Compute prenucleolus

Description

Computes the prenucleolus of a TU game with n players. Note that the prenucleolus is a member of the set of preimputations.

Usage

prenucleolus(v)

Arguments

Value

Numeric vector of length n representing the prenucleolus.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

Daniel Gebele daniel.a.gebele@stud.hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 107–132

Examples

library(CoopGame)
prenucleolus(c(1, 1, 1, 2, 3, 4, 5))


#Example 5.5.12 from Peleg/Sudhoelter, p. 96
library(CoopGame)
prenucleolus(c(0,0,0,10,0,0,2))
#Output
#[1]  3  3 -4
#In the above example nucleolus and prenucleolus do not coincide!

library(CoopGame)
prenucleolus(c(0, 0, 0, 0, 5, 5, 8, 9, 10, 8, 13, 15, 16, 17, 21))
# [1] 3.5 4.5 5.5 7.5

#Final example:
#Estate division problem from Babylonian Talmud with E=200,
#see e.g. seminal paper by Aumann & Maschler from 1985 on
#'Game Theoretic Analysis of a Bankruptcy Problem from the Talmud'
library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=200)
prenucleolus(v)
#[1]  50 75 75
#Note that nucleolus and prenucleolus need to coincide for the above game

Compute absolute Public Good value

Description

absolutePublicGoodValue calculates the absolute Public Good value for a specified nonnegative TU game. Note that in general the absolute Public Good value is not an efficient vector, i.e. the sum of its entries is not always 1.

Usage

absolutePublicGoodValue(v)

Arguments

Value

Absolute Public Good value for specified nonnegative game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Li X. (1995) "From public good index to public value. An axiomatic approach and generalization", Control and Cybernetics 24, pp. 257–270

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v <- c(1,2,3,4,0,0,0)
absolutePublicGoodValue(v)


v=c(0,0,0,0.7,11,0,15)
absolutePublicGoodValue(v) 
#[1] 26.7 15.7 26.0

Compute absolute Public Help value Chi

Description

Calculates the absolute Public Help value Chi for a specified nonnegative TU game. Note that in general the absolute Public Help value Chi is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the absolute Public Help value Chi is provided. Note that the greek letter Xi (instead of Chi) was used in the original paper by Bertini and Stach (2015).

Usage

absolutePublicHelpChiValue(v)

Arguments

Value

Absolute Public Help value Chi for specified nonnegative game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,2,2,0,2)
absolutePublicHelpChiValue(v) 

Compute absolute Public Help value Theta

Description

absolutePublicHelpValue calculates the absolute Public Help value Theta for a specified nonnegative TU game. Note that in general the absolute Public Help value Theta is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the absolute Public Help Value is provided.

Usage

absolutePublicHelpValue(v)

Arguments

Value

Absolute Public Help value Theta for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,0.7,11,0,15)
absolutePublicHelpValue(v) 

Construct an apex game

Description

Create a list containing all information about a specified apex game:
A coalition can only win (and hence obtain the value 1) if it
a) contains both the apex player and one additional player
or
b) contains all players except for the apex player.
Any non-winning coalitions obtain the value 0.
Note that apex games are always simple games.

Usage

apexGame(n, apexPlayer)

Arguments

Value

A list with three elements representing the apex game (n, apexPlayer, Game vector v)

Related Functions

apexGameValue, apexGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 164–165

Examples

library(CoopGame)
apexGameVector(n=3,apexPlayer=2)


library(CoopGame)
#Example with four players, apex player is number 3
(vv<-apexGame(n=4,apexPlayer=3))
#$n
#[1] 4

#$apexPlayer
#[1] 4

#$v
# [1] 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1

Compute value of a coalition for an apex game

Description

Coalition value for an apex game:
For further information see apexGame

Usage

apexGameValue(S, n, apexPlayer)

Arguments

Value

value of coalition S

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 164–165

Examples

library(CoopGame)
apexGameValue(c(1,2),3,2)


library(CoopGame)
apexGameValue(c(1,2,3,4),4,3)
# Output:
# [1] 1

Compute game vector for an apex game

Description

Game vector for an apex game:
For further information see apexGame

Usage

apexGameVector(n, apexPlayer)

Arguments

Value

Game vector for the apex game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 164–165

Examples

library(CoopGame)
apexGameVector(n=3,apexPlayer=2)


library(CoopGame)
(v <- apexGameVector(n=4,apexPlayer=3))
#[1] 0 0 0 0 0 1 0 1 0 1 1 1 1 1 1

Construct a bankruptcy game

Description

Create a list containing all information about a specified bankruptcy game:
The list contains the number of players, the claims vector, the estate and the bankruptcy game vector. Bankruptcy games are defined by a vector of debts d of n creditors (players) and an estate E less than the sum of the debt vector. The roots of bankruptcy games can be traced back to the Babylonian Talmud.

Usage

bankruptcyGame(n, d, E)

Arguments

Value

A list with four elements representing the specified bankruptcy game (n, d, E, Game vector v)

Related Functions

bankruptcyGameValue, bankruptcyGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

O'Neill, B. (1982) "A problem of rights arbitration from the Talmud", Mathematical Social Sciences 4(2), pp. 345 – 371

Aumann R.J. and Maschler M. (1985) "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud", Journal of Economic Theory 36(1), pp. 195 – 213

Aumann R.J. (2002) "Game Theory in the Talmud", Research Bulletin Series on Jewish Law and Economics, 12 pages.

Gura E. and Maschler M. (2008) Insights into Game Theory, Cambridge University Press, pp. 166–204

Examples

library(CoopGame)
bankruptcyGame(n=3, d=c(1,2,3), E=4)


#Estate division problem from Babylonian Talmud 
#from paper by Aumann (2002) with E=300
library(CoopGame)
bankruptcyGame(n=3,d=c(100,200,300),E=300)
#Output
#$n
#[1] 3

#$d
#[1] 100 200 300

#$E
#[1] 300

#$v
#[1]   0   0   0   0 100 200 300

Compute value of a coalition for a bankruptcy game

Description

Coalition value for a specified bankruptcy game:
For further information see bankruptcyGame

Usage

bankruptcyGameValue(S, d, E)

Arguments

Value

A positive value if the sum of the claims outside of coalition S is less than E else 0

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

O'Neill, B. (1982) "A problem of rights arbitration from the Talmud", Mathematical Social Sciences 4(2), pp. 345 – 371

Aumann R.J. and Maschler M. (1985) "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud", Journal of Economic Theory 36(1), pp. 195 – 213

Aumann R.J. (2002) "Game Theory in the Talmud", Research Bulletin Series on Jewish Law and Economics, 12 pages.

Gura E. and Maschler M. (2008) Insights into Game Theory, Cambridge University Press, pp. 166–204

Examples

library(CoopGame)
bankruptcyGameValue(S=c(2,3),d=c(1,2,3),E=4)


#Estate division problem from Babylonian Talmud 
#from paper by Aumann (2002) with E=300
library(CoopGame)
bankruptcyGameValue(S=c(2,3),d=c(100,200,300),E=300)
#Output
#[1] 200

Compute game vector for a bankruptcy game

Description

Game vector for a specified bankruptcy game:
For further information see bankruptcyGame

Usage

bankruptcyGameVector(n, d, E)

Arguments

Value

Game Vector where each element contains a positive value if the sum of the claims outside of coalition 'S' is less than E else 0

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

O'Neill, B. (1982) "A problem of rights arbitration from the Talmud", Mathematical Social Sciences 4(2), pp. 345 – 371

Aumann R.J. and Maschler M. (1985) "Game Theoretic Analysis of a Bankruptcy Problem from the Talmud", Journal of Economic Theory 36(1), pp. 195 – 213

Aumann R.J. (2002) "Game Theory in the Talmud", Research Bulletin Series on Jewish Law and Economics, 12 pages.

Gura E. and Maschler M. (2008) Insights into Game Theory, Cambridge University Press, pp. 166–204

Examples

library(CoopGame)
bankruptcyGameVector(n=3, d=c(1,2,3), E=4)


#Estate division problem from Babylonian Talmud
#from paper by Aumann (2002) with E=300
library(CoopGame)
bankruptcyGameVector(n=3,d=c(100,200,300),E=300)
#Output
#[1] 0   0   0   0 100 200 300

Compute Banzhaf value

Description

banzhafValue computes the Banzhaf value for a specified TU game The Banzhaf value itself is an alternative to the Shapley value.
Conceptually, the Banzhaf value is very similar to the Shapley value. Its main difference from the Shapley value is that the Banzhaf value is coalition based rather than permutation based. Note that in general the Banzhaf vector is not efficient! In this sense this implementation of the Banzhaf value could also be referred to as the non-normalized Banzhaf value, see formula (20.6) in on p. 368 of the book by Hans Peters (2015).

Usage

banzhafValue(v)

Arguments

Value

The return value is a numeric vector which contains the Banzhaf value for each player.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 367–370

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Gambarelli G. (2011) "Banzhaf value", Encyclopedia of Power, SAGE Publications, pp. 53–54

Examples

library(CoopGame)
v=c(0,0,0,1,2,1,4)
banzhafValue(v)


#Example from paper by Gambarelli (2011)
library(CoopGame)
v=c(0,0,0,1,2,1,3)
banzhafValue(v)
#[1] 1.25 0.75 1.25

Compute Barua Chakravarty Sarkar index

Description

Calculates the Barua Chakravarty Sarkar index for a specified simple TU game. Note that in general the Barua Chakravarty Sarkar index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Barua Chakravarty Sarkar index is provided.

Usage

baruaChakravartySarkarIndex(v)

Arguments

Value

Barua Chakravarty Sarkar index for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Barua R., Chakravarty S.R. and Sarkar P. (2012) "Measuring p-power of voting", Journal of Economic Theory and Social Development 1(1), pp. 81–91

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 120–123

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
baruaChakravartySarkarIndex(v) 

Check if point is core element

Description

belongsToCore checks if a given point is in the core

Usage

belongsToCore(x, v)

Arguments

Value

TRUE for a point belonging to the core and FALSE otherwise

Author(s)

Franz Mueller

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gillies D.B. (1953) Some Theorems on n-person Games, Ph.D. Thesis, Princeton University Press.

Aumann R.J. (1961) "The core of a cooperative game without side payments", Transactions of the American Mathematical Society 98(3), pp. 539–552

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–49

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 686–747

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 257–275

Examples

library(CoopGame)
v = c(0,1,2,3,4,5,6)
belongsToCore(c(1,2,3),v)

Check if point is core cover element

Description

belongsToCoreCover checks if the point is in the core cover

Usage

belongsToCoreCover(x, v)

Arguments

Value

TRUE if point belongs to core cover, FALSE otherwise

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tijs S.H. and Lipperts F.A.S. (1982) "The hypercube and the core cover of n-person cooperative games", Cahiers du Centre d' Etudes de Researche Operationelle 24, pp. 27–37

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 45–46

Examples

library(CoopGame)
belongsToCoreCover(x=c(1,1,1), v=c(0,0,0,1,1,1,3))


library(CoopGame)
v <- c(0,0,0,3,3,3,6)
belongsToCoreCover(x=c(2,2,2),v)
#[1] TRUE
belongsToCoreCover(x=c(1,2,4),v)
#[1] FALSE

Check if point is imputation

Description

belongsToImputationset checks if the point belongs to the imputation set

Usage

belongsToImputationset(x, v)

Arguments

Value

TRUE for a point belonging to the imputation set and FALSE otherwise

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 20

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 674

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 278

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 407

Examples

library(CoopGame)
belongsToImputationset(x=c(1,0.5,0.5), v=c(0,0,0,1,1,1,2))


library(CoopGame)
v=c(0,1,2,3,4,5,6)

#Point belongs to imputation set:
belongsToImputationset(x=c(1.5,1,3.5),v)

#Point does not belong to imputation set:
belongsToImputationset(x=c(2.05,2,2),v)

Check if point is element of reasonable set

Description

belongsToReasonableSet checks if the point is in the reasonable set

Usage

belongsToReasonableSet(x, v)

Arguments

Value

TRUE if point belongs to reasonable set, FALSE otherwise

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Milnor J.W. (1953) Reasonable Outcomes for N-person Games, Rand Corporation, Research Memorandum RM 916.

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 43–44

Gerard-Varet L.A. and Zamir S. (1987) "Remarks on the reasonable set of outcomes in a general coalition function form game", Int. Journal of Game Theory 16(2), pp. 123–143

Examples

library(CoopGame)
belongsToReasonableSet(x=c(1,0.5,0.5), v=c(0,0,0,1,1,1,2))


library(CoopGame)
v <- c(0,0,0,3,3,3,6)
belongsToReasonableSet(x=c(2,2,2),v)
#[1] TRUE
belongsToReasonableSet(x=c(1,2,4),v)
#[1] FALSE

Check if point is element of Weber Set

Description

belongsToWeberset checks if the point is in the Weber Set

Usage

belongsToWeberset(x, v)

Arguments

Value

TRUE if point belongs to Weber Set and FALSE otherwise

Author(s)

Johannes Anwander anwander.johannes@gmail.com

References

Weber R.J. (1988) "Probabilistic values for games". In: Roth A.E. (Ed.), The Shapley Value. Essays in in honor of Lloyd S. Shapley, Cambridge University Press, pp. 101–119

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 327–329

Examples

library(CoopGame)
belongsToWeberset(x=c(1,0.5,0.5), v=c(0,0,0,1,1,1,2))


library(CoopGame)
v=c(0,1,2,3,4,5,6)

#Point belongs to Weber Set:
belongsToWeberset(x=c(1.5,1,3.5),v)

#Point does not belong to Weber Set:
belongsToWeberset(x=c(2.05,2,2),v)

Construct a cardinality game

Description

Create a list containing all information about a specified cardinality game:
For a cardinality game the worth of each coalition is simply the number of the members of the coalition.

Usage

cardinalityGame(n)

Arguments

Value

A list with two elements representing the cardinality game (n, Game vector v)

Related Functions

cardinalityGameValue, cardinalityGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

Examples

library(CoopGame)
cardinalityGame(n=3)


library(CoopGame)
#Example: Cardinality function for four players
(vv<-cardinalityGame(n=4))
#$n
#[1] 4

#$v
#[1] 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4

Compute value of a coalition for a cardinality game

Description

Coalition value for a cardinality game:
For further information see cardinalityGame

Usage

cardinalityGameValue(S)

Arguments

Value

The return value is the cardinality, i.e. the number of elements, of coalition S

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
S=c(1,2,4,5)
cardinalityGameValue(S)

Compute game vector for a cardinality game

Description

Game vector for a cardinality game:
For further information see cardinalityGame

Usage

cardinalityGameVector(n)

Arguments

Value

Game vector for the cardinality game

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
cardinalityGameVector(n=3)


library(CoopGame)
(v <- cardinalityGameVector(n=4))
#[1] 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4

Compute centroid of core

Description

Calculates the centroid of the core for specified game.

Usage

centroidCore(v)

Arguments

Value

Calculates the centroid of the core for a game specified by a game vector v.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gillies D.B. (1953) Some Theorems on n-person Games, Ph.D. Thesis, Princeton University Press.

Aumann R.J. (1961) "The core of a cooperative game without side payments", Transactions of the American Mathematical Society 98(3), pp. 539–552

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–49

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 686–747

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 257–275

Examples

library(CoopGame)
v=c(0,0,0,2,2,3,5)
centroidCore(v) 

Compute centroid of the core cover

Description

Calculates the centroid of the core cover for specified game.

Usage

centroidCoreCover(v)

Arguments

Value

Centroid of the core cover for a quasi-balanced game specified by a game vector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tijs S.H. and Lipperts F.A.S. (1982) "The hypercube and the core cover of n-person cooperative games", Cahiers du Centre d' Etudes de Researche Operationelle 24, pp. 27–37

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 45–46

Examples

library(CoopGame)
v=c(0,0,0,2,2,3,5)
centroidCoreCover(v) 

Compute centroid of the imputation set

Description

Calculates the centroid of the imputation set for specified game.

Usage

centroidImputationSet(v)

Arguments

Value

Calculates the centroid of the imputation set for a game specified by a game vector.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 20

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 674

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 278

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 407

Examples

library(CoopGame)
v=c(0,0,0,2,2,3,5)
centroidImputationSet(v) 

Compute centroid of reasonable set

Description

Calculates the centroid of the reasonable set for specified game.

Usage

centroidReasonableSet(v)

Arguments

Value

Calculates the centroid of the reasonable set for a game specified by a game vector.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Milnor J.W. (1953) Reasonable Outcomes for N-person Games, Rand Corporation, Research Memorandum RM 916.

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 43–44

Gerard-Varet L.A. and Zamir S. (1987) "Remarks on the reasonable set of outcomes in a general coalition function form game", Int. Journal of Game Theory 16(2), pp. 123–143

Examples

library(CoopGame)
v=c(0,0,0,2,2,3,5)
centroidReasonableSet(v) 

Compute centroid of Weber set

Description

Calculates the centroid of the Weber set for specified game.

Usage

centroidWeberSet(v)

Arguments

Value

Calculates the centroid of the Weber set for a game specified by a game vector.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Weber R.J. (1988) "Probabilistic values for games". In: Roth A.E. (Ed.), The Shapley Value. Essays in in honor of Lloyd S. Shapley, Cambridge University Press, pp. 101–119

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 327–329

Examples

library(CoopGame)
v=c(0,0,0,2,2,3,5)
centroidWeberSet(v) 

Compute Coleman Power index of a Collectivity to Act

Description

Calculates the Coleman Power index of a Collectivity to Act for a specified simple TU game. Note that in general the Coleman Power index of a Collectivity to Act is not an efficient vector, i.e. the sum of its entries is not always 1. Note also that the the Coleman Power index of a Collectivity to Act is identical for each player, i.e. the result for each player is the number of winning coalitions divided by 2^n. Hence no drawing routine for the Coleman Power index of a Collectivity to Act is provided.

Usage

colemanCollectivityPowerIndex(v)

Arguments

Value

Coleman Power index of a Collectivity to Act for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Coleman J.S. (1971) "Control of collectivities and the power of a collectivity to act". In: Liberman B. (Ed.), Social Choice, Gordon and Breach, pp. 269–300

De Keijzer B. (2008) "A survey on the computation of power indices", Technical Report, Delft University of Technology, p. 18

Bertini C. and Stach I. (2011) "Coleman index", Encyclopedia of Power, SAGE Publications, p. 117–119

Examples

library(CoopGame) 
v=c(0,0,0,1,1,0,1)
colemanCollectivityPowerIndex(v) 

Compute Coleman Initiative Power index

Description

Calculates the Coleman Initiative Power index for a specified simple TU game. Note that in general the Coleman Initiative Power index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Coleman Initiative Power index is provided.

Usage

colemanInitiativePowerIndex(v)

Arguments

Value

Coleman Initiative Power index for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Coleman J.S. (1971) "Control of collectivities and the power of a collectivity to act". In: Liberman B. (Ed.), Social Choice, Gordon and Breach, pp. 269–300

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 120–123

De Keijzer B. (2008) "A survey on the computation of power indices", Technical Report, Delft University of Technology, p. 18

Bertini C. and Stach I. (2011) "Coleman index", Encyclopedia of Power, SAGE Publications, p. 117–119

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
colemanInitiativePowerIndex(v) 

Compute Coleman Preventive Power index

Description

Calculates the Coleman Preventive Power index for a specified simple TU game. Note that in general the Coleman Preventive Power index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Coleman Preventive Power index is provided.

Usage

colemanPreventivePowerIndex(v)

Arguments

Value

Coleman Preventive Power index for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Coleman J.S. (1971) "Control of collectivities and the power of a collectivity to act". In: Liberman B. (Ed.), Social Choice, Gordon and Breach, pp. 269–300

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 120–123

De Keijzer B. (2008) "A survey on the computation of power indices", Technical Report, Delft University of Technology, p. 18

Bertini C. and Stach I. (2011) "Coleman index", Encyclopedia of Power, SAGE Publications, p. 117–119

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
colemanPreventivePowerIndex(v) 

Compute core cover vertices

Description

Calculates the core cover for a given game vector

Usage

coreCoverVertices(v)

Arguments

Value

rows of the matrix are the vertices of the core cover

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tijs S.H. and Lipperts F.A.S. (1982) "The hypercube and the core cover of n-person cooperative games", Cahiers du Centre d' Etudes de Researche Operationelle 24, pp. 27–37

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 45–46

Examples

library(CoopGame)
coreCoverVertices(c(0,0,0,1,1,1,3))


library(CoopGame)
v <- c(0,0,0,3,3,3,6)
coreCoverVertices(v)
#       [,1] [,2] [,3]
# [1,]    3    0    3
# [2,]    0    3    3
# [3,]    3    3    0

Compute core vertices

Description

Calculates the core vertices for given game vector

Usage

coreVertices(v)

Arguments

Value

rows of the matrix are the vertices of the core

Author(s)

Franz Mueller

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gillies D.B. (1953) Some Theorems on n-person Games, Ph.D. Thesis, Princeton University Press.

Aumann R.J. (1961) "The core of a cooperative game without side payments", Transactions of the American Mathematical Society 98(3), pp. 539–552

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–49

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 686–747

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 257–275

Examples

library(CoopGame)
coreVertices(c(0,0,0,1,1,1,3))


#In the following case the core consists of a single point
v1 = c(0,1,2,3,4,5,6)
coreVertices(v1)
#     [,1] [,2] [,3]
#[1,]    1    2    3

#Users may also want to try the following commands:
coreVertices(c(0,0,0,60,80,100,135))
coreVertices(c(5,2,4,7,15,15,15,15,15,15,20,20,20,20,35))
coreVertices(c(0,0,0,0,0,5,5,5,5,5,5,5,5,5,5,15,15,15,15,15,15,15,15,15,15,30,30,30,30,30,60))

Construct a cost sharing game

Description

Create a list containing all information about a specified cost sharing game:
The user may specify the cost function of a cost allocation problem. A corresponding savings game will be calculated. The savings game specified by the game vector v will work like an ordinary TU game.

Usage

costSharingGame(n, Costs)

Arguments

Value

A list with three elements representing the specified cost sharing game (n, Costs, Game vector v)

Related Functions

costSharingGameValue, costSharingGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 14–16

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 667–668

Examples

library(CoopGame)
costSharingGame(n=3, Costs=c(2,2,2,3,3,3,4))


#Example with 3 students sharing an appartment:
#-------------------------------
#| costs     |  A  |  B  |  C  |
#- -----------------------------
#|single     | 300 | 270 | 280 |
#|appartment |     |     |     |
#-------------------------------
#
#Appartment for 2 persons => costs: 410
#Appartment for 3 persons => costs: 550

#Savings for all combinations sharing appartments
library(CoopGame)
(vv <- costSharingGame(n=3, Costs=c(300,270,280,410,410,410,550)))
#Output:
#$n
#[1] 3

#$Costs
#[1] 300 270 280 410 410 410 550

#$v
#[1]   0   0   0 160 170 140 300

Compute value of a coalition for a cost game

Description

Coalition value for a cost sharing game:
For further information see costSharingGame

Usage

costSharingGameValue(S, Costs)

Arguments

Value

Cost savings of coalition S as compared to singleton coalitions

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 14–16

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 667–668

Examples

library(CoopGame)
costSharingGameValue(S=c(1,2), Costs=c(2,2,2,3,3,3,4))


#Example with 3 students sharing an appartment:
#-------------------------------
#| costs     |  A  |  B  |  C  |
#- -----------------------------
#|single     | 300 | 270 | 280 |
#|appartment |     |     |     |
#-------------------------------
#
#Appartment for 2 persons => costs: 410
#Appartment for 3 persons => costs: 550

#Savings when A and B share appartment
library(CoopGame)
costSharingGameValue(S=c(1,2),Costs=c(300,270,280,410,410,410,550))
#Output: 
#[1] 160

Compute game vector for a cost sharing game

Description

Coalition vector for a cost sharing game:
For further information see costSharingGame

Usage

costSharingGameVector(n, Costs)

Arguments

Value

Game vector with cost-savings of each coalition S as compared to singleton coalitions.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 14–16

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 667–668

Examples

library(CoopGame)
costSharingGameVector(n=3, Costs=c(2,2,2,3,3,3,4))


#Example with 3 students sharing an appartment:
#-------------------------------
#| costs     |  A  |  B  |  C  |
#- -----------------------------
#|single     | 300 | 270 | 280 |
#|appartment |     |     |     |
#-------------------------------
#
#Appartment for 2 persons => costs: 410
#Appartment for 3 persons => costs: 550

#Savings for all combinations sharing appartments
library(CoopGame)
(v=costSharingGameVector(n=3, Costs=c(300,270,280,410,410,410,550)))
#Output: 
#[1]   0   0   0 160 170 140 300

create bit matrix

Description

createBitMatrix creates a bit matrix with (numberOfPlayers+1) columns and (2^numberOfPlayers-1) rows which contains all possible coalitions (apart from the null coalition) for the set of all players. Each player is represented by a column which describes if this player is either participating (value 1) or not participating (value 0). The last column (named cVal) contains the values of each coalition. Accordingly, each row of the bit matrix expresses a coalition as a subset of all players.

Usage

createBitMatrix(n, A = NULL)

Arguments

Value

The return is a bit matrix containing all possible coalitions apart from the empty coalition

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
createBitMatrix(3,c(0,0,0,60,60,60,72))


library(CoopGame)
A=weightedVotingGameVector(n=3,w=c(1,2,3),q=5)
bm=createBitMatrix(3,A)
bm
# Output:
#            cVal
# [1,] 1 0 0    0
# [2,] 0 1 0    0
# [3,] 0 0 1    0
# [4,] 1 1 0    0
# [5,] 1 0 1    0
# [6,] 0 1 1    1
# [7,] 1 1 1    1

Compute Deegan-Packel index

Description

deeganPackelIndex calculates the Deegan-Packel index for simple games

Usage

deeganPackelIndex(v)

Arguments

Value

Deegan-Packel index for a specified simple game

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Deegan J. and Packel E.W. (1978) "A new index of power for simple n-person games", Int. Journal of Game Theory 7(2), pp. 151–161

Holler M.J. and Illing G. (2006) "Einfuehrung in die Spieltheorie". 6th Edition (in German), Springer, pp. 323–324

Examples

library(CoopGame)
deeganPackelIndex(c(0,0,0,0,1,0,1))


#Example from HOLLER & ILLING (2006), chapter 6.3.3
#Expected result: dpv=(18/60,9/60,11/60,11/60,11/60)
library(CoopGame)
v1=weightedVotingGameVector(n = 5, w=c(35,20,15,15,15), q=51)
deeganPackelIndex(v1)
#Output (as expected, see HOLLER & ILLING chapter 6.3.3) :
#[1] 0.3000000 0.1500000 0.1833333 0.1833333 0.1833333

Construct a dictator game

Description

Create a list containing all information about a specified dictator game:
Any coalitions including the dictator receive coalition value 1. All the other coalitions, i.e. each and every coalition not containing the dictator, receives coalition value 0.
Note that dictator games are always simple games.

Usage

dictatorGame(n, dictator)

Arguments

Value

A list with three elements representing the dictator game (n, dictator, Game vector v)

Related Functions

dictatorGameValue, dictatorGameVector

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 295

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame) 
dictatorGame(n=3,dictator=2)


library(CoopGame) 
dictatorGame(n=4,dictator=2)
#Output:
#$n
#[1] 4

#$dictator
#[1] 2

#$v
#[1] 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1

Compute value of a coalition for a dictator game

Description

Coalition value for a dictator game:
For further information see dictatorGame

Usage

dictatorGameValue(S, dictator)

Arguments

Value

1 if dictator is involved in coalition, 0 otherwise.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 295

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame)
dictatorGameValue(S=c(1,2,3),dictator=2)

Compute game vector for a dictator game

Description

Game vector for a dictator game:
For further information see dictatorGame

Usage

dictatorGameVector(n, dictator)

Arguments

Value

Game vector where each element contains 1 if dictator is involved, 0 otherwise.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 295

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame)
dictatorGameVector(n=3,dictator=2)

Compute disruption nucleolus

Description

Computes the disruption nucleolus of a balanced TU game with n players. Note that the disruption nucleolus needs to be a member of the core.

Usage

disruptionNucleolus(v)

Arguments

Value

Numeric vector of length n representing the disruption nucleolus of the specified TU game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Examples

library(CoopGame)
v<-c(0, 0, 0, 1, 1, 0, 1)
disruptionNucleolus(v)


library(CoopGame)
exampleVector<-c(0,0,0,0,2,3,4,1,3,2,8,11,6.5,9.5,14)
disruptionNucleolus(exampleVector)
#[1] 3.193548 4.754839 2.129032 3.922581

Construct a divide-the-dollar game

Description

Create a list containing all information about a specified divide-the-dollar game:
Returns a divide-the-dollar game with n players:
This sample game is taken from the book 'Social and Economic Networks' by Matthew O. Jackson (see p. 413 ff.). If coalition S has at least n/2 members it generates a value of 1, otherwise 0.
Note that divide-the-dollar games are always simple games.

Usage

divideTheDollarGame(n)

Arguments

Value

A list with two elements representing the divide-the-dollar game (n, Game vector v)

Related Functions

divideTheDollarGameValue, divideTheDollarGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 413

Examples

library(CoopGame)
divideTheDollarGame(n=3)


#Example with four players
library(CoopGame)
(vv<-divideTheDollarGame(n=4))
#$n
#[1] 4

#$v
#[1] 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1

Compute value of a coalition for a divide-the-dollar game

Description

Coalition value for a divide-the-dollar game:
For further information see divideTheDollarGame

Usage

divideTheDollarGameValue(S, n)

Arguments

Value

value of coalition

Author(s)

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 413

Examples

library(CoopGame)
S <- c(1,2)
divideTheDollarGameValue(S, n = 3)

Compute game vector for a divide-the-dollar game

Description

Game vector for a divide-the-dollar game:
For further information see divideTheDollarGame

Usage

divideTheDollarGameVector(n)

Arguments

Value

Game vector for the specified divide-the-dollar game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 413

Examples

library(CoopGame) 
divideTheDollarGameVector(n=3)


library(CoopGame)
(v <- divideTheDollarGameVector(n=4))
#Output: 
# [1] 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1

draw centroid of the core for 3 or 4 players

Description

drawCentroidCore draws the centroid of the core for 3 or 4 players.

Usage

drawCentroidCore(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "centroid of core"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gillies D.B. (1953) Some Theorems on n-person Games, Ph.D. Thesis, Princeton University Press.

Aumann R.J. (1961) "The core of a cooperative game without side payments", Transactions of the American Mathematical Society 98(3), pp. 539–552

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–49

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 686–747

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 257–275

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawCentroidCore(v,colour="green")

draw centroid of core cover for 3 or 4 players

Description

drawCentroidCoreCover draws the centroid of the core cover for 3 or 4 players.

Usage

drawCentroidCoreCover(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "centroid of core cover"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tijs S.H. and Lipperts F.A.S. (1982) "The hypercube and the core cover of n-person cooperative games", Cahiers du Centre d' Etudes de Researche Operationelle 24, pp. 27–37

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 45–46

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawCentroidCoreCover(v,colour="black")

draw centroid of imputation set for 3 or 4 players

Description

drawCentroidImputationSet draws the centroid of the imputation set for 3 or 4 players.

Usage

drawCentroidImputationSet(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "centroid of imputation set"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 20

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 674

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 278

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 407

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawCentroidImputationSet(v,colour="green")

draw centroid of reasonable set for 3 or 4 players

Description

drawCentroidReasonableSet draws the centroid of the reasonable set for 3 or 4 players.

Usage

drawCentroidReasonableSet(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "centroid of reasonable set"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Milnor J.W. (1953) Reasonable Outcomes for N-person Games, Rand Corporation, Research Memorandum RM 916.

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 43–44

Gerard-Varet L.A. and Zamir S. (1987) "Remarks on the reasonable set of outcomes in a general coalition function form game", Int. Journal of Game Theory 16(2), pp. 123–143

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawCentroidReasonableSet(v,colour="green")

draw centroid of Weber set for 3 or 4 players

Description

drawCentroidWeberset draws the centroid of the Weber set for 3 or 4 players.

Usage

drawCentroidWeberSet(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "centroid of Weber set"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Weber R.J. (1988) "Probabilistic values for games". In: Roth A.E. (Ed.), The Shapley Value. Essays in in honor of Lloyd S. Shapley, Cambridge University Press, pp. 101–119

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 327–329

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawCentroidWeberSet(v,colour="blue")

Draw core for 3 or 4 players

Description

drawCore draws the core for 3 or 4 players.

Usage

drawCore(v, holdOn = FALSE, colour = "red", label = FALSE, name = "Core")

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gillies D.B. (1953) Some Theorems on n-person Games, Ph.D. Thesis, Princeton University Press.

Aumann R.J. (1961) "The core of a cooperative game without side payments", Transactions of the American Mathematical Society 98(3), pp. 539–552

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–49

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 686–747

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 257–275

Examples

library(CoopGame)
v <- c(0,0,0,3,3,3,6)
drawCore(v)

Draw core cover for 3 or 4 players

Description

drawCoreCover draws the core cover for 3 or 4 players.

Usage

drawCoreCover(
  v,
  holdOn = FALSE,
  colour = NA,
  label = FALSE,
  name = "Core Cover"
)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tijs S.H. and Lipperts F.A.S. (1982) "The hypercube and the core cover of n-person cooperative games", Cahiers du Centre d' Etudes de Researche Operationelle 24, pp. 27–37

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 45–46

Examples

library(CoopGame)
v <- c(0,0,0,3,3,3,6)
drawCoreCover(v)

draw Deegan-Packel index for 3 or 4 players

Description

drawDeeganPackelIndex draws the Deegan-Packel index for 3 or 4 players.

Usage

drawDeeganPackelIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Deegan Packel Index"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Deegan J. and Packel E.W. (1978) "A new index of power for simple n-person games", Int. Journal of Game Theory 7(2), pp. 151–161

Holler M.J. and Illing G. (2006) "Einfuehrung in die Spieltheorie". 6th Edition (in German), Springer, pp. 323–324

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawDeeganPackelIndex(v) 

draw disruption nucleolus for 3 or 4 players

Description

drawDisruptionNucleolus draws the disruption nucleolus for 3 or 4 players.

Usage

drawDisruptionNucleolus(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Disruption Nucleolus"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Examples

library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=200)
drawDisruptionNucleolus(v)

draw Gately point for 3 or 4 players

Description

drawGatelyValue draws the Gately point for 3 or 4 players.

Usage

drawGatelyValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Gately Value"
)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gately D. (1974) "Sharing the Gains from Regional Cooperation. A Game Theoretic Application to Planning Investment in Electric Power", International Economic Review 15(1), pp. 195–208

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, pp. 455–456

Examples

library(CoopGame)
drawGatelyValue(c(0,0,0,1,1,1,3.5))


#Example from original paper by Gately (1974):
library(CoopGame)
v=c(0,0,0,1170,770,210,1530)
drawGatelyValue(v)

Draw imputation set for 3 or 4 players

Description

drawImputationset draws the imputation set for 3 or 4 players.

Usage

drawImputationset(v, label = TRUE)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 20

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 674

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 278

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 407

Examples

library(CoopGame)
v=c(0,1,2,3,4,5,6)
drawImputationset(v)

Draw Johnston index for 3 or 4 players

Description

drawJohnstonIndex draws the Johnston index for 3 or 4 players.

Usage

drawJohnstonIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Johnston index"
)

Arguments

Value

None.

References

Johnston R.J. (1978) "On the measurement of power: Some reactions to Laver", Environment and Planning A, pp. 907–914

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, p. 124

Examples

library(CoopGame)
v <- c(0,0,0,1,1,0,1)
drawJohnstonIndex(v)

Draw modiclus for 3 or 4 players

Description

drawModiclus draws the modiclus for 3 or 4 players.

Usage

drawModiclus(v, holdOn = FALSE, colour = NA, label = TRUE, name = "Modiclus")

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 124–132

Sudhoelter P. (1997) "The Modified Nucleolus. Properties and Axiomatizations", Int. Journal of Game Theory 26(2), pp. 147–182

Sudhoelter P. (1996) "The Modified Nucleolus as Canonical Representation of Weighted Majority Games", Mathematics of Operations Research 21(3), pp. 734–756

Examples

library(CoopGame)
v=c(1, 1, 1, 2, 3, 4, 5)
drawModiclus(v)

draw normalized Banzhaf Index for 3 or 4 players

Description

drawNormalizedBanzhafIndex draws the Banzhaf Value for 3 or 4 players.
Drawing any kind of Banzhaf values only makes sense from our point of view for the normalized Banzhaf index for simple games, because only in this case will the Banzhaf index be efficient.

Usage

drawNormalizedBanzhafIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Normalized Banzhaf index"
)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 367–370

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Bertini C. and Stach I. (2011) "Banzhaf voting power measure", Encyclopedia of Power, SAGE Publications, pp. 54–55

Examples

library(CoopGame)
v<-weightedVotingGameVector(n=3,w=c(50,30,20),q=c(67))
drawNormalizedBanzhafIndex(v)

draw normalized Banzhaf value for 3 or 4 players

Description

drawNormalizedBanzhafValue draws the Banzhaf Value for 3 or 4 players.
Drawing any kind of Banzhaf values only makes sense from our point of view for the normalized Banzhaf value, because only in this case will the Banzhaf value be efficient.

Usage

drawNormalizedBanzhafValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Normalized Banzhaf value"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gambarelli G. (2011) "Banzhaf value", Encyclopedia of Power, SAGE Publications, pp. 53–54

Stach I. (2017) "Sub-Coalitional Approach to Values", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXVI, Springer, pp. 74–86

Examples

library(CoopGame)
drawNormalizedBanzhafValue(c(0,0,0,1,2,3,6))


#Example from paper by Gambarelli (2011)
library(CoopGame)
v=c(0,0,0,1,2,1,3)
drawNormalizedBanzhafValue(v)

Draw nucleolus for 3 or 4 players

Description

drawNucleolus draws the nucleolus for 3 or 4 players.

Usage

drawNucleolus(v, holdOn = FALSE, colour = NA, label = TRUE, name = "Nucleolus")

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

References

Schmeidler D. (1969) "The nucleolus of a characteristic function game", SIAM Journal on applied mathematics 17(6), pp. 1163–1170

Kohlberg E. (1971) "On the nucleolus of a characteristic function game", SIAM Journal on applied mathematics 20(1), pp. 62–66

Kopelowitz A. (1967) "Computation of the kernels of simple games and the nucleolus of n-person games", Technical Report, Department of Mathematics, The Hebrew University of Jerusalem, 45 pages.

Megiddo N. (1974) "On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game", SIAM Journal on applied mathematics 27(2), pp. 355–358

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 82–86

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,3)
drawNucleolus(v) 


#Visualization for estate division problem from Babylonian Talmud with E=300,
#see e.g. seminal paper by Aumann & Maschler from 1985 on
#'Game Theoretic Analysis of a Bankruptcy Problem from the Talmud'
library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=300)
drawNucleolus(v)

Draw per capita nucleolus for 3 or 4 players

Description

drawPerCapitaNucleolus draws the per capita nucleolus for 3 or 4 players.

Usage

drawPerCapitaNucleolus(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Per Capita Nucleolus"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Young H.P. (1985) "Monotonic Solutions of cooperative games", Int. Journal of Game Theory 14(2), pp. 65–72

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,3)
drawPerCapitaNucleolus(v) 


#Example from YOUNG 1985, p. 68
library(CoopGame)
v=c(0,0,0,0,9,10,12)
drawPerCapitaNucleolus(v)

Draw prenucleolus for 3 or 4 players

Description

drawPrenucleolus draws the prenucleolus for 3 or 4 players.

Usage

drawPrenucleolus(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Prenucleolus"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 107–132

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,3)
drawPrenucleolus(v) 


#Visualization for estate division problem from Babylonian Talmud with E=200,
#see e.g. seminal paper by Aumann & Maschler from 1985 on
#'Game Theoretic Analysis of a Bankruptcy Problem from the Talmud'
library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=200)
drawPrenucleolus(v)

Draw proportional nucleolus for 3 or 4 players

Description

drawProportionalNucleolus draws the proportional nucleolus for 3 or 4 players.

Usage

drawProportionalNucleolus(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Proportional Nucleolus"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Young H. P., Okada N. and Hashimoto, T. (1982) "Cost allocation in water resources development", Water resources research 18(3), pp. 463–475

Examples

library(CoopGame)
v<-c(0,0,0,48,60,72,140)
drawProportionalNucleolus(v)

Draw Public Good index for 3 or 4 players

Description

drawPublicGoodIndex draws the Public Good Index of a simple game for 3 or 4 players.

Usage

drawPublicGoodIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Public Good Index"
)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Packel E.W. (1983) "Power, luck and the right index", Zeitschrift fuer Nationaloekonomie 43(1), pp. 21–29

Holler M. (2011) "Public Goods index", Encyclopedia of Power, SAGE Publications, pp. 541–542

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawPublicGoodIndex(v)

Draw Public Good value for 3 or 4 players

Description

drawPublicGoodValue draws the (normalized) Public Good value for 3 or 4 players.

Usage

drawPublicGoodValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Normalized Public Good Value"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Li X. (1995) "From public good index to public value. An axiomatic approach and generalization", Control and Cybernetics 24, pp. 257 – 270

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9 – 25

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawPublicGoodValue(v) 

Draw Public Help index Chi for 3 or 4 players

Description

drawPublicHelpChiIndex draws the Public Help index Chi for a simple game with 3 or 4 players.

Usage

drawPublicHelpChiIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Public Help Chi Index"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Stach I. (2016) "Power Measures and Public Goods", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXIII, Springer, pp. 99–110

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawPublicHelpChiIndex(v) 

Draw Public Help value Chi for 3 or 4 players

Description

drawPublicHelpChiValue draws the (normalized) Public Help value Chi for 3 or 4 players.

Usage

drawPublicHelpChiValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Normalized Public Help Value Chi"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,2,2,0,3)
drawPublicHelpChiValue(v) 

Draw Public Help index Theta for 3 or 4 players

Description

drawPublicHelpIndex draws the Public Help index Theta for a simple game with 3 or 4 players.

Usage

drawPublicHelpIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Public Help Index"
)

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C., Gambarelli G. and Stach I. (2008) "A public help index", In: Braham, M. and Steffen, F. (Eds): Power, freedom, and voting: Essays in Honour of Manfred J. Holler, pp. 83–98

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Stach I. (2016) "Power Measures and Public Goods", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXIII, Springer, pp. 99–110

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawPublicHelpIndex(v) 

Draw Public Help value Theta for 3 or 4 players

Description

drawPublicHelpValue draws the (normalized) Public Help value Theta for 3 or 4 players.

Usage

drawPublicHelpValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Normalized Public Help Value"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawPublicHelpValue(v) 

Draw reasonable set for 3 or 4 players

Description

drawReasonableSet draws the reasonable set for 3 or 4 players.

Usage

drawReasonableSet(
  v,
  holdOn = FALSE,
  colour = NA,
  label = FALSE,
  name = "Reasonable Set"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Milnor J.W. (1953) Reasonable Outcomes for N-person Games, Rand Corporation, Research Memorandum RM 916.

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 43–44

Gerard-Varet L.A. and Zamir S. (1987) "Remarks on the reasonable set of outcomes in a general coalition function form game", Int. Journal of Game Theory 16(2), pp. 123–143

Examples

library(CoopGame)
v <- c(0,0,0,3,3,3,6)
drawReasonableSet(v)

Draw Shapley-Shubik index for 3 or 4 players

Description

drawShapleyShubik draws the Shapley-Shubik index simple game with 3 or 4 players.

Usage

drawShapleyShubikIndex(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Shapley-Shubik index"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Shapley L.S. and Shubik M. (1954) "A method for evaluating the distribution of power in a committee system". American political science review 48(3), pp. 787–792

Shapley L.S. (1953) "A value for n-person games". In: Kuhn, H., Tucker, A.W. (Eds.), Contributions to the Theory of Games II, Princeton University Press, pp. 307–317

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 156–159

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 748–781

Stach I. (2011) "Shapley-Shubik index", Encyclopedia of Power, SAGE Publications, pp. 603–606

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawShapleyShubikIndex(v) 

Draw Shapley value for 3 or 4 players

Description

drawShapleyValue draws the Shapley value for 3 or 4 players.

Usage

drawShapleyValue(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Shapley value"
)

Arguments

Value

None.

Author(s)

Alexandra Tiukkel

References

Shapley L.S. (1953) "A value for n-person games". In: Kuhn, H., Tucker, A.W. (Eds.), Contributions to the Theory of Games II, Princeton University Press, pp. 307–317

Aumann R.J. (2010) "Some non-superadditive games, and their Shapley values, in the Talmud", Int. Journal of Game Theory 39(1), pp. 3–10

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 156–159

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 748–781

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
drawShapleyValue(v) 

Draw simplified modiclus for 3 or 4 players

Description

drawSimplifiedModiclus draws the simplified modiclus for 3 or 4 players.

Usage

drawSimplifiedModiclus(
  v,
  holdOn = FALSE,
  colour = NA,
  label = TRUE,
  name = "Simplified Modiclus"
)

Arguments

Value

None.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Tarashnina S. (2011) "The simplified modified nucleolus of a cooperative TU-game", TOP 19(1), pp. 150–166

Examples

library(CoopGame)
v=c(0, 0, 0, 1, 1, 0, 1)
drawSimplifiedModiclus(v)

Draw tau-value for 3 or 4 players

Description

drawTauValue draws the tau-value for 3 or 4 players.

Usage

drawTauValue(v, holdOn = FALSE, colour = NA, label = TRUE, name = "Tau value")

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 32

Tijs S. (1981) "Bounds for the core of a game and the t-value", In: Moeschlin, O. and Pallaschke, D. (Eds.): Game Theory and Mathematical Economics, North-Holland, pp. 123–132

Stach I. (2011) "Tijs value", Encyclopedia of Power, SAGE Publications, pp. 667–670

Examples

library(CoopGame)
v <-c(1,2,3,60,60,60,142)
drawTauValue(v,colour="green")

Draw Weber Set for 3 or 4 players

Description

drawWeberset draws the Weber Set for 3 or 4 players.

Usage

drawWeberset(v, holdOn = FALSE, colour = NA, label = FALSE, name = "Weber Set")

Arguments

Value

None.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Weber R.J. (1988) "Probabilistic values for games". In: Roth A.E. (Ed.), The Shapley Value. Essays in in honor of Lloyd S. Shapley, Cambridge University Press, pp. 101–119

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 327–329

Examples

library(CoopGame)
v = c(0,1,2,3,4,5,6)
drawWeberset(v, colour ="yellow")

Compute equal propensity to disrupt

Description

equalPropensityToDisrupt calculates the equal propensity to disrupt for a TU game with n players and a specified coalition size k. See the original paper by Littlechild & Vaidya (1976) for the formula with general k and the paper by Staudacher & Anwander (2019) for the specific expression for k=1 and interpretations of the equal propensity to disrupt.

Usage

equalPropensityToDisrupt(v, k = 1)

Arguments

Value

the value for the equal propensity to disrupt

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Examples

library(CoopGame)
v=c(0,0,0,4,0,3,6)
equalPropensityToDisrupt(v, k=1)

Compute Gately point

Description

gatelyValue calculates the Gately point for a given TU game

Usage

gatelyValue(v)

Arguments

Value

Gately point of the TU game or NULL in case the Gately point is not defined

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gately D. (1974) "Sharing the Gains from Regional Cooperation. A Game Theoretic Application to Planning Investment in Electric Power", International Economic Review 15(1), pp. 195–208

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, pp. 455–456

Examples

library(CoopGame)
gatelyValue(c(0,0,0,1,1,1,3.5))


library(CoopGame)
v=c(0,0,0,4,0,3,6)
gatelyValue(v)

#Output (18/11,36/11,12/11):
#1.636364 3.272727 1.090909

#Example from original paper by Gately (1974)
library(CoopGame)
v=c(0,0,0,1170,770,210,1530)
gatelyValue(v)

#Output:
#827.7049 476.5574 225.7377 

Compute critical coalitions of a player for simple games

Description

getCriticalCoalitionsOfPlayer identifies all coalitions for one player in which that player is critical (within a simple game). These coalitions are characterized by the circumstance that without this player the other players generate no value (then also called a losing coalition) - therefore this player is also described as a critical player.

Usage

getCriticalCoalitionsOfPlayer(player, v)

Arguments

Value

A data frame containing all minimal winning coalitions for one special player

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Deegan J. and Packel E.W. (1978) "A new index of power for simple n-person games", Int. Journal of Game Theory 7(2), pp. 151–161

Examples

library(CoopGame)
getCriticalCoalitionsOfPlayer(2,v=c(0,0,0,0,0,1,1))


library(CoopGame)
v=c(0,1,0,1,0,1,1)

#Get coalitions where player 2 is critical:
getCriticalCoalitionsOfPlayer(2,v)
#Output are all coalitions where player 2 is involved.
#Observe that player 2 is dictator in this game.
#
#     V1 V2 V3 cVal bmRow
#  2  0  1  0    1     2
#  4  1  1  0    1     4
#  6  0  1  1    1     6
#  7  1  1  1    1     7

Compute dual game vector

Description

Computes the dual game for a given TU game with n players specified by a game vector.

Usage

getDualGameVector(v)

Arguments

Value

Numeric vector of length (2^n)-1 representing the dual game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 125

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 7

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 737

Examples

library(CoopGame)
v<-unanimityGameVector(4,c(1,2))
getDualGameVector(v)

getEmptyParamCheckResult for generating stucture according to parameter check results

Description

Returns a defined data structure which is intended to store an error code and a message after the check of function parameters was executed. In case parameter check was successfull the error code has the value '0' and the message is 'NULL'.

Usage

getEmptyParamCheckResult()

Value

list with 2 elements named errCode which contains an integer representing the error code ('0' if no error) and errMessage for the error message ('NULL' if no error)

Author(s)

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)

initParamCheck_example=function(numberOfPlayers){
 paramCheckResult=getEmptyParamCheckResult()
 if(numberOfPlayers!=3){
   paramCheckResult$errMessage="The number of players is not 3 as expected"
   paramCheckResult$errCode=1
 }
 return(paramCheckResult)
}

initParamCheck_example(3)
#Output:
#$errCode
#[1] 0
#$errMessage
#NULL

Compute excess coefficients

Description

getExcessCoefficients computes the excess coefficients for a specified TU game and an allocation x

Usage

getExcessCoefficients(v, x)

Arguments

Value

numeric vector containing the excess coefficients for every coalition

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 58

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 12

Examples

library(CoopGame)
getExcessCoefficients(c(0,0,0,60,48,30,72), c(24,24,24))

Compute gaining coalitions of a TU game

Description

The function getGainingCoalitions identifies all gaining coalitions. Coalition S is a gaining coalition if there holds: v(S) > 0

Usage

getGainingCoalitions(v)

Arguments

Value

A data frame containing all gaining coalitions.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
getGainingCoalitions(v=c(0,0,0,2,0,2,3))


library(CoopGame)
v <- c(1,2,3,4,0,0,11)
getGainingCoalitions(v)
# Output:
#    V1 V2 V3 cVal
# 1  1  0  0    1
# 2  0  1  0    2
# 3  0  0  1    3
# 4  1  1  0    4
# 7  1  1  1   11

Compute gap function coefficients

Description

getGapFunctionCoefficients computes the gap function coefficients for a specified TU game

Usage

getGapFunctionCoefficients(v)

Arguments

Value

numeric vector containing the gap function coefficients for every coalition

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 57

Examples

library(CoopGame)
getGapFunctionCoefficients(c(0,0,0,60,48,30,72))

Compute marginal contributions

Description

Calculates the marginal contributions for all permutations of the players

Usage

getMarginalContributions(v)

Arguments

Value

a list with given game vector, a matrix of combinations used and a matrix with the marginal contributions

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 156–159

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 6

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
getMarginalContributions(v)

Compute minimal rights vector

Description

Calculates the minimal rights vector.

Usage

getMinimalRights(v)

Arguments

Value

Vector of minimal rights of each player

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, pp. 20–21

Examples

library(CoopGame)
getMinimalRights(c(0,0,0,1,0,1,1))


library(CoopGame)
v1 <- c(0,0,0,60,60,60,72)
getMinimalRights(v1)
#[1] 48 48 48

library(CoopGame)
v2 <- c(2,4,5,18,14,9,24) 
getMinimalRights(v2)
#[1] 8 4 5

Compute minimal winning coalitions in a simple game

Description

The function getMinimumWinningCoalitions identifies all minimal winning coalitions of a specified simple game. These coalitions are characterized by the circumstance that if any player breaks away from them, then the coalition generates no value (then also called a losing coalition) - all players in the coalition can therefore be described as critical players.

Usage

getMinimumWinningCoalitions(v)

Arguments

Value

A data frame containing all minimum winning coalitions for a simple game.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Deegan J. and Packel E.W. (1978) "A new index of power for simple n-person games", Int. Journal of Game Theory 7(2), pp. 151–161

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 295

Bertini C. (2011) "Minimal winning coalition", Encyclopedia of Power, SAGE Publications, pp. 422–423

Examples

library(CoopGame)
getMinimumWinningCoalitions(v=c(0,0,0,0,0,0,1))


library(CoopGame)
v=weightedVotingGameVector(n=3,w=c(1,2,3),q=5)
getMinimumWinningCoalitions(v)
# Output:
#   V1 V2 V3 cVal
# 6  0  1  1    1
# => the coalition containing player 2 and 3 is a minimal winning coalition

Get number of players

Description

Gets the number of players from a game vector

Usage

getNumberOfPlayers(v)

Arguments

Value

Number of players in the game (specified by game vector v)

Author(s)

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
maschlerGame=c(0,0,0,60,60,60,72)
getNumberOfPlayers(maschlerGame)

Compute per capita excess coefficients

Description

getPerCapitaExcessCoefficients computes the per capita excess coefficients for a specified TU game and an allocation x

Usage

getPerCapitaExcessCoefficients(v, x)

Arguments

Value

numeric vector containing the per capita excess coefficients for every coalition

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
getPerCapitaExcessCoefficients(c(0,0,0,60,48,30,72), c(24,24,24))

Extract players from bit matrix row

Description

getPlayersFromBMRow determines players involved in a coalition from the row of a bit matrix

Usage

getPlayersFromBMRow(bmRow)

Arguments

Value

playerVector contains involved players (e.g. c(1,3), see example below for bitIndex=5 and n=3)

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
bm=createBitMatrix(n=3,A=c(0,0,0,1,1,1,2))
getPlayersFromBMRow(bmRow=bm[4,])


library(CoopGame)
bm=createBitMatrix(n=3,A=c(1:7))
#Corresponding bit matrix:
#           cVal
#[1,] 1 0 0    1
#[2,] 0 1 0    2
#[3,] 0 0 1    3
#[4,] 1 1 0    4
#[5,] 1 0 1    5 <=Specified bit index
#[6,] 0 1 1    6
#[7,] 1 1 1    7

#Determine players from bit matrix row by index 5
players=getPlayersFromBMRow(bmRow=bm[5,])
#Result:
players 
#[1] 1 3

Extract players from bit vector

Description

getPlayersFromBitVector determines players involved in a coalition from a binary vector.

Usage

getPlayersFromBitVector(bitVector)

Arguments

Value

playerVector contains the numbers of the players involved in the coalition

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
myBitVector <-c(1,0,1,0)
(players<-getPlayersFromBitVector(myBitVector))

Compute real gaining coalitions of game

Description

The function getRealGainingCoalitions identifies all real gaining coalitions. Coalition S is a real gaining coalition if for any true subset T of S there holds: v(S) > v(T)

Usage

getRealGainingCoalitions(v)

Arguments

Value

A data frame containing all real gaining coalitions.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Li X. (1995) "From public good index to public value. An axiomatic approach and generalization", Control and Cybernetics 24, pp. 257–270

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
getRealGainingCoalitions(v=c(0,0,0,0,0,0,2))


library(CoopGame)
v <- c(1,2,3,4,0,0,0)
getRealGainingCoalitions(v)
# Output:
#    V1 V2 V3 cVal
# 1  1  0  0    1
# 2  0  1  0    2
# 3  0  0  1    3
# 4  1  1  0    4

Compute unanimity coefficients of game

Description

getUnanimityCoefficients calculates to unanimity coefficients of a specified TU game. Note that the unanimity coefficients are also frequently referred to as Harsanyi dividends in the literature.

Usage

getUnanimityCoefficients(v)

Arguments

Value

numeric vector containing the unanimity coefficients

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 153

Gilles R. P. (2015) The Cooperative Game Theory of Networks and Hierarchies, Springer, pp. 15–17

Shapley L.S. (1953) "A value for n-person games". In: Kuhn, H., Tucker, A.W. (Eds.), Contributions to the Theory of Games II, Princeton University Press, pp. 307–317

Examples

library(CoopGame)
getUnanimityCoefficients(c(0,0,0,60,48,30,72))

Compute utopia payoff vector of game

Description

getUtopiaPayoff calculates the utopia payoff vector for each player in a TU game.
The utopia payoff of player i is the marginal contribution of player i to the grand coalition.

Usage

getUtopiaPayoff(v)

Arguments

Value

utopia payoffs for each player

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 31

Examples

library(CoopGame)
maschlerGame <- c(0,0,0,60,60,60,72)
getUtopiaPayoff(maschlerGame)

Compute vector of propensities to disrupt

Description

getVectorOfPropensitiesToDisrupt computes a vector of propensities to disrupt for game vector v and an allocation x

Usage

getVectorOfPropensitiesToDisrupt(v, x)

Arguments

Value

a numerical vector of propensities to disrupt at a given allocation x

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Examples

library(CoopGame)
v=c(0,0,0,4,0,3,6)
x=c(2,3,1)
getVectorOfPropensitiesToDisrupt(v,x)

Compute winning coalitions in a simple game

Description

The function getWinningCoalitions identifies all winning coalitions of a specified simple game.

Usage

getWinningCoalitions(v)

Arguments

Value

A data frame containing all winning coalitions for a simple game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C., Gambarelli G. and Stach I. (2008) "A public help index", In: Braham, M. and Steffen, F. (Eds): Power, freedom, and voting: Essays in Honour of Manfred J. Holler, pp. 83–98

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Stach I. (2016) "Power Measures and Public Goods", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXIII, Springer, pp. 99–110

Examples

library(CoopGame)
getWinningCoalitions(v=c(0,0,0,1,0,1,1))


library(CoopGame)
v=weightedVotingGameVector(n=3,w=c(1,2,3),q=5)
getWinningCoalitions(v)
# Output:
#   V1 V2 V3 cVal
# 6  0  1  1    1
# 7  1  1  1    1
# => the coalition containing player 2 and 3 and 
#    the grand coalition are winning coalitions

Compute 0-normalized game vector

Description

Computes the zero-normalized game for a given game specified by a game vector.

Usage

getZeroNormalizedGameVector(v)

Arguments

Value

Numeric vector of length (2^n)-1 representing the zero-normalized game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 9

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 11

Examples

library(CoopGame)
v<-c(1:7)
getZeroNormalizedGameVector(v)

Compute 0-1-normalized game vector

Description

Computes the zero-one-normalized game for a given game specified by a game vector.

Usage

getZeroOneNormalizedGameVector(v)

Arguments

Value

Numeric vector of length (2^n)-1 representing the zero-one-normalized game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Gilles R. P. (2015) The Cooperative Game Theory of Networks and Hierarchies, Springer, p. 18

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 670

Examples

library(CoopGame)
v<-c(1:7)
getZeroOneNormalizedGameVector(v)

Compute k-cover of game

Description

getkCover returns the k-cover for a given TU game according to the formula on p. 173 in the book by Driessen. Note that the k-cover does not exist if condition (7.2) on p. 173 in the book by Driessen is not satisfied.

Usage

getkCover(v, k)

Arguments

Value

numeric vector containing the k-cover of the given game if the k-cover exists, NULL otherwise

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 173

Examples

library(CoopGame)
getkCover(c(0,0,0,9,9,12,18),k=1)


library(CoopGame)
#Example from textbook by Driessen, p. 175, with alpha = 0.6 and k = 2
alpha = 0.6
getkCover(c(0,0,0,alpha,alpha,0,1), k=2)
#[1] 0.0 0.0 0.0 0.6 0.6 0.0 1.0

Construct a glove game

Description

Create a list containing all information about a specified glove game:
We have a set of players L with left-hand gloves and a set of players R with right-hand gloves. The worth of a coalition S equals the number of pairs of gloves the members of S can make. Note that the sets L and R have to be disjoint.

Usage

gloveGame(n, L, R)

Arguments

Value

A list with four elements representing the glove game (n, L, R, Game vector v)

Related Functions

gloveGameValue, gloveGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 155–156

Examples

library(CoopGame)
gloveGame(n=3,L=c(1,2), R=c(3))


#Example with four players: 
#players 1, 2 and 4 hold a left-hand glove each, 
#player 3 holds a right-hand glove. 
library(CoopGame)
(vv<-gloveGame(n=4,L=c(1,2,4), R=c(3)))
#$n
#[1] 3

#$L
#[1] 1 2 4
#
#$R
#[1] 3
#
#$v
#[1] 0 0 0 0 0 1 0 1 0 1 1 0 1 1 1

Compute value of a coalition for a glove game

Description

Coalition value for a specified glove game:
For further information see gloveGame

Usage

gloveGameValue(S, L, R)

Arguments

Value

Number of matched pairs of gloves for given coalition S

Author(s)

Alexandra Tiukkel

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 155–156

Examples

library(CoopGame)
gloveGameValue(S=c(1,2), L=c(1,2), R=c(3)) 

Compute game vector for glove game

Description

Game vector for glove game:
For further information see gloveGame

Usage

gloveGameVector(n, L, R)

Arguments

Value

Game vector of the specified glove game

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 155–156

Examples

library(CoopGame)
gloveGameVector(3, L=c(1,2), R=c(3))

Compute vertices of imputation set

Description

imputationsetVertices calculates the imputation set vertices for given game vector.

Usage

imputationsetVertices(v)

Arguments

Value

rows of the matrix are the vertices of the imputation set

Author(s)

Michael Maerz

Franz Mueller

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 20

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 674

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 278

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 407

Examples

library(CoopGame)
imputationsetVertices(c(0,0,0,1,1,1,2))


library(CoopGame)
v = c(2, 4, 5, 18, 24, 9, 24)

imputationsetVertices(v)

#      [,1] [,2] [,3]
#[1,]   15    4    5
#[2,]    2   17    5
#[3,]    2    4   18

Check if game is 1-Convex

Description

is1ConvexGame checks if a TU game is 1-convex. A TU game is 1-convex if and only if the following condition holds true: Let S be a nonempty coalition. Whenever all players outside S receive their payoffs according to the utopia payoff of the game, then the remaining part of the total savings is at least v(S).

Usage

is1ConvexGame(v)

Arguments

Value

TRUE if the game is 1-convex, else FALSE

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 73

Examples

library(CoopGame)
is1ConvexGame(c(0,0,0,9,9,12,18))


#1-convex game (taken from book by T. Driessen, p. 75)
library(CoopGame)
v=c(0,0,0,9,9,15,18)
is1ConvexGame(v)

#Example of a game which is not 1-convex 
library(CoopGame)
v=c(1:7)
is1ConvexGame(v)

Check if game is additive

Description

Checks if a TU game with n players is additive.
In an additive game for any two disjoint coalitions S and T the value of the union of S and T equals the sum of the values of S and T. In other words, additive games are constant-sum and the imputation set of an additive game consists of exactly one point.

Usage

isAdditiveGame(v)

Arguments

Value

TRUE if the game is additive, else FALSE

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 11

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 292

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, p. 261

Examples

library(CoopGame)
isAdditiveGame(c(1,1,1,2,2,2,3))


#The following game is not additive
library(CoopGame)
v=c(0,0,0,40,50,20,100)
isAdditiveGame(v) 

#The following game is additive
library(CoopGame)
v=c(1,1,1,1, 2,2,2,2,2,2, 3,3,3,3, 4)
isAdditiveGame(v)

Check if game is balanced

Description

Checks if a game is balanced. A game is balanced if the core is a nonempty set.

Usage

isBalancedGame(v)

Arguments

Value

TRUE if the game is balanced, else FALSE

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bondareva O.N. (1963) "Some applications of linear programming methods to the theory of cooperative games". Problemy kibernetiki 10, pp. 119–139

Shapley L.S. (1967) "On Balanced Sets and Cores". Naval Research Logistics Quarterly 14, pp. 453–460

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 27–32

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 691–702

Slikker M. and van den Nouweland A. (2001) Social and Economic Networks in Cooperative Game Theory, Springer, pp. 6–7

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 262–263

Examples

library(CoopGame)
v=c(0,0,0,40,50,20,100)
isBalancedGame(v)


#Example of an unbalanced game with 3 players
library(CoopGame)
v=c(1,1,1,2,3,4,3)
isBalancedGame(v)

#Example of an unbalanced game with 4 players
library(CoopGame)
v=c(0,0,0,0,1,0,0,0,0,3,3,3,3,3,4)
isBalancedGame(v)

#Example of a balanced game with 4 players
library(CoopGame)
v= c(0,0,0,0,1,0,0,0,0,2,2,2,2,2,4)
isBalancedGame(v)

Check if game is constant-sum

Description

Checks if a TU game with n players is constant-sum.
In a constant-sum game for any coalition S the sums of the values of the coalition S and its complement equal the value of the grand coalition N.

Usage

isConstantSumGame(v)

Arguments

Value

TRUE if the game is constant-sum, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 11

Examples

library(CoopGame)
v=c(0,0,0,2,2,2,2) 
isConstantSumGame(v)


#Example of a game that is not constant-sum 
library(CoopGame)
v=c(0,0,0,40,30,130,100) 
isConstantSumGame(v)

#Another example of a constant-sum game
library(CoopGame)
v=c(1,1,1,2, 2,2,2,2,2,2, 2,3,3,3, 4)
isConstantSumGame(v)

Check if game is convex

Description

isConvexGame checks if a TU game is convex. A TU game is convex if and only if each player's marginal contribution to any coalition is monotone nondecreasing with respect to set-theoretic inclusion.

Usage

isConvexGame(v)

Arguments

Value

TRUE if the game is convex, else FALSE

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 10

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 329

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 717–718

Osborne M.J. and Rubinstein A. (1994) A Course in Game Theory, MIT Press, pp. 260–261

Examples

library(CoopGame)
isConvexGame(c(0,0,0,1,1,1,5))


#Example of a convex game with three players
library(CoopGame) 
v=c(0,0,0,1,2,1,4)
isConvexGame(v)

#Example of a nonconvex game
library(CoopGame) 
v=c(1:7)
isConvexGame(v)

Check if game is degenerate

Description

Checks if a TU game is degenerate. We call a game essential if the value of the grand coalition is greater than the sum of the values of the singleton coalitions. We call a game degenerate (or inessential), if

v(N) = \sum v({i})

.

Usage

isDegenerateGame(v)

Arguments

Value

TRUE if the game is degenerate, else FALSE

Author(s)

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
isDegenerateGame(c(1,2,3,4,4,4,6))


#The following game, i.e. the Maschler game, is not degenerate
library(CoopGame)
v1 <- c(0,0,0,60,60,60,72)
isDegenerateGame(v1) 

#The following game is also not degenerate
library(CoopGame)
v2 <- c(30,30,15,60,60,60,72)
isDegenerateGame(v2)

#The following game is degenerate
library(CoopGame)
v3 <- c(20,20,32,60,60,60,72)
isDegenerateGame(v3)

Check if game is essential

Description

Checks if a TU game with n players is essential. We call a game essential, if the value of the grand coalition is greater than the sum of the values of the singleton coalitions. A game is essential, if

v(N) > \sum v({i})

.
For an essential game the imputation set is nonempty and consists of more than one point.

Usage

isEssentialGame(v)

Arguments

Value

TRUE if the game is essential, else FALSE.

Author(s)

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, p. 23

Gilles R. P. (2015) The Cooperative Game Theory of Networks and Hierarchies, Springer, p. 18

Examples

library(CoopGame)
isEssentialGame(c(1,2,3,4,4,4,7))


# Example of an essential game
library(CoopGame)
v1 <- c(0,0,0,60,60,60,72)
isEssentialGame(v1)

# Example of a game that is not essential  
library(CoopGame)
v2 <- c(30,30,15,60,60,60,72)
isEssentialGame(v2)

# Example of a game that is not essential 
library(CoopGame)
v3 <- c(20,20,32,60,60,60,72)
isEssentialGame(v3)

Check if game is monotonic

Description

Checks if a TU game with n players is monotonic.
For a monotonic game a coalition S can never obtain a larger value than another coalition T if S is contained in T.

Usage

isMonotonicGame(v)

Arguments

Value

TRUE if the game is monotonic, else FALSE

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 12

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 408

Examples

library(CoopGame)
isMonotonicGame(c(0,0,0,1,0,1,1))


#Example of a non-monotonic game
library(CoopGame)
v1=c(4,2,5,2,3,6,10)
isMonotonicGame(v1)

#Example of a monotonic game
library(CoopGame)
v2=c(2,5,7,10, 9, 13,20)
isMonotonicGame(v2)

Check if game is nonnegative

Description

isNonnegativeGame checks if a TU game is a nonnegative game. A TU game is a nonnegative game if the game vector does not contain any negative entries.

Usage

isNonnegativeGame(v)

Arguments

Value

TRUE if the game is nonnegative, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Examples

library(CoopGame)
isNonnegativeGame(c(0,0,0,0.5,0.1,0.4,1))


#Nonnegative game
library(CoopGame) 
v1<-c(0,0,0,0,1,1,1)
isNonnegativeGame(v1)

#Example for game which is not nonnegative 
library(CoopGame)
v2<-c(0,0,0,0,-1.1,1,2)
isNonnegativeGame(v2)

Check if game is quasi-balanced

Description

Checks if a TU game is quasi-balanced.
A TU game is quasi-balanced if
a) the components of its minimal rights vector are less or equal than the components of its utopia payoff vector
and
b) the sum of the components of its minimal rights vector is less or equal the value of the grand coalition which in turn is less or equal than the sum of the components of its utopia payoff vector.
Note that any balanced game is also quasi-balanced, but not vice versa.
Note that the quasi-balanced games are those games with a non-empty core cover. Note also that quasi-balancedness is sometimes in the literature also referred to as compromise-admissibility.

Usage

isQuasiBalancedGame(v)

Arguments

Value

TRUE if the game is quasi-balanced, else FALSE.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 31

Examples

library(CoopGame)
isQuasiBalancedGame(c(0,0,0,1,1,1,4))


#Example of a quasi-balanced game:
library(CoopGame)
v1=c(1,1,2,6,8,14,16)
isQuasiBalancedGame(v1)

#Example of a game which is not quasi-balanced:
library(CoopGame)
v2=c(1:7)
isQuasiBalancedGame(v2)

Check if game is semiconvex

Description

isSemiConvexGame checks if a TU game is semiconvex. A TU game is semiconvex if and only if the following conditions hold true: The gap function of any single player i is minimal among the gap function values of coalitions S containing player i. Also, the gap function itself is required to be nonnegative.

Usage

isSemiConvexGame(v)

Arguments

Value

TRUE if the game is semiconvex, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Driessen T. and Tijs S. (1985) "The tau-value, the core and semiconvex games", Int. Journal of Game Theory 14(4), pp. 229–247

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 76

Examples

library(CoopGame)
isSemiConvexGame(c(0,0,0,1,1,1,4))


#Example of a semiconvex game 
library(CoopGame)
v1<-c(3,4,5,9,10,11,18)
isSemiConvexGame(v1)

#Example of a game which not semiconvex 
library(CoopGame)
v2=c(1:7)
isSemiConvexGame(v2)

Check if game is simple

Description

isSimpleGame checks if a TU game is a simple game. A TU game is a simple game in the sense of the book by Peleg and Sudhoelter (2007), p. 16, if and only if the game is monotonic and the values of all coalitions are either 0 or 1.

Usage

isSimpleGame(v)

Arguments

Value

TRUE if the game is essential, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 16

Examples

library(CoopGame)
isSimpleGame(c(0,0,0,1,0,1,1))


#Example of a simple game
library(CoopGame) 
v1<-c(0,0,0,0,1,1,1)
isSimpleGame(v1)

#Example of a game which not simple 
library(CoopGame)
v2<-c(0,0,0,0,1,1,2)
isSimpleGame(v2)

#Another example of a game which not simple 
#according to our definition
library(CoopGame) 
v3<-c(1,0,0,0,1,1,1)
isSimpleGame(v3)

Check if game is superadditive

Description

Checks if a TU game with n players is superadditive.
In a superadditive game for any two disjoint coalitions S and T the value of the union of S and T is always greater or equal the sum of the values of S and T. In other words, the members of any two disjoint coalitions S and T will never be discouraged from collaborating.

Usage

isSuperadditiveGame(v)

Arguments

Value

TRUE if the game is superadditive, else FALSE.

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 10

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, p. 295

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 671

Narahari Y. (2015) Game Theory and Mechanism Design, World Scientific Publishing, p. 408

Examples

library(CoopGame)
isSuperadditiveGame(c(0,0,0,1,1,1,2))


#Example of a superadditive game
library(CoopGame)
v1=c(0,0,0,40,50,20,100) 
isSuperadditiveGame(v1)

#Example of a game that is not superadditive 
library(CoopGame)
v2=c(0,0,0,40,30,130,100) 
isSuperadditiveGame(v2)

#Another example of a superadditive game
library(CoopGame)
v3=c(1,1,1,1, 2,2,2,2,2,2, 3,3,3,3, 4)
isSuperadditiveGame(v3)

Check if game is symmetric

Description

isSymmetricGame checks if a TU game is symmetric. A TU game is symmetric if and only if the values of all coalitions containing the same number of players are identical.

Usage

isSymmetricGame(v)

Arguments

Value

TRUE if the game is symmetric, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 12

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, p. 26

Examples

library(CoopGame)
isSymmetricGame(c(0,0,0,1,1,1,2))


#Example of a symmetric game
library(CoopGame) 
v1<-c(3,3,3,10,10,10,17)
isSymmetricGame(v1)

#Example of a game which is not symmetric 
library(CoopGame) 
v2=c(1:7)
isSymmetricGame(v2)

Check if game is weakly constant-sum

Description

Checks if a TU game with n players is weakly constant-sum.
In a weakly constant-sum game for any singleton coalition the sums of the values of that singleton coalition and its complement equal the value of the grand coalition N.

Usage

isWeaklyConstantSumGame(v)

Arguments

Value

TRUE if the game is weakly constant-sum, else FALSE.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Examples

library(CoopGame)
v1=c(0,0,0,2,2,2,2) 
isWeaklyConstantSumGame(v1)


#Example of a game that is not weakly constant-sum 
library(CoopGame)
v2=c(0,0,0,40,30,130,100) 
isWeaklyConstantSumGame(v2)

#Another example of a weakly constant-sum game
library(CoopGame)
v3=c(1,1,1,2, 7,7,7,7,7,7, 2,3,3,3, 4)
isWeaklyConstantSumGame(v3)

Check if game is weakly superadditive

Description

Checks if a TU game with n players is weakly superadditive.
Let S be a coalition and i a player not contained in S. Then the TU game is weakly superadditive if for any S and any i the value of the union of S and i is greater or equal the sum of the values of S and i.
Note that weak superadditivity is equivalent to zero-monotonicity.

Usage

isWeaklySuperadditiveGame(v)

Arguments

Value

TRUE if the game is weakly superadditive, else FALSE.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 10

Examples

library(CoopGame)
isWeaklySuperadditiveGame(c(0,0,0,1,1,1,1))


#Example of a weakly superadditive game
library(CoopGame)
v1=c(1:15)
isWeaklySuperadditiveGame(v1)

#Example of a game which is not weakly superadditive
library(CoopGame)
v2=c(1:5,7,7)
isWeaklySuperadditiveGame(v2)

Check if game is k-Convex

Description

iskConvexGame checks if a TU game is k-convex. A TU game is k-convex if and only if its k-cover exists and is convex. See section 7.1 of the book by Driessen for more details

Usage

iskConvexGame(v, k)

Arguments

Value

TRUE if the game is k-convex, else FALSE

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Driessen T. (1998) Cooperative Games, Solutions and Applications, Springer, p. 171–178

Examples

library(CoopGame)
iskConvexGame(v=c(0,0,0,9,9,12,18), k=1)


# Two examples motivated by the book by T. Driessen, p. 175:
#
# The following game is 2-convex 
library(CoopGame)
alpha = 0.4
v=c(0,0,0,alpha,alpha,0,1)
iskConvexGame(v,2)

# The following game is not 2-convex 
library(CoopGame)
alpha = 0.7
v=c(0,0,0,alpha,alpha,0,1)
iskConvexGame(v,2)

Compute Johnston index

Description

johnstonIndex calculates the Johnston index for a simple game.

Usage

johnstonIndex(v)

Arguments

Value

Johnston index for a specified simple game

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Johnston R.J. (1978) "On the measurement of power: Some reactions to Laver", Environment and Planning A, pp. 907–914

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, p. 124

Examples

library(CoopGame)
johnstonIndex(c(0,0,0,1,0,0,1))


#player 1 has 3 votes
#player 2 has 2 votes
#player 3 has 1 vote
#majority for the decision is 4 (quota)

library(CoopGame)
#function call generating the game vector:
v <- weightedVotingGameVector(n = 3, w = c(3,2,1), q = 4)

johnstonIndex(v)
#[1] 0.6666667 0.1666667 0.1666667

Compute Koenig-Braeuninger index

Description

Calculates the Koenig-Braeuninger index for a specified simple TU game. Note that in general the Koenig-Braeuninger index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Koenig-Braeuninger index is provided.

Usage

koenigBraeuningerIndex(v)

Arguments

Value

Koenig-Braeuninger index for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Koenig T. and Braeuninger T. (1998) "The inclusiveness of European decision rules", Journal of Theoretical Politics 10(1), pp. 125–142

Nevison C.H., Zicht, B. and Schoepke S. (1978) "A naive approach to the Banzhaf index of power", Behavioral Science 23(2), pp. 130–131

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
koenigBraeuningerIndex(v) 

Construct a weighted majority game with a single veto player

Description

Create a list containing all information about a specified weighted majority game with a single veto player:
If coalition S has at least 2 members and if the veto player is part of the coalition it generates a value of 1, otherwise 0.
Note that weighted majority games with a single veto player are always simple games.

Usage

majoritySingleVetoGame(n, vetoPlayer)

Arguments

Value

A list with three elements representing the specified weighted majority game with a single veto player (n, vetoPlayer, Game vector v)

Related Functions

majoritySingleVetoGameValue, majoritySingleVetoGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 415

Examples

library(CoopGame) 
majoritySingleVetoGame(n=3, vetoPlayer=1)

Compute value of a coalition for a weighted majority game with a single veto player

Description

Coalition value for a weighted majority game with a single veto player:
For further information see majoritySingleVetoGame

Usage

majoritySingleVetoGameValue(S, vetoPlayer)

Arguments

Value

1 if vetoPlayer is included in S and S is not a singleton coalition, 0 otherwise

Author(s)

Michael Maerz

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 415

Examples

library(CoopGame)
majoritySingleVetoGameValue(S=c(1,2), vetoPlayer=1)

Compute game vector for a weighted majority game with a single veto player

Description

Game vector for a weighted majority game with a single veto player:
For further information see majoritySingleVetoGame

Usage

majoritySingleVetoGameVector(n, vetoPlayer)

Arguments

Value

Game Vector where each elements contains 1 if vetoPlayer is included in S and S is not a singleton coalition, 0 otherwise

Author(s)

Michael Maerz

References

Jackson M.O. (2008) Social and Economic Networks, Princeton University Press, p. 415

Examples

library(CoopGame)
majoritySingleVetoGameVector(n=3, vetoPlayer=1)

Compute modiclus

Description

Calculates the modiclus of a TU game with a non-empty imputation set and n players. Note that the modiclus is also know as the modified nucleolus in the literature. Due to complexity of modiclus computation we recommend to use this function for at most n=11 players. Note that the modiclus is a member of the set of preimputations.

Usage

modiclus(v)

Arguments

Value

Numeric vector of length n representing the modiclus (aka modified nucleolus) of the specified TU game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 124–132

Sudhoelter P. (1997) "The Modified Nucleolus. Properties and Axiomatizations", Int. Journal of Game Theory 26(2), pp. 147–182

Sudhoelter P. (1996) "The Modified Nucleolus as Canonical Representation of Weighted Majority Games", Mathematics of Operations Research 21(3), pp. 734–756

Examples

library(CoopGame)
modiclus(c(1, 1, 1, 2, 3, 4, 5))


library(CoopGame)
modiclus(c(0, 0, 0, 0, 5, 5, 8, 9, 10, 8, 13, 15, 16, 17, 21))
#[1] 4.25 5.25 5.75 5.75

Compute Nevison index

Description

Calculates the Nevison index for a specified simple TU game. Note that in general the Nevison index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Nevison index is provided.

Usage

nevisonIndex(v)

Arguments

Value

Nevison index for a specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Nevison, H. (1979) "Structural power and satisfaction in simple games", In: Applied Game Theory, Springer, pp. 39–57

Examples

library(CoopGame)
v=c(0,0,0,1,1,0,1)
nevisonIndex(v) 

Compute non-normalized Banzhaf index

Description

non-normalized Banzhaf index for a specified simple game, see formula (7.5) on p. 119 of the book by Chakravarty, Mitra and Sarkar

Usage

nonNormalizedBanzhafIndex(v)

Arguments

Value

The return value is a vector which contains the non-normalized Banzhaf index for each player.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Bertini C. and Stach I. (2011) "Banzhaf voting power measure", Encyclopedia of Power, SAGE Publications, pp. 54–55

Examples

library(CoopGame)
nonNormalizedBanzhafIndex(dictatorGameVector(n=3, dictator=1))


library(CoopGame)
v<-weightedVotingGameVector(n=4,w=c(8,6,4,2),q=c(12))
nonNormalizedBanzhafIndex(v)
#[1] 0.625 0.375 0.375 0.125

library(CoopGame)
v<- apexGameVector(n = 4,apexPlayer=3)
nonNormalizedBanzhafIndex(v)
#[1] 0.25 0.25 0.75 0.25

library(CoopGame)
#N=c(1,2,3), w=(50,49,1), q=51   
v=weightedVotingGameVector(n=3, w=c(50,49,1),q=51)
nonNormalizedBanzhafIndex(v)
#[1] 0.75 0.25 0.25

library(CoopGame) 
v<-weightedVotingGameVector(n=3,w=c(50,30,20),q=c(67))
nonNormalizedBanzhafIndex(v)
#[1] 0.75 0.25 0.25

Compute normalized Banzhaf index

Description

Normalized Banzhaf index for a specified simple game, see formula (7.6) on p. 119 of the book by Chakravarty, Mitra and Sarkar

Usage

normalizedBanzhafIndex(v)

Arguments

Value

The return value is a numeric vector which contains the normalized Banzhaf index for each player.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 367–370

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Bertini C. and Stach I. (2011) "Banzhaf voting power measure", Encyclopedia of Power, SAGE Publications, pp. 54–55

Examples

library(CoopGame)
normalizedBanzhafIndex(dictatorGameVector(n=3, dictator=1))


library(CoopGame)
v<-weightedVotingGameVector(n=4,w=c(8,6,4,2),q=c(12))
normalizedBanzhafIndex(v)
#[1] 0.41666667 0.25000000 0.25000000 0.08333333

library(CoopGame) 
v<- apexGameVector(n = 4,apexPlayer=3)
normalizedBanzhafIndex(v)
#[1] 0.1666667 0.1666667 0.5000000 0.1666667

library(CoopGame)
#N=c(1,2,3), w=(50,49,1), q=51   
v=weightedVotingGameVector(n=3, w=c(50,49,1),q=51)
normalizedBanzhafIndex(v)
#[1] 0.6 0.2 0.2

library(CoopGame)
v<-weightedVotingGameVector(n=3,w=c(50,30,20),q=c(67))
normalizedBanzhafIndex(v)
#[1] 0.6 0.2 0.2

Compute normalized Banzhaf value

Description

normalizedBanzhafValue computes the normalized Banzhaf value for a specified TU game. The corresponding formula can e.g. be found in the article by Stach (2017), p. 77.

Usage

normalizedBanzhafValue(v)

Arguments

Value

The return value is a numeric vector which contains the normalized Banzhaf value for each player.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Gambarelli G. (2011) "Banzhaf value", Encyclopedia of Power, SAGE Publications, pp. 53–54

Stach I. (2017) "Sub-Coalitional Approach to Values", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXVI, Springer, pp. 74–86

Examples

library(CoopGame)
normalizedBanzhafValue(c(0,0,0,1,2,3,6))


#Example from paper by Gambarelli (2011)
library(CoopGame)
v=c(0,0,0,1,2,1,3)
normalizedBanzhafValue(v)
#[1] 1.1538462 0.6923077 1.1538462
#Expected Result: 15/13  9/13  15/13

Compute nucleolus

Description

Computes the nucleolus of a TU game with a non-empty imputation set and n players. Note that the nucleolus is a member of the imputation set.

Usage

nucleolus(v)

Arguments

Value

Numeric vector of length n representing the nucleolus.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

Daniel Gebele daniel.a.gebele@stud.hs-kempten.de

References

Schmeidler D. (1969) "The nucleolus of a characteristic function game", SIAM Journal on applied mathematics 17(6), pp. 1163–1170

Kohlberg E. (1971) "On the nucleolus of a characteristic function game", SIAM Journal on applied mathematics 20(1), pp. 62–66

Kopelowitz A. (1967) "Computation of the kernels of simple games and the nucleolus of n-person games", Technical Report, Department of Mathematics, The Hebrew University of Jerusalem, 45 pages.

Megiddo N. (1974) "On the nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game", SIAM Journal on applied mathematics 27(2), pp. 355–358

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, pp. 82–86

Examples

library(CoopGame)
nucleolus(c(1, 1, 1, 2, 3, 4, 5))


library(CoopGame)
nucleolus(c(0, 0, 0, 0, 5, 5, 8, 9, 10, 8, 13, 15, 16, 17, 21))
#[1] 3.5 4.5 5.5 7.5

#Final example:
#Estate division problem from Babylonian Talmud with E=300,
#see e.g. seminal paper by Aumann & Maschler from 1985 on
#'Game Theoretic Analysis of a Bankruptcy Problem from the Talmud'
library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=300)
nucleolus(v)
#[1]  50 100 150

Compute per capita nucleolus

Description

perCapitaNucleolus calculates the per capita nucleolus for a TU game with a non-empty imputation set specified by a game vector. Note that the per capita nucleolus is a member of the imputation set.

Usage

perCapitaNucleolus(v)

Arguments

Value

per capita nucleolus for a specified TU game with n players

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Young H.P. (1985) "Monotonic Solutions of cooperative games", Int. Journal of Game Theory 14(2), pp. 65–72

Examples

library(CoopGame)
perCapitaNucleolus(c(1, 1, 1, 2, 3, 4, 5))


#Example from YOUNG 1985, p. 68
v<-costSharingGameVector(n=3,C=c(15,20,55,35,61,65,78))
perCapitaNucleolus(v)
#[1]  0.6666667  1.1666667 10.1666667

Compute propensity to disrupt

Description

propensityToDisrupt for calculating the propensity of disrupt for game vector v, an allocation x and a specified coalition S

Usage

propensityToDisrupt(v, x, S)

Arguments

Value

propensity to disrupt as numerical value

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Littlechild S.C. and Vaidya K.G. (1976) "The propensity to disrupt and the disruption nucleolus of a characteristic function game", Int. Journal of Game Theory 5(2), pp. 151–161

Staudacher J. and Anwander J. (2019) "Conditions for the uniqueness of the Gately point for cooperative games", arXiv preprint, arXiv:1901.01485, 10 pages.

Examples

library(CoopGame)
v=c(0,0,0,4,0,3,6)
x=c(2,3,1)
propensityToDisrupt(v,x,S=c(1))

Compute proportional nucleolus

Description

proportionalNucleolus calculates the proportional nucleolus for a TU game with a non-empty imputation set and n players specified by game vector. Note that the proportional nucleolus is a member of the imputation set.

Usage

proportionalNucleolus(v)

Arguments

Value

proportional nucleolus for specified TU game with n players

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Young H. P., Okada N. and Hashimoto, T. (1982) "Cost allocation in water resources development", Water resources research 18(3), pp. 463–475

Examples

library(CoopGame)
v<-c(0,0,0,48,60,72,140)
proportionalNucleolus(v)

Compute Public Good index

Description

Calculates the Public Good index (aka Holler index) for a specified simple game.

Usage

publicGoodIndex(v)

Arguments

Value

The return value is a vector containing the Public Good index

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Packel E.W. (1983) "Power, luck and the right index", Zeitschrift fuer Nationaloekonomie 43(1), pp. 21–29

Holler M.J. (1982) "Forming coalitions and measuring voting power", Political Studies 30(2), pp. 262–271

Holler M. (2011) "Public Goods index", Encyclopedia of Power, SAGE Publications, pp. 541–542

Examples

library(CoopGame)
publicGoodIndex(v=c(0,0,0,1,1,0,1))


#Example from Holler (2011) illustrating paradox of weighted voting
library(CoopGame)
v=weightedVotingGameVector(n=5,w=c(35,20,15,15,15), q=51)
publicGoodIndex(v) 
#[1] 0.2666667 0.1333333 0.2000000 0.2000000 0.2000000

Compute (normalized) Public Good value

Description

Calculates the (normalized) Public Good value for a specified nonnegative TU game. Note that the normalized Public Good value is sometimes also referred to as Holler value in the literature. Our function implements the formula from Definition 5.4, p. 19, in the paper by Bertini and Stach from 2015.

Usage

publicGoodValue(v)

Arguments

Value

Public Good value for specified nonnegative TU game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Holler M.J. and Li X. (1995) "From public good index to public value. An axiomatic approach and generalization", Control and Cybernetics 24, pp. 257–270

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,0.7,11,0,15)
publicGoodValue(v)

Compute Public Help index Chi

Description

Calculates the Public Help index Chi for a specified simple TU game. Note that the greek letter Xi (instead of Chi) was used in the original paper by Bertini and Stach (2015).

Usage

publicHelpChiIndex(v)

Arguments

Value

Public Help index Chi for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Stach I. (2016) "Power Measures and Public Goods", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXIII, Springer, pp. 99–110

Examples

library(CoopGame)
publicHelpChiIndex(v=c(0,0,0,0,1,0,1))


#Example from original paper by Stach (2016), p. 105:
library(CoopGame)
v=c(0,0,0,1,1,0,1)
publicHelpChiIndex(v) 
#result: 0.4583333 0.2708333 0.2708333

#Second example from original paper by Stach (2016), p. 105:
library(CoopGame)
v=c(0,0,0,0,1,1,0,0,0,0,1,1,1,0,1)
publicHelpChiIndex(v)
#result: 0.3981481 0.2376543 0.2376543 0.1265432

Compute (normalized) Public Help value Chi

Description

Calculates the (normalized) Public Help value Chi by Bertini & Stach (2015) for a nonnegative TU game. Note that the greek letter Xi (instead of Chi) was used in the original paper by Bertini and Stach (2015).

Usage

publicHelpChiValue(v)

Arguments

Value

Public Help value Chi for specified nonnegative TU game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,2,2,0,2)
publicHelpChiValue(v) 

Compute Public Help index Theta

Description

Calculates the Public Help index Theta for a specified simple TU game. Note that the Public Help index Theta goes back to the paper by Bertini, Gambarelli and Stach (2008) and is frequently simply referred to referred to Public Help index in the literature.

Usage

publicHelpIndex(v)

Arguments

Value

Public Help index Theta for specified simple game

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C., Gambarelli G. and Stach I. (2008) "A public help index", In: Braham, M. and Steffen, F. (Eds): Power, freedom, and voting: Essays in Honour of Manfred J. Holler, pp. 83–98

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Stach I. (2016) "Power Measures and Public Goods", In: Nguyen, N.T. and Kowalczyk, R. (Eds.): Transactions on Computational Collective Intelligence XXIII, Springer, pp. 99–110

Examples

library(CoopGame)
publicHelpIndex(v=c(0,0,0,0,1,0,1))


#Example from paper by Stach (2016), p. 105:
library(CoopGame)
v=c(0,0,0,1,1,0,1)
publicHelpIndex(v) 
#result: 0.4285714 0.2857143 0.2857143

#Second example from paper by Stach (2016), p. 105:
library(CoopGame)
v=c(0,0,0,0,1,1,0,0,0,0,1,1,1,0,1)
publicHelpIndex(v)
#result: 0.3529412 0.2352941 0.2352941 0.1764706

Compute Public Help value Theta

Description

publicHelpValue calculates the (normalized) Public Help value Theta for a specified nonnegative TU game. Our function implements the formula from Definition 5.7, p. 20, in the paper by Bertini and Stach from 2015.

Usage

publicHelpValue(v)

Arguments

Value

Public Help value Theta for specified nonnegative TU game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Bertini C. and Stach I. (2015) "On Public Values and Power Indices", Decision Making in Manufacturing and Services 9(1), pp. 9–25

Examples

library(CoopGame)
v=c(0,0,0,0.7,11,0,15)
publicHelpValue(v) 

Compute Rae index

Description

raeIndex calculates the Rae index for a specified simple TU game. Note that in general the Rae index is not an efficient vector, i.e. the sum of its entries is not always 1. Hence no drawing routine for the Rae index is provided.

Usage

raeIndex(v)

Arguments

Value

Rae index for specified simple game

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Rae, D.W. (1969) "Decision-rules and individual values in constitutional choice", American Political Science Review 63(1), pp. 40–56

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 119–120

Examples

library(CoopGame) 
v=c(0,0,0,1,1,0,1)
raeIndex(v) 


library(CoopGame)
v=c(0,0,0,0,1,1,0,0,0,0,1,1,1,0,1)
raeIndex(v)
#result: [1] 0.875 0.625 0.625 0.500

Compute raw Banzhaf Index

Description

Raw Banzhaf Index for a specified simple game, see formula (7.4) on p. 118 of the book by Chakravarty, Mitra and Sarkar

Usage

rawBanzhafIndex(v)

Arguments

Value

The return value is a numeric vector which contains the raw Banzhaf index for each player.

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Examples

library(CoopGame)
rawBanzhafIndex(apexGameVector(n=3, apexPlayer=1))


v<- apexGameVector(n = 4,apexPlayer=3)
rawBanzhafIndex(v)
#[1] 2 2 6 2

#N=c(1,2,3), w=(50,49,1), q=51   
v=weightedVotingGameVector(n=3, w=c(50,49,1),q=51)
rawBanzhafIndex(v)
#[1] 3 1 1

v<-weightedVotingGameVector(n=3,w=c(50,30,20),q=c(67))
rawBanzhafIndex(v)
#[1] 3 1 1

Compute raw Banzhaf Value

Description

raw Banzhaf Value, i.e. the Banzhaf Value without the division by the scaling factor 2^{(n-1)}

Usage

rawBanzhafValue(v)

Arguments

Value

The return value is a numeric vector which contains the raw Banzhaf value for each player.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 118–119

Examples

library(CoopGame)
v = c(0,0,0,1,1,2,5)
rawBanzhafValue(v)


library(CoopGame)
v = c(0,0,0,2,2,3,5)
rawBanzhafValue(v)
#[1] 6 8 8

Compute vertices of reasonable set

Description

Calculates the vertices of the reasonable set for given game vector.

Usage

reasonableSetVertices(v)

Arguments

Value

rows of the matrix are the vertices of the reasonable set

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Milnor J.W. (1953) Reasonable Outcomes for N-person Games, Rand Corporation, Research Memorandum RM 916.

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 21

Chakravarty S.R., Mitra M. and Sarkar P. (2015) A Course on Cooperative Game Theory, Cambridge University Press, pp. 43–44

Gerard-Varet L.A. and Zamir S. (1987) "Remarks on the reasonable set of outcomes in a general coalition function form game", Int. Journal of Game Theory 16(2), pp. 123–143

Examples

library(CoopGame)
reasonableSetVertices(c(0,0,0,1,1,1,2))


library(CoopGame)
v <- c(0,0,0,3,3,3,6)
reasonableSetVertices(v)
#       [,1] [,2] [,3]
# [1,]    3    0    3
# [2,]    0    3    3
# [3,]    3    3    0

Compute Shapley-Shubik index

Description

Calculates the Shapley-Shubik index for a specified simple game with n players. Note that no separate drawing routine for the Shapley-Shubik index is provide as users can always resort to drawShapleyValue

Usage

shapleyShubikIndex(v)

Arguments

Value

Shapley-Shubik index for given simple game

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Shapley L.S. and Shubik M. (1954) "A method for evaluating the distribution of power in a committee system". American political science review 48(3), pp. 787–792

Shapley L.S. (1953) "A value for n-person games". In: Kuhn, H., Tucker, A.W. (Eds.), Contributions to the Theory of Games II, Princeton University Press, pp. 307–317

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 156–159

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 748–781

Stach I. (2011) "Shapley-Shubik index", Encyclopedia of Power, SAGE Publications, pp. 603–606

Examples

library(CoopGame)
shapleyShubikIndex(v=c(0,0,0,0,1,0,1))


#Example from Stach (2011):
library(CoopGame)
v=weightedVotingGameVector(n=4,q=50,w=c(10,10,20,30))
shapleyShubikIndex(v)
#[1] 0.08333333 0.08333333 0.25000000 0.58333333

Compute Shapley value

Description

Calculates the Shapley value for n players with formula from Lloyd Shapley.

Usage

shapleyValue(v)

Arguments

Value

Shapley value for given game vector with n players

Author(s)

Alexandra Tiukkel

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Shapley L.S. (1953) "A value for n-person games". In: Kuhn, H., Tucker, A.W. (Eds.), Contributions to the Theory of Games II, Princeton University Press, pp. 307–317

Aumann R.J. (2010) "Some non-superadditive games, and their Shapley values, in the Talmud", Int. Journal of Game Theory 39(1), pp. 3–10

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 156–159

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 748–781

Bertini C. (2011) "Shapley value", Encyclopedia of Power, SAGE Publications, p. 600–603

Examples

library(CoopGame)
shapleyValue(v=c(0,0,0,1,2,3,7.5))


#Example of a non-superadditive game,
#i.e. the inheritance problem due to Ibn Ezra (1146),
#from paper by Robert Aumann from 2010 on
#'Some non-superadditive games, and their Shapley values, in the Talmud'
library(CoopGame)
Aumann2010Example<-c(120,60,40,30,120,120,120,60,60,40,120,120,120,60,120)
shapleyValue(Aumann2010Example)
#[1] 80.83333 20.83333 10.83333  7.50000

Compute simplified modiclus

Description

Computes the simplified modiclus of a TU game with a non-empty imputation set and n players. Note that the simplified modiclus is a member of the set of preimputations.

Usage

simplifiedModiclus(v)

Arguments

Value

Numeric vector of length n representing the simplified modiclus of the specified TU game.

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Tarashnina S. (2011) "The simplified modified nucleolus of a cooperative TU-game", TOP 19(1), pp. 150–166

Examples

library(CoopGame)
simplifiedModiclus(c(0, 0, 0, 1, 1, 0, 1))


#Second example:
#Estate division problem from Babylonian Talmud with E=100,
#see e.g. seminal paper by Aumann & Maschler from 1985 on
#'Game Theoretic Analysis of a Bankruptcy Problem from the Talmud'
library(CoopGame)
v<-bankruptcyGameVector(n=3,d=c(100,200,300),E=100)
simplifiedModiclus(v)
#[1]  33.33333 33.33333 33.33333

Parameter Function stopOnInconsistentEstateAndClaimsVector

Description

stopOnInconsistentEstateAndClaimsVector checks if sum of claims is greater or equal estate (in bankruptcy games). Calculation stops with an error message if claims vector and estate are inconsistent.

Usage

stopOnInconsistentEstateAndClaimsVector(paramCheckResult, E, d)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
consistentClaims= c(26,27,55,57)
consistentE = 110
stopOnInconsistentEstateAndClaimsVector(paramCheckResult, d=consistentClaims, E=consistentE)

Parameter Function stopOnInvalidAllocation

Description

stopOnInvalidAllocation checks if allocation is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidAllocation(paramCheckResult, x, n = NULL, v = NULL)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validAllocation=c(1,2,3)
stopOnInvalidAllocation(paramCheckResult,x=validAllocation,n=3)

Parameter Function stopOnInvalidBoolean

Description

stopOnInvalidBoolean checks definition is the parameter a boolean

Usage

stopOnInvalidBoolean(paramCheckResult, boolean)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Franz Mueller

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validBoolean = TRUE
stopOnInvalidBoolean(paramCheckResult, validBoolean)

Parameter Function stopOnInvalidClaimsVector

Description

stopOnInvalidClaimsVector checks if claims vector in a bankruptcy game is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes stop otherwise.

Usage

stopOnInvalidClaimsVector(paramCheckResult, n, d)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validClaimsVector = c(100,150,200)
stopOnInvalidClaimsVector(paramCheckResult, n=3, d=validClaimsVector)

Parameter Function stopOnInvalidCoalitionS

Description

stopOnInvalidCoalitionS checks if coalition S as subset of grand coalition N is specified correctly and causes calculation to stop otherwise.

Usage

stopOnInvalidCoalitionS(paramCheckResult, S, N = NULL, n = NULL, v = NULL)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validCoalition = c(1,2,3)
stopOnInvalidCoalitionS(paramCheckResult, S=validCoalition, N=c(1,2,3,4,5))

Parameter Function stopOnInvalidDictator

Description

stopOnInvalidDictator checks if dictator is specified correctly in a dictator game. Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidDictator(paramCheckResult, dictator, n = NULL)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validDictator = 3
stopOnInvalidDictator(paramCheckResult,dictator=validDictator,n=3)

Parameter Function stopOnInvalidEstate

Description

stopOnInvalidBankruptcy checks if estate is specified correctly (as parameter in a bankruptcy game). Validation result gets stored to object paramCheckResult in case an error occured and causes stop otherwise.

Usage

stopOnInvalidEstate(paramCheckResult, E)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validEstate = 55
stopOnInvalidEstate(paramCheckResult, E=validEstate)

Parameter Function stopOnInvalidGameVector

Description

stopOnInvalidGameVector checks if game vector v is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidGameVector(paramCheckResult, v, n = NULL)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
validGameVector=c(0,0,0,60,60,60,72)
stopOnInvalidGameVector(paramCheckResult,validGameVector)

Parameter Function stopOnInvalidGrandCoalitionN

Description

stopOnInvalidGrandCoalitionN checks if grand coalition N is specified correctly and causes calculation to stop otherwise.

Usage

stopOnInvalidGrandCoalitionN(paramCheckResult, N)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validGrandCoalition = c(1,2,3,4,5)
stopOnInvalidGrandCoalitionN(paramCheckResult, N=validGrandCoalition)

Parameter Function stopOnInvalidIndex

Description

stopOnInvalidIndex checks if coalition function (in the form of either v or A) is specified correctly and causes causes calculation to stop otherwise.

Usage

stopOnInvalidIndex(paramCheckResult, index, n = NULL)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
v=c(1:7)
paramCheckResult=getEmptyParamCheckResult()
validIndex = 5
stopOnInvalidIndex(paramCheckResult, index=validIndex, n=3)

Parameter Function stopOnInvalidLeftRightGloveGame

Description

stopOnInvalidLeftRightGloveGame checks if L (left gloves) and R (right gloves) are specified as parameter correctly (also regarding grand coalition). Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidLeftRightGloveGame(paramCheckResult, L, R, N)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validL=c(1,3)
validR=c(2)
stopOnInvalidLeftRightGloveGame(paramCheckResult, L=validL,R=validR,N=c(1,2,3))

Parameter Function stopOnInvalidNChooseB

Description

stopOnInvalidNChooseB checks if definition of n choose b is specified correctly and causes stop otherwise.

Usage

stopOnInvalidNChooseB(paramCheckResult, n, b)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validN = 3
validAndConsistentB = 2
stopOnInvalidNChooseB(paramCheckResult, n=validN, b=validAndConsistentB)

Parameter Function stopOnInvalidNumber

Description

stopOnInvalidNumber checks definition is the parameter a number

Usage

stopOnInvalidNumber(paramCheckResult, number)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Franz Mueller

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validNumber = 5
stopOnInvalidNumber(paramCheckResult, validNumber)

Parameter Function stopOnInvalidNumberOfPlayers

Description

stopOnInvalidNumberOfPlayers checks if number of players is specified correctly and causes calculation to stop otherwise.

Usage

stopOnInvalidNumberOfPlayers(paramCheckResult, n)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validNumberOfPlayers = 10
stopOnInvalidNumberOfPlayers(paramCheckResult, n=validNumberOfPlayers)

Parameter Function stopOnInvalidQuota

Description

stopOnInvalidQuota checks if qutoa in a weighted voting game is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidQuota(paramCheckResult, q)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validQuota = 3
stopOnInvalidQuota(paramCheckResult, q=validQuota)

Parameter Function stopOnInvalidVetoPlayer

Description

stopOnInvalidVetoPlayer checks if vetoPlayer is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes calculation to stop.

Usage

stopOnInvalidVetoPlayer(paramCheckResult, vetoPlayer)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidWeightVector(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validVetoPlayer = 3
stopOnInvalidVetoPlayer(paramCheckResult, vetoPlayer=validVetoPlayer)

Parameter Function stopOnInvalidWeightVector

Description

stopOnInvalidWeightVector checks if weight vector in a weighted voting game is specified correctly. Validation result gets stored to object paramCheckResult in case an error occured and causes stop otherwise.

Usage

stopOnInvalidWeightVector(paramCheckResult, n, w)

Arguments

Error Code Ranges

Error codes and messages shown to user if error on parameter check occurs

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnParamCheckError()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
validWeightVector = c(1,2,3)
stopOnInvalidWeightVector(paramCheckResult, n=3, w=validWeightVector)

stopOnParamCheckError - stop and create error message on error

Description

stopOnParamCheckError causes and creates error message on base of paramCheckResult parameter where 'errCode' <> '0' in case error occured.

Usage

stopOnParamCheckError(paramCheckResult)

Arguments

Author(s)

Johannes Anwander anwander.johannes@gmail.com

See Also

Other ParameterChecks_CoopGame: getEmptyParamCheckResult(), stopOnInconsistentEstateAndClaimsVector(), stopOnInvalidAllocation(), stopOnInvalidBoolean(), stopOnInvalidClaimsVector(), stopOnInvalidCoalitionS(), stopOnInvalidDictator(), stopOnInvalidEstate(), stopOnInvalidGameVector(), stopOnInvalidGrandCoalitionN(), stopOnInvalidIndex(), stopOnInvalidLeftRightGloveGame(), stopOnInvalidNChooseB(), stopOnInvalidNumberOfPlayers(), stopOnInvalidNumber(), stopOnInvalidQuota(), stopOnInvalidVetoPlayer(), stopOnInvalidWeightVector()

Examples

library(CoopGame)
paramCheckResult=getEmptyParamCheckResult()
stopOnParamCheckError(paramCheckResult)

Compute tau-value

Description

Calculates the tau-value for a quasi-balanced TU game with n players.

Usage

tauValue(v)

Arguments

Value

tau-value for a quasi-balanced TU game with n players

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Branzei R., Dimitrov D. and Tijs S. (2006) Models in cooperative game theory, Springer, p. 32

Tijs S. (1981) "Bounds for the core of a game and the t-value", In: Moeschlin, O. and Pallaschke, D. (Eds.): Game Theory and Mathematical Economics, North-Holland, pp. 123–132

Stach I. (2011) "Tijs value", Encyclopedia of Power, SAGE Publications, pp. 667–670

Examples

library(CoopGame)
tauValue(v=c(0,0,0,0,1,0,1))


#Example from article by Stach (2011)
library(CoopGame)
v=c(0,0,0,1,2,1,3)
tauValue(v) 
#[1] 1.2 0.6 1.2

Construct a unanimity game

Description

Create a list containing all information about a specified unanimity game:
The player in coalition T are the productive players. If all players from T are included, the coalition generates value 1, otherwise 0.
Note that unanimity games are always simple games.

Usage

unanimityGame(n, T)

Arguments

Value

A list with three elements representing the unanimity game (n, T, Game vector v)

Related Functions

unanimityGameValue, unanimityGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 152

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame)
unanimityGame(n=3,T=c(1,2))


library(CoopGame)
unanimityGame(n=4,T=c(1,2))
#Output
#$n
#[1] 4
#
#$T
#[1] 1 2

#$v
#[1] 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1

Compute value of a coalition for a unanimity game

Description

Coalition value for a specified unanimity game:
For further information see unanimityGame

Usage

unanimityGameValue(S, T)

Arguments

Value

1 if all players of coalition T are included in S, else 0

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 152

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame)
unanimityGameValue(S=c(1,2,3),T=c(2))

Compute game vector for a unanimity game

Description

Game Vector for a specified unanimity game:
For further information see unanimityGame

Usage

unanimityGameVector(n, T)

Arguments

Value

Game Vector where each element contains 1 if all players of coalition 'T' are included in 'S' else 0

Author(s)

Johannes Anwander anwander.johannes@gmail.com

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 152

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, p. 764

Examples

library(CoopGame)
unanimityGameVector(n=3,T=c(2))

Compute vertices of Weber Set

Description

Calculates the Weber Set for given game vector with n players.

Usage

webersetVertices(v)

Arguments

Value

rows of the matrix are the vertices of the Weber Set

Author(s)

Anna Merkle

Franz Mueller

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Weber R.J. (1988) "Probabilistic values for games". In: Roth A.E. (Ed.), The Shapley Value. Essays in in honor of Lloyd S. Shapley, Cambridge University Press, pp. 101–119

Peters H. (2015) Game Theory: A Multi-Leveled Approach, 2nd Edition, Springer, pp. 327–329

Examples

library(CoopGame)
webersetVertices(c(0,0,0,1,1,1,2))


#Example of a 3-player TU game (with a Weber Set with 6 vertices)
library(CoopGame)
v = c(0,1,2,3,4,5,6)
webersetVertices(v)

#Example of a 4-player TU game (with a Weber Set with 14 vertices)
library(CoopGame)
v = c(5,2,4,7,15,15,15,15,15,15,20,20,20,20,35)
webersetVertices(v)

Construct a weighted voting game

Description

Create a list containing all information about a specified weighted voting game:
For a weighted voting game we receive a game vector where each element contains 1 if the sum of the weights of coalition S is greater or equal than quota q, else 0.
Note that weighted voting games are always simple games.

Usage

weightedVotingGame(n, w, q)

Arguments

Value

A list with four elements representing the weighted voting game (n, w, q, Game vector v)

Related Functions

weightedVotingGameValue, weightedVotingGameVector

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

References

Peleg B. (2002) "Game-theoretic analysis of voting in committees". in: Handbook of Social Choice and Welfare 1, pp. 195–201

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 17

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 825–831

Examples

library(CoopGame)
weightedVotingGame(n=3,w=c(1,2,3),q=4)


library(CoopGame)
weightedVotingGame(n=4,w=c(1,2,3,4),q=5)

#Output:
#$n
#[1] 4

#$w
#[1] 1 2 3 4
#
#$q
#[1] 5
#
#$v
#[1] 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1

Compute value of a coalition for a weighted voting game

Description

Coalition value for a specified weighted voting game:
For further information see weightedVotingGame

Usage

weightedVotingGameValue(S, w, q)

Arguments

Value

1 if the sum of the weights of coalition S is greater or equal than quota q else 0

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

References

Peleg B. (2002) "Game-theoretic analysis of voting in committees". in: Handbook of Social Choice and Welfare 1, pp. 195–201

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 17

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 825–831

Examples

library(CoopGame)
weightedVotingGameValue(S=c(1,2,3),w=c(1,2,3),q=4)

Compute game vector for a weighted voting game (aka quota game)

Description

Game vector for a specified weighted voting game:
For further information see weightedVotingGame

Usage

weightedVotingGameVector(n, w, q)

Arguments

Value

Game Vector where each element contains 1 if the sum of the weights of coalition S is greater or equal than quota q, else 0

Author(s)

Jochen Staudacher jochen.staudacher@hs-kempten.de

Johannes Anwander anwander.johannes@gmail.com

Michael Maerz

References

Peleg B. (2002) "Game-theoretic analysis of voting in committees". in: Handbook of Social Choice and Welfare 1, pp. 195–201

Peleg B. and Sudhoelter P. (2007) Theory of cooperative games, 2nd Edition, Springer, p. 17

Maschler M., Solan E. and Zamir S. (2013) Game Theory, Cambridge University Press, pp. 825–831

Examples

library(CoopGame)
weightedVotingGameVector(n=3,w=c(1,2,3),q=4)