# proportionalityindex: Proportionality deviation index In ClaimsProblems: Analysis of Conflicting Claims

## Description

This function returns the proportionality deviation index of a rule for a claims problem.

## Usage

 `1` ```proportionalityindex(E, d, Rule) ```

## Arguments

 `E` The endowment. `d` The vector of claims. `Rule` A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

## Details

Let E> 0 be the endowment to be divided and d the vector of claims with d≥ 0 and such that D=∑ di ≥ E, the sum of claims D exceeds the endowment.

Rearrange the claims from small to large, 0 ≤ d1 ≤...≤ dn. The proportionality deviation index of the rule R for the problem (E,d), denoted by I(R,E,d), is the ratio of the area that lies between the identity line and the cumulative claims-awards curve over the total area under the identity line.

Let d0=0 and R0(E,d)=0. For each k=1,…,n define Xk=(d0+…+dk)/D and Yk=(R0+…+Rk)/E. Then

I(R,E,d)=1-∑ (Xk-X(k-1))(Yk+Y(k-1)) where the sum goes from k=1 to n.

The proportionality deviation index of the proportional rule is zero for all claims problems. In general -1 ≤ I(R,E,d) ≤ 1.

## Value

The proportionality deviation index of a rule for a claims problem.

## References

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

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 5 6``` ```E=10 d=c(2,4,7,8) Rule=AA proportionalityindex(E,d,Rule) #The proportionality deviation index of the proportional rule is 0 proportionalityindex(E,d,PRO) ```