CEA: Constrained equal awards rule

Description

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

Usage

CEA(E, d, name = FALSE)

Value

The awards vector selected by the CEA rule. If name = TRUE, the name of the function (CEA) 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 awards rule (CEA) equalizes awards under the constraint that no individual's award exceeds his/her claim. Then, claimant $i$ receives the minimum of the claim and a value $\lambda \ge 0$ chosen so as to achieve balance. That is, for each $i\in N$, $$ \text{CEA}_i(E,d)=\min\{d_i,\lambda\},$$

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

The constrained equal awards rule corresponds to the Dutta-Ray solution to the associated (pessimistic) coalitional game. The CEA and CEL rules are dual.

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.

Examples

Run this code

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

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