NSstructure: Composition of the no-subsidy set

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Composition of the no-subsidy set

Description

NSstructure quantifies the key elements of the structure of the no-subsidy set.

Usage

NSstructure(cw, eta = rep(1, length(cw)))

Arguments

cw	A numeric cost vector, with the same length as eta.
eta	A numeric vector representing the size of each group of cloned agents. All its elements must be positive integers. By default, eta = rep(1, length(cw)), i.e., all groups have size 1.

Details

For any cost vector c, if there are n agents with different cost parameters, the number of faces of the NS set is 2n-2. However, the number of full-dimensional faces is indeed affected by the presence of clones. Let t\in N, (\eta,c)\in\mathcal{A}^N_t, and \eta\ast c \in C^N, \text{NS}(\eta\ast c) has n+t-2 full-dimensional faces if \eta_t=1 and n+t-1 full-dimensional faces otherwise. On the other hand, the number of different extreme points of the set \text{NS}(\eta\ast c) is: \eta_t \prod_{i \in T \setminus \{t\}} (\eta_i + 1) (so, when there are no clones, the \text{NS}(c) has 2^{n-1} extreme points).

Let k\in\mathbb{N} and denote by \lambda_k the k-dimensional Lebesgue measure. If X=(X_1,\dots,X_k) is a random vector with joint density function f and \Omega is a Borel set, then P(X\in\Omega)=\int_{\Omega}f(x)d\lambda_k and the expected value of X is \mathbb{E}[X]=\int_{\mathbb{R}^k}x f(x)d\lambda_k. Given a Borel set \Omega\subset\mathbb{R}^k of positive measure, \lambda_k(\Omega)>0, we say that a random vector U=(U_1,\dots,U_k) has a uniform distribution on \Omega, and we write U\thicksim U(\Omega), if U has a probability density function f(x_1,\dots,x_k)=\frac{1}{\lambda_k(\Omega)} if (x_1,\dots, x_k)\in \Omega and f(x_1,\dots,x_k)=0 otherwise. If a=(a_1,\dots,a_k)\in\mathbb{R}^k with 0<a_1\leq \dots \leq a_k, denote

V_k(a)=\displaystyle\int_0^{a_1}\dots\displaystyle\int_0^{a_k-\sum\limits_{j=1}^{k-1} x_j}d x_k\dots d x_1.

Therefore, for each c\in C^N, the value V_{n-1}(c_{-n}) is the (n-1)-Lebesgue measure of NS_n(c), so \lambda_{n-1}(\text{NS}(c))=\sqrt{n}\lambda_{n-1}(\text{NS}_n(c))=\sqrt{n}V_{n-1}(c_{-n}).

Value

A list containing the following items:

n.faces	A positive integer representing the number of faces that form the NS set.
n.full.dim.faces	A positive integer indicating the number of full-dimensional faces forming the NS set.
n.extreme.points	A positive integer counting the number of extreme points of the NS set.
actual.volume	A positive number representing the volume of the NS set.
projected.volume	A positive number reflecting the projected volume of the NS set.

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., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2016). Airport games: the core and its center. Mathematical Social Sciences, 82, 105–115.

See Also

NScheck, NSfaces, NSset, CCrule

Examples

# Without cloned agents
c <- c(1, 2, 3, 4)
NSstructure(c)

# With cloned agents
c <- c(1, 2)
eta <- c(3, 1)
NSstructure(c, eta)

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