Weighted Voting and Power Indices
A voting arrangement in which voters may control unequal number of votes and decisions are made by forming coalitions with the total of votes equal or in access of an agreed upon quota is called a weighted voting system. The usual notation is
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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 p1, ..., pn is any permutation pi1, ..., pin of that set. An element pik is said to be pivotal to a (sequential) coalition pi1, ..., pin, provided the (regular) coalition pi1, ..., pik-1 is losing, whereas the (regular) coalition pi1, ..., pik 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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