Examples
A simple voting game, taken from Game Theory and Strategy by Phillip 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
Consider the U.S. Electoral College. Each state has more or less power than the next state. There are a total of 538 electoral votes. A majority vote is considered 270 votes. The Banzhaf power index would be a mathematical representation of how likely a single state would be able to swing the vote. For a state such as California, which is allocated 55 electoral votes, they would be more likely to swing the vote than a state such as Montana, which only has 3 electoral votes.
The United States is having a presidential election between a Republican and a Democrat. For simplicity, suppose that only three states are participating: California (55 electoral votes), Texas (34 electoral votes), and New York (31 electoral votes).
The possible outcomes of the election are:
| California (55) | Texas (34) | New York (31) | R votes | D votes | States that could swing the vote |
|---|---|---|---|---|---|
| R | R | R | 120 | 0 | none |
| R | R | D | 89 | 31 | California (D would win 86-34), Texas (D would win 65-55) |
| R | D | R | 86 | 34 | California (D would win 89-31), New York (D would win 65-55) |
| R | D | D | 55 | 65 | Texas (R would win 89-31), New York (R would win 86-34) |
| D | R | R | 65 | 55 | Texas (D would win 89-31), New York (D would win 86-34) |
| D | R | D | 34 | 86 | California (R would win 89-31), New York (R would win 65-55) |
| D | D | R | 31 | 89 | California (R would win 86-34), Texas (R would win 65-55) |
| D | D | D | 0 | 120 | none |
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 Ohio, with only 20 electoral votes, the situation changes dramatically.
| California (55) | Texas (34) | Ohio (20) | R votes | D votes | States that could swing the vote |
|---|---|---|---|---|---|
| R | R | R | 109 | 0 | California (D would win 55-54) |
| R | R | D | 89 | 20 | California (D would win 75-34) |
| R | D | R | 75 | 34 | California (D would win 89-20) |
| R | D | D | 55 | 54 | California (D would win 109-0) |
| D | R | R | 54 | 55 | California (R would win 109-0) |
| D | R | D | 34 | 75 | California (R would win 89-20) |
| D | D | R | 20 | 89 | California (R would win 75-34) |
| D | D | D | 0 | 109 | California (R would win 55-54) |
In this example, the Banzhaf index gives California 1 and the other states 0, since California alone has more than half the votes.
Read more about this topic: Banzhaf Power Index
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