From Discrete Kinetic and Stochastic Game Theory to Modelling Complex Systems in Applied Sciences

Abstract

This paper deals with some methodological aspects related to the discretization of a class of integro-differential equations modelling the evolution of the probability distribution over the microscopic state of a large system of interacting individuals. The microscopic state includes both mechanical and socio-biological variables. The discretization of the microscopic state generates a class of dynamical systems defining the evolution of the densities of the discretized state. In general, this yields a system of partial differential equations replacing the continuous integro-differential equation. As an example, a specific application is discussed, which refers to modelling in the field of social dynamics. The derivation of the evolution equation needs the development of a stochastic game theory.

Cluster formation in opinion dynamics: a qualitative analysis

“…The present assumptions guarantee (see [4]) the well-posedness of the model and the positive invariance of the standard (n − 1)-simplex: for any given set…”

“…In this section, we look for equilibria of systems of form (4). In particular, we find that, if (f 1 , .…”

From the kinetic theory of active particles to the modeling of social behaviors and politics

“…The reader interested to these mathematical aspects is referred to paper Bertotti and Delitala (2004) or to the Chapters 2 and 3 of the book (Bellomo 2007).…”

“…Following this line, a systematic analysis of socio-political systems was developed in three recent papers by Bertotti and Delitala (2004, 2007, within the framework of the kinetic theory for particles with discrete states. These papers, together with Chauviere and Brazzoli (2006), offer the guiding lines of the analysis to investigate the ability of the model proposed in Bertotti and Delitala (2007) in describing how a social politics and the overall wealth disposable may have a relevant influence towards the trend of the wealth distribution.…”

On the qualitative analysis of the solutions of a mathematical model of social dynamics

“…different levels of wealth, was proposed in [10], corresponding to a society where interactions express competition and/or cooperation. The model refers to the generalized kinetic theory for active particles whose microscopic state includes the mechanical variables, typically position and velocity, but also an additional variable, called "activity", corresponding to a non-mechanical function of the particles.…”