MO {ClaimsProblems}

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 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 a claimant i is the minimum of the claim and the endowment:

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 each agent receives the sum of the compensations she/he gets from the various parts that he claimed.

Let d_0=0. The minimal overlap rule is defined, for each problem (E,d) and each claimant i, as:

If E\le d_n then

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}.

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 \sum_{i \in N} \max\{d_i-s,0\} =E-s. Then:

MO_i(E,d)=\frac{d_1}{n}+\frac{d_2-d_1}{n-1}+\dots+\frac{d_i-d_{i-1}}{n-i+1}, \ i\in\{1,\dots,k\}

MO_i(E,d)=MO_i(s,d)+d_i-s, \ i\in\{k+1,\dots,n\}.

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.A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2021). The adjusted proportional and the minimal overlap rules restricted to the lower-half, higher-half, and middle domains. Working paper 2021-02, ECOBAS.

O'Neill, B. (1982). A problem of rights arbitration from the Talmud. Math. Social Sci. 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)