volume: 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)

Value

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

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).

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.

Examples

Run this code

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)

Run the code above in your browser using DataLab