Formalization of the generalized Pareto principle and structural typicality of the 20/80-rule

TL;DR

This paper formalizes a generalized Pareto principle using gain densities and Lorenz curves, deriving explicit formulas for common distributions and analyzing dataset size effects on the 20/80-rule.

Contribution

It introduces a formal framework connecting the Pareto principle with Lorenz curves and provides closed-form expressions for various distributions.

Findings

Datasets of size 10^2 to 10^5 from exponential and normal distributions have p near 0.15 to 0.29.

The p-values are close to but below the 0.2/0.8-rule.

The study discusses the structural ubiquity of Pareto-type imbalances.

Abstract

We formalize a generalized form of the Pareto principle - ``fraction $p$ of inputs yields fraction $1-p$ of outputs'' - as a property of non-negative gain densities $\ell \in L^1([0,1])$, working with the decreasing rearrangement to obtain a unique characterization. For probability distributions, the resulting $p$ coincides with $1 - k_F$, where $k_F$ is the Kolkata index of the corresponding Lorenz curve. Within this framework we analyze both constructed gain densities and commonly encountered distribution families. We derive closed-form expressions for $p$ for truncated power-law, exponential, and normal distribution families. Combining these with estimates of the truncation parameter as a function of sample size $N$, we predict that datasets of size $N \in [10^2, 10^5]$ from exponential and normal families concentrate $p$ near $[0.15, 0.26]$ and $[0.20, 0.29]$ - values close to the canonical 0.2/0.8-rule, and strictly below the saturation $k \approx 0.865$ conjectured earlier by Ghosh and Chakrabarti. We discuss the implications of the structural ubiquity of Pareto-type imbalances for their use as prescriptive targets.