balancedcheck {TUGLab} | R Documentation |
Balanced check
Description
This function checks if the given game is balanced and computes its balanced cover.
Usage
balancedcheck(v, game = FALSE, binary = FALSE, tol = 100 * .Machine$double.eps)
Arguments
v |
A characteristic function, as a vector. |
game |
A logical value. By default, |
binary |
A logical value. By default, |
tol |
A tolerance parameter, as a non-negative number. |
Details
Let v \in G^{N}
. A family F
of non-empty coalitions of N
is balanced if there exists a weight family \delta^{F} = \{ \delta^{F}_{S} \}_{S \in F}
such that
\delta^{F}_{S} > 0
for each S \in F
and \sum_{S \in F} \delta^{F}_{S} e^{S} = e^{N}
,
being e^{S}
the characteristic vector of S
, that is, the vector (e_{i}^{S})_{i \in N}
in which e_{i}^{S}=1
if i \in S
and e_{i}^{S}=0
if i \notin S
).
The game v
is balanced if, for each balanced family F
, it is true that
\sum_{S \in F} \delta^{F}_{S} v(S) \leq v(N).
The balanced cover of v
is the game \tilde{v}
defined by
\tilde{v}(S)=v(S)
for all S \neq N
and
\tilde{v}(N) = \max_{\delta \in P}{\sum_{S \subset N} \delta_{S} v(S)},
being P
the set of the weight families associated with the balanced families of N
.
A game is balanced if and only if it coincides with its balanced cover. By the Bondareva-Shapley Theorem, a game has a non-empty core if and only if it is balanced.
Value
TRUE
if the game is balanced, FALSE
otherwise. If game=TRUE
, the balanced cover of the game is also returned.
References
Maschler, M., Solan, E., & Zamir, S. (2013). Game Theory. Cambridge University Press.
See Also
Examples
balancedcheck(c(12,10,20,20,50,70,70), game=TRUE)
balancedcheck(c(rep(0,4), rep(30,6), rep(0,4), 50))
v <- runif(2^3-1,0,10) # random three-player game
balancedcheck(v, game=TRUE)
balancedcheck(balancedcheck(v, game=TRUE)$game) # balanced cover is indeed balanced
balancedcheck(runif(2^(15)-1,min=10,max=20)) # random game