Banzhaf power index

The Banzhaf power index, named after John F. Banzhaf III (though originally invented by harvtxt|Penrose|1946 and sometimes calledPenrose-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:

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.

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 harvtxt|Penrose|1946 and went largely forgotten. It has been reinvented by harvtxt|Banzhaf|1965, but it had to be reinvented once more by harvtxt|Coleman|1971 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. Fact|date=March 2008

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 harv|Gelman|Katz|2002.

ee also

* Shapley-Shubik power index
* Penrose method

References

*Citation
last = Banzhaf
first = John F.
authorlink = John F. Banzhaf III
title = Weighted voting doesn't work: A mathematical analysis
journal = Rutgers Law Review
volume = 19
issue = 2
pages = 317-343
year = 1965
url =
*Citation
last = Coleman
first = James S.
authorlink = James S. Coleman
contribution =Control of Collectives and the Power of a Collectivity to Act
editor-last =Lieberman | editor-first=Bernhardt
title = Social Choice
publisher= Gordon and Breach
place = New York
pages = 192-225
year = 1971
url =
*Citation
last = Felsenthal
first = Dan S
last2 = Machover
first2 = Moshé
title = The measurement of voting power theory and practice, problems and paradoxes
year = 1998
location = Cheltenham
publisher = Edward Elgar
url =

*Citation
last = Gelman
first = Andrew
last2 = Katz
first2 = Jonathan
title = The Mathematics and Statistics of Voting Power
journal = Statistical Science
volume = 17
issue = 4
pages = 420-435
year = 2002
url =
*Citation
last = Penrose
first = Lionel
authorlink = Lionel Penrose
title = The Elementary Statistics of Majority Voting
journal = Journal of the Royal Statistical Society
volume = 109
issue = 1
pages = 53-57
year = 1946
url =
* Seth J. Chandler (2007), [http://demonstrations.wolfram.com/BanzhafPowerIndex/ "Banzhaf Power Index"] , The Wolfram Demonstrations Project.

External links

* [http://www.cs.unc.edu/~livingst/Banzhaf/ Banzhaf Power Index] Includes power index estimates for the 1990s U.S. Electoral College.