Study of a measure of efficiency as a tool for applying the principle of least effort to the derivation of the Zipf and the Pareto laws

TL;DR

This paper investigates a mathematical efficiency measure derived from the principle of least effort, aiming to better understand its properties and its role in deriving power law distributions like Zipf's and Pareto's laws.

Contribution

It provides a detailed mathematical analysis of the efficiency measure, exploring its properties, robustness, and relationship with system inequality and uncertainty.

Findings

The efficiency measure's sign, uniqueness, and robustness are characterized.

A new method for calculating non-negative continuous entropy is introduced.

Insights into the measure's usefulness for deriving power law distributions are discussed.

Abstract

The principle of least effort is believed to be a universal rule for living systems. Its application to the derivation of the power law probability distributions of living systems has long been challenging. Recently, a measure of efficiency was proposed as a tool of deriving Zipf s and Pareto s laws directly from the principle of least effort. The present work is a further investigation of this efficiency measure from a mathematical point of view. The aim is to get further insight into its properties and usefulness as a metric of performance. We address some key mathematical properties of this efficiency such as its sign, uniqueness and robustness. We also look at the relationship between this measure and other properties of the system of interest such as inequality and uncertainty, by introducing a new method for calculating non-negative continuous entropy.