Banzhaf Power Index

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The Banzhaf index is a measure of how probable it is that someone's vote will change the outcome of an election in settings in which voters may have unequal numbers of votes. This Demonstration examines a scenario with six voters, each of whom have a user‐selectable number of votes. You can also determine the percentage of total votes needed to win. From this data, the Demonstration shows all the winning coalitions that can form and all the members of each coalition ("the swing voters") who, if they had voted differently, would have changed the outcome of the election. The Banzhaf index, which ranges from 0 to 1, is the percentage of times each voter is a swing voter over all possible winning coalitions.

Contributed by: Seth J. Chandler (March 2011)
Open content licensed under CC BY-NC-SA



The Banzhaf index is closely related to the Shapley value, often used in cooperative games.

Snapshot 1: the percentage of votes needed to win affects the power index

Snapshot 2: sometimes, parties with different amounts of raw voting power have the same power index

Snapshot 3: only two parties have any real power

Snapshot 4: a four party system in which the second party has somewhat more votes but has no more power when a two thirds majority is needed

Snapshot 5: a four party system in which the second party has somewhat more votes but has considerably more power

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