CE {ClaimsProblems}

Constrained egalitarian rule

Description

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

Usage

CE(E, d, name = FALSE)

Arguments

E

The endowment.

d

The vector of claims.

name

A logical value.

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.

Assume that the claims are ordered from small to large, 0 \le d_1 \le...\le d_n. The constrained egalitarian rule coincides with the constrained equal awards rule (CEA) applied to the problem (E, d/2) if the endowment is less or equal than the half-sum of the claims D/2. Otherwise, any additional unit is assigned to claimant 1 until she/he receives the minimum of the claim and half of d_2. If this minimun is d_1, she/he stops there. If it is not, the next increment is divided equally between claimants 1 and 2 until claimant 1 receives d_1 (in this case she drops out) or they reach d_3/2. If claimant 1 leaves, claimant 2 receives any aditional increment until she/he reaches d_2 or d_3/2. In the case that claimant 1 and 2 reach d_3/2, any additional unit is divided between claimants 1, 2, and 3 until the first one receives d_1 or they reach d_4/2, and so on.

Therefore:

If E \le D/2 then CE(E,d) = CEA(E,d/2)=(\min\{\frac{d_i}{2},\lambda\})_{i\in N} where \lambda \ge 0 is chosen so as to achieve balance.

If E \ge D/2 then the CE rule assigns to claimant i the maximum of two quantities: the half-claim and the minimum of the claim and a value \lambda \ge 0 chosen so as to achieve balance.

CE_i(E,d)=\max\{\frac{d_i}{2},\min\{d_i,\lambda\}\},\ i=1,\dots,n, \ where \ \sum_{i=1}^{n} CE_i(E,d)=E.

Value

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

References

Chun, Y., Schummer, J., Thomson, W. (2001). Constrained egalitarianism: a new solution for claims problems. Seoul J. Economics 14, 269–297.

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, CEA, Talmud, PIN

Examples

E=10
d=c(2,4,7,8)
CE(E,d)