DT {ClaimsProblems}

Dominguez-Thomson rule

Description

This function returns the awards vector assigned by the Dominguez-Thomson rule (DT) to a claims problem.

Usage

DT(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.

The truncated claim of claimant i in (E,d) is the minimum of the claim and the endowment.

t_i(E,d)=\min\{d_i,E\},\ i=1,\dots,n

Let t(E,d)=(m_1(E,d),\dots,m_n(E,d)) be the vector of truncated claims and b(E,d)=\frac{1}{n}t(E,d)

The DT rule is defined recursively such that, in each step, each claimant receives the n-th part of the truncated claim.

Let (E^1,d^1)=(E,d). For each k\ge 2 define:

(E^k,d^k)=(E^{k-1}-\sum_{i=1}^n b_i(E^{k-1},d^{k-1}),d^{k-1}-b(E^{k-1},d^{k-1})).

In step 1, the endowment is E and the claims vector is d. For k \ge 2, the endowment is the remainder once all the claimants have received the amount of the previous steps and the new vector of claims is readjusted accordingly. Observe that neither the endowment nor each agent's claim ever increases from one step to the next. This recursive process exhausts E in the limit.

For each (E,d) the Dominguez-Thomson rule assigns the awards vector:

DT(E,d)=\sum_{k=1}^{\infty} b(E^k,d^k)

Value

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

References

Domínguez, D. and Thomson, W. (2006). A new solution to the problem of adjudicating conflicting claims. Economic Theory, 28(2), 283-307.

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

Examples

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