Banzhaf power index
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
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.
Examples
Voting game
Simple voting game
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
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.
“Banzhaf power index” enters the record as 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. Crown Archives preserves that source wording while asking what Banzhaf, power and index can confirm, complicate or overturn.
Why this record matters
“Banzhaf power index” is worth following because a concise public description often conceals a longer documentary argument. Here, Banzhaf, power and index provides the most credible route into that argument.
Vocabulary and entity names are the principal evidence signals here, because they determine the precision of every later search. The source revision retrieved here is dated Aug 29, 2026. The linked authority identifier is Q2742375. None of the 0 selected statements returned an explicit reference. The first chronological checks are 1946.
A concise general-reference account can conceal disagreements about scope, terminology or the weight assigned to individual sources. The lead is largely declarative, so disagreement and counter-evidence require a deliberate search beyond the opening account. Authority statements aid reconciliation but still require their own references, qualifiers and ranks to be checked.
How to read it
Use the entry as an orientation point, then follow its citations and revision history. Names, dates and institutional relationships should be checked against the original record.
- Subject orientation
- Search vocabulary
- Locating named sources
The closest primary source, responsible institution and strongest cited specialist reference.
Three-step research path
- Establish the record: confirm the title “Banzhaf power index”, its source revision and the description used here.
- Expand the search: follow Banzhaf power index primary sources, Banzhaf power index archive and Banzhaf research across catalogues and specialist indexes.
- Test the account: compare the strongest cited source with the responsible institution’s current record and note any disagreement.
Questions for further research
- Which source most directly establishes the central claim about “Banzhaf power index”?
- Which cited source is closest to the event, object or claim?
- Which institution is responsible for the underlying evidence?
Search terms from this dossier
This entry incorporates text from “Banzhaf power index” on English Wikipedia. Contributors are listed in the page history. Text is available under the Creative Commons Attribution-ShareAlike 4.0 License. Selected authority identifiers and statements are retrieved from Wikidata under CC0; their references and qualifiers remain part of the verification path.