Shapley Value, Weighted Majority Game n = 5

Question

So I have a weighted majority game with five players. What I don't quite understand is how I can actually then calculate the Shapley value. I have calculated the winning and losing coalitions but instead of going through each permutation, is there another idea how to calculate the values?

Answer 1

Of course, rather than computing the permutations to get the Shapley value, it is more effective to apply the potential approach. This method was invented by Hart and Mas-Colell (1989), Potential, value and consistency, Econometrica 57, 589-614.

The potential of the game $v$ can be computed while applying recursively the following definition for each $S \subseteq N$ \begin{equation*} \rho(S):= \frac{v(S) + \sum_{k \in S}\,\rho(S\backslash\{k\})}{s}, \end{equation*} whereas $\rho(\{k\}):=v(\{k\})$ for all $k \in N$ and $\rho(\emptyset) := 0$. Then we can calculate the Shapley value of each player $k$ by his marginal contribution, defined by \begin{equation*} \phi_{k}(v)= \rho(N) - \rho(N\backslash\{k\}) \qquad\forall\, k \in N. \end{equation*}

Hence, you compute from the game in characteristic function form its potential, and then the Shapley value can be directly computed.

This procedure is implemented in the following software program:

enter link description here