Abstract
In this paper we present mathematical tools inspired by the kinetic theory, which can be used to model the social behaviors of large communities of individuals. The focus is especially on human societies, such as the population of a certain country, and on the interplays between concurrent social dynamics, for instance economic issues linked to the formation of political opinions, which sometimes can even degenerate into dramatic extreme events with massive impact (Black Swans). Starting from Boltzmann-type models, we present an evolution of the classical approach of statistical mechanics, whose hallmark is the use of stochastic game theory for the description of social interactions. By this we mean that the latter are modeled as games whose payoffs, however, are known only in probability. This is consistent with the basic unpredictability of human reactions, which ultimately cannot be compared to deterministic mechanical-like “collisions”.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Unraveling Complex Systems (2011). URL http://www.mathaware.org/mam/2011. Mathematics Awareness Month of the American Mathematical Society, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics
Agrawal, A., Kapur, D., McHale, J.: How do spatial and social proximity influence knowledge flows? Evidence from patent data. J. Urban Econ. 64(2), 258–269 (2008)
Ajmone Marsan, G.: New paradigms towards the modelling of complex systems in behavioral economics. Math. Comput. Modelling 50(3–4), 584–597 (2009)
Ajmone Marsan, G.: On the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition. Comput. Math. Appl. 57(5), 710–728 (2009)
Ajmone Marsan, G., Bellomo, N., Egidi, M.: Towards amathematical theory of complex socioeconomical systems by functional subsystems representation. Kinet. Relat. Models 1(2), 249–278 (2008)
Ajmone Marsan, G., Bellomo, N., Tosin, A.: Complex Systems and Society-Modeling and Simulation. SpringerBriefs in Mathematics. Springer, New York (2013)
Ariel, R.: Modeling Bounded Rationality. MIT Press, Cambridge, MA (1998)
Arlotti, L., Bellomo, N., De Angelis, E.: Generalized kinetic (Boltzmann) models: mathematical structures and applications. Math. Models Methods Appl. Sci. 12(4), 567–591 (2002)
Arlotti, L., De Angelis, E., Fermo, L., Lachowicz, M., Bellomo, N.: On a class of integrodifferential equations modeling complex systems with nonlinear interactions. Appl. Math. Lett. 25(3), 490–495 (2012)
Ballerini, M., Cabibbo, N., Candelier, R., Cavagna, A., Cisbani, E., Giardina, I., Lecomte, V., Orlandi, A., Parisi, G., Procaccini, A., Viale, M., Zdravkovic, V.: Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. USA 105(4), 1232–1237 (2008)
Bellomo, N.: Modeling complex living systems-A kinetic theory and stochastic game approach. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2008)
Bellomo, N., Carbonaro, B.: Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives. Phys. Life Rev. 8(1), 1–18 (2011)
Bellomo, N., Herrero, M.A., Tosin, A.: On the dynamics of social conflicts: Looking for the Black Swan. Kinet. Relat. Models 6(3), 459–479 (2013). Open Access http://dx.doi.org/10.3934/krm.2013.6.459
Bellomo, N., Knopoff, D., Soler, J.: On the difficult interplay between life, “complexity”, and mathematical sciences. Math. Models Methods Appl. Sci. 23(10), 1861–1913 (2013)
Bellomo, N., Soler, J.: On the mathematical theory of the dynamics of swarms viewed as complex systems. Math. Models Methods Appl. Sci. 22(suppl. 1), 1140,006 (29 pages) (2012)
Bertotti, M.L., Delitala, M.: Conservation laws and asymptotic behavior of a model of social dynamics. Nonlinear Anal. Real World Appl. 9(1), 183–196 (2008)
Bertotti, M.L., Delitala, M.: On the existence of limit cycles in opinion formation processes under time periodic influence of persuaders. Math. Models Methods Appl. Sci. 18(6), 913–934 (2008)
Bertotti, M.L., Delitala, M.: Cluster formation in opinion dynamics: a qualitative analysis. Z. Angew. Math. Phys. 61(4), 583–602 (2010)
Bertotti, M.L., Modanese, G.: From microscopic taxation and redistribution models to macroscopic income distributions. Phys. A 390(21–22), 3782–3793 (2011)
Bisi, M., Spiga, G., Toscani, G.: Kinetic models of conservative economies with wealth redistribution. Commun. Math. Sci. 7(4), 901–916 (2009)
Comincioli, V., Della Croce, L., Toscani, G.: A Boltzmann-like equation for choice formation. Kinet. Relat. Models 2(1), 135–149 (2009)
Cristiani, E., Piccoli, B., Tosin, A.: Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In: G. Naldi, L. Pareschi, G. Toscani (eds.) Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, pp. 337–364. Birkhäuser, Boston (2010)
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. Ser. A Math. Phys. Eng. Sci. 465(2112), 3687–3708 (2009)
Goyal, S., Vega-Redondo, F.: Network formation and social coordination. Game Econ. Behav. 50(2), 178–207 (2005)
Helbing, D.: Stochastic and Boltzmann-like models for behavioral changes, and their relation to game theory. Phys. A 193(2), 241–258 (1993)
Helbing, D.: Social Self-Organization. Springer-Verlag, Berlin Heidelberg (2012)
Herbert, S.: Bounded rationality and organizational learning. Organ. Sci. 2(1), 125–134 (1991)
Maldarella, D., Pareschi, L.: Kinetic models for socio-economic dynamics of speculative markets. Phys. A 391(3), 715–730 (2012)
Naldi, G., Pareschi, L., Toscani, G. (eds.): Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston (2010)
Rubinstein, A.: Modeling Bounded Rationality, Zeuthen Lecture Book, vol. 1. MIT Press, Cambridge, MA (1998)
Taleb, N.N.: The Black Swan: The Impact of the Highly Improbable. Random House, New York City (2007)
Toscani, G.: Kinetic models of opinion formation. Commun. Math. Sci. 4(3), 481–496 (2006)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Tosin, A. (2014). Kinetic Equations and Stochastic Game Theory for Social Systems. In: Celletti, A., Locatelli, U., Ruggeri, T., Strickland, E. (eds) Mathematical Models and Methods for Planet Earth. Springer INdAM Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-02657-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-02657-2_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02656-5
Online ISBN: 978-3-319-02657-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)