SMrule: Slack maximizer rule

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Slack maximizer rule

Description

SMrule calculates the contribution vector selected by the SM rule.

Usage

SMrule(c)

Arguments

c	A numeric cost vector.

Details

For each c\in C^N and each i\in N\backslash\{n\}, the slack maximizer rule is defined by

\text{SM}_i(c) = \text{min} \Bigg\{ \dfrac{1}{r-i+2} \Big( c_r - \displaystyle\sum\limits_{j \in N^i_{-}} \text{SM}_j(c) \Big):r=i,\dots,n-1 \Bigg\},\ \text{SM}_n(c)=c_n-\displaystyle\sum\limits^{n-1}_{i=1}SM_i(c)

This rule aims to maximize the 'slacks', that is, the available margin for each agent within the imposed constraints.

The contribution selected by the SM rule for a problem c \in C^N coincides with the payoff vector assigned by the nucleolus to the associated cost game v\in G^N, that is, \text{SM}(c)=\text{Nu}(v).

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

Littlechild, S. C. (1974). A simple expression for the nucleolus in a special case. International Journal of Game Theory, 3(1), 21-29.

Thomson, W. (2024). Cost allocation and airport problems. Mathematical Social Sciences, 31(C), 17–31.

See Also

SIGMArule, basicrule, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
SMrule(c)

AirportProblems documentation built on June 8, 2025, 10:49 a.m.