In the example from the beginning with the assigned probability (W) this means
.
The and would be computed analogically. In general, the normalized revised Banzhaf index looks as follows.


In the revised index, even so, comes immediately the question of how do we compute the probability that a coalition would be formed. This would require additional research as in our available literature the authors do not comment on this problem. But even pure guesses of political observers would add a lot into the approximation to reality.
3.2 Shapley-Shubik index
As we saw, the Banzhaf index counts with all possible combinations of coalitions. Here, Shapley-Shubik index counts with all possible permutations of winning coalitions. In this case the order of parties matters. Obviously, we get many more possibilities of coalitions now. In the same example as in 2.1 [3; 2, 1, 1], all possible orders (namely 3!) are:
{A,B,C}, {A,C,B}, {B,A,C}, {B,C,A}, {C,A,B}, {C,B,A}
In bold are pivotal players – i.e. players turning a loosing coalition into a winning one. Here, following Shapley-Shubik indices result as the probabilities of being randomly chosen from the set of possible outcomes:

For n players, the standard Shapley-Shubik index is defined as

where s denotes the number of players of S. The function ν has value 1 for winning coalitions and 0 for losing .
In Banzhaf index, some coalitions are counted in the denominator several times, as there can be more parties with critical position in one coalition. Whereas in the Shapley-Shubik index, only one party can be the pivotal player, because we assume their voting in an order.
To summarize, the voting power index (as by Shapley and Shubik) is given by the number of times a player is pivotal in all orderings of the players divided by the number of all such orderings .
3.3 Spatial Models
Power index need not to be expressed only as a probability of being chosen from a set of winning coalitions. The intuition can lead us to the minimization of interparty conflict achieved by ‘political closeness’. It’s plausible to assume that the closer are the parties’ preferences on a certain policy scale the more likely they are to form a coalition with each other.
One-dimensional explanation of coalition formation is formally explained by the normalized uni-dimensional scale in an Euclidean space, such that each party can be assigned a value on this interval (usually from 0 to 1).