0
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Hi, I do not understand where the 2 in $2*(x_1+x_2+x_3)$ comes from. Can you clarify? $\endgroup$ – user290335 Nov 30 '17 at 17:12
  • $\begingroup$ I have clarified the answer on your request. $\endgroup$ – Holger I. Meinhardt Nov 30 '17 at 17:16
  • $\begingroup$ Makes sense now, than you very much for your help. $\endgroup$ – user290335 Nov 30 '17 at 17:22

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.