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 \(0\le G(\mathcal{R},E,d) \le 1\).