# cumulativecurve: Cumulative claims-awards curve In ClaimsProblems: Analysis of Conflicting Claims

## Description

This function returns the cumulative claims-awards curve of a rule for a claims problem.

## Usage

 `1` ```cumulativecurve(E, d, Rules, col = NULL, legend = TRUE) ```

## Arguments

 `E` The endowment. `d` The vector of claims. `Rules` The rules: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud. `col` The colors. If col = NULL then the sequence of default colors is: c("red", "blue", "green", "yellow", "pink", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet"). `legend` A logical value. The color legend is shown if legend = TRUE.

## 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 D=∑ di exceeds the endowment.

Rearrange the claims from small to large, 0 ≤ d1 ≤...≤ dn. The cumulative claims-awards curve allows us to compare the division recommended by a specific rule R with the division that preserves the proportions of the claims, the proportional rule.

The cumulative claims-awards curve of a rule R for the claims problem (E,d) is the polygonal path connecting the n+1 points

(0,0) , (d1/D,R1(E,d)/E) , ((d1+d2)/D , (R1(E,d)+R2(E,d))/E ,… , (1,1).

Basically, it represents the percentage of the endowment assigned by the rule to each cumulative proportion of claims.

The cumulative claims-awards curve fully captures the Lorenz ranking of rules: if a rule R Lorenz-dominates a rule R' then, for each claims problem, the cumulative claims-awards curve of R lies above the cumulative curve of R'.

## Value

The graphical representation of the cumulative claims-awards curves of a rule (or several rules) for a claims problem.

## 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.

 ```1 2 3 4``` ```E=10 d=c(2,4,7,8) Rules=c(AA,RA,Talmud,CEA,CEL) cumulativecurve(E,d,Rules) ```