APRO {ClaimsProblems}

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 E\ge 0 be the endowment to be divided and d\in \mathcal{R}^n the vector of claims with d\ge 0 and such that \sum_{i=1}^{n} d_i\ge E, the sum of claims exceeds the endowment.

For each subset S of the set of claimants N, let d(S)=\sum_{j\in S}d_j be the sum of claims of the members of S and let N\backslash S be the complementary coalition of S.

The minimal right of claimant i 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 truncated claim of claimant i in (E,d) is the minimum of the claim and the endowment:

t_i(E,d)=\min\{d_i,E\},\ i=1,\dots,n.

Let t(E,d)=(t_1(E,d),\dots,t_n(E,d)) be the vector of truncated claims.

The adjusted proportional rule first gives to each claimant the 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:

APRO(E,d)=m(E,d)+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. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

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, CD, PRO, coalitionalgame

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)