owenvalue {TUGLab} | R Documentation |
Owen value
Description
Given a game and a partition of the set of players, this function computes the Owen value.
Usage
owenvalue(v, binary = FALSE, partition = NULL, game = FALSE)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
partition |
A partition of the set of players, as a list of vectors. When not specified, it is taken to be the partition whose only element is the set of all players. |
game |
A logical value. By default, |
Details
Let v \in G^{N}
and let C=\{C_{1},\dots,C_{m}\}
be a partition of the set of players.
For each T \in 2^{N} \setminus \emptyset
, let R'_{T}=\{j : C_{j} \cap T \neq \emptyset\}
and R^{T}_{j}=C_{j} \cap T
for each j \in \{1,\dots,m\}
.
Being c_{T}
the Harsanyi dividend of coalition T \in 2^{N}
, the Owen value of each player i \in N
is defined as
O_{i}(v,C)=\sum_{T \in 2^{N}:j \in R'_{T},i \in R^{T}_{j}}\frac{c_{T}}{|R'_{T}||R^{T}_{j}|}.
Value
The corresponding Owen value, as a vector; and, if game=TRUE
, the associated quotient game, as a vector in binary order if binary=TRUE
and in lexicographic order otherwise.
References
Owen, G. (1977). Values of Games with a Priori Unions. In R. Henn and O. Moeschlin (Eds.), Mathematical Economics and Game Theory (pp. 76-88), Springer.
See Also
shapleyvalue, harsanyidividend
Examples
v <- c(0,0,0,0,30,30,40,40,50,50,60,70,80,90,100) # in lexicographic order
owenvalue(v, partition=list(c(1,3),c(2),c(4)))
owenvalue(v)
round(owenvalue(v),10) == round(shapleyvalue(v),10)
w <- c(0,0,0,0,0,10,10,20,10,20,10,20,10,20,10,20,40,20,40,20,40,
20,40,20,20,80,60,80,80,60,100) # in lexicographic order
owenvalue(w, partition=list(c(1,2,3),c(4,5)))