EXAMPLE 3 A Six-Voter System

In 1958, the Board of Supervisors of Nassau County (on Long island, new York) had six members. each was to represent a district in the county. Because the districts had significantly different populations, the supervisors were given different voting weights: 9, 9, 7, 3, 1, 1, with 30 votes in all. The quota was to be a simple majority, 16. in our notation, the voting system was . To pass a measure, two of the three supervisors with voting weight 7 or 9 would be sufficient. The other three supervisors could not cast a decisive vote, even if all three teamed with one of the higher-weight supervisors, because the combined weight of the four would still be less than the quota. if these three joined two of the higher-weight supervisors, a motion would pass, but it would also pass without their votes. in other words, these voters were dummies. This voting system was used by John F. Banzhaf iii to call attention to the need for mathematical analysis of weighted voting. (See Spotlight 11.4 on page 480.)