The function returns the deviation index path or the signed deviation index path for a rule with respect to another rule for a vector of claims.
indexgpath(
d,
Rule = PRO,
Rules,
signed = TRUE,
col = NULL,
points = 201,
legend = TRUE
)This function returns the deviation index path of a rule (or several rules) for a vector of claims.
The vector of claims.
Principal Rule: AA, APRO, CE, CEA, CEL, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud. By default, Rule = PRO.
The rules: AA, APRO, CE, CEA, CEL, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud.
A logical value. If signed = FALSE, it draws the deviation index path and, if signed = TRUE it draws the signed deviation index path. By default, signed = TRUE.
The colours. If col = NULL then the sequence of default colours is:
c("red", "blue", "green", "yellow", "pink", "orange", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
The number of endowment values to be drawn.
A logical value. The legend is shown if legend = TRUE.
Let \(N=\{1,\ldots,n\}\) be the set of claimants, \(d\in \mathbb{R}^N\) a vector of claims rearranged from small to large, \(0 \le d_1 \le...\le d_n\) and \(D=\sum_{i\in N}d_i\).
Given two rules \(\mathcal{R}\) and \(\mathcal{S}\), consider the function \(J\) that assigns to each \(E\in (0,D]\) the value \(J(E)=I\Bigl(\mathcal{R}(E,d),\mathcal{S}(E,d)\Bigr)\), that is, the signed deviation index of the rules \(\mathcal{R}\) and \(\mathcal{S}\) for the problem \((E,d)\). The graph of \(J\) is the signed index path of \(\mathcal{S}\) in function of the rule \(\mathcal{R}\) for the vector of claims \(d\).
Given two rules \(\mathcal{R}\) and \(\mathcal{S}\), consider the function \(J^{+}\) that assigns to each \(E\in (0,D]\) the value \(J^{+}(E)=I^{+}\Bigl(\mathcal{R}(E,d),\mathcal{S}(E,d)\Bigr)\), that is, the deviation index of the rules \(\mathcal{R}\) and \(\mathcal{S}\) for the problem \((E,d)\). The graph of \(J^{+}\) is the index path of \(\mathcal{S}\) in function of the rule \(\mathcal{R}\) for the vector of claims \(d\).
The signed index path and the index path are simple tools to visualize the discrepancy of the divisions recommended by a rule for a vector of claims with respect to the divisions recommended by another rule. If \(\mathcal{R} = \text{PRO}\), the function draws the proportionality deviation index path or the signed proportionality deviation index path.
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.
Thomson, W. (2019). How to divide when there isn't enough. From Aristotle, the Talmud, and Maimonides to the axiomatics of resource allocation. Cambridge University Press.
allrules, cumawardscurve, deviationindex, giniindex, lorenzcurve, lorenzdominance.
d=c(2,4,7,8)
Rule=PRO
Rules=c(Talmud,RA,AA)
col=c("red","green","blue")
indexgpath(d,Rule,Rules,signed=TRUE,col)
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