lorenzdominance: Lorenz-dominance relation

In ClaimsProblems: Analysis of Conflicting Claims

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, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud.
Info	A logical value.

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.

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. A rule is a function that assigns to each claims problem (E,d) an awards vector.

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,

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

Let \mathcal{R} and \mathcal{S} be two rules, we say that \mathcal{R} Lorenz-dominates \mathcal{S} if \mathcal{R}(E,d) Lorenz-dominates \mathcal{S}(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. (2023a). Deviation from proportionality and Lorenz-domination for claims problems. Review of Economic Design 27, 439-467.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023b). Refining the Lorenz‐ranking of rules for claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558

See Also

cumawardscurve, deviationindex, giniindex, indexgpath, lorenzcurve.

Examples

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

ClaimsProblems documentation built on April 4, 2025, 2:21 a.m.