Package 'TUvalues'

Banzhaf value

Description

Calculate the Banzhaf value

Usage

banzhaf(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 10000,
  replace = FALSE,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to calculate the Banzhaf value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling.
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation
replace	Should sampling be with replacement?
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Banzhaf value for each player

Examples

n <- 8
v <- function(coalition) {
if (length(coalition) > n/2) {
   return(1)
 } else {
   return(0)
 }
}
banzhaf(v, method = "exact", n_players = n)
banzhaf(v, method = "appro", n_rep = 2000, n_players = n, replace = TRUE)

v<-c(0,0,0,1,2,1,3)
banzhaf(v, method = "exact")
banzhaf(v, method = "appro", n_rep = 2000, replace = TRUE)

---

Banzhaf Index (approximated)

Description

Calculate the approximated Banzhaf Index based on sampling

Usage

banzhaf_appro(characteristic_func, n_players, n_rep, replace, echo)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	The number of players
n_rep	The number of iterations to perform in the approximated calculation
replace	Should sampling be with replacement?
echo	Show progress of the calculation.

Value

The Shapley value for each player

---

Banzhaf Index (exact)

Description

Calculate the approximated Banzhaf Index

Usage

banzhaf_exact(characteristic_func, n_players)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	The number of players in the game.

Value

The Banzhaf Index for each player

---

Banzhaf-Owen value

Description

Calculate the Banzhaf-Owen value

Usage

banzhaf_owen(
  characteristic_func,
  union,
  n_players = 0,
  method = "exact",
  n_rep = 10000,
  replace = TRUE,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to calculate the Owen value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling proposed by Saavedra-Nieves & Fiestras-Janeiro (2021).
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation
replace	Should sampling be with replacement?
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Banzhaf-Owen value for each player

References

Saavedra-Nieves, A., & Fiestras-Janeiro, M. G. (2021). Sampling methods to estimate the Banzhaf–Owen value. Annals of Operations Research, 301(1), 199-223.

Examples

characteristic_func <- c(0,0,0,0,30,30,40,40,50,50,60,70,80,90,100)
union <- list(c(1,3),c(2),c(4))
banzhaf_owen(characteristic_func, union)
banzhaf_owen(characteristic_func, union, method = "appro", n_rep = 4000)

---

Banzhaf-Owen Value

Description

Calculate the approximated Banzhaf-Owen value using the algorithm proposed by Saavedra-Nieves & Fiestras-Janeiro (2021).

Usage

banzhaf_owen_appro(characteristic_func, union, n_players, n_rep, replace, echo)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players
n_players	The number of players
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
replace	Should sampling be with replacement?
echo	Show progress of the calculation.

Value

The Banzhaf-Owen Index for each player

References

Saavedra-Nieves, A., & Fiestras-Janeiro, M. G. (2021). Sampling methods to estimate the Banzhaf–Owen value. Annals of Operations Research, 301(1), 199-223.

---

Banzhaf-Owen Value

Description

Calculate the approximated Banzhaf-Owen value

Usage

banzhaf_owen_exact(characteristic_func, union, n_players)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players
n_players	The number of players in the game.

Value

The Banzhaf Index for each player

---

Centroid of the core of the game

Description

Calculate the centroid of core of the game if it exits.

Usage

centroid(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 1000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to calculate the core. Valid methods are: exact for the exact calculation or appro for approximated core based on Camacho et al. (2025).
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The centroid of the core if it exists.

References

Camacho, J., Gonçalves-Dosantos, J. C., & Sánchez-Soriano, J. (2025). A Linear Programming Approach to Estimate the Core in Cooperative Games. arXiv preprint arXiv:2510.01766.

Examples

v <- c(2,3,5,5,7,8,10)
centroid(v, method = "exact")
centroid(v, method = "appro", n_rep = 100)

n <- 3
v <- function(coalition) {
 size <- length(coalition)
 if (size <= 1) {
   return(0)
 } else if (size == 2) {
   return(10)
 } else if (size == 3) {
   return(24)
 } else {
   return(0)
 }
}
centroid(v, n, method = "exact")
centroid(v, n, method = "appro", n_rep = 200)

---

coalitions

Description

Create all the possible coalitions given the number of players

Usage

coalitions(n_players)

Arguments

n_players

Number of players

Value

A list containing a data.frame of the binary representation of the coalitions and a vector of the classical representation (as sets) of the coalitions

---

Approximated core of the game

Description

Calculate the vertices of the core of the game following Camacho et al. (2025)

Usage

core_appro(characteristic_func, n_players = 0, n_rep = 1000, echo = TRUE)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
n_rep	The number of iterations to perform in the algorithm.
echo	Show progress of the calculation.

Value

The vertices of the estimated core

References

Camacho, J., Gonçalves-Dosantos, J. C., & Sánchez-Soriano, J. (2025). A Linear Programming Approach to Estimate the Core in Cooperative Games. arXiv preprint arXiv:2510.01766.

---

Exact core of the game

Description

Calculate the vertices of core of the game.

Usage

core_exact(characteristic_func, n_players = 0)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.

Value

The vertices of the core

---

Vertices of the core of the game

Description

Calculate the vertices of core of the game if it exits.

Usage

coreVertex(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 1000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to calculate the core. Valid methods are: exact for the exact calculation or appro for approximated core based on Camacho et al. (2025).
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The vertices of the core if it exists.

References

Camacho, J., Gonçalves-Dosantos, J. C., & Sánchez-Soriano, J. (2025). A Linear Programming Approach to Estimate the Core in Cooperative Games. arXiv preprint arXiv:2510.01766.

Examples

v <- c(2,3,5,5,7,8,10)
coreVertex(v, method = "exact")
coreVertex(v, method = "appro", n_rep = 100)

n <- 3
v <- function(coalition) {
 size <- length(coalition)
 if (size <= 1) {
   return(0)
 } else if (size == 2) {
   return(10)
 } else if (size == 3) {
   return(24)
 } else {
   return(0)
 }
}
coreVertex(v, n, method = "exact")
coreVertex(v, n, method = "appro", n_rep = 200)

---

Egalitarian value

Description

Calculate the egalitarian value

Usage

egalitarian(characteristic_func, n_players = 0)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	Only used if characteristic_func is a function. The number of players in the game.

Value

The egalitarian value for each player

Examples

n <- 10
v <- function(coalition) {
  if (length(coalition) > n/2) {
    return(1)
  } else {
    return(0)
  }
}
egalitarian(v,n)

v <- c(1,1,2,1,2,2,2)
egalitarian(v)

---

Egalitarian value with a priori unions

Description

Calculate the egalitarian value in games with a priori unions

Usage

egalitarian_unions(characteristic_func, union, n_players = 0)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players.
n_players	Only used if characteristic_func is a function. The number of players in the game.

Value

The egalitarian value for each player

Examples

n <- 10
v <- function(coalition) {
  if (length(coalition) > n/2) {
    return(1)
  } else {
    return(0)
  }
}
union <- list(1:4,5:n)
egalitarian_unions(v,union,n)

v <- c(1,1,2,1,2,2,2)
union <- list(c(1,2),c(3))
egalitarian_unions(v, union)

---

Equal Surplus Division value

Description

Calculate the equal surplus division value

Usage

equal_surplus_division(characteristic_func, n_players = 0)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	Only used if characteristic_func is a function. The number of players in the game.

Value

The equal surplus division value for each player

Examples

n <- 10
v <- function(coalition) {
  if (length(coalition) > n/2) {
    return(1)
  } else {
    return(0)
  }
}
equal_surplus_division(v,n)

v <- c(1,1,2,1,2,2,2)
equal_surplus_division(v)

---

Equal Surplus Division value with a priori unions

Description

Calculate the equal surplus division value in games with a priori unions

Usage

equal_surplus_division_unions(
  characteristic_func,
  union,
  n_players = 0,
  type = 1
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
type	Number indicating the type of equal surplus division value to compute following Alonso-Meijide et al. (2020). Values 1, 2 and 3 are implemented.

Value

The equal surplus division value for each player

References

Alonso-Meijide, J. M., Costa, J., García-Jurado, I., & Gonçalves-Dosantos, J. C. (2020). On egalitarian values for cooperative games with a priori unions. Top, 28(3), 672-688.

Examples

n <- 3
v <- function(coalition) {
 size <- length(coalition)
 if (size <= 1) {
   return(0)
 } else if (size == 2) {
   return(10)
 } else if (size == 3) {
   return(24)
 } else {
   return(0)
 }
}
union <- list(1:4,5:n)
equal_surplus_division_unions(v,union,n,type = 1)

v <- c(1,1,2,1,2,2,2)
union <- list(c(1,2),c(3))
equal_surplus_division_unions(v, union, type = 2)

---

Myerson value

Description

Calculate the Myerson value in a communication game.

Usage

myerson(
  characteristic_func,
  graph_edges,
  n_players = 0,
  method = "exact",
  n_rep = 10000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players. It can be provided as a vector or as a function.
graph_edges	Edges of the communication graph of the game. It must be a list of pairs indicating the connected players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to calculate the Myerson value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling proposed.
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Myerson value for each player.

Examples

characteristic_func <- c(
1, 2, 0, 3,
3, 1, 4, 2, 5, 3,
3, 6, 4, 5,
15
)
graph_edges <- list(c(1, 2), c(2, 4))
myerson(characteristic_func, graph_edges, method = "exact")
myerson(characteristic_func, graph_edges, method = "appro", n_rep = 1000)

v <- function(S) {
  if (length(S) == 2) {
    return(1)
   }
   return(0)
}
n <- 3
graph_edges <- list(c(1, 2))
myerson(v, graph_edges, n_players = n, method = "exact")
myerson(v, graph_edges, n_players = n, method = "appro", n_rep = 2000)

---

Myerson value with a priori unions

Description

Calculate the Myerson value in a communication game with a priori unions.

Usage

myerson_unions(
  characteristic_func,
  n_players = 0,
  unions,
  graph_edges,
  method = "exact",
  n_rep = 10000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players. It can be provided as a vector or as a function.
n_players	Only used if characteristic_func is a function. The number of players in the game.
unions	List of vectors indicating the a priori unions between the players.
graph_edges	Edges of the communication graph of the game. It must be a list of pairs indicating the connected players.
method	Method used to calculate the Myerson value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling proposed.
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Myerson value for each player.

Examples

v <- c(
  0, 0, 0, 0,
  1, 1, 1, 0, 0, 0,
  1, 1, 1, 1,
  1
)
graph_edges <- list(c(2,3),c(3,1),c(1,4))
unions <- list(c(2,3),c(1),c(4))
myerson_unions(v, unions = unions, graph_edges = graph_edges, method = "exact")
myerson_unions(v, unions = unions, graph_edges = graph_edges, method = "appro", n_rep = 2000)

---

Owen value

Description

Calculate the Owen value

Usage

owen(
  characteristic_func,
  union,
  n_players = 0,
  method = "exact",
  n_rep = 10000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
union	List of vectors indicating the a priori unions between the players.
n_players	The number of players in the game.
method	Method used to calculate the Owen value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling proposed by Saavedra-Nieves et al. (2018).
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Owen value for each player.

References

Saavedra-Nieves, A., García-Jurado, I., & Fiestras-Janeiro, M. G. (2018). Estimation of the Owen value based on sampling. In The mathematics of the uncertain: A tribute to Pedro Gil (pp. 347-356). Cham: Springer International Publishing.

Examples

n <- 10
v <- function(coalition) {
  if (length(coalition) > n/2) {
    return(1)
  } else {
    return(0)
  }
}
u <- lapply(1:(n/2), function(i) c(2*i - 1, 2*i))
owen(v, union = u, method = "appro", n_rep = 4000, n_players = n)

characteristic_func <- c(1,1,2,1,2,2,2)
union <- list(c(1,2),c(3))
owen(characteristic_func, union)
owen(characteristic_func, union, method = "appro", n_rep = 4000)

---

Owen value (approximation)

Description

Calculate the approximated Owen value based on sampling using the algorithm proposed by Saavedra-Nieves et al. (2018).

Usage

owen_appro(characteristic_func, union, n_players, n_rep, echo)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players
n_players	The number of players
n_rep	The number of iterations to perform in the approximated calculation
echo	Show progress of the calculation.

Value

The Owen value for each player

References

Saavedra-Nieves, A., García-Jurado, I., & Fiestras-Janeiro, M. G. (2018). Estimation of the Owen value based on sampling. In The mathematics of the uncertain: A tribute to Pedro Gil (pp. 347-356). Cham: Springer International Publishing.

---

Owen value (exact)

Description

Calculate the exact Owen

Usage

owen_exact(characteristic_func, union, n_players = NULL)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
union	List of vectors indicating the a priori unions between the players
n_players	The number of players

Value

The Owen value for each player

---

Predecessor

Description

Given a permutation 0 of players and a player i, calculate the set of predecessors of the player i in the order 0

Usage

predecessor(permutation, player, include_player = FALSE)

Arguments

permutation	A permutation of the players
player	Number of the player i
include_player	Whether the player i is included as predecessor of itself or not

Value

The set of predecessors of the player i in the order 0

---

Shapley value

Description

Calculate the Shapley value

Usage

shapley(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 10000,
  echo = TRUE
)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players.
n_players	Only used if characteristic_func is a function. The number of players in the game.
method	Method used to alculate the Shapley value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation based on sampling proposed by Castro et al. (2009).
n_rep	Only used if method is appro. The number of iterations to perform in the approximated calculation.
echo	Only used if method is appro. Show progress of the approximated calculation.

Value

The Shapley value for each player.

References

Castro, J., Gómez, D., & Tejada, J. (2009). Polynomial calculation of the Shapley value based on sampling. Computers & operations research, 36(5), 1726-1730.

Examples

n <- 10
v <- function(coalition) {
if (length(coalition) > n/2) {
   return(1)
 } else {
   return(0)
 }
}
shapley(v, method = "appro", n_rep = 4000, n_players = n)

n <- 3
v <- c(1,1,2,1,2,2,2)
shapley(v, method = "exact")
shapley(v, method = "appro", n_rep = 4000)

---

Shapley value (approximation)

Description

Calculate the approximated Shapley value based on sampling using the algorithm proposed by Castro et al. (2009).

Usage

shapley_appro(characteristic_func, n_players, n_rep, echo)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	The number of players
n_rep	The number of iterations to perform in the approximated calculation
echo	Show progress of the calculation.

Value

The Shapley value for each player

References

Castro, J., Gómez, D., & Tejada, J. (2009). Polynomial calculation of the Shapley value based on sampling. Computers & operations research, 36(5), 1726-1730.

---

Shapley value (exact)

Description

Calculate the exact Shapley value

Usage

shapley_exact(characteristic_func, n_players)

Arguments

characteristic_func	The valued function defined on the subsets of the number of players
n_players	The number of players

Value

The Shapley value for each player

---