This function returns the proportionality deviation index of a rule for a claims problem.
proportionalityindex(E, d, Rule)
The vector of claims.
A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.
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.
The proportionality deviation index of a rule for a claims problem.
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.
indexpath, cumulativecurve, lorenzcurve, giniindex, lorenzdominance, PRO.
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