giniindex: Gini index
| giniindex | R Documentation |
Gini index
Description
This function returns the Gini index of any rule for a claims problem.
Usage
giniindex(E, d, Rule)
Arguments
E |
The endowment. |
d |
The vector of claims. |
Rule |
A rule: AA, APRO, CE, CEA, CEL, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud. |
Details
Let N=\{1,\ldots,n\} be the set of claimants, E\ge 0 the endowment to be divided and d\in \mathbb{R}_+^N the vector of claims
such that \sum_{i \in N} d_i\ge E.
Rearrange the claims from small to large, 0 \le d_1 \le...\le d_n.
The Gini index is a number aimed at measuring the degree of inequality in a distribution.
The Gini index of the rule \mathcal{R} for the problem (E,d), denoted by G(\mathcal{R},E,d), is
the ratio of the area that lies between the identity line and the Lorenz curve of the rule over the total area under the identity line.
Let \mathcal{R}_0(E,d)=0. For each k=0,\dots,n define
X_k=\frac{k}{n} and Y_k=\frac{1}{E} \sum_{j=0}^{k} \mathcal{R}_j(E,d). Then,
G(\mathcal{R},E,d)=1-\sum_{k=1}^{n}\Bigl(X_{k}-X_{k-1}\Bigr)\Bigl(Y_{k}+Y_{k-1}\Bigr).
In general 0\le G(\mathcal{R},E,d) \le 1.
Value
The Gini index of a rule for a claims problem and the Gini index of the vector of claims.
References
Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality 10(3), 421-443.
Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2023). Deviation from proportionality and Lorenz-domination for claims problems. Review of Economic Design 27, 439-467.
See Also
cumawardscurve, deviationindex, indexgpath, lorenzcurve, lorenzdominance.
Examples
E=10
d=c(2,4,7,8)
Rule=AA
giniindex(E,d,Rule)
# The Gini index of the proportional awards coincides with the Gini index of the vector of claims
giniindex(E,d,PRO)
