Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Formalization and inevitability of the Pareto principle

Antti Hippeläinen

TL;DR

This work asks whether the Pareto principle is a structural property of bounded cumulative processes rather than a universal empirical rule. It formalizes a generalized Pareto principle using a non-negative gain density on the unit interval and a decreasing rearrangement to define a unique, domain-order-independent criterion via $\\mathcal{L}^{(*)}(p) = 1 - p$. The authors establish existence results and analyze both constructed densities and common distribution families (power-law, exponential, normal) to map the attainable $p$ values as functions of distributional form and dataset size, highlighting how tails and truncation shape imbalance. They conclude that Pareto-like imbalances are structural features of gain distributions, with exponential and normal cases typically yielding $p$ near the canonical 0.2, while power-laws yield more extreme imbalances; this cautions against using 0.2/0.8 as a universal benchmark and emphasizes the distribution-specific interpretation of inequality and resource allocation.

Abstract

We formalize and study a generalized form of the Pareto principle or "20/80-rule" as a property of bounded cumulative processes. Modeling such processes by non-negative gain densities, we first show that any such process satisfies a generalized Pareto principle of the form "fraction $p$ of inputs yields fraction $1-p$ of outputs". To obtain a non-trivial and unique characterization, we define the generalized Pareto principle via the decreasing rearrangement of the gain density function. Within this framework, we analyze both constructed gain densities that exemplify the framework and its imposed restrictions, as well as distribution families commonly encountered in datasets, including power-law, exponential, and normal distributions. Finally, we predict commonly encountered ranges for the generalized Pareto principle and discuss the implications of elevating a structural property into a prescriptive role.

Formalization and inevitability of the Pareto principle

TL;DR

This work asks whether the Pareto principle is a structural property of bounded cumulative processes rather than a universal empirical rule. It formalizes a generalized Pareto principle using a non-negative gain density on the unit interval and a decreasing rearrangement to define a unique, domain-order-independent criterion via . The authors establish existence results and analyze both constructed densities and common distribution families (power-law, exponential, normal) to map the attainable values as functions of distributional form and dataset size, highlighting how tails and truncation shape imbalance. They conclude that Pareto-like imbalances are structural features of gain distributions, with exponential and normal cases typically yielding near the canonical 0.2, while power-laws yield more extreme imbalances; this cautions against using 0.2/0.8 as a universal benchmark and emphasizes the distribution-specific interpretation of inequality and resource allocation.

Abstract

We formalize and study a generalized form of the Pareto principle or "20/80-rule" as a property of bounded cumulative processes. Modeling such processes by non-negative gain densities, we first show that any such process satisfies a generalized Pareto principle of the form "fraction of inputs yields fraction of outputs". To obtain a non-trivial and unique characterization, we define the generalized Pareto principle via the decreasing rearrangement of the gain density function. Within this framework, we analyze both constructed gain densities that exemplify the framework and its imposed restrictions, as well as distribution families commonly encountered in datasets, including power-law, exponential, and normal distributions. Finally, we predict commonly encountered ranges for the generalized Pareto principle and discuss the implications of elevating a structural property into a prescriptive role.
Paper Structure (25 sections, 5 theorems, 63 equations, 10 figures, 1 table)

This paper contains 25 sections, 5 theorems, 63 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Formalization
  3. Preliminary formulation of the generalized principle
  4. Final formulation of the generalized principle
  5. Example distributions and existence of generalized principles
  6. Constructed examples
  7. Step function
  8. Divergent profiles and rearrangements
  9. Polynomial densities
  10. Distribution families
  11. Power-law distribution
  12. Exponential distribution
  13. Normal distribution
  14. Common generalized principles and social discussion
  15. Common generalized Pareto principles
  16. ...and 10 more sections

Key Result

Theorem 1

For any continuous gain function and for all $k \in \mathbb{R}^+$, some form of the asymmetric generalized Pareto principle "fraction $p$ of inputs yields fraction $1 - k p$ of outputs" is satisfied; That is, there exists $p^* \in (0,1)$ such that $\mathcal{L}(p^*) = 1 - k p^*$. In particular, with $k = 1$, there exists $p^*$ such

Figures (10)

  • Figure 1: Step function densities and their related cumulative distributions with $p = 0.1, 0.2$ and $0.4$.
  • Figure 2: Minimal inequality index or ratio of gain densities a distribution must have to satisfy a given $p/(1-p)$-principle.
  • Figure 3: An $L^1$-integrable divergent density with its decreasing rearrangement, and their related cumulative gain functions. As must be, for all $t \in [0,1]$, $\mathcal{L}^*(t) \geq \mathcal{L}(t)$.
  • Figure 4: A periodic gain density with its decreasing rearrangement, and their related cumulative gain functions.
  • Figure 5: Polynomial densities with $\alpha = \frac{1}{4}, 1, 3$ and $10$, and their related cumulative gain functions.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem