RA: Random arrival rule

In ClaimsProblems: Analysis of Conflicting Claims

Random arrival rule

Description

This function returns the awards vector assigned by the random arrival rule (RA) to a claims problem.

Usage

RA(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. For each coalition S\in 2^N, d(S)=\sum_{j\in S}d_j.

The random arrival rule (RA) considers all the possible arrivals of the claimants and applies the principle “first to arrive, first to be served". Then, for each order, the corresponding marginal worth vector assigns to each claimant the minimum of her/his claim and what remains of the endowment. The rule averages all the marginal worth vectors considering all the permutations of the elements of N.

Let \Pi^N denote the set of permutations of the set of claimants N and |\Pi^N| its cardinality. Given a permutation \pi \in \Pi and a claimant i\in N let \pi_{\le i} be the set of claimants that precede i in the order \pi, that is, \pi_{\le i}=\{ j \in N :\pi(j)<\pi(i) \}.

The random arrival rule assigns to each (E,d) and each i\in N,

\text{RA}_i(E,d)=\frac{1}{|\Pi^N|}\sum_{\pi\in \Pi^N}\min\Bigl\{d_i,\max\{ 0,E-d(\pi_{\le i}) \}\Bigr\}.

The random arrival rule corresponds to the Shapley value of the associated (pessimistic) coalitional game.

This function is programmed following the algorithm of Le Creurer et al. (2022).

Value

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

References

Le Creurer, I.J, Mirás Calvo, M. A., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). On the computation of the Shapley value and the random arrival rule. Available at https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4293746.

O’Neill, B. (1982) A problem of rights arbitration from the Talmud. Mathematical 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

AA, allrules, APRO, axioms, CD, setofawards, Talmud.

Examples

E=10
d=c(2,4,7,8)
RA(E,d)
D=sum(d)
#The random arrival rule is self-dual: RA(E,d)= d-RA(D-E,d)
d-RA(D-E,d)

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