belong2corecheck {TUGLab} | R Documentation |
Belong to core
Description
This function checks if an allocation belongs to the core of a game.
Usage
belong2corecheck(v, binary = FALSE, x, instance = FALSE)
Arguments
v |
A characteristic function, as a vector. |
binary |
A logical value. By default, |
x |
An allocation, as a vector. |
instance |
A logical value. By default, |
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
.
Value
TRUE
if x
belongs to the core of v
, FALSE
otherwise. If instance=TRUE
and x
does not belong to the core of v
, a justification is also provided: if efficiency is violated, not efficient
is returned; if efficiency is not violated, the position (binary order position if binary=TRUE
; lexicographic order position otherwise) of a coalition for which rationality is violated is returned.
References
Gillies, D. (1953). Some theorems on n-person games. PhD thesis, Princeton, University Press Princeton, New Jersey.
Examples
v <- c(0, 0, 0, 2, 1, 4, 6)
a <- c(3, 1, 2) # an allocation for v
b <- c(2, 2, 2) # egalitarian solution for v
belong2corecheck(v = v, binary = TRUE, x = a, instance = TRUE)
belong2corecheck(v = v, binary = FALSE, x = b, instance = TRUE)
# What if the game is a cost game?
cost.v <- c(2,2,2,3,4,4,5) # cost game
cost.x <- c(1,2,2) # core allocation of cost.v
belong2corecheck(v = -cost.v, x = -cost.x)