show inequality holds for a non-empty core

Question

I am having difficulties in understanding how can I show this:

Consider a cooperative game with 3 players and a superadditive function $v(S)$

If the game has non empty core the inequality holds:

$v({1,2})+v(1,3)+v(2,3)\ge2v(1,2,3)$

I cannot understand how can I show the above without having values.

Can anyone give me a tip in how to start?

Thanks

Answer 1

If the core is nonempty for a $n$-person game, then it holds

$$v(S) \ge x(S) = \sum_{i \in S}\,x_{i} \qquad S \subseteq N.$$

Thus, we have

$$v(1,2) \ge x(1,2) = x_{1}+x_{2}\qquad v(1,3) \ge x(1,3) = x_{1}+x_{3}\qquad v(2,3) \ge x(2,3) = x_{2}+x_{3}.$$

Adding up these relations, we have

$$v(1,2)+v(1,3)+v(2,3) \ge 2*(x_{1}+x_{2}+x_{3})= 2*v(N),$$

with $x_{1}+x_{2}+x_{3} = x(N) = v(N)$.