# lorenzdominance: Lorenz-dominance relation In ClaimsProblems: Analysis of Conflicting Claims

## Description

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

## Usage

 `1` ```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≥ 0 be the endowment to be divided and d the vector of claims with d≥ 0 and such that the sum of claims exceeds the endowment.

A vector x=(x1,...,xn) is an awards vector for the claims problem (E,d) if 0≤ x ≤ d and satisfies the balance requirement, that is, x1+…+xn=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,…,n-1 we have that

x1+…+xk ≥ y1+…+yk.

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.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). Deviation from proportionality and Lorenz-dominance between the average of awards and the standard rules for claims problems. Working paper 2021-01, ECOBAS.

Mirás Calvo, M.A., 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.

 ```1 2 3 4``` ```E=10 d=c(2,4,7,8) Rules=c(AA,CEA) lorenzdominance(E,d,Rules) ```