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Analogies and Relations between Non-Additive Entropy Formulas and Gintropy.

Tamás S Biró1,2,3, András Telcs1, Antal Jakovác1

  • 1HUN-REN Wigner Research Centre for Physics, 1121 Budapest, Hungary.

Entropy (Basel, Switzerland)
|March 28, 2024
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Summary
This summary is machine-generated.

This study reveals mathematical links between trace-form entropies and the Gini index for inequality measurement. It introduces gintropy, connecting traditional entropy to non-additive formulas and the Pareto distribution.

Keywords:
Gini indexLorenz curveentropynon-extensive

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Area of Science:

  • Information Theory
  • Econometrics
  • Statistical Mechanics

Background:

  • The Gini index is a standard measure for income and wealth inequality.
  • Trace-form entropies are fundamental concepts in information theory and statistical mechanics.
  • Gini index and entropy measures have distinct mathematical frameworks.

Purpose of the Study:

  • To explore formal similarities and mathematical transformations between general trace-form entropies and the Gini index.
  • To introduce and utilize the concept of 'gintropy' derived from the Lorenz curve.
  • To establish novel connections between traditional entropy, gintropy, and non-additive formulas.

Main Methods:

  • Mathematical analysis of formal similarities between entropy and inequality measures.
  • Utilizing the concept of gintropy based on Lorenz curve properties.
  • Deriving transformation formulas between different entropy forms and the Gini index.
  • Revisiting Tsallis' q-entropy in relation to the Pareto distribution.

Main Results:

  • Formal mathematical connections established between trace-form entropies and the Gini index.
  • Gintropy is defined and used to link Lorenz curve properties to entropy.
  • Tsallis' q-entropy formula is re-derived in the context of the Pareto distribution.
  • Novel expressions for traditional entropy in terms of gintropy and new non-additive formulas are presented.

Conclusions:

  • The study provides a unified mathematical framework for understanding inequality and entropy measures.
  • Gintropy offers a novel perspective for analyzing economic inequalities and their relationship to information theory.
  • The findings facilitate the reconstruction of non-additive entropy formulas and dynamical models for inequality.