# American Institute of Mathematical Sciences

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March  2009, 2(1): 135-149. doi: 10.3934/krm.2009.2.135

## A Boltzmann-like equation for choice formation

 1 Department of Mathematics at the University of Pavia, via Ferrata 1, 27100 Pavia, Italy, Italy 2 Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia

Received  September 2008 Revised  October 2008 Published  January 2009

We describe here a possible approach to the formation of choice in a society by methods borrowed from the kinetic theory of rarefied gases. It is shown that the evolution of the continuous density of opinions obeys a linear Boltzmann equation where the background density represents the fixed distribution of possible choices. The binary interactions between individuals are in general non-local, and take into account both the compromise propensity and the self-thinking. In particular regimes, the linear Boltzmann equation is well described by a Fokker-Planck type equation, for which in some cases the steady states (distribution of choices) can be obtained in analytical form. This Fokker-Planck type equation generalizes analogous one obtained by mean field approximation of the voter model in [27]. Numerical examples illustrate the influence of different model parameters in the description both of the shape of the distribution of choices, and in its mean value.
Citation: Valeriano Comincioli, Lucia Della Croce, Giuseppe Toscani. A Boltzmann-like equation for choice formation. Kinetic and Related Models, 2009, 2 (1) : 135-149. doi: 10.3934/krm.2009.2.135
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