AA | R Documentation |

This function returns the awards vector assigned by the average of awards rule (AA) to a claims problem.

AA(E, d, name = FALSE)

`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.

A vector *x=(x1,...,xn)* is an awards vector for the claims problem *(E,d)* if *0≤ x ≤ d*
and satisfies the balance requirement, that is, *x1+…+xn=E* the sum of its coordinates is equal to *E*.
Let *X(E,d)* be the set of awards vectors for *(E,d)*.

The average of awards rule assigns to each claims problem *(E,d)*
the expectation of the uniform distribution defined over the set of awards vectors, that is,
the centroid of *X(E,d)*.

Let *μ* be the (n-1)-dimensional Lebesgue measure and *V(E,d)=μ (X(E,d))* the measure (volume) of the set of awards *X(E,d)*.
The average of awards rule assigns to each problem *(E,d)* the awards vector given by:

*AA(E,d)=1/V(E,d) ∫xdμ, where the integral is taken over X(E,d).*

The average of awards rule corresponds to the core-center of the associated coalitional (pessimistic) game.

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

Gonzalez-Díaz, J. and Sánchez-Rodríguez, E. (2007). A natural selection from the core of a TU game: the core-center. International Journal of Game Theory, 36(1), 27-46.

Mirás Calvo, M.Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2022). The average-of-awards rule for claims problems. Soc Choice Welf. doi: 10.1007/s00355-022-01414-6

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2020). An algorithm to compute the core-center rule of a claims problem with an application to the allocation of CO2 emissions. Working paper.

allrules, CD, setofawards, coalitionalgame

E=10 d=c(2,4,7,8) AA(E,d) #The average of awards rule is self-dual: AA(E,d)=d-AA(D-E,d) D=sum(d) d-AA(D-E,d)

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