NSfaces: Face games associated with an airport problem

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Face games associated with an airport problem

Description

NSfaces determines the airport problems defining the two components of the decomposition of one specific face game.

Usage

NSfaces(c, R)

Arguments

c	A numeric cost vector.
R	A numeric vector representing the agents forming the coalition

Details

Let c\in C^N be an airport problem and v\in G^N its associated cost game, for each non-empty proper coalition R\in 2^N\backslash\{\emptyset, N\} define the N\backslash R-face of Core(v) as the set

F_{N\backslash R}(c)=\text{Core}(v)\cap\{x\in \mathbb{R}^N:x(R)=v(R)\}.

Also, the N\backslash R-face game v_{F_{N\backslash R}}\in G^N is given by v_{F_{N\backslash R}}(S)=v(S\cup R)-v(R)+v(S\cap R),\ S\in 2^N. It turns out that F_{N\backslash R}(c)=\text{Core}(v_{F_{N\backslash R}}), the N\backslash R-face of the core of the associated cost game is the core of the N\backslash R-face game. Let r\in N such that c_r=v(R)=\text{max}\{c_i:i\in R\} and denote R_+=\{k\in N:k>r\}. Consider the airport problems

c_{|R}=(0_{N\backslash R}, c_R)\in C^N \text{ and } c_{|R_+}=(0_{N\backslash R_+},c_{r+1}-c_r,\dots,c_n-c_r)\in C^N

with associated games v^{|R}\in G^N and v^{|R_+}\in G^N, respectively. It is easy to see that the N\backslash R-face game v_{F_{N\backslash R}}\in G^N is decomposable with respect to the partition R,N\backslash R and its components are v^{|R} and v^{|R_+}. Moreover,

F_{N\backslash R}(c)=NS(c_R)\times NS(c_{r+1}-c_r,\dots,c_n-c_r)\times 0_{N\backslash(R\cup R_+)}.

Therefore, the components of the face games of the associated cost game are associated cost games themselves. In the N\backslash R-face game, the players of R play the game associated with the problem where the agents of R keep their cost parameters while the cost parameter of the agents in N\backslash R is null. On the other hand, since the players of R already share c_r=v(R) among themselves, the players of N\backslash R with an initial cost lower than c_r now have their cost parameter equal to zero while the others see their cost reduced by c_r.

Value

A numeric matrix with two rows representing the decomposition of the R-face game:

[1,]	The first row is obtained by setting the cost of a specific coalition to zero while retaining the cost parameters of the complementary coalition.
[2,]	The second row is derived by subtracting the highest cost in the complementary coalition from each agent's cost, or setting it to zero if the result is negative.

References

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

González-Díaz, J. and Sánchez-Rodríguez, E. (2008). Cores of convex and strictly convex games. Games and Economic Behavior, 62, 100-105.

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

NScheck, NSstructure, NSset, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
R <- c(3, 4) # Coalition of agents
NSfaces(c, R) # Components of the face game

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