September  2019, 14(3): 617-632. doi: 10.3934/nhm.2019024

Opinion formation in voting processes under bounded confidence

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

Received  October 2018 Revised  March 2019 Published  May 2019

Fund Project: 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

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.

Citation: Sergei Yu. Pilyugin, M. C. Campi. Opinion formation in voting processes under bounded confidence. Networks & Heterogeneous Media, 2019, 14 (3) : 617-632. doi: 10.3934/nhm.2019024
References:
[1]

M. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis, Z. Angew. Math. Phys., 61 (2010), 583-602. doi: 10.1007/s00033-009-0040-0. Google Scholar

[2]

D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. Google Scholar

[3]

F. Ceragioli and P. Frasca, Continuous and discontinuous opinion dynamics with bounded confidence, Nonlinear Anal. Real World Appl., 13 (2012), 1239-1251. doi: 10.1016/j.nonrwa.2011.10.002. Google Scholar

[4]

S. Chatterjee, Reaching a consensus: Some limit theorems, Proc. Int. Statist. Inst., 159–164.Google Scholar

[5]

S. Chatterjee and E. Seneta, Toward consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97. doi: 10.2307/3213262. Google Scholar

[6]

J. H. J. Cohen and C. Newman, Approaching consensus can be delicate when positions harden, Stochastic Proc. and Appl., 22 (1986), 315-322. doi: 10.1016/0304-4149(86)90008-6. Google Scholar

[7]

G. DeffuantD. Neau and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. Google Scholar

[8]

J. C. Dittmer, Consensus formation under bounded confidence, Nonlinear Analysis, 47 (2001), 4615-4621. doi: 10.1016/S0362-546X(01)00574-0. Google Scholar

[9]

S. Fortunato, The Krause – Hegselmann consensus model with discrete opinions, Internat. J. Modern Phys. C, 15 (2004), 1021-1029. doi: 10.1142/S0129183104006479. Google Scholar

[10]

J. French, A formal theory of social power, Psychological Review, 63 (1956), 181-194. doi: 10.1016/B978-0-12-442450-0.50010-9. Google Scholar

[11]

M. H. De Groot, Reaching a consensus, J. Amer. Statist. Assoc., 69 (1974), 118-121. Google Scholar

[12]

F. Harary, A criterion for unanimity in French's theory of social power, in Studies in Social Power (ed. D. Cartwright), Institute for Social Research, Ann Arbor, 1959.Google Scholar

[13]

P. Hegarty and E. Wedin, The Hegselmann – Krause dynamics for equally spaced agents, J. Difference Equ. Appl., 22 (2016), 1621-1645. doi: 10.1080/10236198.2016.1234611. Google Scholar

[14]

R. Hegselmann and A. Flache, Understanding complex social dynamics: A plea for cellular automata based modelling, Journal of Artificial Societies and Social Simulation, 1 (1998).Google Scholar

[15]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), 1-33. Google Scholar

[16]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model, Networks and Heterogeneous Media, 10 (2015), 477-509. doi: 10.3934/nhm.2015.10.477. Google Scholar

[17]

P. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, J. Differential Equations, 257 (2014), 4165-4187. doi: 10.1016/j.jde.2014.08.005. Google Scholar

[18]

U. Krause, A discrete nonlinear and non - autonomous model of consensus formation, in Communications in Difference Equations (ed. S. Elaydi et al.), Gordon and Breach Publ., Amsterdam, 2000. Google Scholar

[19]

U. Krause, Soziale dynamiken mit vielen interakteuren. eine problemskizze, in Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren (eds. U. Krause and M. Stockler), Universitat Bremen, 1997.Google Scholar

[20]

U. Krause, Compromise, consensus, and the iteration of means, Elem. Math., 64 (2009), 1-8. doi: 10.4171/EM/109. Google Scholar

[21]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models, and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696. Google Scholar

[22]

S. Kurz and J. Rambau, On the Hegselmann – Krause conjecture in opinion dynamics, Journal of Difference Equations and Applications, 17 (2011), 859-876. doi: 10.1080/10236190903443129. Google Scholar

[23]

K. Lehrer, Social consensus and rational agnoiology, Synthese, 31 (1975), 141-160. doi: 10.1007/BF00869475. Google Scholar

[24]

K. Lehrer and C. Wagner, Rational Consensus in Science and Society, D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1981. Google Scholar

[25]

J. Lorenz, A stabilization theorem for dynamics of continuous opinions, Physica A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086. Google Scholar

[26]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. Google Scholar

[27]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks. Emergent Problems, Models, and Issues, Springer, 2011.Google Scholar

[28]

C. Wagner, Consensus through respect: A model of rational group decision-making, Philosophical Studies, 34 (1978), 335-349. doi: 10.1007/BF00364701. Google Scholar

[29]

E. Wedin and P. Hegarty, The Hegselmann – Krause dynamics for the continuous-agent model and a regular opinion function do not always lead to consensus, IEEE Trans. Automat. Control, 60 (2015), 2416-2421. doi: 10.1109/TAC.2015.2396643. Google Scholar

[30]

G. Weisbuch, G. Deffuant and G. Nadal, Interacting agents and continuous opinion dynamics, in Heterogenous Agents, Interactions and Economic Performance (eds. R. Cowan and N. Jonard), Lecture Notes in Economics and Mathematical Systems, 521, Springer, Berlin, 2003.Google Scholar

[31]

S. WongkaewM. Caponigro and A. Borzi, On the control through leadership of the Hegselmann – Krause opinion formation model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 565-585. doi: 10.1142/S0218202515400060. Google Scholar

show all references

References:
[1]

M. Bertotti and M. Delitala, Cluster formation in opinion dynamics: A qualitative analysis, Z. Angew. Math. Phys., 61 (2010), 583-602. doi: 10.1007/s00033-009-0040-0. Google Scholar

[2]

D. Borra and T. Lorenzi, Asymptotic analysis of continuous opinion dynamics models under bounded confidence, Commun. Pure Appl. Anal., 12 (2013), 1487-1499. doi: 10.3934/cpaa.2013.12.1487. Google Scholar

[3]

F. Ceragioli and P. Frasca, Continuous and discontinuous opinion dynamics with bounded confidence, Nonlinear Anal. Real World Appl., 13 (2012), 1239-1251. doi: 10.1016/j.nonrwa.2011.10.002. Google Scholar

[4]

S. Chatterjee, Reaching a consensus: Some limit theorems, Proc. Int. Statist. Inst., 159–164.Google Scholar

[5]

S. Chatterjee and E. Seneta, Toward consensus: Some convergence theorems on repeated averaging, J. Appl. Prob., 14 (1977), 89-97. doi: 10.2307/3213262. Google Scholar

[6]

J. H. J. Cohen and C. Newman, Approaching consensus can be delicate when positions harden, Stochastic Proc. and Appl., 22 (1986), 315-322. doi: 10.1016/0304-4149(86)90008-6. Google Scholar

[7]

G. DeffuantD. Neau and G. Weisbuch, Mixing beliefs among interacting agents, Advances in Complex Systems, 3 (2000), 87-98. Google Scholar

[8]

J. C. Dittmer, Consensus formation under bounded confidence, Nonlinear Analysis, 47 (2001), 4615-4621. doi: 10.1016/S0362-546X(01)00574-0. Google Scholar

[9]

S. Fortunato, The Krause – Hegselmann consensus model with discrete opinions, Internat. J. Modern Phys. C, 15 (2004), 1021-1029. doi: 10.1142/S0129183104006479. Google Scholar

[10]

J. French, A formal theory of social power, Psychological Review, 63 (1956), 181-194. doi: 10.1016/B978-0-12-442450-0.50010-9. Google Scholar

[11]

M. H. De Groot, Reaching a consensus, J. Amer. Statist. Assoc., 69 (1974), 118-121. Google Scholar

[12]

F. Harary, A criterion for unanimity in French's theory of social power, in Studies in Social Power (ed. D. Cartwright), Institute for Social Research, Ann Arbor, 1959.Google Scholar

[13]

P. Hegarty and E. Wedin, The Hegselmann – Krause dynamics for equally spaced agents, J. Difference Equ. Appl., 22 (2016), 1621-1645. doi: 10.1080/10236198.2016.1234611. Google Scholar

[14]

R. Hegselmann and A. Flache, Understanding complex social dynamics: A plea for cellular automata based modelling, Journal of Artificial Societies and Social Simulation, 1 (1998).Google Scholar

[15]

R. Hegselmann and U. Krause, Opinion dynamics and bounded confidence: Models, analysis and simulation, Journal of Artificial Societies and Social Simulation, 5 (2002), 1-33. Google Scholar

[16]

R. Hegselmann and U. Krause, Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model, Networks and Heterogeneous Media, 10 (2015), 477-509. doi: 10.3934/nhm.2015.10.477. Google Scholar

[17]

P. Jabin and S. Motsch, Clustering and asymptotic behavior in opinion formation, J. Differential Equations, 257 (2014), 4165-4187. doi: 10.1016/j.jde.2014.08.005. Google Scholar

[18]

U. Krause, A discrete nonlinear and non - autonomous model of consensus formation, in Communications in Difference Equations (ed. S. Elaydi et al.), Gordon and Breach Publ., Amsterdam, 2000. Google Scholar

[19]

U. Krause, Soziale dynamiken mit vielen interakteuren. eine problemskizze, in Modellierung und Simulation von Dynamiken mit vielen interagierenden Akteuren (eds. U. Krause and M. Stockler), Universitat Bremen, 1997.Google Scholar

[20]

U. Krause, Compromise, consensus, and the iteration of means, Elem. Math., 64 (2009), 1-8. doi: 10.4171/EM/109. Google Scholar

[21]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models, and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696. Google Scholar

[22]

S. Kurz and J. Rambau, On the Hegselmann – Krause conjecture in opinion dynamics, Journal of Difference Equations and Applications, 17 (2011), 859-876. doi: 10.1080/10236190903443129. Google Scholar

[23]

K. Lehrer, Social consensus and rational agnoiology, Synthese, 31 (1975), 141-160. doi: 10.1007/BF00869475. Google Scholar

[24]

K. Lehrer and C. Wagner, Rational Consensus in Science and Society, D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1981. Google Scholar

[25]

J. Lorenz, A stabilization theorem for dynamics of continuous opinions, Physica A, 355 (2005), 217-223. doi: 10.1016/j.physa.2005.02.086. Google Scholar

[26]

J. Lorenz, Continuous opinion dynamics under bounded confidence: A survey, International Journal of Modern Physics C, 18 (2007), 1819-1838. doi: 10.1142/S0129183107011789. Google Scholar

[27]

W. Ren and Y. Cao, Distributed Coordination of Multi-agent Networks. Emergent Problems, Models, and Issues, Springer, 2011.Google Scholar

[28]

C. Wagner, Consensus through respect: A model of rational group decision-making, Philosophical Studies, 34 (1978), 335-349. doi: 10.1007/BF00364701. Google Scholar

[29]

E. Wedin and P. Hegarty, The Hegselmann – Krause dynamics for the continuous-agent model and a regular opinion function do not always lead to consensus, IEEE Trans. Automat. Control, 60 (2015), 2416-2421. doi: 10.1109/TAC.2015.2396643. Google Scholar

[30]

G. Weisbuch, G. Deffuant and G. Nadal, Interacting agents and continuous opinion dynamics, in Heterogenous Agents, Interactions and Economic Performance (eds. R. Cowan and N. Jonard), Lecture Notes in Economics and Mathematical Systems, 521, Springer, Berlin, 2003.Google Scholar

[31]

S. WongkaewM. Caponigro and A. Borzi, On the control through leadership of the Hegselmann – Krause opinion formation model, Mathematical Models and Methods in Applied Sciences, 25 (2015), 565-585. doi: 10.1142/S0218202515400060. Google Scholar

Figure 1.  Initial distribution of opinions for the first example
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 $
Figure 3.  Initial distribution of opinions for the second example
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 $
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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