# Finding the core of a coalition game

I need to find the core of a 3-player coalition game graphically, given that $v(\phi)=0$, $v(1) = 9, v(2)=8, v(3) = 9, v({1,2}) = 14, v({1,3})=15, v({2,3}) = 13, v({1,2,3}) = 21$ So I'm following the methods outlined here, and so I start by saying that the core is the set of values $x_1, x_2, x_3$ such that $x_1 + x_2 + x_3 = 21, x_1 \geq 9, x_2 \geq 8, x_3 \geq 9, x_1+x_2 \geq 14, x_2+x_3 \geq 13, x_1+x_3 \geq 15$

Then, from the inequalities, I get that $x_3 = 21 - (x_1 + x_2) \Rightarrow x_3 \leq 21-(14) = 7$ Since we also have $x_3 \geq 9$, this gives me that the core is empty.

I'm not sure if I'm doing anything wrong, but I don't know how to show this graphically at all. I can't draw a triangle for this at all for $x_1+x_2+x_3 = 21$.

• This is correct, the core is empty. This can be immediately seen by the fact, that the game is not essential, that is, v(1)+v(2)+v(3)=9+8+9>21=v(N). – Holger I. Meinhardt Apr 17 '15 at 15:24