Until early 1990’s Finland also needed even five-sixths majorities for certain types of economic legislation, which account for its tendency to forming oversized cabinets.
3 Mathematics view
Trying to describe social processes with pure mathematics is a Sisyphean task. The more one tries to incorporate everything that influences the social outcome, the more it becomes difficult and problematic. This all applies also, as we will see, to such a complex problem as the coalition formation. Manfred Holler argues that there is “no obvious answer to the problem about the right power measure”. Nevertheless, the discussion over power indexes is characterized mainly by the search for the “right index”. At a first sight, coalition building can be seen only as a process of aggregating enough voting power required to legitimately make political decisions. In other words, this means aquiring the majority of votes. This basic principle is used as a premise for all mathematics attempts to formulate a power index. We come to problems, when approaching the notion of power. Is power in social context a probability, capacity, potential or just a purely theoretical concept? The latter was rejected, at least by the authors concerned with mathematical expression of the concept. Another fundamental question would be whether the power depends on preferences of players. As we will see now, the two basic indices take the probability (or the share of the total) as its’ core principle. 3.1 Banzhaf index
This index reflects the number of coalitions where the party A holds a critical position as a share of the sum of all parties’ number of critical position coalitions.
Let us assume three players (parties A, B and C) in a voting game with this profile:
[3; 2, 1, 1]
Here, the first number indicates the vote quota (in our example majority voting), the following numbers represent the amount of players’ votes. Winning coalitions are only three:
{A,B}, {A,C} and {A,B,C}
where players holding critical position (position of pivotal or swing voter) are in bold. This means, that if party in bold leaves the coalition it loses its’ majority.
Now, as we can see, party A has three critical positions, B and C one each. Therefore, the Banzhaf index will look like this:
(or 60%),
(or 20%)
and (20% as well).
Consequently, Banzhaf index in general can be expressed as:

where denotes the number of winning coalitions in which party i (i=1 to n) holds the critical position and is the sum of all this values for all parties (not only the total number of possible coalitions).

However, there can be many remarks raised against the Banzhaf index.