SIGMArule {AirportProblems} | R Documentation |
Parametric family of rules
Description
SIGMArule
calculates the contribution vector selected by a SIGMA rule.
Usage
SIGMArule(c, a = 0.5)
Arguments
c |
A numeric cost vector. |
a |
A numeric value in the range [0,1], controlling the parameterization of the rule.
By default, |
Details
Let N^i_{-}=\{j\in N:j<i\}
. For each a \in [0,1]
, each c \in C^N
, and each i \in N\backslash \{n\}
, a \sigma^a
rule is defined by
\sigma^{a}_i(c)=\text{min}\Bigg\{\text{min}\bigg\{\dfrac{1}{r-(i-1)+a}\Big(c_r-\displaystyle\sum\limits_{j\in N^i_-}\sigma^{a}_{j}(c)\Big):r=i,\dots,n-1\bigg\},
\ \dfrac{1}{n+1-i}\Big(c_n-\displaystyle\sum\limits_{j\in N^i_-}\sigma_j^{a}(c)\Big)\Bigg\},
\text{and} \ \ \sigma_n^{a}(c)=c_n-\displaystyle\sum\limits_{i=1}^{n-1}\sigma_i^{a}(c).
In this rule, the closer the parameter a
is to 0, the more equal the distribution of payments among the agents will be, and vice versa.
In fact, it is easy to verify that \sigma^0=\text{CEC}\ \ \text{and} \ \ \sigma^1=\text{SM}
.
Value
A numeric contribution vector, where each element represents the payment of the different agents.
References
Thomson, W. (2024). Cost allocation and airport problems. Mathematical Social Sciences, 31(C), 17–31.
van Gellekom, J. R. G. and Potters, J. A. M. (1999). Consistent solution rules for standard tree enterprises. Technical Report 9910, University of Nijmegen.
See Also
CECrule
, SMrule
, basicrule
, hierarchicalrule
Examples
c <- c(1, 3, 7, 10) # Cost vector
SIGMArule(c) # a=0.5
# The SIGMA rule with a=0 is the CEC rule
a <- 0
all.equal(SIGMArule(c, a), CECrule(c))
# The SIGMA rule with a=1 is the SM rule
a <- 1
all.equal(SIGMArule(c, a), SMrule(c))