SIGMArule: Parametric family of rules

In AirportProblems: Analysis of Cost Allocation for Airport Problems

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, a = 0.5.

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))

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