The Banzhaf Index of Power

The Banzhaf index is a quantity to measure the political power of each member of a voting system. A member in a voting system is, e.g., a party in a parliament or a country in a confederation. In general, each member will have a certain number of votes, and so its power will be different.

The Banzhaf index is derived by simply counting, for each member, the number of winning coalitions it can participate but which are not winning if it does not participate. Suppose the simple case of three parties A, B, and C in a parliament with the following distribution of votes:

Obviously, the majority of 51 votes can only be achieved with a coalition. In this simple case the Banzhaf index is determined by simply counting:

• A is in three winning coalitions: AB, AC, and ABC
• B is in one winning coalition: AB
• C is in one winning coalition: AC

Each of the listed winning coalitions is not winning anymore, if the respective party leaves it. (Note that A is essential for the coalition ABC, but B and C, rsp., are not!) The "Banzhaf power" of A thus is 3, whereas the one of B and C each is 1. This yields the "total Banzhaf power", or "Penrose number", of 3+1+1 = 5. The Banzhaf index is the "normalized" Banzhaf power:

At first glance it is a bit surprising: From the point of view of Banzhaf power, B has no more power than C!

The situation may be generalized to the case where the mere majority of votes does not suffice to yield a winning coalition, but where there has to be achieved a predefined vote quota (e.g. 2/3 majority) or a qualified majority. Sometimes additionally there has to be satisfied a second quota, depending on the population quantity or a veto possibility. Examples are the Council of Europe or the Security Council of the United Nations.

For details see, e.g.,

• Alan D. Taylor (1995): Mathematics and Politics. Strategy, Voting, Power and Proof. Springer-Verlag New York Berlin
• Werner Kirsch et al.: Double Majority and Generalized Brexit: Explaining Counterintuitive Results, arxiv:1812.07048

Some results for different scenarios concerning the Council of the European Union (e.g., Treaties of Nice and Lisbon, with or without UK ...) are documented here: Banzhaf_Council-EU.xls.

Software. You can download a Java application calculating the Banzhaf indices of a voting system with different numbers of members and different vote quota: Banzhaf.jar. It requires a Java Runtime Environment ≥ version 14. Start the program by the command line java -jar Banzhaf.jar.

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