Package {TUvalues}

Type: Package
Title: Tools for Calculating Allocations in Game Theory using Exact and Approximated Methods
Version: 1.3.0
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 sampling-based approximation methods, enabling the analysis of large-scale games. The characteristic function can be supplied as a vector or as an R function, so that coalition values are evaluated only when required. Additionally, it supports core set-based solutions, allowing computation of the vertices and the centroid of the core and checking whether an allocation belongs to it.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding: UTF-8
RoxygenNote: 7.3.3
URL: https://github.com/mariaguilleng/TUvalues
BugReports: https://github.com/mariaguilleng/TUvalues/issues
Imports: utils, stats, ROI, ROI.plugin.glpk, rcdd
NeedsCompilation: no
Packaged: 2026-11-06 13:54:58 UTC; maria.guilleng
Author: Maria D. Guillen ORCID iD [cre, aut], Juan Carlos Gonçalves ORCID iD [aut], Jesús Camacho ORCID iD [aut]
Maintainer: Maria D. Guillen <maria.guilleng@umh.es>
Repository: CRAN
Date/Publication: 2026-11-07 23:40:10 UTC

TUvalues: Tools for Calculating Allocations in Game Theory using Exact and Approximated Methods

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 sampling-based approximation methods, enabling the analysis of large-scale games. The characteristic function can be supplied as a vector or as an R function, so that coalition values are evaluated only when required. Additionally, it supports core set-based solutions, allowing computation of the vertices and the centroid of the core and checking whether an allocation belongs to it.

Acknowledgements

This research has been partially supported by Grants PID2021-124030NB-C31, PID2025-172680NB-I00 and PID2025-170122NB-C32 from MICINN, Spain, and ERDF, "A way to make Europe", European Union. This research was also partially supported by the CIGE/2026/47 grant, funded by the Conselleria de Educación, Cultura, Universidades y Empleo, Generalitat Valenciana.

Author(s)

Maintainer: Maria D. Guillen maria.guilleng@umh.es (ORCID)

Authors:

See Also

Useful links:

Banzhaf value

Description

Calculate the Banzhaf value

Usage

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

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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

Only used if method is appro. Should sampling be with replacement? Without replacement, n_rep is reduced, if necessary, to the number of different coalitions that can be sampled for each player.

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-Owen value

Description

Calculate the Banzhaf-Owen value

Usage

banzhaf_owen(
  characteristic_func,
  union,
  n_players = 0,
  method = "exact",
  n_rep = 1000,
  replace = TRUE,
  echo = TRUE
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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 Banzhaf-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

Only used if method is appro. Should sampling be with replacement? Without replacement, n_rep is reduced, if necessary, to the number of different coalitions that can be sampled for each player.

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)

Centroid of the core of the game

Description

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

Usage

centroid(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 500,
  echo = TRUE,
  arithmetic = c("double", "rational"),
  tol = 1e-10
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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.

arithmetic

Only used if method is exact. Arithmetic used in the vertex enumeration: "double" (floating point) or "rational" (exact rational arithmetic).

tol

Only used if method is exact. Positive tolerance used in floating point comparisons.

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)

Core membership of an allocation

Description

Check whether an allocation belongs to the core of the game, that is, whether it is efficient (the players share exactly v(N)) and coalitionally rational (every coalition S receives at least v(S)).

Usage

check_core(allocation, characteristic_func, details = FALSE)

Arguments

allocation

Numeric vector with the payoff of each player. Its length is the number of players of the game.

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on the number of players.

details

If FALSE, only TRUE or FALSE is returned and the check stops at the first violated constraint. If TRUE, every constraint is checked and reported.

Details

All comparisons are exact: integer allocations are summed exactly in double precision and any other allocation is handled with exact rational arithmetic (package rcdd). Values are interpreted as the binary numbers stored by R. The value of the empty coalition is taken as 0.

When characteristic_func is a function, there is no limit on the number of players, but up to 2^n coalitions must be evaluated.

Value

If details is FALSE, TRUE if the allocation belongs to the core and FALSE otherwise. If details is TRUE, a list with:

in_core

Whether the allocation belongs to the core.

efficient

Whether the allocation is efficient.

coalitionally_rational

Whether every coalition receives at least its value.

violations

The rows of constraints that are not satisfied.

constraints

A data.frame with one row per coalition (in the order of coalitions): the payoff assigned to the coalition, its value, the slack, the relation required and whether it is satisfied.

Examples

v <- c(0, 0, 0, 2, 2, 2, 3)
check_core(c(1, 1, 1), v)
check_core(c(0, 1, 2), v)
check_core(c(0, 1, 2), v, details = TRUE)

v <- function(coalition) length(coalition)^2
check_core(rep(4, 4), v)

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

Examples

# Order in which the values of a game given as a vector must be supplied
coalitions(3)

Communication game

Description

Build the communication (graph-restricted) game of a communication situation, in which only connected players can cooperate (Myerson, 1977).

Usage

communication_game(characteristic_func, graph_edges, n_players = 0)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on the number of players.

graph_edges

Edges of the communication graph of the game: a list of pairs of players (for example list(c(1, 2), c(2, 3))), or a two-column matrix or data frame. Each edge must join two different players.

n_players

Only used if characteristic_func is a function. The number of players in the game.

Details

The worth of a coalition S in the communication game is the sum of the worths of the connected components of the subgraph induced by S:

v^G(S) = \sum_{C \in C^G(S)} v(C).

The Myerson value is the Shapley value of this game, so shapley(communication_game(v, graph_edges)) equals myerson(v, graph_edges).

The value of the empty coalition is taken as 0.

Value

If characteristic_func is a vector, a vector with the value of every nonempty coalition of the communication game, in the order given by coalitions. If characteristic_func is a function, a function that receives a coalition (a vector of players) and returns its value in the communication game. It evaluates characteristic_func only on the connected components of the coalitions it receives, so there is no limit on the number of players, and it can be passed directly to the other functions of the package.

References

Myerson, R. B. (1977). Graphs and Cooperation in Games. Mathematics of Operations Research, 2(3), 225–229. doi:10.1287/moor.2.3.225

See Also

myerson, myerson_unions

Examples

v <- c(1, 2, 3, 10, 20, 30, 100)
communication_game(v, graph_edges = list(c(1, 3)))

v <- function(coalition) length(coalition)^2
graph_edges <- list(c(1, 2), c(2, 3))
vG <- communication_game(v, graph_edges, n_players = 4)
vG(c(1, 2, 4))
shapley(vG, n_players = 4)
myerson(v, graph_edges, n_players = 4)

Vertices of the core of the game

Description

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

Usage

coreVertex(
  characteristic_func,
  n_players = 0,
  method = "exact",
  n_rep = 500,
  echo = TRUE,
  arithmetic = c("double", "rational"),
  tol = 1e-10
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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.

arithmetic

Only used if method is exact. Arithmetic used in the vertex enumeration: "double" (floating point) or "rational" (exact rational arithmetic).

tol

Only used if method is exact. Positive tolerance used in floating point comparisons.

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 rule

Description

Calculate the egalitarian rule

Usage

egalitarian(characteristic_func, n_players = 0)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on the number of players.

n_players

Only used if characteristic_func is a function. The number of players in the game.

Value

The egalitarian rule 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 rule with a priori unions

Description

Calculate the egalitarian rule in games with a priori unions

Usage

egalitarian_unions(characteristic_func, union, n_players = 0)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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 rule 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 characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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 characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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(1:2,3)
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 = 1000,
  echo = TRUE
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on the number of players.

graph_edges

Edges of the communication graph of the game: a list of pairs of players (for example list(c(1, 2), c(2, 3))), or a two-column matrix or data frame. Each edge must join two different 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, applying to the communication game the sampling procedure of the Shapley value 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 Myerson value for each player.

References

Myerson, R. B. (1977). Graphs and Cooperation in Games. Mathematics of Operations Research, 2(3), 225–229. doi:10.1287/moor.2.3.225

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

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,
  union,
  graph_edges,
  method = "exact",
  n_rep = 1000,
  echo = TRUE
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on the number of players.

n_players

Only used if characteristic_func is a function. The number of players in the game.

union

List of vectors indicating the a priori unions between the players.

graph_edges

Edges of the communication graph of the game: a list of pairs of players (for example list(c(1, 2), c(2, 3))), or a two-column matrix or data frame. Each edge must join two different players.

method

Method used to calculate the Myerson value. Valid methods are: exact for the exact calculation or appro for approximated polynomial calculation, applying to the communication game the sampling procedure of the Owen value 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 Myerson value for each player.

References

Myerson, R. B. (1977). Graphs and Cooperation in Games. Mathematics of Operations Research, 2(3), 225–229. doi:10.1287/moor.2.3.225

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

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, union = unions, graph_edges = graph_edges, method = "exact")
myerson_unions(v, union = 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 = 1000,
  echo = TRUE
)

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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 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)

Shapley value

Description

Calculate the Shapley value

Usage

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

Arguments

characteristic_func

The characteristic function of the game. It can be a numeric vector with the value of every coalition, in the order given by coalitions: of length 2^n - 1, without the empty coalition (whose value is taken as 0), or of length 2^n, with the empty coalition first. Vectors support up to 30 players. It can also be a function that receives a coalition (a vector of players) and returns its value; in that case there is no limit on 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 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)