hierarchicalrule: Allocation rule according to the agents' hierarchical order

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Allocation rule according to the agents' hierarchical order

Description

hierarchicalrule calculates the contribution vector resulting from the payment allocation among the different agents using one of the various predefined rules in relation to the agents' hierarchical order.

Usage

hierarchicalrule(c, P, rule, a = NULL)

Arguments

c	A numeric cost vector.
P	A list showing the agents involved in the different distribution stages.
rule	A character string specifying the rule to apply.The rules that can be selected are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC" and "SIGMA".
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when rule = "SIGMA". By default, a = 0.5.

Details

Let N=\{1,\dots,n\} be a finite set of agents and let P=\{P_1,\dots,P_{m+1}\} be a partition of N, with m\leq n-1. So, the hierarchical structure, P_{>}=\{P_1>P_2>\dots>P_m>P_{m+1}\}, implies that agents in P_1 have priority over agents in N\backslash P_1, agents in P_2 have priority over agents in N\backslash P_1\cup P_2, and so on. Let \mathcal{P}(N) denote the family of all hierarchical structures over N.

A hierarchical rule is a mapping \mathcal{R}_P:C^N\rightarrow \mathbb{R}^N that, based on a hierarchical structure P_{>}, associates with each problem c\in C^N a cost allocation vector \mathcal{R}_P(c) such that 0\leq\mathcal{R}_P(c)\leq c. In other words, the allocation proceeds by assigning costs to agents in higher-hierarchy coalitions before those in lower-hierarchy ones, through the successive application of the rule to the no-subsidy faces of coalitions obtained from P_{>}. These rules are on the boundary of the no-subsidy set.

In each stage, the agents share the accumulated costs up to that point. However, the remaining costs are allocated in subsequent stages.

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. (2025). Airport problems with cloned agents. [Preprint manuscript].

Faigle, U. and Kern, W. (1992). The Shapley value for cooperation games under precedence constraints. International Journal of Game Theory, 21, 249-266.

Fiestras-Janeiro, M. G., Sánchez-Rodríguez, E., and Schuster, M. (2016). A precedence constraint value revisited. TOP, 24, 156-179.

Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2020). The boundary of the core of a balanced game: faces games. International Journal of Game Theory, 49(2), 579-599.

See Also

NSfaces, PRIORrule, basicrule

Examples

# Two stages
c <- c(1, 3, 7, 10) # Cost vector
P <- list(c(1, 2), c(3, 4)) # Agents' hierarchical order
hierarchicalrule(c, P, rule = "SEC") # SEC rule

# Three stages
c <- c(1, 1, 3, 3, 7, 10) # Cost vector
P <- list(2, c(1, 3), c(4, 5, 6)) # Agents' hierarchical order
hierarchicalrule(c, P, "CEC") # CEC rule

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