I need help understanding the shapely value formula for voting games A company has 4 shareholders with 10 20 30 40 stock respectively, a decision can be made by the shareholders with the majority of the shares ie over 50 percent, determine the voting power of each person. 
I have done this by pure intuition, I understand you work out the value of each player recursively when adding other players into the fold then divide by the number of coalitions but I am required to actually input this into the shapely value formula, I’m struggling to grasp what all of the symbols represent in this case, thanks.
 A: The starting point of your problem is a voting problem with four shareholders, where we formalize the set of shareholders by $N=\{1,2,3,4\}$. However,before we can calculate the Shapley value, we have to represent the voting problem into a cooperative game.   
The four shareholders hold a share of $10,20,30$ and $40$ percent of the equity capital of a company respectively. To pass a decision in the executive board of this firm $51$ shares are necessary. Thus, the voting problem can be represented in form of weighted majority game. 
For doing so, let us denote the total number of shares as $w(N) \in \mathbb{N}$. For passing a decision at least $0 < qt \le w(N)$ shares 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 generically represented as $[qt; w_{1}, \ldots, w_{n}]$. The weights vector is in the example game $w =\{10,20,40,40 \} $, and the quota to pass a decision is fixed at $51$ shares. Hence, shareholders have to form coalitions to pass a decision.  
If we calculate the simple game, we get the following characteristic values for each coalition:
$v(S) = 1$ if $S \in \{\{2,4\},\{3,4\},\{1,2,3\},\{1,2,4\},\{1,3,4\},\{2,3,4\},N \}$,
otherwise $v(S)=0$.
To get the Shapley value we apply its formula on the above game, so that we get the following index of voting power w.r.t. the Shapley value:
$$shv=\{1,3,3,5\}/12.$$
