Opinion formation in voting processes under bounded confidence

Sergei Yu Pilyugin,M C Campi

doi:10.3934/nhm.2019024

Abstract

In recent years, opinion dynamics has received an increasing attention and various models have been introduced and evaluated mainly by simulation. In this study, we introduce a model inspired by the so-called bounded confidence approach where voters engaged in an electoral decision with two options are influenced by individuals sharing an opinion similar to their own. This model allows one to capture salient features of the evolution of opinions and results in final clusters of voters. We provide a detailed study of the model, including a complete taxonomy of the equilibrium points and an analysis of their stability. The model highlights that the final electoral outcome depends on the level of interaction in the society, besides the initial opinion of each individual, so that a strongly interconnected society can reverse the electoral outcome as compared to a society with looser exchange.

Highlights

Studies on opinion dynamics aim to describe the processes by which opinions develop and take form in social systems, and research in this field goes back to the early fifties, [10, 12]
Consensus in opinion dynamics has been the object of several contributions such as [11, 23, 24, 28, 4, 5, 6, 14]
Some basic models for opinion dynamics are described in the recent monographs [27] and [21]

Summary

Introduction

Studies on opinion dynamics aim to describe the processes by which opinions develop and take form in social systems, and research in this field goes back to the early fifties, [10, 12]. We are especially interested in the dynamics of voters that have to choose between two alternatives In this context, a natural assumption is that voters are more influenced by individuals sharing a similar opinion, which, when taken to its extreme, leads to models with bounded confidence. We shall discuss more in detail this aspect below after introducing the model We contend that this situation leads to fixed points in the dynamics that correspond to the formation of opinion clusters. We start with an initial array V = V 0 with the following property: v10 ≤ · · · ≤ vN0 This choice is without loss of generality because we can always arrange initial opinions in non-decreasing order and the dynamics described in the previous section does not depend on the order, it only depends on the values.

We claim that

Now we prove that if