This function returns the deviation index and the signed deviation index for a rule with respect to another rule.
deviationindex(E, d, R, S)The deviation index and the signed deviation index of a rule for a claims problem.
The endowment.
The vector of claims.
A rule : AA, APRO, CE, CEA, CEL, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud.
A rule: AA, APRO, CE, CEA, CEL, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud.
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 \(\mathcal{R} = \text{PRO}\).
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.
allrules, cumawardscurve, giniindex, indexgpath, lorenzcurve, lorenzdominance.
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)
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