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Published in: Dynamic Games and Applications 1/2021

26-03-2020

A Differential Game Model of Opinion Dynamics: Accord and Discord as Nash Equilibria

Authors: Muhammad Umar B. Niazi, A. Bülent Özgüler

Published in: Dynamic Games and Applications | Issue 1/2021

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Abstract

This paper presents a noncooperative differential (dynamic) game model of opinion dynamics with open-loop information structure. In this game, the agents’ motives are shaped by their expectations of the nature of others’ opinions as well as how susceptible they are to get influenced by others, how stubborn they are, and how quick they are willing to change their opinions on a set of issues in a prescribed time interval. These motives are independently formed by all agents. The existence of a Nash equilibrium in the network means that a collective behavior emerges out of local interaction rules and these individual motives. We prove that a unique Nash equilibrium may exist in the game under quite different circumstances. It may exist not only if there is a harmony of perceptions among the agents of the network, but also when agents have different views about the correlation among issues. The first leads to an accord in the network usually expressed as a partial consensus, and the second to a discord in the form of oscillating opinions. In the case of an accord, the harmony in the network may be in the form of similarity in pairwise conceptions about the issues but may also be an agreement on the status of a “leader” in the network. A Nash equilibrium may fail to exist only if the network is in a state of discord.

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Appendix
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Literature
1.
2.
go back to reference Albi G, Pareschi L, Toscani G, Zanella M (2017) Recent advances in opinion modeling: control and social influence. Act Part 1:49–98 MathSciNet Albi G, Pareschi L, Toscani G, Zanella M (2017) Recent advances in opinion modeling: control and social influence. Act Part 1:49–98 MathSciNet
3.
go back to reference Altafini C (2013) Consensus problems on networks with antagonistic interactions. IEEE Trans Autom Control 58(4):935–946 MathSciNetCrossRef Altafini C (2013) Consensus problems on networks with antagonistic interactions. IEEE Trans Autom Control 58(4):935–946 MathSciNetCrossRef
4.
go back to reference Amelkin V, Bullo F, Singh AK (2017) Polar opinion dynamics in social networks. IEEE Trans Autom Control 62(11):5650–5665 MathSciNetCrossRef Amelkin V, Bullo F, Singh AK (2017) Polar opinion dynamics in social networks. IEEE Trans Autom Control 62(11):5650–5665 MathSciNetCrossRef
5.
go back to reference Axelrod R (1997) The dissemination of culture: a model with local convergence and global polarization. J Conflict Resolut 41(2):203–226 CrossRef Axelrod R (1997) The dissemination of culture: a model with local convergence and global polarization. J Conflict Resolut 41(2):203–226 CrossRef
6.
go back to reference Başar T, Olsder GJ (1998) Dynamic noncooperative game theory. SIAM, Philadelphia, PA CrossRef Başar T, Olsder GJ (1998) Dynamic noncooperative game theory. SIAM, Philadelphia, PA CrossRef
7.
go back to reference Bikhchandani S, Hirshleifer D, Welch I (1992) A theory of fads, fashion, custom, and cultural change as informational cascades. J Polit Econ 100(5):992–1026 CrossRef Bikhchandani S, Hirshleifer D, Welch I (1992) A theory of fads, fashion, custom, and cultural change as informational cascades. J Polit Econ 100(5):992–1026 CrossRef
9.
go back to reference Blondel VD, Hendrickx JM, Tsitsiklis JN (2010) Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM J Control Optim 48(8):5214–5240 MathSciNetCrossRef Blondel VD, Hendrickx JM, Tsitsiklis JN (2010) Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM J Control Optim 48(8):5214–5240 MathSciNetCrossRef
10.
go back to reference Boccaletti S, Bianconi G, Criado R, Del Genio CI, Gómez-Gardenes J, Romance M, Sendina-Nadal I, Wang Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544(1):1–122 MathSciNetCrossRef Boccaletti S, Bianconi G, Criado R, Del Genio CI, Gómez-Gardenes J, Romance M, Sendina-Nadal I, Wang Z, Zanin M (2014) The structure and dynamics of multilayer networks. Phys Rep 544(1):1–122 MathSciNetCrossRef
11.
go back to reference Deffuant G, Neau D, Amblard F, Weisbuch G (2000) Mixing beliefs among interacting agents. Adv Complex Syst 3:87–98 CrossRef Deffuant G, Neau D, Amblard F, Weisbuch G (2000) Mixing beliefs among interacting agents. Adv Complex Syst 3:87–98 CrossRef
12.
13.
go back to reference Engwerda J (2005) LQ dynamic optimization and differential games. Wiley, West Sussex Engwerda J (2005) LQ dynamic optimization and differential games. Wiley, West Sussex
14.
go back to reference Etesami SR, Başar T (2015) Game-theoretic analysis of the Hegselmann-Krause model for opinion dynamics in finite dimensions. IEEE Trans Autom Control 60(7):1886–1897 MathSciNetCrossRef Etesami SR, Başar T (2015) Game-theoretic analysis of the Hegselmann-Krause model for opinion dynamics in finite dimensions. IEEE Trans Autom Control 60(7):1886–1897 MathSciNetCrossRef
15.
go back to reference Ferraioli D, Goldberg PW, Ventre C (2016) Decentralized dynamics for finite opinion games. Theoret Comput Sci 648:96–115 MathSciNetCrossRef Ferraioli D, Goldberg PW, Ventre C (2016) Decentralized dynamics for finite opinion games. Theoret Comput Sci 648:96–115 MathSciNetCrossRef
16.
go back to reference Friedkin NE, Johnsen EC (1990) Social influence and opinions. J Math Sociol 15(3–4):193–206 CrossRef Friedkin NE, Johnsen EC (1990) Social influence and opinions. J Math Sociol 15(3–4):193–206 CrossRef
17.
go back to reference Gabbay M (2007) The effects of nonlinear interactions and network structure in small group opinion dynamics. Physica A 378(1):118–126 CrossRef Gabbay M (2007) The effects of nonlinear interactions and network structure in small group opinion dynamics. Physica A 378(1):118–126 CrossRef
18.
go back to reference Ghaderi J, Srikant R (2014) Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate. Automatica 50(12):3209–3215 MathSciNetCrossRef Ghaderi J, Srikant R (2014) Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate. Automatica 50(12):3209–3215 MathSciNetCrossRef
19.
go back to reference Groeber P, Lorenz J, Schweitzer F (2014) Dissonance minimization as a microfoundation of social influence in models of opinion formation. J Math Sociol 38(3):147–174 MathSciNetCrossRef Groeber P, Lorenz J, Schweitzer F (2014) Dissonance minimization as a microfoundation of social influence in models of opinion formation. J Math Sociol 38(3):147–174 MathSciNetCrossRef
20.
go back to reference Hegselmann R, Krause U (2002) Opinion dynamics and bounded confidence models, analysis, and simulation. J Art Soc Soc Simulat 5(3) Hegselmann R, Krause U (2002) Opinion dynamics and bounded confidence models, analysis, and simulation. J Art Soc Soc Simulat 5(3)
21.
go back to reference Heider F (1946) Attitudes and cognitive organization 21(1):107–112 Heider F (1946) Attitudes and cognitive organization 21(1):107–112
22.
go back to reference Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge CrossRef Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge CrossRef
23.
go back to reference Horn RA, Johnson CR (2002) Matrix analysis. Cambridge University Press, Cambridge Horn RA, Johnson CR (2002) Matrix analysis. Cambridge University Press, Cambridge
24.
go back to reference Huckfeldt R (2007) Unanimity, discord, and the communication of public opinion. Am J Polit Sci 51(4):978–995 CrossRef Huckfeldt R (2007) Unanimity, discord, and the communication of public opinion. Am J Polit Sci 51(4):978–995 CrossRef
25.
go back to reference Iñiguez G, Török J, Yasseri T, Kaski K, Kertész J (2014) Modeling social dynamics in a collaborative environment. EPJ Data Sci 3:7 CrossRef Iñiguez G, Török J, Yasseri T, Kaski K, Kertész J (2014) Modeling social dynamics in a collaborative environment. EPJ Data Sci 3:7 CrossRef
26.
go back to reference Jackson MO (2010) Social and economic networks. Princeton University Press, Princeton CrossRef Jackson MO (2010) Social and economic networks. Princeton University Press, Princeton CrossRef
27.
go back to reference Kirk DE (1970) Optimal control theory: an introduction. Prentice-Hall, Englewood Cliffs, NJ Kirk DE (1970) Optimal control theory: an introduction. Prentice-Hall, Englewood Cliffs, NJ
28.
29.
go back to reference Lorenz J (2007) Continuous opinion dynamics under bounded confidence: a survey. Int J Mod Phys C 18(12):1819–1838 CrossRef Lorenz J (2007) Continuous opinion dynamics under bounded confidence: a survey. Int J Mod Phys C 18(12):1819–1838 CrossRef
30.
go back to reference Macy MW, Flache A (2009) Social dynamics from the bottom up: Agent-based models of social interaction. In: The Oxford handbook of analytical sociology. Oxford University Press, pp 245–268 Macy MW, Flache A (2009) Social dynamics from the bottom up: Agent-based models of social interaction. In: The Oxford handbook of analytical sociology. Oxford University Press, pp 245–268
31.
go back to reference Mirtabatabaei A, Bullo F (2012) Opinion dynamics in heterogeneous networks: convergence conjectures and theorems. SIAM J Control Optim 50(5):2763–2785 MathSciNetCrossRef Mirtabatabaei A, Bullo F (2012) Opinion dynamics in heterogeneous networks: convergence conjectures and theorems. SIAM J Control Optim 50(5):2763–2785 MathSciNetCrossRef
32.
go back to reference Morarescu IC, Girard A (2011) Opinion dynamics with decaying confidence: application to community detection in graphs. IEEE Trans Autom Control 56(8):1862–1873 MathSciNetCrossRef Morarescu IC, Girard A (2011) Opinion dynamics with decaying confidence: application to community detection in graphs. IEEE Trans Autom Control 56(8):1862–1873 MathSciNetCrossRef
33.
go back to reference Niazi MUB (2017) A noncooperative dynamic game model of opinion dynamics in multilayer social networks. Dissertation, Bilkent University, Turkey Niazi MUB (2017) A noncooperative dynamic game model of opinion dynamics in multilayer social networks. Dissertation, Bilkent University, Turkey
34.
go back to reference Niazi MUB,Özgüler AB, Yildiz A (2016) Consensus as a Nash equilibrium of a dynamic game. In: 2016 12th international conference on signal-image technology and internet-based systems (SITIS), IEEE, pp 365–372 Niazi MUB,Özgüler AB, Yildiz A (2016) Consensus as a Nash equilibrium of a dynamic game. In: 2016 12th international conference on signal-image technology and internet-based systems (SITIS), IEEE, pp 365–372
35.
go back to reference Olfati-Saber R (2007) Consensus and cooperation in multi-agent networked systems. Proc IEEE 95(1):215–233 CrossRef Olfati-Saber R (2007) Consensus and cooperation in multi-agent networked systems. Proc IEEE 95(1):215–233 CrossRef
36.
go back to reference Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge MATH Osborne MJ, Rubinstein A (1994) A course in game theory. MIT Press, Cambridge MATH
37.
go back to reference Özgüler AB, Yıldız A (2014) Foraging swarms as Nash equilibria of dynamic games. IEEE Trans Cybern 44(6):979–987 CrossRef Özgüler AB, Yıldız A (2014) Foraging swarms as Nash equilibria of dynamic games. IEEE Trans Cybern 44(6):979–987 CrossRef
38.
go back to reference Parsegov SE, Proskurnikov AV, Tempo R, Friedkin NE (2017) Novel multidimensional models of opinion dynamics in social networks. IEEE Trans Autom Control 62(5):2270–2285 MathSciNetCrossRef Parsegov SE, Proskurnikov AV, Tempo R, Friedkin NE (2017) Novel multidimensional models of opinion dynamics in social networks. IEEE Trans Autom Control 62(5):2270–2285 MathSciNetCrossRef
39.
go back to reference Proskurnikov AV, Matveev AS, Cao M (2016) Opinion dynamics in social networks with hostile camps: consensus versus polarization. IEEE Trans Autom Control 61(6):1524–1536 CrossRef Proskurnikov AV, Matveev AS, Cao M (2016) Opinion dynamics in social networks with hostile camps: consensus versus polarization. IEEE Trans Autom Control 61(6):1524–1536 CrossRef
40.
go back to reference Ren W, Cao Y (2011) Distributed coordination of multi-agent networks: emergent problems, models, and issues. Springer, New York CrossRef Ren W, Cao Y (2011) Distributed coordination of multi-agent networks: emergent problems, models, and issues. Springer, New York CrossRef
41.
go back to reference Ren W, Beard RW, Atkins EM (2005) A survey of consensus problems in multi-agent coordination. In: Proceedings of the American control conference, IEEE, pp 1859–1864 Ren W, Beard RW, Atkins EM (2005) A survey of consensus problems in multi-agent coordination. In: Proceedings of the American control conference, IEEE, pp 1859–1864
42.
go back to reference Shi G, Proutiere A, Johansson M, Baras JS, Johansson KH (2015) Emergent behaviors over signed random dynamical networks: state-flipping model. IEEE Trans Control Netw Syst 2(2):142–153 MathSciNetCrossRef Shi G, Proutiere A, Johansson M, Baras JS, Johansson KH (2015) Emergent behaviors over signed random dynamical networks: state-flipping model. IEEE Trans Control Netw Syst 2(2):142–153 MathSciNetCrossRef
43.
go back to reference Smith DD (1968) Cognitive consistency and the perception of others’ opinions. Public Opin Q 32(1):1–15 CrossRef Smith DD (1968) Cognitive consistency and the perception of others’ opinions. Public Opin Q 32(1):1–15 CrossRef
44.
go back to reference Yıldız A, Özgüler AB (2015) Partially informed agents can form a swarm in a Nash equilibrium. IEEE Trans Autom Control 60(11):3089–3094 MathSciNetCrossRef Yıldız A, Özgüler AB (2015) Partially informed agents can form a swarm in a Nash equilibrium. IEEE Trans Autom Control 60(11):3089–3094 MathSciNetCrossRef
45.
go back to reference Yıldız A, Özgüler AB (2016) Foraging motion of swarms with leaders as Nash equilibria. Automatica 73:163–168 MathSciNetCrossRef Yıldız A, Özgüler AB (2016) Foraging motion of swarms with leaders as Nash equilibria. Automatica 73:163–168 MathSciNetCrossRef
Metadata
Title
A Differential Game Model of Opinion Dynamics: Accord and Discord as Nash Equilibria
Authors
Muhammad Umar B. Niazi
A. Bülent Özgüler
Publication date
26-03-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 1/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00350-3

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