# APRO: Adjusted proportional rule In ClaimsProblems: Analysis of Conflicting Claims

## Description

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

## Usage

 `1` ```APRO(E, d, name = FALSE) ```

## Arguments

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

## Details

Let E≥ 0 be the endowment to be divided and d the vector of claims with d≥ 0 and such that the sum of claims exceeds the endowment.

For each subset S of the set of claimants N, let d(S) be the sum of claims of the members of S and let N-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:

mi = max{ 0 , E-d(N-{i}) }, i=1,…,n.

Let m(E,d)=(m1,…,mn) be the vector of minimal rights.

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

ti = min{di,E}, i=1,…,n.

Let t(E,d)=(t1,…,tn) 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 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-∑ mj,di-mi}, i=1,…,n . Therefore:

APRO(E,d)=m(E,d)+PRO(E-∑ mi,d').

The adjusted proportional rule corresponds to the τ-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.A., 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.

 ```1 2 3 4 5 6``` ```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) ```