deviationindex {ClaimsProblems}

Deviation index

Description

This function returns the deviation index and the signed deviation index for a rule with respect to another rule.

Usage

deviationindex(E, d, R, S)

Arguments

E

The endowment.

d

The vector of claims.

R

A rule : AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

S

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\in \mathcal{R}^n the vector of claims with d\ge 0 and such that D=\sum_{i=1}^{n} d_i\ge E, the sum of claims D exceeds the endowment.

Rearrange the claims from small to large, 0 \le d_1 \le...\le d_n. The signed deviation index of the rule S with respect to the rule R for the problem (E,d), denoted by I(R(E,d),S(E,d)), is the ratio of the area that lies between the identity line and the cumulative curve over the total area under the identity line.

Let R_0=0 and S_0=0. For each k=1,\dots,n define X_k=\frac{1}{E} \sum_{j=0}^{k} R_j and Y_k=\frac{1}{E} \sum_{j=0}^{k} S_j. Then

I(R(E,d),S(E,d))=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1}).

In general -1 \le I(R(E,d),S(E,d)) \le 1.

The deviation index of the rule S with respect to the rule R for the problem (E,d), denoted by I^{+}(R(E,d),S(E,d)), is the ratio of the area between the line of the cumulative sum of the distribution proposed by the rule R and the cumulative curve over the area under the line x=y.

In general 0 \le I^{+}(R(E,d),S(E,d)) \le 1.

The proportionality deviation index is the deviation index when R = PRO. The proportionality deviation index of the proportional rule is zero for all claims problems. The signed proportionality deviation index is the signed deviation index with R = PRO.

proportionalityindex function of version 0.1.0 returned the the signed proportionality index.

Value

The deviation index and the signed 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.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi: 10.1007/s10058-022-00300-y

See Also

indexgpath, cumawardscurve, lorenzcurve, giniindex, lorenzdominance, allrules.

Examples

E=10
d=c(2,4,7,8)
R=CEA
S=AA
deviationindex(E,d,R,S)
#The deviation index of rule R with respect of the rule R is 0.
deviationindex(E,d,PRO,PRO)