MO: Minimal overlap rule

In ClaimsProblems: Analysis of Conflicting Claims

Minimal overlap rule

Description

This function returns the awards vector assigned by the minimal overlap rule rule (MO) to a claims problem.

Usage

MO(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 \sum_{i \in N} d_i\ge E.

The truncated claim of a claimant i\in N is the minimum of the claim and the endowment, that is, t_i(E,d)=t_i=\min\{d_i,E\},\ i=1,\dots,n.

Suppose that each agent claims specific parts of E equal to her/his claim. After arranging which parts agents claim so as to “minimize conflict”, equal division prevails among all agents claiming a specific part and the minimal overlap rule (MO) assigns the sum of the compensations she/he gets from the various parts that he claimed.

Let d_0=0. For each problem (E,d) and each claimant i\in N,

1) If E\le d_n then

\text{MO}_i(E,d)=\frac{t_1}{n}+\frac{t_2-t_1}{n-1}+\dots+\frac{t_i-t_{i-1}}{n-i+1}.

2) If E>d_n, let s'\in (d_{k'},d_{k'+1}], with k' \in \{0,1,\dots,n-2\}, be the unique solution to the equation \underset{j \in N}{\sum} \max\{d_j-s,0\} =E-s. Then,

\text{MO}_i(E,d)= \begin{cases} \frac{d_1}{n}+\frac{d_2-d_1}{n-1}+\dots+\frac{d_i-d_{i-1}}{n-i+1}& \text{if } i\in \{1,\dots , k'\} \\[4pt] \text{MO}_i(s',d)+d_i-s' & \text{if } i\in \{k'+1,\dots, n\} \end{cases}.

Value

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

References

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023). Refining the Lorenz‐ranking of rules for claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558.

O'Neill, B. (1982). A problem of rights arbitration from the Talmud. Mathemathical Social Sciences. 2, 345-371.

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, CD.

Examples

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

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