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 |
tol |
A tolerance parameter, as a non-negative number. |
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