Social climbing and Amoroso distribution

TL;DR

This paper models social status dynamics using kinetic equations, showing that the equilibrium distribution of social status follows Amoroso distributions with Pareto tails, indicating social stratification.

Contribution

It introduces a kinetic model based on prospect theory to derive Amoroso distribution as the steady state of social status, linking microscopic interactions to macro-level social hierarchies.

Findings

Steady states follow Amoroso distributions with Pareto tails.

Kinetic model captures social elite formation.

Numerical results confirm Amoroso distribution approximation.

Abstract

We introduce a class of one-dimensional linear kinetic equations of Boltzmann and Fokker--Planck type, describing the dynamics of individuals of a multi-agent society questing for high status in the social hierarchy. At the Boltzmann level, the microscopic variation of the status of agents around a universal desired target, is built up introducing as main criterion for the change of status a suitable value function in the spirit of the prospect theory of Kahneman and Twersky. In the asymptotics of grazing interactions, the solution density of the Boltzmann type kinetic equation is shown to converge towards the solution of a Fokker--Planck type equation with variable coefficients of diffusion and drift, characterized by the mathematical properties of the value function. The steady states of the statistical distribution of the social status predicted by the Fokker--Planck equations belong to the class of Amoroso distributions with Pareto tails, which correspond to the emergence of a \emph{social elite}. The details of the microscopic kinetic interaction allow to clarify the meaning of the various parameters characterizing the resulting equilibrium. Numerical results then show that the steady state of the underlying kinetic equation is close to Amoroso distribution even in an intermediate regime in which interactions are not grazing.