nucleoluspcvalue {TUGLab} | R Documentation |
Per capita nucleolus
Description
Given a game, this function computes its per capita nucleolus.
Usage
nucleoluspcvalue(v, binary = FALSE, tol = 100 * .Machine$double.eps)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
tol |
A tolerance parameter, as a non-negative number. |
Details
Given a game v\in G^N
and an allocation x \in I(v)
, the per capita excess
of each coalition S \in 2^{N}
with respect to x
is defined as
e^{p}(v,x,S) = \frac{v(S)-\sum_{i \in S}x_{i}}{|S|}.
The per capita excesses of all non-empty coalitions, sorted in non-increasing order, are stored
in the per capita excesses vector, \theta^{p}(x)
.
For any game v\in G^N
with a non-empty set of imputations, the per capita nucleolus
is defined as the only imputation pcn(v) \in I(v)
that satisfies
\theta^{p}(pcn(v))_{i} \leqslant \theta^{p}(y)_{i}
for each i \in \{1,\dots,2^{N}-1\}
and for all y \in I(v)
.
This function is programmed following the algorithm of Potters, J.A., et al. (1996).
Value
The per capita nucleolus of the game, as a vector.
References
Grotte, J. (1970). Computation of and Observations on the Nucleolus, the Normalized Nucleolus and the Central Games. Master’s thesis), Cornell University, Ithaca.
Potters, J. A., Reijnierse, J. H., & Ansing, M. (1996). Computing the nucleolus by solving a prolonged simplex algorithm. Mathematics of Operations Research, 21(3), 757-768.
See Also
excesses, leastcore, nucleolusvalue, prenucleolusvalue
Examples
nucleoluspcvalue(c(1,5,10,6,11,15,16))
nucleoluspcvalue(c(0,0,0,30,30,80,100))
# Computing the per capita nucleolus of a random essential game:
n <- 10 # number of players in the game
v <- c(rep(0,n),runif(2^n-(n+1),min=10,max=20)) # random essential game
nucleoluspcvalue(v)
# What if the game is a cost game?
cost.v <- airfieldgame(c(1,5,10,15)) # cost game
-nucleoluspcvalue(-cost.v) # per capita nucleolus of the cost game