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.
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:
$$shv=\{1,3,3,5\}/12.$$