kohlbergcriterion {TUGLab} | R Documentation |
Kohlberg criterion for the prenucleolus
Description
This function applies the Kohlberg criterion to check if the given efficient allocation is the prenucleolus of the given game.
Usage
kohlbergcriterion(v, x, binary = FALSE, tol = 100 * .Machine$double.eps)
Arguments
v |
A characteristic function, as a vector. |
x |
An efficient allocation, as a vector. |
binary |
A logical value. By default, |
tol |
A tolerance parameter, as a non-negative number. |
Details
Given v \in G^{N}
and x \in \mathbb{R}^{n}
with \sum_{i \in N} x_{i} = v(N)
,
let k(x)
be the number of different excesses in x
.
According to the Kohlberg criterion for the prenucleolus, x
is the prenucleolus of v
if and only if, for each j \in \{1,\dots,k(x)\}
, \bigcup_{t=1}^{j} F^{t}
is a balanced family, being F^{t}
the set of coalitions associated with the excess
that occupies position t
when excesses are arranged in decreasing order.
Value
TRUE
if x
is the prenucleolus of v
, FALSE
otherwise.
References
Kohlberg, E. (1971). On the Nucleolus of a Characteristic Function Game. SIAM Journal on Applied Mathematics, 20(1), 62–66.
See Also
balancedfamilycheck, excesses, prenucleolusvalue
Examples
v <- c(0,0,0,0,10,40,30,60,10,20,90,90,90,130,160)
x <- prenucleolusvalue(v)
kohlbergcriterion(v, x) # x is the prenucleolus of v
y <- prenucleolusvalue(v) + c(1,-1,0,0)
kohlbergcriterion(v, y) # y is not the prenucleolus of v
# If the game is 0-monotonic, its nucleolus coincides with its prenucleolus,
# and therefore must pass the Kohlberg criterion for the prenucleolus:
v4 <- c(-2,-2,-2,7,7,7,6)
zeromonotoniccheck(v4)
kohlbergcriterion(v4, nucleolusvalue(v4))