balancedfamilycheck {TUGLab}R Documentation

Balanced family check

Description

This function checks if the given family is balanced.

Usage

balancedfamilycheck(Fam, n = NULL, tol = 100 * .Machine$double.eps)

Arguments

Fam

A vector containing the binary order positions of a family of coalitions.

n

The number of players in the set of players from which Fam is taken. When not specified, n is assumed to be the the number of players present in Fam.

tol

A tolerance parameter, as a non-negative number.
By default, tol=100*.Machine$double.eps.

Details

A family F of non-empty coalitions of a set of players N is balanced if there exists a weight family \delta^{F} = \{ \delta^{F}_{S} \}_{S \in F} such that \delta^{F}_{S} > 0 for each S \in F and \sum_{S \in F} \delta^{F}_{S} e^{S} = e^{N}, being e^{S} the characteristic vector of S, that is, the vector (e_{i}^{S})_{i \in N} in which e_{i}^{S}=1 if i \in S and e_{i}^{S}=0 if i \notin S).

A balanced family F is said to be minimal if there does not exist a balanced family F' such that F' \subsetneq F.

Value

This function returns three outputs: check, minimal and delta. If Fam is not a balanced family: check=FALSE and both minimal and delta are NULL. If Fam is a balanced family: check=TRUE, minimal=TRUE if Fam is minimal (minimal=FALSE otherwise), and delta returns an associated weight family.

References

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

See Also

balancedcheck, kohlbergcriterion, totallybalancedcheck

Examples

balancedfamilycheck(c(3,6,13,8)) # balanced and minimal
balancedfamilycheck(c(3,5,9,4,8,14)) # balanced but not minimal
balancedfamilycheck(c(1,2,4,12,13)) # not balanced

[Package TUGLab version 0.0.1 Index]