This function returns the awards vector assigned by the adjusted proportional rule (APRO) to a claims problem.
APRO(E, d, name = FALSE)The awards vector selected by the APRO rule. If name = TRUE, the name of the function (APRO) as a character string.
The endowment.
The vector of claims.
A logical value.
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 \(\sum_{i \in N} d_i\ge E\). For each coalition \(S\in 2^N\), let \(d(S)=\sum_{j\in S}d_j\) and \(N\backslash S\) be the complementary coalition of \(S\).
The minimal right of claimant \(i\in N\) in \((E,d)\) is whatever is left after every other claimant has received his claim, or 0 if that is not possible: $$m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\},\ i=1,\dots,n.$$ Let \(m(E,d)=(m_1(E,d),\dots,m_n(E,d))\) be the vector of minimal rights.
The adjusted proportional rule (APRO) first assigns to each claimant its minimal right, and then divides the remainder of the endowment \(E'=E-\sum_{i=1}^n m_i(E,d)\) proportionally with respect to the new claims. The vector of the new claims \(d'\) is determined by the minimum of the remainder and the lowered claims, \(d_i'=\min\{E-\sum_{j=1}^n m_j(E,d),d_i-m_i\},\ i=1,\dots,n\). Therefore, $$\text{APRO}(E,d)=m(E,d)+\text{PRO}(E',d').$$
The adjusted proportional rule corresponds to the \(\tau\)-value of the associated (pessimistic) coalitional game.
Curiel, I. J., Maschler, M., and Tijs, S. H. (1987). Bankruptcy games. Zeitschrift für operations research 31(5), A143-A159.
Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023). Refining the Lorenz‐ranking of rules for claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558.
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, axioms, CD, coalitionalgame, PRO.
E=10
d=c(2,4,7,8)
APRO(E,d)
#The adjusted proportional rule is self-dual: APRO(E,d)=d-APRO(D-E,d)
D=sum(d)
d-APRO(D-E,d)
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