Talmud {ClaimsProblems} | R Documentation |
This function returns the awards vector assigned by the Talmud rule to a claims problem.
Talmud(E, d, name = FALSE)
E |
The endowment. |
d |
The vector of claims. |
name |
A logical value. |
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 D=\sum_{i=1}^{n} d_i\ge E
, the sum of claims D
exceeds the endowment.
The Talmud 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, 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:
If E \le \frac{D}{2}
then:
Talmud(E,d) = CEA(E,d/2).
If E \ge \frac{D}{2}
then:
Talmud(E,d) =d/2+ CEL(E-D/2,d/2) = d-CEA(D-E,d/2).
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.
The awards vector selected by the Talmud rule. If name = TRUE, the name of the function (Talmud) as a character string.
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.
allrules, CEA, CEL, AA, APRO, RA, CD.
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)