0
$\begingroup$

Please help me to calculate the core of this easy coalitional game. I really didn't get it from my game theory course but want to understand the mechanism of calculating, describe it in detail please! Thank you!

The task is:

*Three players together can obtain $1$ to share, any two players can obtain $0.8$, and one player by herself can obtain zero.

*Then, $N=3$ and $v(1)=v(2)=v(3)=0$, $v(1,2)=v(2,3)=v(3,1)=0.8$, $v(1,2,3)=1$. Which allocation is in the core of this coalitional game?

a) $(0,0,0)$;

b) $(0.4, 0.4, 0)$;

c) $(1/3, 1/3, 1/3)$;

d) The core is empty;

$\endgroup$
  • $\begingroup$ Hint: the payoff a player receives in the core should be greater than the payoff he could receive as a member of any other group. $\endgroup$ – Ben Feb 19 '13 at 15:35
  • $\begingroup$ An almost idential example in Coase theorm and Empty Core,bbs.cenet.org.cn/uploadimages/200642414363295143.pdf $\endgroup$ – Metta World Peace Feb 19 '13 at 16:00
1
$\begingroup$

The sum of the payoffs in an imputation in the core must be the value of the grand coalition. In this case that's $1$, so we can eliminate answers a) and b) on that basis alone. Answer c) does fulfil that requirement, but any pair of players can get $4/5$ instead of the $2/3$ allocated to them by forming a smaller coalition, so this allocation isn't in the core either.

If this is a multiple choice question, that would be enough to deduce that the answer must be d). To prove that d) is in fact correct, consider any imputation $(a,b,1-a-b)$ whose payoffs sum to $1$. For this not to be dominated by the coalition $(2,3)$, we must have $b+(1-a-b)=1-a\ge0.8$, and thus $a\le0.2$. But the same reasoning applies to all three payoffs, so their sum is at most $0.6\lt1$. It follows that the core is empty.

$\endgroup$
1
$\begingroup$

Let x_i be the allocation for player i. The allocation conditions are going to be:

x_1 + x_2 >= 0.8
x_2 + x_3 >= 0.8
x_1 + x_3 >= 0.8
x_1 + x_2 + x_3 = 1

Summing the first 3 constraints, we get:

x_1 + x_2 + x_3 >= 3*0.8/2 = 1.2

This clearly doesn't satisfy the 4th condition and hence the core is empty.

$\endgroup$

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.