volume: Volume of the set of awards vectors

In ClaimsProblems: Analysis of Conflicting Claims

Volume of the set of awards vectors

Description

This function computes the volume of the set of award vectors of a claims problem and the projected volume.

Usage

volume(E, d, real = TRUE)

Arguments

E	The endowment.
d	The vector of claims.
real	Logical parameter. By default, real = TRUE.

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.

A vector x=(x_1,\dots,x_n) is an awards vector for the claims problem (E,d) if 0\le x \le d and satisfies the balance requirement, that is, \sum_{i=1}^{n}x_i=E. Let X(E,d) be the set of awards vectors for (E,d).

Let \mu be the (n-1)-dimensional Lebesgue measure. We define by V(E,d)=\mu (X(E,d)) the measure (volume) of the set of awards X(E,d) and \hat{V}(E,d) the volume of the projection onto an (n-1)-dimensional space.

V(E,d)=\sqrt{n}\hat{V}(E,d).

The function is programmed following the procedure explained in Mirás Calvo et al. (2024b).

Value

The volume of the set of awards vectors. If real = FALSE, it returns the volume of the projection into the last coordinate.

References

Mirás Calvo, M.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2024a). An algorithm to compute the average-of-awards rule for claims problems with an application to the allocation of CO_2 emissions. Annals of Operations Research, 336: 1435-1459.

Mirás Calvo, M.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2024b). On properties of the set of awards vectors for a claims problem. TOP, 32: 137-167.

See Also

setofawards.

Examples

E=10
d=c(2,4,7,10)
volume(E,d)
#The volume function is a symmetric function.
D=sum(d)
volume(D-E,d)

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