Measuring inequality in society-oriented Lotka--Volterra-type kinetic equations

TL;DR

This paper introduces a method to measure inequality in systems modeled by coupled Fokker-Planck equations with oscillatory dynamics, using the coefficient of variation to track inequality changes over time.

Contribution

It proposes a novel approach to quantify inequality in complex kinetic systems with oscillatory behavior, focusing on the coefficient of variation as an insightful measure.

Findings

Inequality initially decreases despite oscillations.

The coefficient of variation effectively tracks inequality dynamics.

Numerical experiments validate the approach.

Abstract

We present a possible approach to measuring inequality in a system of coupled Fokker-Planck-type equations that describe the evolution of distribution densities for two populations interacting pairwise due to social and/or economic factors. The macroscopic dynamics of their mean values follow a Lotka-Volterra system of ordinary differential equations. Unlike classical models of wealth and opinion formation, which tend to converge toward a steady-state profile, the oscillatory behavior of these densities only leads to the formation of local equilibria within the Fokker-Planck system. This makes tracking the evolution of most inequality measures challenging. However, an insightful perspective on the problem is obtained by using the coefficient of variation, a simple inequality measure closely linked to the Gini index. Numerical experiments confirm that, despite the system's oscillatory nature, inequality initially tends to decrease.