Given two awards vectors, this function returns the Lorenz dominance relation between them.
lorenzdominancerelation(x, y)There are four possible outputs:
if the introduced vectors are not Lorenz-comparable.
if the vectors are Lorenz-equal.
if the vectors are not Lorenz-equal and the first one Lorenz-dominates the second one.
if the vectors are not Lorenz-equal and the second one Lorenz-dominates the first one.
A vector.
A vector.
In order to compare two vectors \(x,y\in \mathbb{R}^n\) through the Lorenz criterion, both of them must be rearranged in non-decreasing order; thus, let \(\bar{x}\) and \(\bar{y}\) be the vectors obtained by rearranging \(x\) and \(y\), respectively, in non-decreasing order. It is said that \(x\) Lorenz-dominates \(y\) (or that \(y\) is Lorenz-dominated by \(x\)) if all the cumulative sums of \(\bar{x}\) are not less than those of \(\bar{y}\). That is, \(x\) Lorenz-dominates \(y\) if \(\sum_{j=1}^{n}\bar{x}_j=\sum_{j=1}^{n}\bar{y}_j\) and, for each \(k=1,\dots,n-1\), $$\sum_{j=1}^{k}\bar{x}_j \geq \sum_{j=1}^{k}\bar{y}_j.$$
If \(x\) Lorenz-dominates \(y\) and \(y\) Lorenz-dominates \(x\), then \(x\) and \(y\) are said to be Lorenz-equal.
If \(x\) does not Lorenz-dominate \(y\) and \(y\) does not Lorenz-dominate \(x\), then \(x\) and \(y\) are not Lorenz-comparable.
Lorenz, M. O. (1905). Methods of Measuring the Concentration of Wealth. Publications of the American Statistical Association, 9(70), 209-219.
lorenzdominancerelation(c(1,2,3), c(1,1,4))
lorenzdominancerelation(c(1,2,7,2), c(1,1,4,6))
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