I am facing a problem to understand the calculation of shapley value on the below example:
Question: The parliament of Micronesia is made up of four political parties, $A$, $B$, $C$, and $D$, which have $45$, $25$, $15$, and $15$ representatives, respectively. They are to vote on whether to pass a $\$100$ million spending bill and how much of this amount should be controlled by each of the parties. A majority vote, that is, a minimum of $51$ votes, is required in order to pass any legislation, and if the bill does not pass then every party gets zero to spend.
Solution: we get the payoff division $(50, 16.66, 16.66, 16.66)$, which adds up to the entire $\$100$ million.
I didn't understand the solution of it how those value comes? Need Help! Thanks in advance!
The four parties $A, B, C$ and $D$ have $45,25,15$ and $15$ representatives in the parliament respectively. To pass a bill $51$ votes are necessary, called a quota. This is a weighted majority game. Let us denote the total number of votes as $w(N) = \sum_{k \in N}\, w_{k} \in \mathbb{N}$, hence $w(N)=100$. Here $N$ denotes the player set, that is, $\{A,B,C,D\}$. For passing a bill at least $0 < qt \le w(N)$ votes are needed. A simple game is referred to a weighted majority game, if there exists a quota $ qt > 0$ and weights $w_{k} \ge 0$ for all $k \in N$ such that for all $S \subseteq N$ it holds either $v(S) = 1$ if $w(S) \ge qt$, or $v(S) = 0$ otherwise. Such a game is formally represented as $[qt; w_{1}, \ldots, w_{n}]$. The weights vector in this example game is given by $\vec{w} =\{45,25,15,15\}$ and the quota by $qt=51$. Then the weighted majority game (TU game) with its coalitional values is given by
$$ sv =\{0,0,0,0,1,1,1,0,0,0,1,1,1,1,1\},$$
whereas the set of all permissible coalitions are $$\{\{1\},\{2\},\{3\},\{4\},\{1,2\},\{1,3\},\{1,4\},\{2,3\},\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},N\}.$$
From this game the Shapley value is computed, which is
$$sh_{sv}=\{1/2,1/6,1/6,1/6\}, $$
multiplying this result by $100$ million gives the "payoff division" $\{50,16.66,16.66,16.66\}$. This indicates the power of the parties in the parliament.