Talmud: Talmud rule

In ClaimsProblems: Analysis of Conflicting Claims

Talmud rule

Description

This function returns the awards vector assigned by the Talmud rule to a claims problem.

Usage

Talmud(E, d, name = FALSE)

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 D=\sum_{i \in N} d_i\ge E.

The Talmud rule (Talmud) 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, the Talmud rule assigns d/2 and the remainder, E-D/2, is awarded with the constrained equal losses rule with claims d/2. Therefore, for each i\in N,

\text{Talmud}_i(E,d) = \begin{cases} \min\{\frac{d_i}{2},\lambda\} & \text{if } E\leq \tfrac{1}{2}D\\[3pt] d_i-\min\{\frac{d_i}{2},\lambda\} & \text{if } E \geq \tfrac{1}{2}D \end{cases},

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

The Talmud rule when applied to a two-claimant problem is often referred to as the contested garment rule and coincides with concede-and-divide rule. The Talmud rule corresponds to the nucleolus of the associated (pessimistic) coalitional game.

Value

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

References

Aumann, R. and Maschler, M. (1985). Game theoretic analysis of a bankruptcy problem from the Talmud. Journal of Economic Theory 36, 195-213.

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

AA, allrules, APRO, axioms, CEA, CEL, CD, RA, RTalmud.

Examples

E=10
d=c(2,4,7,8)
Talmud(E,d)
D=sum(d)
#The Talmud rule is self-dual
d-Talmud(D-E,d)

ClaimsProblems documentation built on April 4, 2025, 2:21 a.m.