Opinion formation in voting processes under bounded confidence

Sergei Yu. Pilyugin; M. C. Campi

doi:10.3934/nhm.2019024

Article Contents

2019, Volume 14, Issue 3: 617-632. Doi: 10.3934/nhm.2019024

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Opinion formation in voting processes under bounded confidence

Sergei Yu. Pilyugin^1, and
M. C. Campi^2, ,

1.
St. Petersburg State University, Universitetskaya nab., 7-9, St.Petersburg, 199034, Russia
2.
Dipartimento di Ingegneria dell'Informazione, Università di Brescia, via Branze 38, 25123 Brescia, Italia

^* Corresponding author: M. C. Campi
^* Corresponding author: M. C. Campi

Received: September 30, 2018

Revised: February 28, 2019

Published: August 2019

The work of S. Y. Pilyugin was supported by the Russian Foundation for Basic Research, grant 18-01-00230, and the work of M.C. Campi was supported by the Russian Science Foundation, grant 16-19-00057, and by the HW project of the University of Brescia CLAFITE

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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.

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Mathematics Subject Classification: 90B10, 91D30, 91B12, 34D20, 15B51.

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Figure 1. Initial distribution of opinions for the first example

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Figure 2. Opinions' evolution for first example at steps $ 10 $, $ 20 $, $ 30 $ and $ 34 $, when the equilibrium is reached; $ {\epsilon} = 0.3 $, $ h = 0.1 $

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Figure 3. Initial distribution of opinions for the second example

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Figure 4. Opinions' evolution for second example at steps $ 5 $, $ 10 $, $ 20 $ and $ 27 $, when the equilibrium is reached; $ {\epsilon} = 0.45 $, $ h = 0.1 $

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Figure 5. Opinions' evolution for second example at steps $ 10 $, $ 20 $, $ 30 $ and $ 49 $, when the equilibrium is reached; $ {\epsilon} = 0.05 $, $ h = 0.1 $

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