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Computational Economics

, Volume 53, Issue 2, pp 479–505 | Cite as

Opinion Formation with Imperfect Agents as an Evolutionary Process

  • Matjaž Steinbacher
  • Mitja SteinbacherEmail author
Article

Abstract

We develop and simulate an interaction-based model of continuous opinion formation under bounded confidence to identify conditions and understand circumstances that lead a society into either a consensus, multiple opinion classes or perpetual opinion dynamics. The society is modeled as a social network and random meetings are presumed. When only regular agents are present, we have shown that the small world networks may bring the society very close to consensus for even small threshold levels, but require higher tolerance than the complete network to reach consensus. We have identified the conditions under which the process with stubborn agents generates long-run consensus, permanent disagreement or permanent fluctuation in opinions. There cannot be a persistent fluctuation in opinions in the environment of regular agents nor in the presence of a single group of stubborn agents. In the runs with a single group of stubborn extremists, we have identified the Popper paradox despite the existence of a tolerance span in which the proportion of extremism decreases as the tolerance level increases. Further, in a highly tolerant society with two competing extremist groups, they have no supporters among the regular agents whose opinions are oscillating around the center of the opinion space. The influence of inconsistent agents is persistent and induces a perpetual opinion dynamics. The model is non-equilibrium and emerging, while consensus, if attainable, can be reached in a finite time.

Keywords

Opinion formation Continuous opinions Consensus Social networks Bounded confidence Stubborn agents Insincere agents 

Notes

Acknowledgements

The algorithm is written in C++ (compiled for 64-bit Visual Studio 12). The source code is available at https://github.com/pixlifai/opiform. The authors would like to thank seminar participants at the Fifth World Congress of the Game Theory Society (GAMES 2016), Maastricht, the Netherlands, July 24–28, 2016, and the 22nd Annual Workshop on Economic Science with Heterogeneous Interacting Agents (WEHIA 2017), Milan, Italy, June 12–14, 2017, for helpful comments and suggestions. We also thank two anonymous referees who made a number of helpful comments and suggestions. All errors remain the responsibility of the authors.

References

  1. Acemoglu, D., & Ozdaglar, A. (2011). Opinion dynamics and learning in social networks. Dynamic Games and Applications, 1(1), 3–49.Google Scholar
  2. Acemoglu, D., Ozdaglar, A., & ParandehGheibi, A. (2010). Spread of (mis) information in social networks. Games and Economic Behavior, 70(2), 194–227.Google Scholar
  3. Acemoglu, D., Como, G., Fagnani, F., & Ozdaglar, A. (2013). Opinion fluctuations and disagreement in social networks. Mathematics of Operations Research, 38(1), 1–27.Google Scholar
  4. Albert, R., Jeong, H., & Barabási, A. L. (2000). Error and attack tolerance of complex networks. Nature, 406(6794), 378–382.Google Scholar
  5. Altafini, C. (2013). Consensus problems on networks with antagonistic interactions. IEEE Transactions on Automatic Control, 58(4), 935–946.Google Scholar
  6. Axelrod, R. (1997). The dissemination of culture a model with local convergence and global polarization. Journal of Conflict Resolution, 41(2), 203–226.Google Scholar
  7. Bala, V., & Goyal, S. (1998). Learning from neighbours. Review of Economic Studies, 65(3), 595–621.Google Scholar
  8. Banerjee, A., & Fudenberg, D. (2004). Word-of-mouth learning. Games and Economic Behavior, 46(1), 1–22.Google Scholar
  9. Barabási, A. L., & Albert, R. (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512.Google Scholar
  10. Baronchelli, A., Loreto, V., & Steels, L. (2008). In-depth analysis of the naming game dynamics: The homogeneous mixing case. International Journal of Modern Physics C, 19(05), 785–812.Google Scholar
  11. Ben-Naim, E. (2005). Opinion dynamics: Rise and fall of political parties. EPL (Europhysics Letters), 69(5), 671–677.Google Scholar
  12. Blondel, V. D., Hendrickx, J. M., & Tsitsiklis, J. N. (2009). On Krause’s multi-agent consensus model with state-dependent connectivity. IEEE Transactions on Automatic Control, 54(11), 2586–2597.Google Scholar
  13. Blume, L. E. (1993). The statistical mechanics of strategic interaction. Games and Economic Behavior, 5(3), 387–424.Google Scholar
  14. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., & Hwang, D. U. (2006). Complex networks: Structure and dynamics. Physics Reports, 424(4), 175–308.Google Scholar
  15. Boccaletti, S., Bianconi, G., Criado, R., Del Genio, C. I., Gómez-Gardenes, J., Romance, M., et al. (2014). The structure and dynamics of multilayer networks. Physics Reports, 544(1), 1–122.Google Scholar
  16. Burnside, C., Eichenbaum, M., & Rebelo, S. (2011). Understanding booms and busts in housing markets. Working Paper 16734, National Bureau of Economic Research.Google Scholar
  17. Castellano, C., Fortunato, S., & Loreto, V. (2009). Statistical physics of social dynamics. Reviews of Modern Physics, 81, 591–646.Google Scholar
  18. Deffuant, G., Neau, D., Amblard, F., & Weisbuch, G. (2000). Mixing beliefs among interacting agents. Advances in Complex Systems, 3(01n04), 87–98.Google Scholar
  19. Deffuant, G., Amblard, F., Weisbuch, G., & Faure, T. (2002). How can extremism prevail? A study based on the relative agreement interaction model. Journal of Artificial Societies and Social Simulation, 5(4). http://jasss.soc.surrey.ac.uk/5/4/1.html.
  20. DeGroot, M. H. (1974). Reaching a consensus. Journal of the American Statistical Association, 69(345), 118–121.Google Scholar
  21. DeMarzo, P. M., Vayanos, D., & Zwiebel, J. (2003). Persuasion bias, social influence, and unidimensional opinions. Quarterly Journal of Economics, 118(3), 909–968.Google Scholar
  22. Dutta, B., & Sen, A. (2012). Nash implementation with partially honest individuals. Games and Economic Behavior, 74(1), 154–169.Google Scholar
  23. Eliaz, K. (2002). Fault tolerant implementation. Review of Economic Studies, 69(3), 589–610.Google Scholar
  24. Fischer, M. J., Lynch, N. A., & Paterson, M. S. (1985). Impossibility of distributed consensus with one faulty process. Journal of the ACM (JACM), 32(2), 374–382.Google Scholar
  25. Galam, S. (2004). Contrarian deterministic effects on opinion dynamics:the hung elections scenario. Physica A: Statistical Mechanics and its Applications, 333, 453–460.Google Scholar
  26. Galam, S. (2008). Sociophysics: A review of galam models. International Journal of Modern Physics C, 19(3), 409–440.Google Scholar
  27. Gale, D., & Kariv, S. (2003). Bayesian learning in social networks. Games and Economic Behavior, 45(2), 329–346.Google Scholar
  28. Gentzkow, M., & Shapiro, J. M. (2011). Ideological segregation online and offline. Quarterly Journal of Economics, 126(4), 1799–1839.Google Scholar
  29. Glaeser, E. L., Sacerdote, B., & Scheinkman, J. A. (1996). Crime and social interactions. Quarterly Journal of Economics, 111(2), 507–548.Google Scholar
  30. Golub, B., & Jackson, M. O. (2010). Naïve learning in social networks and the wisdom of crowds. American Economic Journal Microeconomics, 2(1), 112–149.Google Scholar
  31. Granovetter, M. (1978). Threshold models of collective behavior. American Journal of Sociology, 83(6), 1420–1443.Google Scholar
  32. Granovetter, M. (1995). Getting a job: A study of contacts and careers. Chicago: University of Chicago Press.Google Scholar
  33. Gruhl, D., Guha, R., Liben-Nowell, D., & Tomkins, A. (2004). Information diffusion through blogspace. In Proceedings of the 13th international conference on world wide web, ACM, WWW ’04 (pp. 491–501).Google Scholar
  34. Hayek, F. (1945). The use of knowledge in society. American Economic Review, 35(4), 519–530.Google Scholar
  35. Hegselmann, R., & Krause, U. (2002). Opinion dynamics and bounded confidence models, analysis, and simulation. Journal of Artificial Societies and Social Simulation, 5(3). http://jasss.soc.surrey.ac.uk/5/3/2.html.
  36. Hegselmann, R., & Krause, U. (2005). Opinion dynamics driven by various ways of averaging. Computational Economics, 25(4), 381–405.Google Scholar
  37. Hirshleifer, D. (2001). Investor psychology and asset pricing. Journal of Finance, 56(4), 1533–1597.Google Scholar
  38. Jackson, M. O. (2010). Social and economic networks. Princeton: Princeton University Press.Google Scholar
  39. Jadbabaie, A., Lin, J., & Morse, A. S. (2003). Coordination of groups of mobile autonomous agents using nearest neighbor rules. IEEE Transactions on Automatic Control, 48(6), 988–1001.Google Scholar
  40. Jadbabaie, A., Molavi, P., Sandroni, A., & Tahbaz-Salehi, A. (2012). Non-Bayesian social learning. Games and Economic Behavior, 76(1), 210–225.Google Scholar
  41. Katz, E., & Lazarsfeld, P. F. (1955). Personal influence, the part played by people in the flow of mass communications. New York, NY: The Free Press.Google Scholar
  42. Kauffman, S. (1993). The origins of order: Self-organization and selection in evolution. Oxford: Oxford University Press.Google Scholar
  43. Kempe, D., Kleinberg, J., & Tardos, É. (2003). Maximizing the spread of influence through a social network. In Proceedings of the ninth ACM SIGKDD international conference on knowledge discovery and data mining, ACM (pp. 137–146).Google Scholar
  44. Krapivsky, P. L., & Redner, S. (2003). Dynamics of majority rule in two-state interacting spin systems. Physical Review Letters, 90(23), 238,701.Google Scholar
  45. Kwak, H., Lee, C., Park, H., & Moon, H. (2010). What is twitter, a social network or a news media? In Proceedings of the 19th international conference on world wide web, ACM, WWW ’10 (pp. 591–600).Google Scholar
  46. Lamport, L., Shostak, R., & Pease, M. (1982). The byzantine generals problem. ACM Transactions on Programming Languages and Systems (TOPLAS), 4(3), 382–401.Google Scholar
  47. LeBlanc, H. J., Zhang, H., Sundaram, S., & Koutsoukos, X. (2012). Consensus of multi-agent networks in the presence of adversaries using only local information. In Proceedings of the 1st international conference on high confidence networked systems, ACM (pp. 1–10).Google Scholar
  48. Leskovec, J., Adamic, L. A., & Huberman, B. A. (2007). The dynamics of viral marketing. ACM Transactions on the Web (TWEB), 1(1), 5.Google Scholar
  49. Lorenz, J. (2010). Heterogeneous bounds of confidence: Meet, discuss and find consensus!. Complexity, 15(4), 43–52.Google Scholar
  50. Marti, S., Giuli, T.J., Lai, K., & Baker, M. (2000). Mitigating routing misbehavior in mobile ad hoc networks. In Proceedings of the 6th annual international conference on mobile computing and networking, ACM (pp. 255–265).Google Scholar
  51. Motsch, S., & Tadmor, E. (2014). Heterophilious dynamics enhances consensus. SIAM Review, 56(4), 577–621.Google Scholar
  52. Newman, M. E. (2002). Spread of epidemic disease on networks. Physical Review E, 66(1), 016,128.Google Scholar
  53. Nowak, M. A., & May, R. M. (1992). Evolutionary games and spatial chaos. Nature, 359(6398), 826–829.Google Scholar
  54. Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533.Google Scholar
  55. Pasqualetti, F., Bicchi, A., & Bullo, F. (2012). Consensus computation in unreliable networks: A system theoretic approach. IEEE Transactions on Automatic Control, 57(1), 90–104.Google Scholar
  56. Pastor-Satorras, R., & Vespignani, A. (2001). Epidemic spreading in scale-free networks. Physical Review Letters, 86, 3200–3203.Google Scholar
  57. Popper, K. (1945). The open society and its enemies. London: Routledge.Google Scholar
  58. Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50(5), 655–661.Google Scholar
  59. Schelling, T. (1971). Dynamic models of segregation. Journal of Mathematical Sociology, 1(2), 143–186.Google Scholar
  60. Sood, V., & Redner, S. (2005). Voter model on heterogeneous graphs. Physical Review Letters, 94(17), 178,701.Google Scholar
  61. Szabó, G., & Fath, G. (2007). Evolutionary games on graphs. Physics Reports, 446(4), 97–216.Google Scholar
  62. Sznajd-Weron, K., & Sznajd, J. (2000). Opinion evolution in closed community. International Journal of Modern Physics C, 11(06), 1157–1165.Google Scholar
  63. Tesfatsion, L., & Judd, K. (2006). Handbook of computational economics: Agent-based computational economics. Amsterdam: North Holland.Google Scholar
  64. Valente, T. W. (1995). Network models of the diffusion of innovations. New York: Hampton Press Cresskill.Google Scholar
  65. Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I., & Shochet, O. (1995). Novel type of phase transition in a system of self-driven particles. Physical Review Letters, 75(6), 1226.Google Scholar
  66. Ward, A. J., Sumpter, D. J., Couzin, I. D., Hart, P. J., & Krause, J. (2008). Quorum decision-making facilitates information transfer in fish shoals. Proceedings of the National Academy of Sciences, 105(19), 6948–6953.Google Scholar
  67. Wasserman, S., & Faust, K. (1994). Social network analysis: Methods and applications. Cambridge: Cambridge University Press.Google Scholar
  68. Watts, D. J., & Dodds, P. S. (2007). Influentials, networks, and public opinion formation. Journal of Consumer Research, 34(4), 441–458.Google Scholar
  69. Watts, D. J., & Strogatz, S. H. (1998). Collective dynamics of small-world networks. Nature, 393(6684), 440–442.Google Scholar
  70. Weisbuch, G., Deffuant, G., Amblard, F., & Nadal, J. P. (2002). Meet, discuss, and segregate!. Complexity, 7(3), 55–63.Google Scholar
  71. Young, H. P., & Zamir, S. (2015). Handbook of game theory with economic applications (Vol. 4). Amsterdam: Elsevier.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Faculty of Business StudiesLjubljanaSlovenia
  2. 2.Kiel Institute for the World EconomyKielGermany

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