ClaimsProblems (version 1.0.0)

CEL: Constrained equal losses rule

Description

This function returns the awards vector assigned by the constrained equal losses rule (CEL) to a claims problem.

Usage

CEL(E, d, name = FALSE)

Value

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

Arguments

E

The endowment.

d

The vector of claims.

name

A logical value.

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 \(\sum_{i \in N} d_i\ge E\).

The constrained equal losses rule (CEL) equalizes losses under the constraint that no award is negative. Then, claimant \(i\) receives the maximum of zero and the claim minus a number \(\lambda \ge 0\) chosen so as to achieve balance. That is, for each \(i\in N\), $$\text{CEL}_i(E,d)=\max\{0,d_i-\lambda\},$$

where \(\lambda\geq 0\) is chosen such that \(\sum_{i\in N} \text{CEL}_i(E,d)=E.\)

CEA and CEL are dual rules.

References

Maimonides, Moses, [1135-1204], Book of Judgements (translated by Rabbi Elihahu Touger, 2000), New York and Jerusalem: Moznaim Publishing Corporation, 2000.

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

allrules, axioms, CE, CEA, AV, PIN, Talmud, RTalmud.

Examples

E=10
d=c(2,4,7,8)
CEL(E,d)
# CEL and CEA are dual: CEL(E,d)=d-CEA(D-E,d)
D=sum(d)
d-CEA(D-E,d)