weightedrule: Weighted allocation rule

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Weighted allocation rule

Description

weightedrule calculates the contribution vector resulting from the payment allocation among the different agents using one of the various predefined weighted rules.

Usage

weightedrule(c, w, rule)

Arguments

c	A numeric cost vector.
w	A numeric weight vector.
rule	A character string specifying the rule to apply.The rules that can be selected are: "SFC", "SEC", "CSEC", "CEC", "CP" and "CEB".

Details

Let w=(w_i)_{i\in N}\in\mathbb{R}^n be a positive weight vector, satisfying w_i> 0 for all i\in N and w(N)=1. Consider the (n-1)-standard simplex, defined as \Delta_n=\{x\in\mathbb{R}^n:x\geq 0, \ x_1+\dots+x_n=1\}. Then, the set of all positive weight vectors corresponds to \text{Int}(\Delta_n), the interior of the (n-1)-standard simplex.

A weighted rule is a mapping \mathcal{R}:C^N\times \text{Int}(\Delta_N)\rightarrow \mathbb{R}^N which associates with a problem c\in C^N and a positive weight vector w\in \text{Int}(\Delta_n) a contribution vector \mathcal{R}(c,w)\in X(c).

It is possible to define weighted versions of the rules: SFC, SEC, CEC, CP and CEB. In fact, two different rules emerge from the standard SEC rule: the weighted SEC rule and the coalition-weighted SEC rule. If w_i=w_j for all i,j \in N, then the solution of weighted SEC(c) and weighted CSEC(c) coincides.

In all the rules, the higher the weight w_i, the more the corresponding agent will have to pay, except for the weighted CEB rule (the construction of this rule is based on the concept of allocating 'benefits', so it is logical that it is set up this way). Furthermore, as previously stated, all the rules, except for the weighted CSEC rule, require the weights to be positive. Although the weighted CSEC rule allows zero weights, it requires that at least one weight must be positive.

The weighted version of the SFC, SEC, CEC and CP rules is equal to their respective versions for clones, so the formulation of these rules will be the same for the version with clones. Only the CEB rule has a weighted version and a clone version that are different.

Value

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

References

Bernárdez Ferradás, A., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2025a). Airport problems with cloned agents. [Preprint manuscript].

Bernárdez Ferradás, A., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2025b). A characterization of the CSEC rule for airport problems. [Prepint manuscript].

Sánchez-Rodríguez, E., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Núñez Lugilde, I. (2024). Coalition-weighted Shapley values. International Journal of Game Theory, 53, 547-577.

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

See Also

NScheck, NSset, basicrule, clonesrule

Examples

c <- c(1, 3, 3, 7, 10) # Cost vector
w <- c(1, 4, 1, 2, 8)  # Weight vector

# Weighted SFC rule
weightedrule(c, w, "SFC")

# Weighted CEB rule
weightedrule(c, w, "CEB")

# Weighted SEC rule
weightedrule(c, w, "SEC")

# Weighted CSEC rule
w <- c(0, 4, 1, 2, 8) # New weight vector
weightedrule(c, w, "CSEC")

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