Type:	Package
Title:	Tools for Calculating Allocations in Game Theory using Exact and Approximated Methods
Version:	1.1.0
Description:	The main objective of cooperative Transferable-Utility games (TU-games) is to allocate a good among the agents involved. The package implements major solution concepts including the Shapley value, Banzhaf value, and egalitarian rules, alongside their extensions for structured games: the Owen value and Banzhaf-Owen value for games with a priori unions, and the Myerson value for communication games on networks. To address the inherent exponential computational complexity of exact evaluation, the package offers both exact algorithms and linear approximation methods based on sampling, enabling the analysis of large-scale games. Additionally, it supports core set-based solutions, allowing computation of the vertices and the centroid of the core.
License:	AGPL (≥ 3)
Encoding:	UTF-8
RoxygenNote:	7.3.3
URL:	https://github.com/mariaguilleng/TUvalues
BugReports:	https://github.com/mariaguilleng/TUvalues/issues
Imports:	utils, gtools, highs
NeedsCompilation:	no
Packaged:	2025-12-16 08:17:23 UTC; meryg
Author:	Maria D. Guillen [cre, aut], Juan Carlos Gonçalves [aut]
Maintainer:	Maria D. Guillen <maria.guilleng@umh.es>
Repository:	CRAN
Date/Publication:	2025-12-16 09:30:16 UTC

Banzhaf value

Description

Calculate the Banzhaf value

Usage

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

Arguments

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

Value

The Shapley value for each player

Banzhaf Index (exact)

Description

Calculate the approximated Banzhaf Index

Usage

banzhaf_exact(characteristic_func, n_players)

Arguments

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

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

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

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

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

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

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

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)

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)

Arguments

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

Value

The vertices of the core

Egalitarian value

Description

Calculate the egalitarian value

Usage

egalitarian(characteristic_func, n_players = 0)

Arguments

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

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

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

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

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

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

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

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

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

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

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

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

Value

The Shapley value for each player