EXAMPLE 5 Pivotal Voters in the Film Selection Committee

Allen, Betty, and Cao vote to approve film selections by using the voting system . We observed in Example 4 that the weight-5 voter (Allen) has veto power. Betty and Cao have voting weights 3 and 1, respectively. Let’s consider their voting permutations, shown in Table 11.2. (Voters are identified by initials.) Next to each voting permutation, the total weights of the first voter, the first two voters, and all three voters are shown in sequence. The first number in the sequence that equals or exceeds the quota (6) is underlined, and the corresponding pivotal voter’s symbol is circled. We see that Allen is pivotal in four permutations, while Betty and Cao are each pivotal in one. Hence Allen’s share of the voting permutations is , while Betty and Cao each have a share—even though their voting weights differ.