setofawards {ClaimsProblems}

Set of awards vectors for a claims problem

Description

This function plots the set of awards vectors for a claims problem with 2, 3, or 4 claimants and returns its vertices for any problem.

Usage

setofawards(E, d, draw = TRUE, col = NULL)

Arguments

E

The endowment.

d

The vector of claims.

draw

A logical value.

col

The colour.

Details

Let E\ge 0 be the endowment to be divided and d\in \mathcal{R}^n the vector of claims with d\ge 0 and such that \sum_{i=1}^{n} d_i\ge E,\; the sum of claims exceeds the endowment.

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 the sum of its coordinates is equal to E. Let X(E,d) be the set of awards vectors for the problem (E,d).

For each subset S of the set of claimants N, let d(S)=\sum_{j\in S}d_j be the sum of claims of the members of S and let N\backslash 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:

m_i(E,d)=\max\{0,E-d(N\backslash\{i\})\},\ i=1,\dots,n.

Let m(E,d)=(m_1(E,d),\dots,m_n(E,d)) be the vector of minimal rights.

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

t_i(E,d)=\min\{d_i,E\},\ i=1,\dots,n.

Let t(E,d)=(t_1(E,d),\dots,t_n(E,d)) be the vector of truncated claims.

A vector x is efficient if the sum of its coordinates coincides with the endowment. The set of awards is the the set of all efficient vectors bounded by the minimal right and trucated claim vectors.

The set of awards vectors for the claims problem (E,d) can be given in terms of the minimal rights and truncated claims vectors:

X(E,d)=\bigl\{x \in \mathcal{R}^n: \sum_{i=1}^n x_i=E, m_i(E,d) \le x_i \le t_i(E,d),\ i=1,\dots,n \bigr\}

The set of awards vectors for a problem coincides with the core of its associated coalitional (pessimistic) game.

The vertices of the set of awards are the marginal worth vectors. For each order of the claimants, the marginal worth vectors are obtained applying the principle “first to arrive, first to be served". Then, for each order, the corresponding marginal worth vector assigns to each claimant the minimum of her/his claim and what remains of the endowment.

Value

The vertices of the set of awards vectors for any claims problem. For two-claimant and three-claimant problems, if draw = TRUE it plots the set of awards vectors. For a four-claimant problem, if draw = TRUE, it plots the projection of the set of awards vector over the euclidean space of the first three coordinates. For a claims problem with more than four claimants, it only displays the vertices of the set of awards. The default colours (col = NULL) are: red for two-claimant problems, beige for three-claimant problems, and white for four-claimant problems.

References

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.

See Also

plotrule, problemdata, AA, RA

Examples

E=10
d=c(2,4,7,8)
setofawards(E,d,col="darkgreen")