Suppose we have the following recursive definition:
A coalition is stable if and only if it has no proper stable sub-coalition holding greater than $50\%$ of the total weight.
Now there occur four questions:
a)When a coalition of two agents is stable?
b)When a coalition of three agents is stable?
c)Prove that the coalition of 5 agents with weights 3, 4, 5, 10, 20 is stable. What happens if agent 20 suddenly disappears? (The speculation by the authors is that agent 20 is Stalin and agent 10 is Beria — after the exam google who he was and what happened with him in 1953 if you are not familiar).
d)Consider a coalition of $n$ agents with equal weights. Prove that it is stable if and only if $ n = 2^k - 1$ for some $k$.
If I am not wrong, I think for the first two parts we have to find the core for $\{x_1 \leq \frac{1}{2} , x_2 \leq \frac{1}{2} ~ and ~ x_1 + x_2 = 1 \} $ and $\{ x_1 \leq \frac{1}{2}, x_2 \leq \frac{1}{2}, x_3 \leq \frac{1}{2} ~and~ x_1+x_2+x_3 = 1 \} $ respectively?
In the second one we will get $x_1 + x_2 \geq \frac{1}{2} $ and $x_2 + x_3 \geq \frac{1}{2}$ and $x_2 + x_3 \geq \frac{1}{2}$ and to draw this is not so difficult and will be a triangle.
Please correct me if I am wrong, I am new in game theory! Thanks!