deviationindex: Deviation index

In ClaimsProblems: Analysis of Conflicting Claims

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

Details

Let N=\{1,\ldots,n\} be the set of claimants, E\ge 0 the endowment to be divided and d\in \mathbb{R}_+^N the vector of claims such that \sum_{i \in N} d_i\ge E.

Rearrange the claims from small to large, 0 \le d_1 \le...\le d_n. The signed deviation index of the rule \mathcal{S} with respect to the rule \mathcal{R} for the problem (E,d), denoted by I(\mathcal{R}(E,d),\mathcal{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 \mathcal{R}_0=0 and \mathcal{S}_0=0. For each k=1,\dots,n define X_k=\frac{1}{E} \sum_{j=0}^{k}\mathcal{R}_j and Y_k=\frac{1}{E} \sum_{j=0}^{k} \mathcal{S}_j. Then,

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

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

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

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

The proportionality deviation index is the deviation index when \mathcal{R} = \text{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 \mathcal{R} = \text{PRO}.

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. (2023). Deviation from proportionality and Lorenz-domination for claims problems. Review of Economic Design 27, 439-467.

See Also

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

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)

ClaimsProblems documentation built on April 4, 2025, 2:21 a.m.