmetrical relationships reduced redundancy while testing, where, with respect to political alignment, relationships tended to be symmetrical (requiring duplicate entries of the same values). This symmetry has limitations for applying the model to other areas like health, where a factor like ‘exercise’ might increase ‘tiredness’ (suggesting a positive valence) but ‘tiredness’ might decrease ‘exercise’ (suggesting a negative valence in the other direction). This limitation could be addressed by having a full matrix (rather than a triangular half-matrix) where the top/left triangle of values auto-fills the bottom/right values, but the bottom/right values can be edited manually without affecting the top/left values—balancing efficiency with flexibility. Something to look for in correlations is if there are any consistent asymmetrical correlations that occur despite the symmetrical valences.
  2. Binary Beliefs: Representing binary beliefs allows for numerous data points to be added to the simulation from limited entries (6 relationships rather than 1 for each valence value). This does allow for nuance but requires careful application of values and consideration of the actual relationship of one binary belief to another. For a belief like ‘Agnosticism’, which is by definition not on an extreme in the dimension of religious belief, it can be paired with ‘Dogmatism’, which is its opposite in the dimension of certainty. This can be a difficult process, and it might be a worthwhile feature to allow for the entry of individual factors that are not binary.
  3. Ignores Social Pressure: While the simulation ignores social pressure, the physical nature of the map goes some way to allowing for social pressure in the form of repeated transmission of ideas by neighbours. So that, as soon as a vulnerability in an agent’s belief coalition appears, their neighbours may influence them. I am satisfied with the level at which this simulation aligns with social pressure. To develop a system that enables avoidance might add nuance but would also introduce complications to the model, making result less clear.
  4. Doesn't explicitly address Bias: While the model doesn’t explicitly address bias, it is actually founded on the principle of cognitive bias. By having pre-existing beliefs determine the adoption or rejection of incoming beliefs, the rules of the simulation model cognitive bias as a feature of belief adoption rather than a bug (although the more its more buggy attributes are also an emergent property of the model). So, the model doesn't need a separate explicit factoring of cognitive bias.

Conclusion

This simulation offers a framework for exploring how beliefs interact and spread within a population. By utilising a symmetrical valence matrix and a hexagonal grid, we can observe patterns of belief clustering and propagation. While the model abstracts many real-world complexities, it serves as a valuable tool for visualising ideological dynamics and fostering a deeper understanding of belief systems.

Note: This methodology outlines a simplified model designed for exploratory purposes. Real-world belief dynamics involve asymmetries, degrees of belief, and complex network structures not fully captured in this simulation.

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