clonesproblem: Cost vector with cloned agents

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Cost vector with cloned agents

Description

clonesproblem determines the new cost vector after disaggregating the original groups of clones.

Usage

clonesproblem(cw, eta)

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.

Details

In an airport problem, agents i,j \in N are clones if both have the same cost parameter, that is, if c_i=c_j.

If a problem has cloned agents, then the agent set N can be divided into several non-overlapping groups such that any pair of agents that belong to the same group are clones, but any two agents from two different groups have different cost parameters.

For each t \in N, let T=\{1,\dots,t\} and let \mathcal{A}_t^N be the set of pairs (\eta, c)\in\mathbb{N}^t\times\mathbb{R}^t such that:

\quad \space \space \space \space \eta=(\eta_1,\dots,\eta_t)\in \mathbb{N}^t \; \text{with} \; \eta_1+\dots+\eta_t=n

c=(c_1,\dots,c_t) \in C^T \; \text{with} \; c_1<\dots< c_t

Given t\in N and (\eta,c)\in \mathcal{A}_t^N we define the cost problem \eta\ast c \in C^N as:

\eta\ast c=(\eta_1\ast c_1,\dots,\eta_t\ast c_t)=(c_1,\overset{\eta_1}{\dots},c_1,\dots,c_t,\overset{\eta_t}{\dots},c_t)\in C^N

Given a problem d\in C^N there are unique t\in N and (\eta,c)\in \mathcal{A}_t^N such that d=n\ast c. We refer to the problem c\in C^T, formed by the different cost parameters of problem d\in C^N, as the reduced problem without clones associated with d\in C^N. Clearly, d\in C^N is obtained from c\in C^T by adding, for each i\in T, \eta_i, clones of agent i.

Let t\in N and (\eta,c)\in \mathcal{A}_t^{\mathcal{N}}. For each s\in T=\{1,\dots,t\}, let M_s^\eta=\eta_1+\dots+\eta_s and, to simplify the notation, write N^\eta_s=N^{\eta\ast c}_s=\{j\in N:(\eta\ast c)_j=c_s\}. Therefore, N_1^\eta=\{1,\dots,M^\eta_1\} and N_s^\eta=\{M^\eta_{s-1}+1,\dots,M^\eta_s\} if s\in T\backslash\{1\}. Obviously, the family \{N^\eta_1,\dots,N^\eta_t\} is a partition of N and |N^\eta_s|=\eta_s for all s\in T. Moreover, all the agents that belong to N^\eta_s have the same cost parameter c_s, i.e., (\eta\ast c)_j=c_s for all s\in T and j\in N^\eta_s. So, each agent in the reduced problem c\in C^T can be seen as a representative agent of the corresponding group of clones in the original problem d=\eta\ast c \in C^N.

Value

A numeric cost vector, where each element corresponds to a different agent's cost.

References

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

Littlechild, S. C. and Thompson, G. F. (1977). Aircraft landing fees: a game theory approach. The Bell Journal of Economics, 8, 186-204.

See Also

clonesgroups

Examples

# 4 groups of cloned agents
cw <- c(2, 5, 7, 12) # Different costs
eta <- c(3, 2, 3, 1) # Size of each group of clones
clonesproblem(cw, eta) # General cost vector

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