SMrule {AirportProblems} | R Documentation |
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)