1.k people produce $kf(n)/n$ amount of good. Define a CG and find $C(v)$ and SV.
How can I define the CG? My problem in this is the coalitions, I have to define the value function, my idea is:
$v(i_1,..i_k)=\sum_{j=1}^{k}i_k f(n)/n$ This will be the amount for each one, or divided by n again?.
If I understand your post correctly, then in case one I would define the game as
\begin{equation*} v(S):= |S|*f(n)/n \qquad \forall S \in 2^{N}, \end{equation*}
where $k$ is replaced by the number of players belonging to coalition $S$ (cardinality of $S$).
For case two, I would define your game as $v(S):=f(n)$ if $|S| \ge n/2$, and $v(S):=0$ if $|S| < n/2$.
To simplify matters, I recommend to set for case one $f(n)=n$, and for case two $f(n)=1$, and then compute the resulting symmetric game as an exercise, for instance, for three and four players. Finally, both games are symmetric, the Shapley value gives for both games $sh_{i}=v(N)/n$ for all $i \in N$. However, for game one the core is a single point, namely the Shapley value, whereas for the second game, the core is empty. Why?