clonesgroups: Cloned agents in an airport problem

In AirportProblems: Analysis of Cost Allocation for Airport Problems

Cloned agents in an airport problem

Description

clonesgroups determines a new cost vector that excludes cloned agents, and calculates the size of each clone group that shares the same cost.

Usage

clonesgroups(c)

Arguments

c	A numeric cost vector.

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 \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 list containing the following items:

cw			A numeric cost vector, with the same length as eta.
eta			A numeric vector representing the size of each group of cloned agents.

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

clonesproblem

Examples

# 9 different agents
c <- c(2, 2, 2, 5, 5, 7, 7, 7, 12) # Cost vector
clonesgroups(c) # 4 groups of cloned agents

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