Package 'TUvalues' reference manual

Title:	Tools for Calculating Allocations in Game Theory using Exact and Approximated Methods
Description:	The main objective of cooperative games is to allocate a good among the agents involved. This package includes the most well-known allocation rules, i.e., the Shapley value, the Banzhaf value, the egalitarian rule, and the equal surplus division value. In addition, it considers the point of view of a priori unions (situations in which agents can form coalitions). For this purpose, the package includes the Owen value, the Banzhaf-Owen value, and the corresponding extensions of the egalitarian rules. All these values can be calculated exactly or estimated by sampling.
Authors:	Maria D. Guillen [cre, aut] , Juan Carlos Gonçalves [aut]
Maintainer:	Maria D. Guillen <[email protected]>
License:	AGPL (>= 3)
Version:	0.1.0
Built:	2025-01-09 06:57:44 UTC
Source:	CRAN

Banzhaf value

Description

Calculate the Banzhaf value

Usage

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

Arguments

`characteristic_func`	The valued function defined on the subsets of the number of players.
`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
`n_players`	Only used if `characteristic_func` is a `function`. The number of players in the game.
`replace`	should sampling be with replacement?

Value

The Banzhaf value for each player

Examples

n <- 10
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 = 4000, n_players = n, replace = TRUE)

v<-c(0,0,0,1,2,1,3)
banzhaf(v, method = "exact")
banzhaf(v, method = "appro", n_rep = 4000, replace = TRUE)

n <- 10
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 = 4000, n_players = n, replace = TRUE)

v<-c(0,0,0,1,2,1,3)
banzhaf(v, method = "exact")
banzhaf(v, method = "appro", n_rep = 4000, replace = TRUE)

Banzhaf Index (approximated)

Description

Calculate the approximated Banzhaf Index based on sampling

Usage

banzhaf_appro(characteristic_func, n_players, n_rep, replace = TRUE)
banzhaf_appro(characteristic_func, n_players, n_rep, replace = TRUE)

Arguments

`characteristic_func`	The valued function defined on the subsets of the number of players
`n_players`	Only used if `value_func` is a `function`. The number of players in the game
`n_rep`	The number of iterations to perform in the approximated calculation
`replace`	should sampling be with replacement?

Value

The Banzhaf Index for each player

Banzhaf Index (approximation)

Description

Calculate the approximated Banzhaf Index based on sampling

Usage

banzhaf_appro_func(value_func, n_rep, n_players, replace = TRUE)
banzhaf_appro_func(value_func, n_rep, n_players, replace = TRUE)

Arguments

`value_func`	The valued function defined on the subsets of the number of players
`n_rep`	The number of iterations to perform in the approximated calculation
`n_players`	Only used if `value_func` is a `function`. The number of players in the game.
`replace`	should sampling be with replacement?

Value

The Banzhaf Index for each player

Banzhaf Index (approximated)

Description

Calculate the approximated Banzhaf Index based on sampling

Usage

banzhaf_appro_vector(value_func, n_rep)
banzhaf_appro_vector(value_func, n_rep)

Arguments

`value_func`	The valued function defined on the subsets of the number of players
`n_rep`	The number of iterations to perform in the approximated calculation

Value

The Banzhaf Index for each player

Banzhaf Index (exact)

Description

Calculate the approximated Banzhaf Index

Usage

banzhaf_exact(characteristic_func, n_players)
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,
  method = "exact",
  n_rep = 10000,
  n_players = 0,
  replace = TRUE
)
banzhaf_owen(
  characteristic_func,
  union,
  method = "exact",
  n_rep = 10000,
  n_players = 0,
  replace = 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
`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.
`n_rep`	Only used if `method` is `appro`. The number of iterations to perform in the approximated calculation
`n_players`	Only used if `characteristic_func` is a `function`. The number of players in the game.
`replace`	should sampling be with replacement?

Value

The Banzhaf-Owen value for each player

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)

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

Usage

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

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?

Value

The Banzhaf-Owen Index for each player

Banzhaf-Owen Value

Description

Calculate the approximated Banzhaf-Owen value

Usage

banzhaf_owen_exact(characteristic_func, union, n_players)
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

coalitions

Description

Create all the possible coalitions given the number of players

Usage

coalitions(n_players)
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

Egalitarian value

Description

Calculate the egalitarian value

Usage

egalitarian(characteristic_func, n_players = 0)
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)

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

Equal Surplus Division value

Description

Calculate the equal surplus division value

Usage

equal_surplus_division(characteristic_func, n_players = 0)
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)

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

Owen value

Description

Calculate the Owen value

Usage

owen(
  characteristic_func,
  union,
  method = "exact",
  n_rep = 10000,
  n_players = 0
)
owen(
  characteristic_func,
  union,
  method = "exact",
  n_rep = 10000,
  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.
`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.
`n_rep`	Only used if `method` is `appro`. The number of iterations to perform in the approximated calculation.
`n_players`	The number of players in the game.

Value

The Owen value for each player.

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)

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

Usage

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

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

Value

The Owen value for each player

Owen value (exact)

Description

Calculate the exact Owen

Usage

owen_exact(characteristic_func, union, n_players = NULL)
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)
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, method = "exact", n_rep = 10000, n_players = 0)
shapley(characteristic_func, method = "exact", n_rep = 10000, n_players = 0)

Arguments

`characteristic_func`	The valued function defined on the subsets of the number of players.
`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.
`n_rep`	Only used if `method` is `appro`. The number of iterations to perform in the approximated calculation.
`n_players`	Only used if `characteristic_func` is a `function`. The number of players in the game.

Value

The Shapley value for each player.

Examples

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

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

Usage

shapley_appro(characteristic_func, n_players, n_rep)
shapley_appro(characteristic_func, n_players, n_rep)

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

Value

The Shapley value for each player

Shapley value (exact)

Description

Calculate the exact Shapley value

Usage

shapley_exact(characteristic_func, n_players)
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

Package 'TUvalues'

Help Index

Banzhaf value

Description

Usage

Arguments

Value

Examples

Banzhaf Index (approximated)

Description

Usage

Arguments

Value

Banzhaf Index (approximation)

Description

Usage

Arguments

Value

Banzhaf Index (approximated)

Description

Usage

Arguments

Value

Banzhaf Index (exact)

Description

Usage

Arguments

Value

Banzhaf-Owen value

Description

Usage

Arguments

Value

Examples

Banzhaf-Owen Value

Description

Usage

Arguments

Value

Banzhaf-Owen Value

Description

Usage

Arguments

Value

coalitions

Description

Usage

Arguments

Value

Egalitarian value

Description

Usage

Arguments

Value

Examples

Equal Surplus Division value

Description

Usage

Arguments

Value

Examples

Owen value

Description

Usage

Arguments

Value

Examples

Owen value (approximation)

Description

Usage

Arguments

Value

Owen value (exact)

Description

Usage

Arguments

Value

Predecessor

Description

Usage