Kinetic equations and stochastic game theory for social systems

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".