airportgame: Coalitional game associated with an airport problem

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Coalitional game associated with an airport problem

Description

airportgame computes the coalitional game for cost-sharing in an airport problem.

Usage

airportgame(c, lex = TRUE)

Arguments

c	A numeric cost vector.
lex	A logical value indicating the output order of the game. By default, lex = TRUE returns the game in lexicographic order. However, if lex = FALSE, the game is returned in binary order.

Details

Let N = \{1, \dots, n\} denote the set of agents, and let c \in \mathbb{R}_+^N be the cost vector such that c \geq 0. The value c_i should be interpreted as the associated cost for each agent i in the context of the problem, i.e., every component represents the cost of the facility required by an agent. Segmental costs are defined as the difference between a given cost and the first immediately lower cost: c_i - c_{i-1} for i \in N \backslash \{1\}. Therefore, C^N represents the domain of all problems.

Given an airport problem c \in \mathbb{R}^N_{+}, the corresponding coalitional game is defined, for each S \subseteq N, as:

v(S) = \max\{c_j : j \in S\}.

It is easy to check that this class of games associated with airport problems is always concave, since for any pair of coalitions S \subseteq T \subseteq N, it is verified that:

v(S \cup \{i\}) - v(S) \geq v(T \cup \{i\}) - v(T) \quad \text{for all} \quad i \notin T

An efficient way to represent a nonempty coalition S \in 2^N is by identifying it with the binary sequence a_n,a_{n-1},\dots a_1, where a_i=1 if i \in S and a_i=0 otherwise. Consequently, each coalition S is represented by the number associated with its binary representation: \sum_{i \in S} 2^{i-1}. Then coalitions can be ordered by their associated numbers.

Alternatively, coalitions can be ordered lexicographically, meaning they are first sorted by increasing size, and then by lexicographic order among coalitions of the same size.

Value

A numeric vector representing the associated coalitional game.

References

Littlechild, S. C. and Owen, G. (1973). A simple expression for the Shapley value in a special case. Management Science, 23, 370-372.

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

See Also

airportvector

Examples

# 4 agents
(c <- c(1, 3, 7, 10)) # Vector of costs
airportgame(c, lex = TRUE)  # Game in lexicographic order
airportgame(c, lex = FALSE) # Game in binary order

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