Description Usage Arguments Details Value References See Also Examples

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

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`E` |
The endowment. |

`d` |
The vector of claims. |

`name` |
A logical value. |

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.

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

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.

allrules, CD, PRO, coalitionalgame

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