isEssentialGame: Check if game is essential

In CoopGame: Important Concepts of Cooperative Game Theory

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) > ∑ v({i})

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

Usage

isEssentialGame(v)

Arguments

v	Numeric vector of length 2^n - 1 representing the values of the coalitions of a TU game with n players

Value

TRUE if the game is essential, else FALSE.

Author(s)

Michael Maerz

Jochen Staudacher [e-mail]

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)

CoopGame documentation built on Aug. 24, 2021, 1:07 a.m.