Banzhaf Index

In weighted voting, each voter has a certain number of votes, and a certain number of votes, called the quota, is needed for each measure to pass. This is summarized in the following notation:

For example, [21: 10, 7, 7, 6, 3] has a quota of 21 and the five voters have 10, 7, 7, 6, and 3 votes respectively. The voters are usually referred to as A, B, C, etc. So in this example A has 10 votes, B and C have 7 each, D has 6 and E has 3.

The Banzhaf index is an attempt to measure how much power each of the voters has. It does so by listing all the winning coalitions, and then looks to see which voters in a coalition are critical to its success; that is, which voters would convert the coalition to a losing one if they were to defect.

The applet below illustrates the process of calculating a Banzhaf Index. For a given vote click all the winning coalitions. Selected coalitions will change color. To check your answers click the Check answers button.

Once the winning coalitions have been selected click the Calculate index tab. You will see a list of all the winning coalitions, with a capital leter indicating a critical voter. Count the number of times each voter is critical to the outcome and enter these values. Click the Check answers button to check your entries and calculate the Index. Since each defection from a winning coalition forms a blocking coalition the Banzhaf Index is 2 times the critical value count.