# The core/Shapley value

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!

*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;

• 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. – Ben Feb 19 '13 at 15:35
• An almost idential example in Coase theorm and Empty Core，bbs.cenet.org.cn/uploadimages/200642414363295143.pdf – Metta World Peace Feb 19 '13 at 16:00

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.

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.