# Gintropy: Gini Index Based Generalization of Entropy

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Basics

## 2. Gross Inequality in General

**gintropy**, and define as the difference between the rich-end-cumulative Lorenz curve and the diagonal:

- The gintropy is never negative: $\sigma =\overline{F}-\overline{C}\ge 0$ is proven by inspecting the integral$$\sigma \left(x\right)\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\underset{x}{\overset{\infty}{\int}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(y/\u2329x\u232a-1)\rho \left(y\right)dy\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\underset{0}{\overset{x}{\int}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}(1-y/\u2329x\u232a)\rho \left(y\right)dy\phantom{\rule{0.222222em}{0ex}}\ge \phantom{\rule{0.222222em}{0ex}}0,$$
- The gintropy is maximal at $x=\u2329x\u232a$, ${\sigma}_{max}=\sigma \left(\langle x\rangle \right)$, since $d\sigma /dx=(1-x/\u2329x\u232a)\rho \left(x\right)$ changes its sign exactly there and only there.
- According to Equation (8) at the Pareto-point, the gintropy equals $\sigma \left({x}_{P}\right)=1-2p$, and therefore, for the Pareto point, $p\le 1/2$ holds for the rich fraction. Since ${\sigma}_{max}\ge \sigma \left({x}_{P}\right)$, in order to get a Pareto point: $\sigma (\langle x\rangle )\ge 1-2p$, i.e., the maximum of the gintropy has to be bigger than this difference value. As a consequence for the Pareto Point, we have a restriction imposed by the maximal gintropy $(1-\sigma \left(\u2329x\u232a\right))/2\le p\le 1/2$.
- The expectation value of gintropy is the half of the gini index: $\underset{0}{\overset{\infty}{\int}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\sigma \left(x\right)\phantom{\rule{0.166667em}{0ex}}\rho \left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\underset{0}{\overset{1}{\int}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\sigma \left(\overline{C}\right)\phantom{\rule{0.166667em}{0ex}}d\overline{C}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}\Sigma \phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}G/2$.
- The integral of gintropy over the base value x is the non-Poissonity index, $\underset{0}{\overset{\infty}{\int}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\sigma \left(x\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{\mathrm{Var}\left(x\right)}{\u2329x\u232a}$, with $\mathrm{Var}\left(x\right)=\left(\right)open="\langle "\; close="\rangle ">{x}^{2}$ being the variance of x. The proof of this statement uses the same mathematical trick as the one in Equation (12).
- For some particular PDFs, $\sigma \left(\overline{C}\right)$ looks like an entropy density formula, $s\left({p}_{i}\right)$. We present important examples in the next section.

## 3. Important Examples

#### 3.1. Communism

#### 3.2. Communism++

#### 3.3. Eco-Window

#### 3.4. Natural Distribution

#### 3.5. Capitalism

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**left**) The rich end Lorenz curve, and connection with the Gini index. (

**right**) Visual illustration of the gintropy, the Pareto Point and the maximal gintropy.

**Figure 2.**The $\overline{F}$ vs. $\overline{C}$ cumulative maps (Lorenz curves) for the (

**a**) communism++ ($a=1$, $b=4$ and $w=0.8$), (

**b**) eco-window ($a=1,b=5$), (

**c**) natural exponential ($\u2329x\u232a=1$) and for the capitalism (

**d**) ($A=1,B=3$→$q=3/4$) distributions. The corresponding Gini index are $G=0.3$, $G=2/9$, $G=1/2$ and $G=4/7$, respectively.

**Figure 3.**(

**a**) $\overline{F}$ – $\overline{C}$ Lorenz curves in one comparison and (

**b**) the corresponding

**gintropy**curves, $\sigma \left(\overline{C}\right)$, for the communism++, eco-window, natural and capitalism models. The Gini indices are $G=0.3$, $G=2/9$, $G=1/2$ and $G=4/7$, respectively.

**Table 1.**Summary of PDFs, the gintropy formulas and Gini index values for some ideal income/wealth distribution schemes.

$\mathit{\rho}\mathbf{\left(}\mathit{x}\mathbf{\right)}$ | $\mathit{\sigma}\mathbf{\left(}\overline{\mathit{C}}\mathbf{\right)}$ | G | |
---|---|---|---|

natural | $\frac{1}{\u2329x\u232a}}{\mathrm{e}}^{-x/\u2329x\u232a$ | $-\overline{C}ln\overline{C}$ | $\frac{1}{2}$ |

capitalism | $\frac{A}{1-q}{(1+Ax)}^{{\displaystyle \frac{-1}{1-q}}}$ | $\frac{1}{1-q}}\left(\right)open="("\; close=")">{\overline{C}}^{q}-\overline{C$ | $\frac{1}{q+1}}\ge {\displaystyle \frac{1}{2}$ |

eco-window | $\frac{1}{b-a}}[\Theta (b-x)-\Theta (a-x)]$ | $3G\phantom{\rule{0.166667em}{0ex}}\overline{C}(1-\overline{C})$ | $\frac{1}{3}}\phantom{\rule{0.166667em}{0ex}}{\displaystyle \frac{b-a}{b+a}}\le \frac{1}{3$ |

communism++ | $w\phantom{\rule{0.166667em}{0ex}}\delta (x-a)+(1-w)\phantom{\rule{0.166667em}{0ex}}\delta (x-b)$ | $G\phantom{\rule{0.166667em}{0ex}}[\Theta (\overline{C}\left(a\right)-\overline{C})-\Theta (\overline{C}\left(b\right)-\overline{C})]$ | $\frac{(b-a)q\phantom{\rule{0.166667em}{0ex}}(1-w)}{w\phantom{\rule{0.166667em}{0ex}}a+(1-w)\phantom{\rule{0.166667em}{0ex}}b}$ |

communism | $\delta (x-a)$ | 0 | 0 |

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Biró, T.S.; Néda, Z.
Gintropy: Gini Index Based Generalization of Entropy. *Entropy* **2020**, *22*, 879.
https://doi.org/10.3390/e22080879

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Biró TS, Néda Z.
Gintropy: Gini Index Based Generalization of Entropy. *Entropy*. 2020; 22(8):879.
https://doi.org/10.3390/e22080879

**Chicago/Turabian Style**

Biró, Tamás S., and Zoltán Néda.
2020. "Gintropy: Gini Index Based Generalization of Entropy" *Entropy* 22, no. 8: 879.
https://doi.org/10.3390/e22080879