## Weighted Voting and Power Indices

A voting arrangement in which voters may control unequal number of votes and decisions are made by forming *coalition*s with the total of votes equal or in access of an agreed upon *quota* is called a *weighted voting system*. The usual notation is _{1}, w_{2}, ..., w_{n}],_{i}, *players*.

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In the system [5: 3, 2, 1], all decisions could be made by the two principal players (3 and 2 votes, respectively), while that last player (the one with the single vote) has no influence whatsoever on the decision process. It is then clear that having a vote does not endow its owner with any real power in making decisions. (The last player in this example is known as a *dummy*.)

Banzhaf's is one possible indicator of the relevance of a particular player. Shapley-Shubik's is another. In both cases, the power wielded by a player is determined by the number of coalitions in which his or her role is important. However, the two indices formalize the notions of coalition and importance in different ways.

*Coalition* is any (non-empty) combination of the players. A coalition is *winning* provided the cumulative vote of its members is equal to or greater than the quota. A coalition is losing if it's not winning. A player is called *critical* to a winning coalition, if his or her removal from the coalition renders it losing. *Banzhaf's index* of a player p is the ratio of the number of winning coalitions to which p is critical to the total number of times the players are critical.

A coalition is just a set of its elements. No order is specified in which the players enter the coalition. Not so with the *sequential coalition*, used to define the Shapley-Shubik index. A sequential coalition of n players p_{1}, ..., p_{n} is any permutation p_{i1}, ..., p_{in} of that set. An element p_{ik} is said to be *pivotal* to a (sequential) coalition p_{i1}, ..., p_{in}, provided the (regular) coalition p_{i1}, ..., p_{ik-1} is losing, whereas the (regular) coalition p_{i1}, ..., p_{ik} is winning. Assuming that the quota does not exceed the total number of votes, every sequential coalition has a unique pivotal element. *Shapley-Shubik's index* of power of a player p is the ratio of the number of sequential coalitions for which p is pivotal to the total number of sequential coalitions, which is always n!.

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