EXAMPLE 18 Majority Rule
A committee uses majority rule. What is the probability that a member of the committee will cast a critical vote in a winning coalition? Since every voter has weight 1, no one is a critical voter in a winning coalition unless the coalition has 0 extra votes; it must be a minimal winning coalition.
If there are three members, , , and , then the minimal winning coalitions with are and . The Banzhaf power index of is 2, so the probability that will be critical is
With four voters, the minimal winning coalitions will have three members. would have to be joined by two of the other three members, so ’s Banzhaf power index is . The probability that will be critical is .
If the number of voters is even, say , then a minimal winning coalition has members. Voter would be joined by of the other members to form a minimal winning coalition. The Banzhaf power index of would be and the probability that will be a critical voter is
With an odd number of voters a minimal winning coalition would still have members. Voter would be joined by of the other voters to form a minimal winning coalition, so the Banzhaf power index of would be , and the probability that is critical would be
Table 11.6 shows a few of these probabilities.
| 1 | 100.0% | 10 | 24.61% | 1,000 | 2.523% |
| 2 | 50.0% | 11 | 24.61% | 2,000 | 1.784% |
| 3 | 50.0% | 20 | 17.62% | 10,000 | 0.564% |
| 4 | 37.5% | 21 | 17.62% | 20,000 | 0.399% |
| 5 | 37.5% | 100 | 7.96% | 100,000 | 0.252% |
| 6 | 31.2% | 101 | 7.96% | 1,000,000 | 0.056% |