Boltzmann Games in Heterogeneous Consensus Dynamics

  • Giacomo AlbiEmail author
  • Lorenzo Pareschi
  • Mattia Zanella


We consider a constrained hierarchical opinion dynamics in the case of leaders’ competition and with complete information among leaders. Each leaders’ group tries to drive the followers’ opinion towards a desired state accordingly to a specific strategy. By using the Boltzmann-type control approach we analyze the best-reply strategy for each leaders’ population. Derivation of the corresponding Fokker–Planck model permits to investigate the asymptotic behaviour of the solution. Heterogeneous followers populations are then considered where the effect of knowledge impacts the leaders’ credibility and modifies the outcome of the leaders’ competition.


Multi-agent systems Differential games Boltzmann equation Opinion leaders Consensus dynamics Knowledge 

Mathematics Subject Classification

35Q20 35Q91 49N70 



MZ was supported by the “Compagnia di San Paolo” (Torino, Italy). GA, LP, and MZ thank the “INDAM-GNCS 2018” (CBGNCSALBI2018) contribution.


  1. 1.
    Albi, G., Choi, Y.-P., Fornasier, M., Kalise, D.: Mean field control hierarchy. Appl. Math. Optim. 76(1), 93–135 (2016)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Albi, G., Pareschi, L., Zanella, M.: Boltzmann-type control of opinion consensus through leaders. Philos. Trans. R. Soc. Lond. A 372(2028), 20140138 (2014)ADSMathSciNetzbMATHGoogle Scholar
  3. 3.
    Albi, G., Herty, M., Pareschi, L.: Kinetic description of optimal control problems and applications to opinion consensus. Commun. Math. Sci. 13(6), 1407–1429 (2015)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Albi, G., Pareschi, L., Toscani, G., Zanella, M.: Recent advances in opinion modeling: control and social influence. In: Bellomo, N., Degond, P., Tadmor, E. (eds.) Active Particles Volume 1, Theory, Methods, and Applications. Springer, New York (2017)Google Scholar
  5. 5.
    Albi, G., Pareschi, L., Zanella, M.: Opinion dynamics over complex networks: kinetic modelling and numerical methods. Kinet. Relat. Models 10(1), 1–32 (2017)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Amblard, F., Deffuant, G.: The role of network topology on extremism propagation with the relative agreement opinion dynamics. Phys. A 343, 725–738 (2004)Google Scholar
  7. 7.
    Bahrami, B., Olsen, K., Latham, P.E., Roepstorff, A., Rees, G., Frith, C.D.: Optimally interacting minds. Science 329(5995), 1081–1085 (2010)ADSGoogle Scholar
  8. 8.
    Bertozzi, A.L., Rosado, J., Short, M.B., Wang, L.: Contagion shocks in one dimension. J. Stat. Phys. 158(3), 647–664 (2015)ADSMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bongini, M., Fornasier, M., Kalise, D.: (UN)conditional consensus emergence under perturbed and decentralized feedback controls. Discret. Contin. Dyn. Syst. Ser. A 35, 4071–4094 (2015)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Brugna, C., Toscani, G.: Kinetic models of opinion formation in the presence of personal conviction. Phys. Rev. E 92(5), 052818 (2015)ADSGoogle Scholar
  11. 11.
    Bürger, M., Lorz, A., Wolfram, M.-T.: Balanced growth path solutions of a Boltzmann mean field game model for knowledge growth. Kinet. Relat. Models 10(1), 117–140 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Cañizo, J.A., Carrillo, J.A., Rosado, J.: A well-posedness theory in measures for some kinetic models of collective behavior. Math. Models Methods Appl. Sci. 21(3), 515–539 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cardaliaguet, P.: A differential game with two players and one target. SIAM J. Control Optim. 34(4), 1441–1460 (1996)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Carrillo, J.A., Fornasier, M., Rosado, J., Toscani, G.: Asymptotic flocking dynamics for the kinetic Cucker–Smale model. SIAM J. Math. Anal. 42(1), 218–236 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Carrillo, J.A., Fornasier, M., Toscani, G., Vecil, F.: Particle, kinetic and hydrodynamic models of swarming. In: Naldi, G., Pareschi, L., Toscani, G. (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, pp. 297–336. Birkhäuser, Boston (2010)Google Scholar
  16. 16.
    Carrillo, J.A., Pareschi, L., Zanella, M.: Particle based gPC methods for mean-field models of swarming with uncertainty. Commun. Comput. Phys. 25(2), 508–531 (2019)MathSciNetGoogle Scholar
  17. 17.
    Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81(2), 591 (2009)ADSGoogle Scholar
  18. 18.
    Cordier, S., Pareschi, L., Toscani, G.: On a kinetic model for a simple market economy. J. Stat. Phys. 120(1–2), 253–277 (2005)ADSMathSciNetzbMATHGoogle Scholar
  19. 19.
    Camerer, C.: Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  20. 20.
    Cristiani, E., Piccoli, B., Tosin, A.: Multiscale Modeling of Pedestrian Dynamics, MS&A: Modeling, Simulation and Applications, vol. 12. Springer International Publishing, New York (2014)zbMATHGoogle Scholar
  21. 21.
    Cristiani, E., Tosin, A.: Reducing complexity of multiagent systems with symmetry breaking: an application to opinion dynamics with polls. Multiscale Model. Simul. 16(1), 528–549 (2018)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Degond, P., Liu, J.-G., Ringhofer, C.: Evolution of wealth in a nonconservative economy driven by local Nash equilibria. Philos. Trans. R. Soc. Lond. A 372, 20130394 (2014)ADSMathSciNetzbMATHGoogle Scholar
  23. 23.
    Degond, P., Herty, M., Liu, J.-G.: Meanfield games and model predictive control. Commun. Math. Sci. 15(5), 1403–1422 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Düring, B., Markowich, P., Pietschmann, J.F., Wolfram, M.-T.: Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. Lond. A 465(2112), 3687–3708 (2009)ADSMathSciNetzbMATHGoogle Scholar
  25. 25.
    Düring, B., Wolfram, M.-T.: Opinion dynamics: inhomogeneous Boltzmann-type equations modelling opinion leadership. Proc. R. Soc. Lond. A 471, 20150345 (2015)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Esmaeili, M., Aryanezhad, M.B., Zeephongsekul, P.: A game theory approach in seller–buyer supply chain. Eur. J. Op. Res. 195(2), 442–448 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Fornasier, M., Piccoli, B., Rossi, F.: Mean-field sparse optimal control. Philos. Trans. R. Soc. Lond. A 372(2028), 20130400 (2014)ADSMathSciNetzbMATHGoogle Scholar
  28. 28.
    Furioli, G., Pulvirenti, A., Terraneo, E., Toscani, G.: The grazing collision limit of the inelastic Kac model around a Lévy-type equilibrium. SIAM J. Math. Anal. 44, 827–850 (2012)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Furioli, G., Pulvirenti, A., Terraneo, E., Toscani, G.: Fokker–Planck equations in the modeling of socio-economic phenomena. Math. Models Methods Appl. Sci. 27(1), 115–158 (2017)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Giesecke, K., Weber, S.: Credit contagion and aggregate losses. J. Econ. Dyn. Control 30(5), 741–767 (2006)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Hegselmann, R., Krause, U.: Opinion dynamics and bounded confidence models, analysis, and simulation. J. Artif. Soc. Soc. Simul. 5(3), 2–33 (2002)Google Scholar
  32. 32.
    Herty, M., Zanella, M.: Performance bounds for the mean-field limit of constrained dynamics. Discret. Contin. Dyn. Syst. Ser. A 37(4), 2023–2043 (2017)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Herty, M., Steffensen, S., Pareschi, L.: Mean-field control and Riccati equations. Netw. Heterog. Media 10, 699–715 (2015)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Isaacs, R.: Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit Control and Optimization. Wiley, New York (1965)zbMATHGoogle Scholar
  35. 35.
    Jørgensen, S., Zaccour, G.: Differential Games in Marketing, vol. 15. Springer Science & Business Media, New York (2012)Google Scholar
  36. 36.
    Hovland, C.I., Weiss, W.: The influence of source credibility on communication effectiveness. Pub. Opin. Quart. 15(4), 635–650 (1951)Google Scholar
  37. 37.
    Motsch, S., Tadmor, E.: Heterophilious dynamics enhances consensus. SIAM Rev. 56(4), 577–621 (2014)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Nitzan, S.: Modelling rent-seeking contests. Eur. J. Pol. Econ. 10(1), 41–60 (1994)Google Scholar
  39. 39.
    Pareschi, L., Russo, G.: An introduction to Monte Carlo methods for the Boltzmann equation. ESAIM Proc. 10, 35–75 (2001)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Pareschi, L., Toscani, G.: Interacting Multiagent Systems. Kinetic Equations and Monte Carlo Methods. Oxford University Press, Oxford (2013)zbMATHGoogle Scholar
  41. 41.
    Pareschi, L., Toscani, G.: Wealth distribution and collective knowledge: a Boltzmann approach. Philos. Trans. R. Soc. A 372(2028), 20130396 (2014)ADSMathSciNetzbMATHGoogle Scholar
  42. 42.
    Pareschi, L., Vellucci, P., Zanella, M.: Kinetic models of collective decision-making in the presence of equality bias. Phys. A Stat. Mech. Appl. 467, 201–217 (2017)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Sanfey, A.G.: Social decision-making: insights from game theory and neuroscience. Science 318(5850), 598–602 (2007)ADSGoogle Scholar
  44. 44.
    Sznajd-Weron, K., Sznajd, J.: Opinion evolution in closed community. Int. J. Mod. Phys. C 11(06), 1157–1165 (2000)ADSzbMATHGoogle Scholar
  45. 45.
    Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Tosin, A., Zanella, M.: Boltzmann-type models with uncertain binary interactions. Commun. Math. Sci. 16(4), 962–984 (2018)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Villani, C.: On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143(3), 273–307 (1998)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Watts, D.J., Dodds, P.S.: Influentials, networks, and public opinion formation. J. Consum. Res. 34(4), 441–458 (2007)Google Scholar
  49. 49.
    Wirl, F.: The dynamics of lobbying: a differential game. Pub. Choice 80(3–4), 307–323 (1994)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer SciencesUniversity of VeronaVeronaItaly
  2. 2.Department of Mathematics and Computer ScienceUniversity of FerraraFerraraItaly
  3. 3.Department of Mathematical SciencesPolitecnico di TorinoTorinoItaly

Personalised recommendations