Dissonance Minimization as a Microfoundation of Social Influence in Models of Opinion Formation

Abstract

Models of opinion formation are used to investigate many collective phenomena. While social influence often constitutes a basic mechanism, its implementation differs between the models. In this article, we provide a general framework of social influence based on dissonance minimization. We only premise that individuals strive to minimize dissonance resulting from different opinions compared to individuals in a given social network. Within a game theoretic context, we show that our concept of dissonance minimization resembles a coordination process when interactions are homogeneous. We further show that different models of opinion formation can be represented as best response dynamics within our framework. Thus, we offer a unifying perspective on these heterogeneous models and link them to rational choice theory.

Keywords:

• conventions
• coordination
• opinion dynamics
• social influence

Notes

¹For a more detailed overview of applications of opinion formation, see Lorenz (2007b, Section 2.1).

²Dissonance reduction is a motivation which one could assume also in the balance theory (Heider, 1946) in triadic relations.

³Of course, in these models, dissonance minimization is a rational choice of the agent, which is not the idea of the psychological concept of cognitive dissonance. The mental process of reducing dissonance is not necessarily a conscious one. Nevertheless it is interesting to study group constellations where dissonance becomes minimal, regardless of whether the adjustment mechanisms are consciously applied by the agents or not.

⁴A set X ⊂ ℝ ^m is convex if for any x, y ∈ X and α ∈ [0, 1] it holds that αx + (1 − α)y ∈ X; that is, for any two points form the set, the line between them is also part of the set.

⁵The convex hull of a set of points is the smallest possible convex set which contains all points.

⁶A function f: ℝ → ℝ is strictly convex when it holds for any x, y ∈ ℝ and α ∈ ]0, 1[that f(αx + (1 − α)y) < αf(x) + (1 − α)f(y); that is, the graph of f is below the line between any two points on the graph.

⁷In the context of games with a finite number of strategies, alternative dynamics that also include the probability of random mistakes by the agents are provided by Young (1993). Ellison (1993), and Blume (1995) investigate the effect of local interactions in this context.

⁸A set is convex if the line between any two points of the set is contained in the set.

⁹See the proof of Proposition 2.

¹⁰External influence as in Friedkin and Johnsen (1990) is not a component of bounded confidence models.

¹¹Sometimes, the stochastic process is defined on I ²\{(i, i)|i ∈ I}; that is, only distinct agent are chosen.

¹²Note that A _ij (W ^HK(t)) is constant for all agents j in agent i's in-group within the HK model. Hence, this factor can be omitted in the utility function as strictly increasing transformations do not affect the induced preferences.

¹³Axelrod (1997), however, uses the term “culture” instead of “opinion.”

¹⁴Related two-state model are the social impact model (Latanè, 1981) which are analyzed within the framework of cellular automata (Lewenstein, Nowak, & Latané, 1992) and models of social pressure and polarization (Macy, Kitts, Flache, & Steve, 2003) which is analyzed within the framework of Hopfield networks. In both models social impact (respectively social pressure) adds up as the opinions of neighbors.

¹⁵An exception is the model of Vazquez et al. (2008).

¹⁶Although the basic opinion space X = {0, 1} is not convex, we can fulfill the requirements of Proposition 5 by extending it to the interval [0, 1]. Here, we consider only best responses to “pure” opinion profiles (x ₁, …, x _n ) with x _i  ∈ {0, 1}.

¹⁷If there is a risk dominant equilibrium, the process corresponds to a biased voter model (Schwartz, 1977).

¹⁸The continuity of u _i (·, x) on the interior of X is already guaranteed by the convexity of the distance dissonance function.