Determining the core of a permutation game

Question

For my game theory class I have to determine the core of the following cooperative game:

$\mathrm{N}={1,2,3}$

$S=\{1\}$ gives $v(S)=2$

$S=\{2\}$ gives $v(S)=5$

$S=\{3\}$ gives $v(S)=4$

$S=\{1,2\}$ gives $v(S)=14$

$S=\{1,3\}$ gives $v(S)=18$

$S=\{2,3\}$ gives $v(S)=9$

$S=\{1,2,3\}$ gives $v(S)=24$

I don't know what I should do to find the core properly. I also have to draw it in a triangle.

What I found up until now is the following:

$$C=\{ (x_1,x_2,x_3) \in \mathbb{R}^3 | x_1 \geq 2,x_2 \geq 5, x_3 \geq 4, x_1+x_2 \geq 14,\\ x_1+x_3 \geq 18, x_2 + x_3 \geq 9, x_1+x_2+x_3=24 \}$$

Answer 1

The core of the permutation game is given by the convex combination of the four extreme points:

9    5   10  
8    6   10
14    6    4
15    5    4

Since, it is a homework, check out by yourself that the result is correct! Moreover, you will find out that the Shapley value is located outside of the core. Why?