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Hydrodynamic Models of Preference Formation in Multi-agent Societies

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Abstract

In this paper, we discuss the passage to hydrodynamic equations for kinetic models of opinion formation. The considered kinetic models feature an opinion density depending on an additional microscopic variable, identified with the personal preference. This variable describes an opinion-driven polarisation process, leading finally to a choice among some possible options, as it happens, e.g. in referendums or elections. Like in the kinetic theory of rarefied gases, the derivation of hydrodynamic equations is based on the computation of the local equilibrium distribution of the opinions from the underlying kinetic model. Several numerical examples validate the resulting model, shedding light on the crucial role played by the distinction between opinion and preference formation on the choice processes in multi-agent societies.

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Notes

  1. In view of the scaling (16), as \(\epsilon \rightarrow 0^+\), the function (13) converges uniformly to \(\sqrt{1-w^2}\), which can therefore be chosen as diffusion coefficient in the Fokker–Planck equation (18) after performing the quasi-invariant limit.

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Acknowledgements

This research was partially supported by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018–2022)—Department of Mathematics “F. Casorati”, University of Pavia and Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP: E11G18000350001) and through the PRIN 2017 Project (No. 2017KKJP4X) “Innovative numerical methods for evolutionary partial differential equations and applications”. This work is also part of the activities of the Starting Grant “Attracting Excellent Professors” funded by “Compagnia di San Paolo” (Torino) and promoted by Politecnico di Torino. L.P. is member of GNCS (Gruppo Nazionale per il Calcolo Scientifico) of INdAM (Istituto Nazionale di Alta Matematica), Italy. G.T, A.T. and M.Z. are members of GNFM (Gruppo Nazionale per la Fisica Matematica) of INdAM, Italy.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, University of Ferrara, Via Machiavelli 35, 44121, Ferrara, Italy

    Lorenzo Pareschi

  2. Department of Mathematics “F. Casorati”, University of Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Giuseppe Toscani

  3. Department of Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

    Andrea Tosin & Mattia Zanella

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  1. Lorenzo Pareschi
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  2. Giuseppe Toscani
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  4. Mattia Zanella
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Correspondence to Andrea Tosin.

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Communicated by Dr. Paul Newton.

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Pareschi, L., Toscani, G., Tosin, A. et al. Hydrodynamic Models of Preference Formation in Multi-agent Societies. J Nonlinear Sci 29, 2761–2796 (2019). https://doi.org/10.1007/s00332-019-09558-z

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