lorenzdominance {ClaimsProblems}

Lorenz-dominance relation

Description

This function checks whether or not the awards assigned by two rules to a claims problem are Lorenz-comparable.

Usage

lorenzdominance(E, d, Rules, Info = FALSE)

Arguments

E

The endowment.

d

The vector of claims.

Rules

The two rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

Info

A logical value.

Details

Let E\ge 0 be the endowment to be divided and d\in \mathcal{R}^n the vector of claims with d\ge 0 and such that \sum_{i=1}^{n} d_i\ge E,\; the sum of claims exceeds the endowment.

A vector x=(x_1,\dots,x_n) is an awards vector for the claims problem (E,d) if 0\le x \le d and satisfies the balance requirement, that is, \sum_{i=1}^{n}x_i=E the sum of its coordinates is equal to E. Let X(E,d) be the set of awards vectors for (E,d).

Given a claims problem (E,d), in order to compare a pair of awards vectors x,y\in X(E,d) with the Lorenz criterion, first one has to rearrange the coordinates of each allocation in a non-decreasing order. Then we say that x Lorenz-dominates y (or, that y is Lorenz-dominated by x) if all the cumulative sums of the rearranged coordinates are greater with x than with y. That is, x Lorenz-dominates y if for each k=1,\dots,n-1 we have that

\sum_{j=1}^{k}x_j \geq \sum_{j=1}^{k}y_j

Let R and R' be two rules. We say that R Lorenz-dominates R' if R(E,d) Lorenz-dominates R'(E,d) for all (E,d).

Value

If Info = FALSE, the Lorenz-dominance relation between the awards vectors selected by both rules. If both awards vectors are equal then cod = 2. If the awards vectors are not Lorenz-comparable then cod = 0. If the awards vector selected by the first rule Lorenz-dominates the awards vector selected by the second rule then cod = 1; otherwise cod = -1. If Info = TRUE, it also gives the corresponding cumulative sums.

References

Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association, 9(70), 209-219.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi: 10.1007/s10058-022-00300-y

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

See Also

cumawardscurve, deviationindex, indexgpath, lorenzcurve, giniindex.

Examples

E=10
d=c(2,4,7,8)
Rules=c(AA,CEA)
lorenzdominance(E,d,Rules)