Banzhaf power index

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The Banzhaf power index, named after John F. Banzhaf III, is the probability of changing an outcome of a vote where power is not 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 swing voters. A "swing voter" is a voter who, if he changed his 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 he could cast.

[edit] Examples

A simple voting game, taken from Game Theory and Strategy by Phillip 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

Consider the U.S. Electoral College. Each state has more or less power than the next state. There are a total of 538 electoral votes. A majority vote is considered 270 votes. The Banzhaf Power Index would be a mathematical representation of how likely a single state would be able to swing the vote. For a state such as California, which is allocated 55 electoral votes, they would be more likely to swing the vote than a state such as Montana, which only has 3 electoral votes.

The United States is having a presidential election between a Republican and a Democrat. For simplicity, suppose that only three states are participating: California (55 electoral votes), Texas (34 electoral votes), and New York (31 electoral votes).

The possible outcomes of the election are:

California (55)	Texas (34)	New York (31)	R votes	D votes	States that could swing the vote
R	R	R	120	0	none
R	R	D	89	31	California (D would win 86-34), Texas (D would win 65-55)
R	D	R	86	34	California (D would win 89-31), New York (D would win 65-55)
R	D	D	55	65	Texas (R would win 89-31), New York (R would win 86-34)
D	R	R	65	55	Texas (D would win 89-31), New York (D would win 86-34)
D	R	D	34	86	California (R would win 89-31), New York (R would win 65-55)
D	D	R	31	89	California (R would win 86-34), Texas (R would win 65-55)
D	D	D	0	120	none

The Banzhaf Power Index of a state is the proportion of the possible outcomes in which that state could swing the election. In this example, all three states have the same index: 4/12 or 1/3.

However, if New York is replaced by Ohio, with only 20 electoral votes, the situation changes dramatically.

California (55)	Texas (34)	Ohio (20)	R votes	D votes	States that could swing the vote
R	R	R	109	0	California (D would win 55-54)
R	R	D	89	20	California (D would win 75-34)
R	D	R	75	34	California (D would win 89-20)
R	D	D	55	54	California (D would win 109-0)
D	R	R	54	55	California (R would win 109-0)
D	R	D	34	75	California (R would win 89-20)
D	D	R	20	89	California (R would win 75-34)
D	D	D	0	109	California (R would win 55-54)

In this example, the Banzhaf index gives California 1 and the other states 0, since California alone has more than half the votes.

[edit] History

The Banzhaf Power Index was invented by John F. Banzhaf III in 1965 when Banzhaf decided to prove objectively that the Nassau County Board's voting system was unfair. Votes were divided as follows (this information also taken from Game Theory and Strategy, though it can probably be verified elsewhere): [Hempstead #1, Hempstead #2, North Hempstead, Oyster Bay, Glen Cove, Long Beach] are A-F in [16; 9, 9, 7, 3, 1, 1]

The winning coalitions and their swing voters are:

AB AC BC ABC ABD ABE ABF ACD ACE ACF BCD BCE BCF ABCD ABCE ABCF ABDE ABDF ABEF ACDE ACDF ACEF BCDE BCDF BCEF ABCDE ABCDF ABCEF ABDEF ACDEF BCDEF ABCDEF

The Banzhaf index gives these values. Hempstead #1 = 16/48 Hempstead #2 = 16/48 North Hempstead = 16/48 Oyster Bay = 0/48 Glen Cove = 0/48 Long Beach = 0/48

Obviously, a voting arrangement that gives 0% of the power to 16% of the population is unfair, and Banzhaf sued the board.

Today, the Banzhaf power index is an accepted way to measure voting power, along with the alternative Shapley-Shubik power index.