Misplaced Pages

Latest revision as of 22:43, 19 November 2024

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations. (May 2016) (Learn how and when to remove this message) This article cites its sources but does not provide page references. You can help by providing page numbers for existing citations. (August 2017) (Learn how and when to remove this message) (Learn how and when to remove this message)

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 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. There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.

Examples

Voting game

Simple voting game

A simple voting game, taken from Game Theory and Strategy by Philip D. Straffin:

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

U.S. Electoral College

Consider the United States Electoral College. Each state has different levels of voting power. There are a total of 538 electoral votes. A majority vote is 270 votes. The Banzhaf power index would be a mathematical representation of how likely a single state would be able to swing the vote. A state such as California, which is allocated 55 electoral votes, would be more likely to swing the vote than a state such as Montana, which has 3 electoral votes.

Assume the United States is having a presidential election between a Republican (R) and a Democrat (D). For simplicity, suppose that only three states are participating: California (55 electoral votes), Texas (38 electoral votes), and New York (29 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 Georgia, with only 16 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 was originally introduced by Lionel Penrose in 1946 and went largely forgotten. It was reinvented by John F. Banzhaf III in 1965, but it had to be reinvented once more by James Samuel Coleman in 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, are A-F in

There are 32 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.

Today, the Banzhaf power index is an accepted way to measure voting power, along with the alternative Shapley–Shubik power index. Both measures have been applied to the analysis of voting in the Council of the European Union.

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.

See also

• Game theory
• Shapley–Shubik power index
• Penrose method
• Penrose square root law

Notes

1. Banzhaf did not understand how voting in Nassau County actually worked. Initially 24 votes were apportioned to Hempstead, resulting in 36 total votes. Hempstead was then limited to half of the total, or 18, or 9 for each supervisor. The six eliminated votes were not voted, and the majority required to pass a measure remained at 19.
2. Many sources claim that Banzhaf sued (and won). In the original Nassau County litigation, Franklin v. Mandeville 57 Misc.2d 1072 (1968), a New York court ruled that voters in Hempstead were denied equal protection equal because while the town had a majority of the population, they did not have a majority of the weighted vote. Weighted voting would be litigated in Nassau County for the next 25 years, until it was eliminated.

References

Footnotes

1. Matsui & Matsui 2000.
2. ^ Straffin 1993.
3. Penrose 1946.
4. Felsenthal & Machover 1998, p. 5.
5. Banzhaf 1965.
6. Coleman 1971.
7. Varela & Prado-Dominguez 2012.
8. Gelman, Katz & Tuerlinckx 2002.

Bibliography

• Banzhaf, John F. (1965). "Weighted Voting Doesn't Work: A Mathematical Analysis". Rutgers Law Review. 19 (2): 317–343. ISSN 0036-0465.
• Coleman, James S. (1971). "Control of Collectives and the Power of a Collectivity to Act". In Lieberman, Bernhardt (ed.). Social Choice. New York: Gordon and Breach. pp. 192–225.
• Felsenthal, Dan S.; Machover, Moshé (1998). The Measurement of Voting Power Theory and Practice, Problems and Paradoxes. Cheltenham, England: Edward Elgar.
• Felsenthal, Dan S.; Machover, Moshé (2004). "A Priori Voting Power: What is it All About?" (PDF). Political Studies Review. 2 (1): 1–23. doi:10.1111/j.1478-9299.2004.00001.x. ISSN 1478-9302. S2CID 145284470.
• Gelman, Andrew; Katz, Jonathan; Tuerlinckx, Francis (2002). "The Mathematics and Statistics of Voting Power". Statistical Science. 17 (4): 420–435. doi:10.1214/ss/1049993201. ISSN 0883-4237.
• Lehrer, Ehud (1988). "An Axiomatization of the Banzhaf Value" (PDF). International Journal of Game Theory. 17 (2): 89–99. CiteSeerX 10.1.1.362.9991. doi:10.1007/BF01254541. ISSN 0020-7276. S2CID 189830513. Retrieved 30 August 2017.
• Matsui, Tomomi; Matsui, Yasuko (2000). "A Survey of Algorithms for Calculating Power Indices of Weighted Majority Games" (PDF). Journal of the Operations Research Society of Japan. 43 (1): 71–86. doi:10.15807/jorsj.43.71. ISSN 0453-4514. Retrieved 30 August 2017.
• Penrose, Lionel (1946). "The Elementary Statistics of Majority Voting". Journal of the Royal Statistical Society. 109 (1): 53–57. doi:10.2307/2981392. ISSN 0964-1998. JSTOR 2981392.
• Straffin, Philip D. (1993). Game Theory and Strategy. New Mathematical Library. Vol. 36. Washington: Mathematical Association of America.
• Varela, Diego; Prado-Dominguez, Javier (2012). "Negotiating the Lisbon Treaty: Redistribution, Efficiency and Power Indices". Czech Economic Review. 6 (2): 107–124. ISSN 1802-4696. Retrieved 30 August 2017.

External links

This article's use of external links may not follow Misplaced Pages's policies or guidelines. Please improve this article by removing excessive or inappropriate external links, and converting useful links where appropriate into footnote references. (May 2016) (Learn how and when to remove this message)

• Online Power Index Calculator (by Tomomi Matsui)
• Banzhaf Power Index Includes power index estimates for the 1990s U.S. Electoral College.
• Voting Power Perl calculator for the Penrose index.
• Computer Algorithms for Voting Power Analysis Web-based algorithms for voting power analysis
• Power Index Calculator Computes various indices for (multiple) weighted voting games online. Includes some examples.
• Computing Banzhaf power index and Shapley–Shubik power index with Python and R (by Frank Huettner)
• Banzhaf Power Index at the Wolfram Demonstrations Project

• Cooperative games
• Game theory
• Political science theories
• Voting theory