TUGLab (version 0.0.1)

belong2corecheck: 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)

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.

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.

x

An allocation, as a vector.

instance

A logical value. By default, instance=FALSE.

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

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)