Package 'AirportProblems'

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

---

Cost vector associated with an airport game

Description

airportvector computes the cost vector corresponding to an airport problem.

Usage

airportvector(v, lex = TRUE)

Arguments

v	A numeric vector that represents the characteristic function of the airport game.
lex	A logical value indicating the input order of the game. By default, if lex = TRUE, the game has been introduced in lexicographic order. However, if lex = FALSE, the game has been established in binary order.

Details

A cooperative game $(N, v)$ is considered an airport game provided that its characteristic function $v$ satisfies:

$v(S)=\text{max}\{c_j:j \in S\}, \quad \forall S \subseteq N,S \neq \emptyset$

where $c_j$ represents the individual cost associated with each agent $j$.

Evidently, this property implies that the cost assigned to a coalition is determined by the most expensive cost for its members. It is for this reason that this class of games is always concave.

The airport game can be given in lexicographic order or binary order. For instance, if $n=3$, the characteristic function of the associated airport game in lexicographic order is:

$v=[v(\{1\}),v(\{2\}),v(\{3\}),v(\{1,2\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]$

On the other hand, in binary order, it would be:

$v=[v(\{1\}),v(\{2\}),v(\{1,2\}),v(\{3\}),v(\{1,3\}),v(\{2,3\}),v(\{1,2,3\})]$

Anyway, in both cases, we have that $v(\{2\})=v(\{1,2\})$ and $v(\{3\})=v(\{1,3\})=v(\{2,3\})=v(\{1,2,3\})$.

Given an airport game $(N, v)$, it is possible to extract the corresponding cost vector $c$ by setting:

$c_j=\text{min}\{v(S):j\in S \},\quad \text{for all} \quad j \in N.$

Value

A numeric vector representing the cost for each agent of an airport 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

airportgame

Examples

# 4 agents in lexicographic order
v <- c(1, 3, 7, 10, 3, 7, 10, 7, 10, 10, 7, 10, 10, 10, 10)
airportvector(v, lex = TRUE)

# 4 agents in binary order
u <- c(1, 3, 3, 7, 7, 7, 7, 10, 10, 10, 10, 10, 10, 10, 10)
airportvector(u, lex = FALSE)

---

Allocation rule

Description

basicrule calculates the contribution vector resulting from the payment allocation among the different agents using one of the various predefined rules.

Usage

basicrule(c, rule, a = NULL, order = NULL)

Arguments

c	A numeric cost vector.
rule	A character string specifying the rule to apply.The rules that can be selected are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC", "SIGMA" and "PRIOR".
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when rule = "SIGMA". By default, a = 0.5.
order	A numeric vector indicating the priority order of agents when making contributions. It can only be defined when rule = "PRIOR". By default, agents follow their original indexing and contribute accordingly.

Details

A rule is a mapping $\mathcal{R}:C^N\rightarrow \mathbb{R}^N$ which associates with each problem $c\in C^N$ a cost allocation vector $\mathcal{R}(c)$ such that $0\leq\mathcal{R}(c)\leq c$. In other words, a rule is a mechanism that selects for each airport problem an allocation vector.

The various proposed rules, despite their differences, share a key characteristic: for any given problem, each rule selects an allocation vector that belongs to its no-subsidy set. Although these rules have been individually characterized in different functions, the one in question encompasses all of them.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

Note

When rule = "CC", the execution time of the function may significantly increase if the number of agents exceeds 150.

References

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

Potters, J. and Sudhölter, P. (1999). Airport problems and consistent allocation rules. Mathematical Social Sciences, 38, 83–102.

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

See Also

NSset, weightedrule, clonesrule, hierarchicalrule, multibasicrules

Examples

c <- c(1, 3, 7, 10) # Cost vector

# SEC rule
basicrule(c, rule = "SEC")

# PRIOR rule
order <- c(1, 3, 2, 4)
basicrule(c, "PRIOR", order = order)

---

Birmingham Airport data

Description

The data includes several variables describing 13,572 aircraft operations — 6,876 take-offs and 6,876 landings — involving 11 different aircraft types at Birmingham Airport during the period 1968–1969.

Usage

Birmingham

Format

A data frame with 11 rows and 9 columns (variables):

type

Aircraft type

i

Aircraft index (subscript)

eta

Number of movements

p

Average fee per movement (in pounds)

d

Estimated diversion cost per movement (in pounds)

b

Average benefit per movement (in pounds)

l

Runway user index

c

Maintenance cost per movement (in pounds)

g

Assumed capital cost (in pounds)

Source

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

Examples

# Based on capital cost (g)
G <- multiclonesrules(Birmingham$g, Birmingham$eta, c("SEC", "CEC", "SM"), 
group_contribution = FALSE, agents_names = Birmingham$i, labels = FALSE)
(cost_allocation <- round(G, 2))

# Based on capital cost (g) + add maintenance cost per movement (c)
sweep(cost_allocation, MARGIN = 2, Birmingham$c, FUN = "+")

---

Core-center rule

Description

CCrule calculates the contribution vector selected by the CC rule.

Usage

CCrule(c)

Arguments

c	A numeric cost vector.

Details

The core-center rule, CC, assigns to each $c\in C^N$ the contribution vector given by the mean value of $U(c)\thicksim U(\text{NS}(c))$, that is,

$\text{CC}(c)=\mathbb{E}[U(\text{NS}(c))].$

Therefore, this rule is the center of gravity of the set of allocations satisfying the no-subsidy constraints. It coincides with the core-center of the cooperative game $v$ associated with $c\in C^N$.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

Note

The execution time of the function may significantly increase if the number of agents exceeds 150.

References

González-Díaz, J. and Sánchez-Rodríguez, E. (2007). A natural selection from the core of a TU game: the core-center. International Journal of Game Theory, 36, 27-26.

González-Díaz, J., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez (2015). Monotonicity of the core-center of the airport game. TOP, 23, 773-798.

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

See Also

NSset, basicrule, clonesrule hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
CCrule(c)

---

Constrained equal benefits rule

Description

CEBrule calculates the contribution vector selected by the CEB rule.

Usage

CEBrule(c)

Arguments

c	A numeric cost vector.

Details

For each $c\in C^N$ and each $i\in N$, the constrained equal benefits rule is defined by

$\text{CEB}_i(c)=\text{max}\{c_i-\beta,\ 0\}$

where $\beta>0$ is chosen so that $\sum\limits^n_{i=1}\text{CEB}_i(c)=c_n$.

This rule focuses on the benefits each agent receives from not having to fully cover their own needs, aiming to distribute them as equitably as possible, without any agent subsidizing another.

The contribution selected by the CEB rule for a problem $c \in C^N$ coincides with the payoff vector assigned by the modified nucleolus.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

Hu, C.-C., Tsay, M.-H., and Yeh, C.-H. (2012). Axiomatic and strategic justifications for the constrained equal benefits rule in the airport problem. Games and Economic Behavior, 75, 185-197.

Potters, J. and Sudhölter, P. (1999). Airport problems and consistent allocation rules. Mathematical Social Sciences, 38, 83–102.

Sudhölter, P. (1997). The modified nucleolus: Properties and axiomatizations. International Journal of Game Theory, 26, 146-182.

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

See Also

basicrule, weightedrule, clonesrule, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
CEBrule(c)

---

Constrained equal contributions rule

Description

CECrule calculates the contribution vector selected by the CEC rule.

Usage

CECrule(c)

Arguments

c	A numeric cost vector.

Details

Let $N^i_{-}\{j\in N:c_j<c_i\}$. For each $c\in C^N$ and each $i\in N$, the constrained equal contributions rule is defined by

$\text{CEC}_i(c)=\text{min}\left\{\dfrac{1}{r-i+1}\left(c_r-\sum\limits_{j\in N^i_{-}}\text{CEC}_j(c)\right):r=1,\dots,n\right\}.$

This rule offers a different approach to achieving equality. Contributions are distributed as evenly as possible while ensuring compliance with the no-subsidy constraints.

The contribution selected by the CEC rule for a problem $c \in C^N$ coincides with the payoff vector assigned by the Dutta-Ray solution (denoted by $\text{EA}$) to the associated airport game $v\in G^N$, that is, $\text{CEC}(c)=\text{EA}(v)$.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

Aadland, D. and Kolpin, V. (1998). Shared irrigation costs: an empirical and axiomatic analysis. Mathematical Social Sciences, 35, 203-218.

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

See Also

SIGMArule, basicrule, weightedrule, clonesrule, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
CECrule(c)

---

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

---

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

---

Allocation rule with cloned agents

Description

clonesrule calculates the contribution vector resulting from the payment allocation among the different agents using one of the versions for clones of the various predefined rules.

Usage

clonesrule(cw, eta, rule, group_contribution = TRUE, a = NULL, order = NULL)

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.
rule	A character string specifying the rule to apply. The rules that can be selected are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC", "SIGMA" and "PRIOR".
group_contribution	A logical value. By default, if group_contribution = TRUE, the cost allocation vector stores the aggregated contribution for each group of clones. However, if group_contribution = FALSE, the cost allocation vector represents the individual contribution of one of the agents in the group of clones.
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when rule = "SIGMA". By default, a = 0.5.
order	A numeric vector indicating the priority order of agents when making contributions. It can only be defined when rule = "PRIOR". By default, agents follow their original indexing and contribute accordingly.

Details

Let $\mathcal{R}$ be a rule, $t\in N$, and $(\eta,c)\in\mathcal{A}^N_t$. For each $i\in T=\{1,\dots,t\}$, the sum of the contributions requested by $\mathcal{R}$ from the group of clones $N^{\eta}_i$ is $\mathcal{R}(\eta\ast c, N_i^{\eta})=\sum\limits_{j\in N^\eta_i}\mathcal{R}_j(n\ast c)$.

The computation of the cost allocation selected by a rule for a given problem can be substantially simplified where there are cloned agents. Through this function, a direct method is proposed to obtain either the aggregate contribution or the individual contribution of each group of cloned agents, based on the associated reduced problem and the number of clones in each group.

The version for clones of the SFC, SEC, CEC and CP rules is equal to their respective weighted version, so the formulation of these rules will be the same for the weighted version. Only the CEB rule has a weighted version and a clone version that are different.

If a rule $\mathcal{R}$ satisfies equal treatment of equals (cloned agents pay equal amounts), then $\mathcal{R}(\eta\ast c,N_i^\eta)=\eta_i\mathcal{R}_s(n\ast c)$ for any $s\in N_i^\eta$. All the rules listed, with the exception of the PRIOR rule, satisfy equal treatment of equals. Therefore, the contribution demanded by these rules from a group of clones is divided equally among them.

Finally, we define a $k$-replica of an airport problem as the problem in which every agent is replaced by $k$ clones of itself. If $k$ increases, the number of groups of cloned agents does not change, but the number of agents is large. Thus, we say that a rule satisfies replication invariance if for each $c\in C^N$, each $i \in N$, and each $k\in \mathbb{N}$, we have $\mathcal{R}(k\ast c,N_i^k)=\mathcal{R}_i(c)$; i.e., if in any $k$-replica of a problem each group of cloned agents contributes an amount independent of $k$. The SFC, SEC, CEC, CP, and PRIOR rules verify this property, while the others do not.

Value

A numeric contribution vector. By default, if group_contribution = TRUE, each element represents the payment made by each group of cloned agents. However, if group_contribution = FALSE, each element reflects the individual payment made by a representative agent from each group.

Note

When rule = "CC", the execution time of the function may significantly increase if the number of individual agents exceeds 150.

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, clonesproblem, multiclonesrules, basicrule, weightedrule

Examples

# Clones SEC rule
cw <- c(1, 3, 7, 10) # Different costs
eta <- c(3, 4, 1, 2)  # Size of each groups of clones
clonesrule(cw, eta, "SEC")

# CEC rule satisfies replication invariance
cw <- c(1, 5, 12) # Different costs
eta <- rep(8, 3) # Size of each groups of clones
all.equal(clonesrule(cw, eta, "CEC"), CECrule(cw))

---

Graphical evaluation of the contribution vectors

Description

comparisonallocations generates a graphical representation in which, for each agent or group of clones, both the maximum cost they can bear and their corresponding marginal contribution are displayed.

Usage

comparisonallocations(
  c,
  contributions,
  col = NULL,
  colors = NULL,
  agents_names = NULL,
  labels = TRUE,
  legend = NULL,
  tol = 1e-06
)

Arguments

c	A numeric cost vector.
contributions	A list containing the different cost allocation vectors to be compared. It is required that the sum of the coordinates of each vector equals the total cost to be allocated.
col	A character string reflecting the color of the NS constraint for each agent. By default, the color "dodgerblue" is used.
colors	A vector that indicates the colors used to represent each contribution vector. By default, a color palette of different shades is used.
agents_names	A vector defining the name assigned to each agent. By default, the names follow a sequence of natural numbers, starting from 1.
labels	A logical value indicating whether the labels and the title of the plot should be displayed. By default, labels = TRUE.
legend	A vector or list where each of its elements represents a different contribution vector. By default, the coordinates of each contribution vector are displayed with two decimal places.
tol	Tolerance level for evaluating compliance with the NS constraint.

Details

For each $c\in C^N$ let $H(c)=\{x\in\mathbb{R}:x(N)=c_n\}$ be the hyperplane of $\mathbb{R}^N$ given by all the vectors whose coordinates add up to $c_n$. A cost allocation for $c\in C^N$ is a vector $x\in H(c)$ such that $0\leq x\leq c$. The component $x_i$ is the contribution requested from agent $i$. Let $X(c)$ be the set of cost allocations for $c\in C^N$.

A basic requirement is that at an allocation $x\in X(c)$ on group $N'\subset N$ of agents would subsidize the other agents by contributing more than what the group would have to pay on its own. The no-subsidy constraint for the group $N'\subset N$ is $x(N')\geq \text{max}\{c_j:j\in N'\}$. The set of cost allocations for $c\in C^N$ that satisfy the no-subsidy constraints, the no-subsidy set for short, is given by:

$NS(c)=\{x\in X(c):x(N')\leq\text{max}\{c_j:j\in N'\}, \text{ for all } N'\subset N\}$

$= \{x\in \mathbb{R}^N:x\geq 0, \ x(N)=c_n, \ x_1+\dots+x_i\leq c_i,\text{ for all }i\in N\backslash \{n\}\}$

Thus, the no-subsidy correspondence NS assigns to each $c\in C^N$ the set $NS(c)$.

A rule is a mapping $\mathcal{R}:C^N\rightarrow \mathbb{R}^N$ which associates with each problem $c\in C^N$ a contribution vector $\mathcal{R}(c)\in X(c)$. In other words, a rule is a mechanism that, for each airport problem, selects an allocation vector belonging to its no-subsidy set.

Value

A vertical line plot in which each line represents the maximum amount an agent can pay without violating the NS constraint, while the points along the lines indicate the contributions made by the agent.

References

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

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

See Also

NScheck, basicrule, plotallocations

Examples

# CEB rule vs weighted CEB rule vs clones CEB rule
c <- c(1, 3, 7, 10) # Cost vector
w <- c(1, 4, 8, 2) # Weight vector
eta <- w # Size of each groups of clones
CEB <- basicrule(c, "CEB")
wCEB <- weightedrule(c, w, "CEB")
clCEB <- clonesrule(c, w, "CEB")
comparisonallocations(c, list(CEB, wCEB, clCEB))

# SEC rule vs CEC rule vs SM rule
c <- c(5, 10, 20) # Cost vector
comparisonallocations(c, list(SECrule(c), CECrule(c), SMrule(c)),
col = "green", agents_names = c("Alex", "Estela", "Carmen"), labels = FALSE,
legend = list("SEC", "CEC", "SM"))

---

Constrained proportional rule

Description

CPrule calculates the contribution vector selected by the CP rule.

Usage

CPrule(c)

Arguments

c	A numeric cost vector.

Details

For each $c\in C^N$, let $c_0=0$ and $(q_0,q_1,\dots,q_s)\in\mathbb{N}^{s+1}$, with $0=q_0<q_1<\dots<q_s=n$, defined recursively, for $j\geq 0$, by

$q_{j+1}=\text{max}\Bigg\{q\in N_{+}^{q_j}:\dfrac{c_q-c_{q_j}}{c_{q_j+1}+\dots+c_q}=\text{min}\bigg\{\dfrac{c_r-c_{q_j}}{c_{q_j+1}+\dots+c_r}:r\in N_{+}^{q_j}\bigg\}\Bigg\}.$

Then, for each $j\in\{0,\dots,s-1\}$ and each $i\in Q_j=\{q_j+1,\dots,q_{j+1}\}$,

$\text{CP}_i(c)=\dfrac{c_i}{c(Q_j)}(c_{q_{j+1}}-c_{q_j}).$

With this rule, calculating each agent's contribution is not always straightforward. When a coalition of agents violates the NS constraint, it becomes necessary to proceed in two or more steps.

The core idea of this rule is proportionality, aiming to ensure that agents' contributions are as close as possible to being proportional to the cost parameters, while respecting these constraints.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

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

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

See Also

basicrule, weightedrule, clonesrule, hierarchicalrule

Examples

c <- c(1, 3, 7, 10) # Cost vector
CPrule(c)

---

Elevator data

Description

The data includes 4 variables related to the 14 apartments in a building, within the context of distributing the costs associated with installing an elevator in that building.

Usage

elevator

Format

A data frame with 14 rows and 4 columns (variables):

id

Unit identifier

floor

Floor number

c

Cumulative elevator installation cost for each dwelling (in euros).

area

Area of each dwelling (in square meters).

Source

Own elaboration.

Examples

# By number of floors (floor)
cw <- clonesgroups(elevator$c)$cw
eta <- clonesgroups(elevator$c)$eta
rules <- c("SFC", "SEC", "CEC", "CP", "CC")
by_floor <- list()
for (rule in rules){
  by_floor[[rule]] <- clonesrule(cw, eta, rule)
}
comparisonallocations(cw, by_floor, agents_names = unique(elevator$floor),
legend = rules, labels = FALSE)

# Based on the area of each dwelling (area), by unit identifier (id)
c <- elevator$c
w <- elevator$area
by_id <- multiweightedrules(c, w, rules = c("SFC", "SEC", "CSEC", "CEC", "CP"),
         draw = TRUE, agents_names = elevator$id, labels = FALSE)
round(by_id, 2)

---

Allocation rule according to the agents' hierarchical order

Description

hierarchicalrule calculates the contribution vector resulting from the payment allocation among the different agents using one of the various predefined rules in relation to the agents' hierarchical order.

Usage

hierarchicalrule(c, P, rule, a = NULL)

Arguments

c	A numeric cost vector.
P	A list showing the agents involved in the different distribution stages.
rule	A character string specifying the rule to apply.The rules that can be selected are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC" and "SIGMA".
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when rule = "SIGMA". By default, a = 0.5.

Details

Let $N=\{1,\dots,n\}$ be a finite set of agents and let $P=\{P_1,\dots,P_{m+1}\}$ be a partition of $N$, with $m\leq n-1$. So, the hierarchical structure, $P_{>}=\{P_1>P_2>\dots>P_m>P_{m+1}\}$, implies that agents in $P_1$ have priority over agents in $N\backslash P_1$, agents in $P_2$ have priority over agents in $N\backslash P_1\cup P_2$, and so on. Let $\mathcal{P}(N)$ denote the family of all hierarchical structures over $N$.

A hierarchical rule is a mapping $\mathcal{R}_P:C^N\rightarrow \mathbb{R}^N$ that, based on a hierarchical structure $P_{>}$, associates with each problem $c\in C^N$ a cost allocation vector $\mathcal{R}_P(c)$ such that $0\leq\mathcal{R}_P(c)\leq c$. In other words, the allocation proceeds by assigning costs to agents in higher-hierarchy coalitions before those in lower-hierarchy ones, through the successive application of the rule to the no-subsidy faces of coalitions obtained from $P_{>}$. These rules are on the boundary of the no-subsidy set.

In each stage, the agents share the accumulated costs up to that point. However, the remaining costs are allocated in subsequent stages.

Value

A numeric contribution vector, where each element represents the payment of the different agents.

References

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

Faigle, U. and Kern, W. (1992). The Shapley value for cooperation games under precedence constraints. International Journal of Game Theory, 21, 249-266.

Fiestras-Janeiro, M. G., Sánchez-Rodríguez, E., and Schuster, M. (2016). A precedence constraint value revisited. TOP, 24, 156-179.

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

NSfaces, PRIORrule, basicrule

Examples

# Two stages
c <- c(1, 3, 7, 10) # Cost vector
P <- list(c(1, 2), c(3, 4)) # Agents' hierarchical order
hierarchicalrule(c, P, rule = "SEC") # SEC rule

# Three stages
c <- c(1, 1, 3, 3, 7, 10) # Cost vector
P <- list(2, c(1, 3), c(4, 5, 6)) # Agents' hierarchical order
hierarchicalrule(c, P, "CEC") # CEC rule

---

Overview of the allocation rules

Description

multibasicrules calculates the contribution vectors resulting from the allocation of payments among different agents, applying various predefined rules. It also generates a graphical representation of the allocations based on the implemented rules.

Usage

multibasicrules(
  c,
  rules = c("SFC", "SEC", "CEC", "CEB", "CP", "SM", "CC", "SIGMA", "PRIOR"),
  draw = TRUE,
  col = NULL,
  a = NULL,
  order = NULL,
  agents_names = NULL,
  labels = TRUE
)

Arguments

c	A numeric cost vector.
rules	A character vector specifying the allocation rules. The available rules are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC", "SIGMA" and "PRIOR". By default, all the rules are selected.
draw	A logical value indicating whether or not the plot should be generated. By default, draw = TRUE.
col	A vector that indicates the colors used to represent each agent in the graphical representation. By default, the colors are selected by the function rainbow().
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when "SIGMA" is included in rules. By default, a = 0.5.
order	A numeric vector indicating the priority order of agents when making contributions. It can only be defined when "PRIOR" is included in rules. By default, agents follow their original indexing and contribute accordingly.
agents_names	A vector defining the name assigned to each agent. By default, the names follow a sequence of natural numbers, starting from 1.
labels	A logical value indicating whether the labels and the title of the plot should be displayed. By default, labels = TRUE.

Details

Let $X(c)$ be the set of cost allocations for $c \in C^N$. A rule is a mapping $\mathcal{R}: C^N \rightarrow \mathbb{R}^N$ that associates with each problem a contribution vector $\mathcal{R}_P(c)$ such that $0\leq\mathcal{R}_P(c)\leq c$. In other words, a rule is a mechanism that selects an allocation vector for each airport problem.

The various proposed rules, despite their differences, share a key characteristic: for any given problem, each rule selects an allocation vector that belongs to its no-subsidy set. Although these rules have been individually characterized in different functions, the one in question allows for the calculation of all of them at once.

Value

A data frame containing the contribution vectors determined by the selected allocation rules. Additionally, if draw = TRUE, a mosaic plot displaying the allocations obtained for the different rules.

Note

When "CC" is included in the rules argument, the execution time of the function may significantly increase if the number of agents exceeds 150.

References

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

Potters, J. and Sudhölter, P. (1999). Airport problems and consistent allocation rules. Mathematical Social Sciences, 38, 83–102.

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

See Also

basicrule, comparisonallocations, plotallocations

Examples

# All rules with graphical representation
c <- c(1, 3, 7, 10) # Cost vector
multibasicrules(c)

# SEC, CEC and SIGMA rule without plot
c <- c(5, 10, 20) # Cost vector
multibasicrules(c, rules = c("SEC", "CEC", "SIGMA"), draw = FALSE, a = 0.75,
agents_names = c("Alex", "Estela", "Carmen"))

---

Overview of the allocation rules with cloned agents

Description

multiclonesrules calculates the contribution vectors resulting from the allocation of payments among different agents, applying the versions for clones of various predefined rules. It also generates a graphical representation of the allocations based on the implemented rules.

Usage

multiclonesrules(
  cw,
  eta,
  rules = c("SFC", "SEC", "CEC", "CEB", "CP", "SM", "CC", "SIGMA", "PRIOR"),
  group_contribution = TRUE,
  draw = TRUE,
  col = NULL,
  a = NULL,
  order = NULL,
  agents_names = NULL,
  labels = TRUE
)

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.
rules	A character vector specifying the allocation rules. The available rules are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC", "SIGMA" and "PRIOR". By default, all the rules are selected.
group_contribution	A logical value. By default, if group_contribution = TRUE, the cost allocation vector stores the aggregated contribution for each group of clones. However, if group_contribution = FALSE, the cost allocation vector represents the individual contribution of one of the agents in the group of clones.
draw	A logical value indicating whether the plot should be generated. By default, draw = TRUE.
col	A vector that indicates the colors used to represent each agent in the graphical representation. By default, the colors are selected by the function rainbow().
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when "SIGMA" is included in rules. By default, a = 0.5.
order	A numeric vector indicating the priority order of agents when making contributions. It can only be defined when "PRIOR" is included in rules. By default, agents follow their original indexing and contribute accordingly.
agents_names	A vector defining the name assigned to each group of clones. By default, the names follow a sequence of natural numbers, starting from 1.
labels	A logical value indicating whether the labels and the title of the plot should be displayed. By default, labels = TRUE.

Details

Let $\mathcal{R}$ be a rule, $t\in N$, and $(\eta,c)\in\mathcal{A}^N_t$. For each $i\in T=\{1,\dots,t\}$, the sum of the contributions requested by $\mathcal{R}$ from the group of clones $N^{\eta}_i$ is $\mathcal{R}(\eta\ast c, N_i^{\eta})=\sum\limits_{j\in N^\eta_i}\mathcal{R}_j(n\ast c)$.

The computation of the cost allocation selected by a rule for a given problem can be substantially simplified where there are cloned agents. Through this function, a direct method is proposed to obtain either the aggregate contribution or the individual contribution of each group of cloned agents, based on the associated reduced problem and the number of clones in each group.

The version for clones of the SFC, SEC, CEC and CP rules is equal to their respective weighted version, so the formulation of these rules will be the same for the weighted version. Only the CEB rule has a weighted version and a clone version that are different.

If a rule $\mathcal{R}$ satisfies equal treatment of equals (cloned agents pay equal amounts), then $\mathcal{R}(\eta\ast c,N_i^\eta)=\eta_i\mathcal{R}_s(n\ast c)$ for any $s\in N_i^\eta$. All the rules listed, with the exception of the PRIOR rule, satisfy equal treatment of equals. Therefore, the contribution demanded by these rules from a group of clones is divided equally among them.

Finally, we define a $k$-replica of an airport problem as the problem in which every agent is replaced by $k$ clones of itself. If $k$ increases, the number of groups of cloned agents does not change, but the number of agents is large. Thus, we say that a rule satisfies replication invariance if for each $c\in C^N$, each $i \in N$, and each $k\in \mathbb{N}$, we have $\mathcal{R}(k\ast c,N_i^k)=\mathcal{R}_i(c)$; i.e., if in any $k$-replica of a problem each group of cloned agents contributes an amount independent of $k$. The SFC, SEC, CEC, CP, and PRIOR rules verify this property, while the others do not.

Value

A data frame containing the contribution vectors determined by the selected allocation rules. Additionally, if draw = TRUE, a mosaic plot displaying the allocations obtained for the different rules.

Note

When "CC" is included in the rules argument, the execution time of the function may significantly increase if the number of individual agents exceeds 150.

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, clonesproblem, clonesrule, comparisonallocations, plotallocations

Examples

# All rules for clones with graphical representation
cw <- c(1, 3, 7, 10) # Different costs
eta <- c(3, 4, 1, 2)  # Size of each groups of clones
multiclonesrules(cw, eta)

# SEC, CEC and CP rule for clones without plot
cw <- c(5, 10, 20) # Different costs
eta <- c(8, 2, 4) # Size of each groups of clones
multiclonesrules(cw, eta, rules = c("SEC", "CEC", "CP"),
group_contribution = FALSE, draw = FALSE,
agents_names = c("Suppliers", "Wholesalers", "Retailers"))

---

Overview of the allocation rules according to the agents' hierarchical order

Description

multihierarchicalrules calculates the contribution vectors resulting from the allocation of payments among different agents, applying various predefined rules in relation to the agents' hierarchical order. It also generates a graphical representation of the allocations based on the implemented rules.

Usage

multihierarchicalrules(
  c,
  P,
  rules = c("SFC", "SEC", "CEC", "CEB", "CP", "SM", "CC", "SIGMA"),
  a = NULL,
  draw = TRUE,
  col = NULL,
  agents_names = NULL,
  labels = TRUE
)

Arguments

c	A numeric cost vector.
P	A list showing the agents involved in the different distribution stages.
rules	A character vector specifying the allocation rules. The available rules are: "SFC", "SEC", "CEC", "CP", "CEB", "SM", "CC" and "SIGMA". By default, all the rules are selected.
a	A numeric value in the range [0,1], controlling the parameterization of the rule. It can only be defined when "SIGMA" is included in rules. By default, a = 0.5.
draw	A logical value indicating whether the plot should be generated. By default, draw = TRUE.
col	A vector that indicates the colors used to represent each agent in the graphical representation. By default, the colors are selected by the function rainbow().
agents_names	A vector defining the name assigned to each agent. By default, the names follow a sequence of natural numbers, starting from 1.
labels	A logical value indicating whether the labels and the title of the plot should be displayed. By default, labels = TRUE.

Details

Let $N=\{1,\dots,n\}$ be a finite set of agents and let $P=\{P_1,\dots,P_{m+1}\}$ be a partition of $N$, with $m\leq n-1$. So, the hierarchical structure, $P_{>}=\{P_1>P_2>\dots>P_m>P_{m+1}\}$, implies that agents in $P_1$ have priority over agents in $N\backslash P_1$, agents in $P_2$ have priority over agents in $N\backslash P_1\cup P_2$, and so on. Let $\mathcal{P}(N)$ denote the family of all hierarchical structures over $N$.

A hierarchical rule is a mapping $\mathcal{R}_P:C^N\rightarrow \mathbb{R}^N$ that, based on a hierarchical structure $P_{>}$, associates with each problem $c\in C^N$ a cost allocation vector $\mathcal{R}_P(c)$ such that $0\leq\mathcal{R}_P(c)\leq c$. In other words, the allocation proceeds by assigning costs to agents in higher-hierarchy coalitions before those in lower-hierarchy ones, through the successive application of the rule to the no-subsidy faces of coalitions obtained from $P_{>}$. These rules are on the boundary of the no-subsidy set.

In each stage, the agents share the accumulated costs up to that point. However, the remaining costs are allocated in subsequent stages.

Value

A data frame containing the contribution vectors determined by the selected allocation rules. Additionally, if draw = TRUE, a mosaic plot displaying the allocations obtained for the different rules.

References

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

Faigle, U. and Kern, W. (1992). The Shapley value for cooperation games under precedence constraints. International Journal of Game Theory, 21, 249-266.

Fiestras-Janeiro, M. G., Sánchez-Rodríguez, E., and Schuster, M. (2016). A precedence constraint value revisited. TOP, 24, 156-179.

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

NSfaces, PRIORrule, basicrule, comparisonallocations, plotallocations

Examples

# All rules in two stages with graphical representation
c <- c(1, 3, 3, 7, 10) # Cost vector
P <- list(c(1, 2, 3), c(4, 5)) # Agents' hierarchical order
multihierarchicalrules(c, P)

# SEC, CEC and SM rule in three stages without plot
c <- c(5, 10, 20, 20, 30, 50) # Cost vector
P <- list(c(1, 2), c(3, 4), c(5, 6)) # Agents' hierarchical order
multihierarchicalrules(c, P, rules = c("SEC", "CEC", "SM"), draw = FALSE,
agents_names = c("Alex", "Estela", "Carmen", "Miguel", "Gloria", "Brais"))

---

Overview of the weighted allocation rules

Description

multiweightedrules calculates the contribution vectors resulting from the allocation of payments among different agents, applying various predefined weighted rules. It also generates a graphical representation of the allocations based on the implemented weighted rules.

Usage

multiweightedrules(
  c,
  w,
  rules = c("SFC", "SEC", "CSEC", "CEC", "CEB", "CP"),
  draw = TRUE,
  col = NULL,
  agents_names = NULL,
  labels = TRUE
)

Arguments

c	A numeric cost vector.
w	A numeric weight vector.
rules	A character vector specifying the allocation rules. The available rules are: "SFC", "SEC", "CSEC", "CEC", "CP" and "CEB". By default, all the rules are selected.
draw	A logical value indicating whether the plot should be generated. By default, draw = TRUE.
col	A vector that indicates the colors used to represent each agent in the graphical representation. By default, the colors are selected by the function rainbow().
agents_names	A vector defining the name assigned to each agent. By default, the names follow a sequence of natural numbers, starting from 1.
labels	A logical value indicating whether the labels and the title of the plot should be displayed. By default, labels = TRUE.

Details

Let $w=(w_i)_{i\in N}\in\mathbb{R}^n$ be a positive weight vector, satisfying $w_i> 0$ for all $i\in N$ and $w(N)=1$. Consider the $(n-1)$-standard simplex, defined as $\Delta_n=\{x\in\mathbb{R}^n:x\geq 0, \ x_1+\dots+x_n=1\}$. Then, the set of all positive weight vectors corresponds to $\text{Int}(\Delta_n)$, the interior of the $(n-1)$-standard simplex.

A weighted rule is a mapping $\mathcal{R}:C^N\times \text{Int}(\Delta_N)\rightarrow \mathbb{R}^N$ which associates with a problem $c\in C^N$ and a positive weight vector $w\in \text{Int}(\Delta_n)$ a contribution vector $\mathcal{R}(c,w)\in X(c)$.

It is possible to define weighted versions of the rules: SFC, SEC, CEC, CP and CEB. In fact, two different rules emerge from the standard SEC rule: the weighted SEC rule and the coalition-weighted SEC rule. If $w_i=w_j$ for all $i,j \in N$, then the solution of weighted SEC$(c)$ and weighted CSEC$(c)$ coincides.

In all the rules, the higher the weight $w_i$, the more the corresponding agent will have to pay, except for the weighted CEB rule (the construction of this rule is based on the concept of allocating 'benefits', so it is logical that it is set up this way). Furthermore, as previously stated, all the rules, except for the weighted CSEC rule, require the weights to be positive. However, to standardize the criterion, we establish that the CSEC rule also demands positive weights.

The weighted version of the SFC, SEC, CEC and CP rules is equal to their respective versions for clones, so the formulation of these rules will be the same for the version with clones. Only the CEB rule has a weighted version and a clone version that are different.

Value

A data frame containing the contribution vectors determined by the selected allocation rules. Additionally, if draw = TRUE, a mosaic plot displaying the allocations obtained for the different weighted rules.

References

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

Bernárdez Ferradás, A., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2025b). A characterization of the CSEC rule for airport problems. [Prepint manuscript].

Sánchez-Rodríguez, E., Mirás Calvo, M. Á., Quinteiro Sandomingo, C., and Núñez Lugilde, I. (2024). Coalition-weighted Shapley values. International Journal of Game Theory, 53, 547-577.

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

See Also

weightedrule, comparisonallocations, plotallocations

Examples

# All weighted rules with graphical representation
c <- c(1, 3, 3, 7, 10) # Cost vector
w <- c(1, 4, 1, 2, 8) # Weight vector
multiweightedrules(c, w)

# Weighted SEC, CEC and CP rule without plot
c <- c(5, 10, 20)
w <- c(3, 2, 1)
multiweightedrules(c, w, rules = c("SEC", "CEC", "CP"), draw = FALSE,
agents_names = c("Alex", "Estela", "Carmen"))

---

Verification of compliance with the no-subsidy constraints

Description

NScheck evaluates whether or not the no-subsidy constraint is satisfied and, if not, it can also determine one of the coalitions of agents that violates it, as long as the user requests it.

Usage

NScheck(
  c,
  x,
  eta = rep(1, length(x)),
  group_contribution = TRUE,
  coalition = FALSE,
  tol = 1e-06
)

Arguments

c	A numeric cost vector.
x	A numeric cost allocation vector.
eta	A numeric vector representing the size of each group of cloned agents. All its elements must be positive integers. By default, all components of eta are set to 1.
group_contribution	A logical value. By default, if group_contribution = TRUE, x stores the aggregated contribution for each group of clones. However, if group_contribution = FALSE, x represents the individual contribution of one of the agents in the group of clones.
coalition	A logical value. By default, if coalition = FALSE, the function only returns whether the NS constraint is satisfied or not. However, if coalition = TRUE, the function also returns the coalition of agents that breach the NS constraint, if it is violated.
tol	Tolerance level for evaluating compliance with the NS constraint.

Details

For each $c\in C^N$ let $H(c)=\{x\in\mathbb{R}:x(N)=c_n\}$ be the hyperplane of $\mathbb{R}^N$ given by all the vectors whose coordinates add up to $c_n$. A cost allocation for $c\in C^N$ is a vector $x\in H(c)$ such that $0\leq x\leq c$. The component $x_i$ is the contribution requested from agent $i$. Let $X(c)$ be the set of cost allocations for $c\in C^N$. Given $x\in X(c)$, the difference $c_i-x_i$ is the benefit of agent $i$ at $x$.

A basic requirement is that at an allocation $x\in X(c)$ on group $N'\subset N$ of agents would subsidize the other agents by contributing more than what the group would have to pay on its own. The no-subsidy constraint for the group $N'\subset N$ is $x(N')\geq \text{max}\{c_j:j\in N'\}$. The set of cost allocations for $c\in C^N$ that satisfy the no-subsidy constraints, the no-subsidy set for short, is given by:

$NS(c)=\{x\in X(c):x(N')\leq\text{max}\{c_j:j\in N'\}, \;\text{for all}\; N'\subset N\}$

$= \{x\in \mathbb{R}^N:x\geq 0, \ x(N)=c_n, \ x_1+\dots+x_i\leq c_i,\;\text{for all}\;i\in N\backslash \{n\}\}$

Thus, the no-subsidy correspondence NS assigns to each $c\in C^N$ the set $NS(c)$.

Nevertheless, when a problem has group of cloned agents, the structure of its no-subsidy set is simpler than when all the cost parameters are different. Let $t\in N$, $\mathcal{A}_t^N$ be the set of pairs $(\eta,c)\in \mathbb{N}^t\times\mathbb{R}^t$ and $N_s^{\eta}=N_s^{\eta\ast c}=\{j\in N:(n\ast c)_j=c_s\}$. Then the no subsidy set for $\eta\ast c \in C^N$ is:

$NS(\eta\ast c)=\{x\in\mathbb{R}:x\geq 0,\ x(N)=c_t,\ x(N_1^{\eta})+\dots+x(N^{\eta}_s)\leq c_s, \;\text{for all}\; s<t\}.$

It is worth noting that all allocation rules proposed in this package satisfy this property.

Value

If coalition = TRUE, a logical value (TRUE or FALSE) indicating compliance with the NS constraint.

Otherwise, if coalition = FALSE, a list containing the following items:

flag		A logical value (TRUE or FALSE) indicating compliance with the NS constraint.
eta		If the NS constraint is violated, the coalition of agents that breach it will be returned.

References

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

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

See Also

NSfaces, NSstructure, NSset, comparisonallocations

Examples

# Compliance with the NS constraint
c <- c(2, 5, 9)
x <- SECrule(c)
NScheck(c, x)
# Non-compliance with the NS constraint
c <- c(2, 3, 7, 10)
x <- c(1, 2, 5, 2)
NScheck(c, x, coalition = TRUE)

---

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

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