This function checks if the given family is balanced.
Usage
balancedfamilycheck(Fam, n = NULL, tol = 100 * .Machine$double.eps)
Value
This function returns three outputs: check, minimal and delta.
If Fam is not a balanced family: check=FALSE and both minimal and delta are NULL.
If Fam is a balanced family: check=TRUE, minimal=TRUE if Fam is minimal (minimal=FALSE otherwise), and delta returns an associated weight family.
Arguments
Fam
A vector containing the binary order positions of a family of coalitions.
n
The number of players in the set of players from which Fam is taken. When not specified, n is assumed to be the the number of players present in Fam.
tol
A tolerance parameter, as a non-negative number.
By default, tol=100*.Machine$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 \(F' \subsetneq F\).
References
Maschler, M., Solan, E., & Zamir, S. (2013). Game Theory. Cambridge University Press.
balancedfamilycheck(c(3,6,13,8)) # balanced and minimalbalancedfamilycheck(c(3,5,9,4,8,14)) # balanced but not minimalbalancedfamilycheck(c(1,2,4,12,13)) # not balanced