APRO: Adjusted proportional rule

In ClaimsProblems: Analysis of Conflicting Claims

Adjusted proportional rule

Description

This function returns the awards vector assigned by the adjusted proportional rule (APRO) to a claims problem.

Usage

APRO(E, d, name = FALSE)

Arguments

E	The endowment.
d	The vector of claims.
name	A logical value.

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. 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.

Value

The awards vector selected by the APRO rule. If name = TRUE, the name of the function (APRO) as a character string.

References

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.

See Also

allrules, axioms, CD, coalitionalgame, PRO.

Examples

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)

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