Given a game, this function computes its core vertices.
Usage
corevertices(v, binary = FALSE)
Value
If the core of v is non-empty, the core vertices are returned, as a matrix in which each row is a vertex.
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.
Details
The core of a game \(v\in G^N\) is the set of all its stable imputations:
$$C(v)=\{x\in\mathbb{R}^n : x(N)=v(N), x(S)\ge v(S)\ \forall S \in 2^N\},$$
where \(x(S)=\sum_{i\in S} x_i\).
References
Edgeworth, F. Y. (1881). Mathematical psychics: An essay on the application of mathematics to the moral sciences. CK Paul.
Gillies, D. (1953). Some theorems on n-person games. PhD thesis, Princeton, University Press Princeton, New Jersey.
v=c(0,0,0,0,0,0,0,0,1,4,1,3,6,8,10)
corevertices(v)
# What if the game is a cost game?cost.v <- c(2,2,2,3,4,4,5) # cost game-corevertices(-cost.v) # core vertices of the cost game