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ClaimsProblems (version 1.0.0)

lorenzdominance: 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)

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.

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 \((E,d)\).

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

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

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