Analogies and Relations between Non-Additive Entropy Formulas and Gintropy

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.