I'm trying to find the core of this cooperative game:
$N = \{1,2,3\}$ and $v(\{1\})=24$, $v(\{2\})=24$, $v(\{3\})=26$, $v(\{1,2\})=42$, $v(\{1,3\})=44$, $v(\{N\})=52$.
My solution:
$x_1 \ge 24$
$x_2 \ge 24$
$x_3 \ge 26$
$x_1+x_2 \ge 42$
$x_1+x_3 \ge 34$
$x_2+x_3 \ge 44$
$x_1+x_2+x_3=52$
Manipulating the last four expressions I obtained the following inequalities:
$x_1 \le 10$
$x_2 \le 18$
$x_3 \le 8$
Which (I think) are in contrast with the first three, so the core should be empty (?).
My book instead gives as a solution: $co \{(24, 18, 10), (18, 24, 10), (8, 24, 20), (8, 18, 26) \} $.
Solution diagram
What am I missing?
You are completely right, the core must be empty. The quickest way to see that is to use the inequalites of the singleton coalitions, adding them up gives $\sum_{i \in N} x_{i} \ge 24 + 24 +26 = 74 > 52 = v(N)$. Hence, the Pareto-efficiency constraint is violated, the core must be empty.
I guess that this is a misprint in the book. By the way, due to the above constraints set the value of coalition $\{1,3\}$ should be $34$ and that of coalition $\{2,3\}$ must be $44$.