The mathematician L. S. Shapley invented a formula for estimating the "real" power of a player in a coalition voting game.
First some notation: P is the set of all players, S is any coalition (subset of players), and s is any player. |S| is the number of players in the coalition S. n! is the factorial of the number n.
v(S)=1 if S is a winning coalition, and v(S)=0 if S is a losing coalition.
The formula for the Shapley value of a player s is a sum over all
- coalitions containing s
\ (|S|-1)!(|P|-|S|)!
(Øv)(s) = > (v(S) - v(S\{s})) ------------------
/ |P|!
s in S
Note that v(S) - v(S\{s}) is 1 if s contributes power to the coalition S, but 0 if s makes no difference. Smaller and larger coalitions are weighted less than medium-sized ones.
If you use the formula on the examples in the CoalitionTheory page, you get the reasonable result: in the first example, all 3 players A, B and C get value 1/3, while in the second example, A, B and C get value 1/3 while D gets value 0.