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Banzhaf power index
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Banzhaf power index
The Banzhaf power index, named after John Banzhaf (originally invented by Lionel Penrose in 1946 and sometimes called Penrose–Banzhaf index; also known as the Banzhaf–Coleman index after James Samuel Coleman), is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders.
To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A critical voter is a voter who, if changing their vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that they could cast. There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.
A simple voting game, taken from Game Theory and Strategy by Philip D. Straffin:
[6; 4, 3, 2, 1]
The numbers in the brackets mean a measure requires 6 votes to pass, and voter A can cast four votes, B three votes, C two, and D one. The winning groups, with underlined swing voters, are as follows:
AB, AC, ABC, ABD, ACD, BCD, ABCD
There are 12 total swing votes, so by the Banzhaf index, power is divided thus:
A = 5/12, B = 3/12, C = 3/12, D = 1/12
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Banzhaf power index
The Banzhaf power index, named after John Banzhaf (originally invented by Lionel Penrose in 1946 and sometimes called Penrose–Banzhaf index; also known as the Banzhaf–Coleman index after James Samuel Coleman), is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders.
To calculate the power of a voter using the Banzhaf index, list all the winning coalitions, then count the critical voters. A critical voter is a voter who, if changing their vote from yes to no, would cause the measure to fail. A voter's power is measured as the fraction of all swing votes that they could cast. There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.
A simple voting game, taken from Game Theory and Strategy by Philip D. Straffin:
[6; 4, 3, 2, 1]
The numbers in the brackets mean a measure requires 6 votes to pass, and voter A can cast four votes, B three votes, C two, and D one. The winning groups, with underlined swing voters, are as follows:
AB, AC, ABC, ABD, ACD, BCD, ABCD
There are 12 total swing votes, so by the Banzhaf index, power is divided thus:
A = 5/12, B = 3/12, C = 3/12, D = 1/12
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