TUGLab (version 0.0.1)

owenvalue: 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)

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.

Arguments

v

A characteristic function, as a vector.

binary

A logical value. By default, binary=FALSE. Should be set to TRUE if v is introduced in binary order instead of lexicographic order.

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, game=FALSE. If set to TRUE, the associated quotient game is also returned.

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 $$O_{i}(v,C)=\sum_{T \in 2^{N}:j \in R'_{T},i \in R^{T}_{j}}\frac{c_{T}}{|R'_{T}||R^{T}_{j}|}.$$

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)))