In a given country, a new law is adopted if either: $(a)$ Strictly more than one half of the parliament members votes for it and the president approves it or $(b)$ strictly more then $\frac{2}{3}$ of the parliament members votes for it, without the presidential approval.
Now suppose that the members of parliament are organized in $4$ different parties $A, B, C$ and $D$ with $185, 120, 110$ and $35$ votes, respectively. Where all members of one party cast the same votes. Consider a coalitional game with $5$ players (that is, these $4$ parties and the president $P$).
How do I find the Core and the Shapley value? And also is the game supera-dditive or Super-modular ?
Given a coalition game $(N,v)$, I understand that we can compute the shapley value of player $i$ with the formular $$\phi_i (N,v)=\frac{1}{N!} \sum_{S\subseteq N \setminus \{i\}} |S|! (|N|-|S|-1)! \bigg [ v(S \cup \{ i\}) - v(S) \bigg ]$$ But I am having trouble on how to approach this game. Please any hint or help is appreciated. Thanks in Advance!