# MO: Minimal overlap rule In ClaimsProblems: Analysis of Conflicting Claims

## Description

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

## Usage

 `1` ```MO(E, d, name = FALSE) ```

## Arguments

 `E` The endowment. `d` The vector of claims. `name` A logical value.

## Details

Let E≥ 0 be the endowment to be divided and d the vector of claims with d≥ 0 and such that the sum of claims exceeds the endowment.

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

ti=min{di,E}, i=1,…,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 d0=0. The minimal overlap rule is defined, for each problem (E,d) and each claimant i, as:

If E≤ d_n then

MOi(E,d) = t1/n + (t2-t1)/(n-1) + … + (ti-t(i-1))/(n-i+1).

If E>d_n let dk<s ≤ d(k+1), with 0≤ k ≤ n-2, be the unique solution to the equation max{d1-s,0}+…+max{dn-s,0}=E-s. Then:

MOi(E,d) = d1/n + (d2-d1)/(n-1) + … + (di-d(i-1))/(n-i+1), if i≤ k

MOi(E,d) = MOi(s,d)+di-s, if i>k+1.

## 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.

 ```1 2 3``` ```E=10 d=c(2,4,7,9) MO(E,d) ```