The graphical representation of the cumulative curves of a rule (or several rules) with respect to a given rule, for a claims problem.
cumawardscurve(E, d, Rule = PRO, Rules, col = NULL, legend = TRUE)The graphical representation of the cumulative curves of a rule (or several rules) for a claims problem.
The endowment.
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.
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").
A logical value. The colour legend is shown if legend = TRUE.
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 cumulative curve allows us to compare the division recommended by a specific rule \(\mathcal{R}\) with the division the division recommended by another specific rule \(\mathcal{S}\).
The cumulative awards curve of a rule \(\mathcal{S}\) with respect of a rule \(\mathcal{R}\) for the claims problem \((E,d)\) is the polygonal path connecting the \(n+1\) points $$(0,0), \Bigl(\frac{\mathcal{R}_1}{E},\frac{\mathcal{S}_1}{E}\Bigr),\dots,\Bigl(\frac{\sum_{i=1}^{n-1}\mathcal{R}_i}{E},\frac{\sum_{i=1}^{n-1}\mathcal{S}_i}{E}\Bigr),(1,1).$$
The cumulative awards curve fully captures the Lorenz ranking of rules: if a rule \(\mathcal{R}\) Lorenz-dominates a rule \(\mathcal{S}\) then, for each claims problem, the cumulative curve of \(\mathcal{R}\) lies above the cumulative curve of \(\mathcal{S}\). If \(\mathcal{R} = \text{PRO}\), the cumulative curve coincides with the cumulative claims-awards curve.
Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association 9(70), 209-219.
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, deviationindex, giniindex, indexgpath, lorenzcurve, lorenzdominance.
E=10
d=c(2,4,7,8)
Rule=PRO
Rules=c(AA,RA,Talmud,CEA,CEL)
cumawardscurve(E,d,Rule,Rules)
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