Definitions

# Banzhaf power index

The Banzhaf power index, named after John F. Banzhaf III (though originally invented by and sometimes called Penrose-Banzhaf index), 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 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.

The index is also known as the Banzhaf-Coleman index. See History.

## 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.

## History

What is known today as the Banzhaf Power Index has originally been introduced by and went largely forgotten. It has been reinvented by , but it had to be reinvented once more by before it became part of the mainstream literature.

Banzhaf wanted to prove objectively that the Nassau County Board's voting system was unfair. As given in Game Theory and Strategy, votes were allocated as follows:

• Hempstead #1: 9
• Hempstead #2: 9
• North Hempstead: 7
• Oyster Bay: 3
• Glen Cove: 1
• Long Beach: 1

This is 30 total votes, and a simple majority of 16 votes was required for a measure to pass.

In Banzhaf's notation, [Hempstead #1, Hempstead #2, North Hempstead, Oyster Bay, Glen Cove, Long Beach] are A-F in [16; 9, 9, 7, 3, 1, 1]

There are 33 winning coalitions, and 48 swing votes:

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

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

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

However, Banzhaf's analysis has been critiqued as treating votes like coin-flips, and an empirical model of voting rather than a random voting model as used by Banzhaf brings different results .