The Banzhaf power index, named after John F. Banzhaf III (originally invented by

Lionel Penrose Lionel Sharples Penrose, FRS (11 June 1898 – 12 May 1972) was an English psychiatrist, medical geneticist, paediatrician, mathematician and chess theorist, who carried out pioneering work on the genetics of intellectual disability. Penrose w ...

in 1946 and sometimes called Penrose–Banzhaf index; also known as the Banzhaf–Coleman index after

James Samuel Coleman James Samuel Coleman (May 12, 1926 – March 25, 1995) was an American sociologist, theorist, and empirical researcher, based chiefly at the University of Chicago. He was elected president of the American Sociological Association in 1991. He stud ...

), is a

power Power most often refers to: * Power (physics), meaning "rate of doing work" ** Engine power, the power put out by an engine ** Electric power * Power (social and political), the ability to influence people or events ** Abusive power Power may a ...

index defined by the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

of changing an outcome of a

vote Voting is a method by which a group, such as a meeting or an electorate, can engage for the purpose of making a collective decision or expressing an opinion usually following discussions, debates or election campaigns. Democracies elect holde ...

where voting rights are not necessarily equally divided among the voters or

shareholder A shareholder (in the United States often referred to as stockholder) of a corporation is an individual or legal entity (such as another corporation, a body politic, a trust or partnership) that is registered by the corporation as the legal own ...

s. 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 Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. I ...

techniques, enumeration methods and

Monte Carlo method Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

Examples

Voting game

Simple voting game

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

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 The United States Electoral College is the group of presidential electors required by the Constitution to form every four years for the sole purpose of appointing the president and vice president. Each state and the District of Columbia appo ...

. Each state has more or less power than the next state. There are a total of 538 electoral votes. A

majority vote A majority, also called a simple majority or absolute majority to distinguish it from related terms, is more than half of the total.Dictionary definitions of ''majority'' aMerriam-Webster California California is a U.S. state, state in the Western United States, located along the West Coast of the United States, Pacific Coast. With nearly 39.2million residents across a total area of approximately , it is the List of states and territori ...

, which is allocated 55 electoral votes, would be more likely to swing the vote than a state such as

Montana Montana () is a state in the Mountain West division of the Western United States. It is bordered by Idaho to the west, North Dakota and South Dakota to the east, Wyoming to the south, and the Canadian provinces of Alberta, British Columbi ...

, which has 3 electoral votes. Assume the United States is having a

presidential election A presidential election is the election of any head of state whose official title is President. Elections by country Albania The president of Albania is elected by the Assembly of Albania who are elected by the Albanian public. Chile The pre ...

between a

Republican Republican can refer to: Political ideology * An advocate of a republic, a type of government that is not a monarchy or dictatorship, and is usually associated with the rule of law. ** Republicanism, the ideology in support of republics or agains ...

(R) and a

Democrat Democrat, Democrats, or Democratic may refer to: Politics *A proponent of democracy, or democratic government; a form of government involving rule by the people. *A member of a Democratic Party: **Democratic Party (United States) (D) **Democratic ...

(D). For simplicity, suppose that only three states are participating: California (55 electoral votes),

Texas Texas (, ; Spanish language, Spanish: ''Texas'', ''Tejas'') is a state in the South Central United States, South Central region of the United States. At 268,596 square miles (695,662 km2), and with more than 29.1 million residents in 2 ...

(38 electoral votes), and

New York New York most commonly refers to: * New York City, the most populous city in the United States, located in the state of New York * New York (state), a state in the northeastern United States New York may also refer to: Film and television * '' ...

(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.

Cartel game

Five companies (A, B, C, D, E) sign an agreement for the creation of a

monopoly A monopoly (from Greek language, Greek el, μόνος, mónos, single, alone, label=none and el, πωλεῖν, pōleîn, to sell, label=none), as described by Irving Fisher, is a market with the "absence of competition", creating a situati ...

. The size of the market is ''X'' = 54 million units per year (e.g. petroleum barrels) for a monopoly. The maximum production capacity of these companies is A = 44, B = 32, C = 20, D = 8 and E = 4 million units per year. Therefore, there is a set of coalitions able to provide the 54 million units necessary for the monopoly, and a set of coalitions unable to provide that number. In each of the sufficient coalitions one may have necessary members (for the coalition to provide the required production) and unnecessary members (underlined in the table below). Even when ''one'' of these unnecessary members goes out of the sufficient coalition that coalition is able to provide the required production. However, when ''one'' necessary member leaves, the sufficient coalition becomes insufficient. The monopoly's profit to be distributed among the coalition's members is 100 million dollars per year. The Penrose–Banzhaf index may be applied to the calculation of the

Shapley value The Shapley value is a solution concept in cooperative game theory. It was named in honor of Lloyd Shapley, who introduced it in 1951 and won the Nobel Memorial Prize in Economic Sciences for it in 2012. To each cooperative game it assigns a uniq ...

, which provides a basis for a distribution of the profit for each player in the game in proportion to the number of sufficient coalitions in which that player is necessary. The player A is necessary for 10 of the 16 sufficient coalitions, B is necessary for 6, C also for 6, D for 2 and E for 2. Therefore, A is necessary in 38.5% of the total cases (26 = 10 + 6 + 6 + 2 + 2, so 10/26 = 0.385), B in 23.1%, C in 23.1%, D in 7.7% and E in 7.7% (these are the Banzhaf indexes for each company). The distribution of the 100 million of monopoly profits under the Shapley value's criterion has to follow those proportions.

History

What is known today as the Banzhaf power index was originally introduced by

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

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, empstead #1, Hempstead #2, North Hempstead, Oyster Bay, Glen Cove, Long Beachare A-F in 6; 9, 9, 7, 3, 1, 1 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 The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game. The index often reveals surprising power distribution that is not obvious on the surface. The constituent ...

. Both measures have been applied to the analysis of voting in the

Council of the European Union The Council of the European Union, often referred to in the treaties and other official documents simply as the Council, and informally known as the Council of Ministers, is the third of the seven Institutions of the European Union (EU) as ...

. 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.

Notes

References

Footnotes

Bibliography

* * * * * * * * * *

External links

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

with

Python Python may refer to: Snakes * Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia ** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia * Python (mythology), a mythical serpent Computing * Python (pro ...

and R (by Frank Huettner)
Banzhaf Power Index
at the

Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

{{Use Oxford spelling, date=August 2017 Cooperative games Game theory Political science theories Voting theory de:Machtindex#Banzhaf-Index