W3: Opinion Dynamics

CS300b – Agent Modeling

Eric Araújo

Week 3 – Agent Modeling of Opinion Dynamics

AI Prompt: Create an image representing polarization

Exploring the Social Dynamics of Beliefs, Attitudes, and Opinions

Introduction

  • Humans are inherently social creatures, learning and shaping our beliefs, attitudes, and opinions through interactions with others.
  • This class focuses on modeling the social dynamics of opinions, using agent-based models to understand how opinions change and spread within populations.

What are Opinions?

  • For this class, we will treat opinions, beliefs, and attitudes equivalently, referring to them as “opinions”.
  • We can represent an opinion as a position on an issue, formalized as a number on a scale between two extremes.
  • Examples:
    • Is cake delicious?
    • Is Batman a hero?
    • Is climate change real?

Key Concepts in Opinion Dynamics

  • Consensus: When everyone in a population holds the same opinion on an issue.
  • Polarization: When individuals’ opinions gravitate towards two opposing extremes, creating a divided population.
  • Extremism: Holding opinions at the far ends of the spectrum.

Modeling Opinion Dynamics

  • Agent-based models (ABMs) help us understand how simple interactions between individuals can lead to complex patterns of opinion dynamics at the population level.

Modeling Opinion Dynamics

  • ABMs of opinion dynamics require three key components:
    • Representation of opinions.
    • Mechanism for social influence.
    • Population structure.

Positive Influence Model

  • Assumption: Every interaction between individuals with differing opinions brings them closer together.
  • Mechanism: Agents update their opinions by moving a fraction (learning rate, \(\gamma\)) of the difference between their current opinion and their interaction partner’s opinion.
  • Outcome: This model always leads to consensus around the initial mean opinion.

Positive Influence Model

\[x_1 = x_1 + \gamma (x_2 - x_1)\] \[x_2 = x_2 + \gamma (x_1 - x_2)\]

Positive Influence Model

Coding caveats:

people-own [opinion]
...
to setup
    ...
    set opinion (random-float 2) - 1
    ...
    update-colors
end

\[color = \frac{x+1}{2} \times 9.9\]

to update-colors
    ask people [
        set color [opinion + 1] * 9.9 / 2
    ]
end

Bounded Confidence Model

Bounded Confidence Model

  • Assumption: Individuals are only influenced by others whose opinions are sufficiently similar to their own, defined by a confidence threshold (\(d\)).
  • Mechanism: Agents update their opinions as in the positive influence model, but only if the difference between their opinions is less than \(d\).
  • Outcome: Can lead to persistent cliques of like-minded individuals, with the number of cliques increasing as the confidence threshold decreases.

Bounded Confidence Model

Coding caveats:

to go
    ask one-of-people [
        let x1 opinion
        let other-person one-of other people
        ...
        let x2 [opinion] of other-person

        if (abs (x1 - x2) < confidence-threshold) [
            let x1-new (x1 + learning-rate * (x2 - x1))
            let x2-new (x2 + learning-rate * (x1 - x2))
            set opinion x1-new
            ask other-person [ set opinion x2-new ]
        ]
    ]
    update-colors
    tick
end

Bounded Confidence Model

Reporting number of cliques:

to-report num-cliques
    let cliques 1
    let d confidence-threshold
    let min-value min [opinion] of people with [opinion] > (-1 + (d / 2))

    while [any? people with 
        [opinion > (min-value + d) and opinion < (1 - (d / 2))]] [
            set min-value min [opinion] of 
                (people with [opinion > (min-value + (d / 2))])
            set cliques cliques + 1
    ]
    report cliques
end

Negative Influence Model

Negative Influence Model

  • Assumption: Individuals are negatively influenced by those with significantly different opinions, leading to increased divergence in opinions.
  • Mechanism: Agents’ opinions become more dissimilar after interacting with those whose opinions are outside their confidence threshold (\(d\)).
  • Outcome: Can lead to polarization, with opinions moving towards opposite extremes, especially when the confidence threshold is low.

Negative Influence Model

Fig. 5.6 - Modeling Social Behavior

Negative Influence Model

\[x_1 = \begin{cases} x_1 + \frac{\gamma}{2}(x_1 - x_2)(1 - x_1),\ if\ x_1 >x_2 \\ x_1 + \frac{\gamma}{2}(x_1 - x_2)(1 + x_1),\ otherwise \end{cases} \]

\[x_2 = \begin{cases} x_2 + \frac{\gamma}{2}(x_2 - x_1)(1 + x_2),\ if\ x_1 > x_2 \\ x_2 + \frac{\gamma}{2}(x_2 - x_1)(1 - x_2),\ otherwise \end{cases} \]

Spatial Structure in Opinion Dynamics

  • Assumption: Interactions are constrained by spatial proximity or social networks.
  • Mechanism: Agents only interact with their immediate neighbors on a spatial grid or within their defined social network.
  • Outcome:
    • Can slow down consensus formation in the positive influence model.
    • Can lead to the persistence of diverse opinions in the bounded confidence model, with larger clusters of majority opinions and smaller, isolated groups with extreme opinions.
    • Can create distinct demographic zones of opinions in the negative influence model.

Fall 2024