Nonlinear Dynamics

, Volume 92, Issue 1, pp 25–32 | Cite as

Herding as a consensus problem

  • Franco Garofalo
  • Francesco Lo Iudice
  • Elena Napoletano
Original Paper


In this paper, we show that herding phenomena in financial markets can be interpreted using the theoretical tools of pinning control. This is accomplished by viewing herding as a diffusion of a certain opinion in a network of financial agents, whose trading strategies dynamically depend on that of their neighbors according to a nonlinear state-dependent law. The interaction among the agents is modeled through a directed weighted graph, and following the logic of pinning control, we model the generic exogenous information triggering herding behavior as a control signal fed by an external entity to a subset of agents that, by virtue of the received information, can play the correct trading action. The topological conditions of partial pinning control theory enable us to predict the number of agents reaching consensus, i.e., the diffusion of information through the network, and thus the magnitude of the herding phenomenon triggered by the informed/pinned nodes. By testing our model of opinion dynamics in an artificial agent-based financial market, we prove that it is capable of replicating herding phenomena of different and predictable intensities.


Opinion dynamics Informational cascades Herding intensity Partial pinning control 


  1. 1.
    Avery, C., Zemsky, P.: Multidimensional uncertainty and herd behavior in financial markets. Am. Econ. Rev. 88, 724–748 (1998)Google Scholar
  2. 2.
    Banerjee, A.V.: A simple model of herd behavior. Q. J. Econ. 107, 797–817 (1992)CrossRefGoogle Scholar
  3. 3.
    Bargigli, L., Tedeschi, G.: Interaction in agent-based economics: a survey on the network approach. Phys. A Stat. Mech. Appl. 399, 1–15 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Barmish, B.R., Primbs, J.A.: On market-neutral stock trading arbitrage via linear feedback. In: 2012 American Control Conference (ACC), pp. 3693–3698. IEEE (2012)Google Scholar
  5. 5.
    Bikhchandani, S., Hirshleifer, D., Welch, I.: A theory of fads, fashion, custom, and cultural change as informational cascades. J. Political Econ. 100, 992–1026 (1992)CrossRefGoogle Scholar
  6. 6.
    DeGroot, M.H.: Reaching a consensus. J. Am. Stat. Assoc. 69(345), 118–121 (1974)CrossRefzbMATHGoogle Scholar
  7. 7.
    DeLellis, P., di Bernardo, M., Russo, G.: On quad, lipschitz, and contracting vector fields for consensus and synchronization of networks. IEEE Trans. Circuits Syst. I Regul. Papers 58(3), 576–583 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    DeLellis, P., Garofalo, F., Iudice, F.L.: Partial pinning control of complex networks. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 7398–7403. IEEE (2016)Google Scholar
  9. 9.
    DeLellis, P., Garofalo, F., Iudice, F.L., Napoletano, E.: Wealth distribution across communities of adaptive financial agents. New J. Phys. 17(8), 083003 (2015)CrossRefGoogle Scholar
  10. 10.
    DeLellis, P., Garofalo, F., Lo Iudice, F.: The partial pinning control strategy for large complex networks. Automatica 89, 111–116 (2018)MathSciNetCrossRefGoogle Scholar
  11. 11.
    DeLellis, P., Meglio Anna Di, F., Iudice, F.L.: Overconfident agents and evolving financial networks. Nonlinear Dyn. 1–8 (2017)Google Scholar
  12. 12.
    Devenow, A., Welch, I.: Rational herding in financial economics. Eur. Econ. Rev. 40(3), 603–615 (1996)CrossRefGoogle Scholar
  13. 13.
    Estrada, E., Vargas-Estrada, E., Ando, H.: Communicability angles reveal critical edges for network consensus dynamics. Phys. Rev. E 92(5), 052809 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Farmer, J.D., Foley, D.: The economy needs agent-based modelling. Nature 460(7256), 685–686 (2009)CrossRefGoogle Scholar
  15. 15.
    Friedkin, N.E., Johnsen, E.: Social influence networks and opinion change. Adv. Group Process. 16, 1–29 (1999)Google Scholar
  16. 16.
    Friedkin, N.E., Proskurnikov, A.V., Tempo, R., Parsegov, S.E.: Network science on belief system dynamics under logic constraints. Science 354(6310), 321–326 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gleason, K.C., Lee, C.I., Mathur, I.: Herding behavior in european futures markets. Finance Lett. 1(1), 5–8 (2003)Google Scholar
  18. 18.
    Hegselmann, R., Krause, U., et al.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5(3), 1–33 (2002)Google Scholar
  19. 19.
    Hirshleifer, D., Hong Teoh, S.: Herd behaviour and cascading in capital markets: a review and synthesis. Eur. Financ. Manag. 9(1), 25–66 (2003)CrossRefGoogle Scholar
  20. 20.
    Holt, C.A., Laury, S.K., et al.: Risk aversion and incentive effects. Am. Econ. Rev. 92(5), 1644–1655 (2002)CrossRefGoogle Scholar
  21. 21.
    Huang, M., Manton, J.H.: Coordination and consensus of networked agents with noisy measurements: stochastic algorithms and asymptotic behavior. SIAM J. Control Optim. 48(1), 134–161 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ingersoll, J.E.: Theory of Financial Decision Making, vol. 3. Rowman & Littlefield, Lanham, Maryland, USA (1987)Google Scholar
  23. 23.
    Kaltwasser, P.R.: Uncertainty about fundamentals and herding behavior in the FOREX market. Phys. A Stat. Mech. Appl. 389(6), 1215–1222 (2010)CrossRefGoogle Scholar
  24. 24.
    Klamser, P.P., Wiedermann, M., Donges, J.F., Donner, R.V.: Zealotry effects on opinion dynamics in the adaptive voter model (2016). Preprint. arXiv:1612.06644
  25. 25.
    Kononovicius, A., Gontis, V.: Three-state herding model of the financial markets. EPL (Europhys. Lett.) 101(2), 28001 (2013)CrossRefGoogle Scholar
  26. 26.
    Kremer, S., Nautz, D.: Causes and consequences of short-term institutional herding. J. Bank. Finance 37(5), 1676–1686 (2013)CrossRefGoogle Scholar
  27. 27.
    Lakonishok, J., Shleifer, A., Vishny, R.W.: The impact of institutional trading on stock prices. J. Financ. Econ. 32(1), 23–43 (1992)CrossRefGoogle Scholar
  28. 28.
    Li, K., Sun, W., Small, M., Fu, X.: Practical synchronization on complex dynamical networks via optimal pinning control. Phys. Rev. E 92(1), 010903 (2015)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Lillo, F., Moro, E., Vaglica, G., Mantegna, R.N.: Specialization and herding behavior of trading firms in a financial market. New J. Phys. 10(4), 043019 (2008)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Lotero, L., Hurtado, R.G., Floría, L.M., Gómez-Gardeñes, J.: Rich do not rise early: spatio-temporal patterns in the mobility networks of different socio-economic classes. R. Soc. Open Sci. 3(10), 150654 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Mai, V.S., Abed, E.H.: Opinion dynamics with persistent leaders. In: 2014 IEEE 53rd Annual Conference on Decision and Control (CDC), pp. 2907–2913. IEEE (2014)Google Scholar
  32. 32.
    Mirtabatabaei, A., Jia, P., Friedkin, N.E., Bullo, F.: On the reflected appraisals dynamics of influence networks with stubborn agents. In: American Control Conference (ACC), pp. 3978–3983. IEEE (2014)Google Scholar
  33. 33.
    Parker, W.D., Prechter, R.R.: Herding: an interdisciplinary integrative review from a socioeconomic perspective. Available at SSRN 2009898 (2005)Google Scholar
  34. 34.
    Parsegov, S.E., Proskurnikov, A.V., Tempo, R., Friedkin, N.E.: Novel multidimensional models of opinion dynamics in social networks. IEEE Trans. Autom. Control 62, 2270 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Proskurnikov, A.V., Tempo, R.: A tutorial on modeling and analysis of dynamic social networks. Part I. Ann. Rev. Control 43, 65 (2017)CrossRefGoogle Scholar
  36. 36.
    Shapira, Y., Berman, Y., Ben-Jacob, E.: Modelling the short term herding behaviour of stock markets. New J. Phys. 16(5), 053040 (2014)CrossRefGoogle Scholar
  37. 37.
    Sorrentino, F., di Bernardo, M., Garofalo, F., Chen, G.: Controllability of complex networks via pinning. Phys. Rev. E 75(4), 046103 (2007)Google Scholar
  38. 38.
    Tian, Y., Wang, L.: Opinion consensus in social networks with stubborn agents: an issue-based perspective (2016). Preprint. arXiv:1609.03465
  39. 39.
    Von Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (2007)zbMATHGoogle Scholar
  40. 40.
    Welch, I.: Herding among security analysts. J. Financ. Econ. 58(3), 369–396 (2000)CrossRefGoogle Scholar
  41. 41.
    Yang, C., Hu, S., Xia, B.: The endogenous dynamics of financial markets: interaction and information dissemination. Physica A 391(12), 3513–3525 (2012)CrossRefGoogle Scholar
  42. 42.
    Zhang, J.: Strategic delay and the onset of investment cascades. RAND J. Econ. 28, 188–205 (1997)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Electrical Engineering and Information TechnologyUniversity of Naples Federico IINaplesItaly

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