In a weighted voting system, each participant has a specified number of votes, their voting weight. If the voting weight of Voter is more than that of Voter , then may have more power than to influence the outcome and certainly won’t have less power. (We will see that voters with different voting weights may actually have equal power.)
A weighted voting system is provided with a quota. If the sum of the voting weights of voters who favor a motion is greater than or equal to the quota, then “yes” wins. If the total voting weight of voters favoring the motion is less than the quota, then “no” wins.
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EXAMPLE 1 The Film Selection Committee
The Classic Film Society presents a film each month. A three-member committee—whose members are a faculty adviser, Allen, and two students, Betty and Cao—selects the films. A selection must be approved by Allen and at least one of the students.
The selection committee could make its decisions by using a weighted voting system in which Allen has a voting weight of 2, and each student has a voting weight of 1. Thus, Allen, together with one student, would have a total voting weight of 3, while the two students without Allen, or Allen by himself, would have a voting weight of 2. Therefore if we set the quota to be 3, selections will be approved if and only if the committee’s criteria for approval are satisfied.
We have to be careful when setting a quota. For example, if the Film Selection Committee described in Example 1 had a quota more than 4, a film could not be approved even if the members were unanimously in favor. In general, the quota should not be greater than the total weight of all the voters.
The quota for the Film Selection Committee should also not be 2 (or less). With 2 as the quota, Betty and Cao could approve Superman while Allen could approve The Great Dictator. Which would be shown? In general, the quota should be greater than half of the total weight of all the voters, to avoid situations where contradictory motions can pass.
In the United States, some county legislatures use weighted voting systems (see Spotlight 11.4 on page 480). Although the framers of the Constitution intended otherwise, the U.S. Electoral College functions as a weighted voting system when electing the president. The voters are the states (see Spotlight 11.1 on page 465).
Notation for Weighted Voting Systems DEFINITION
A weighted voting system is described by specifying the voting weights of the participants, and the quota, . The following notation is a shorthand way of making these specifications:
For example, the weighted voting system we found for the Film Selection Committee (Example 1) would be .
For the system , make a list of all whole numbers that could be the quota, .
Dictator DEFINITION
A voter is a dictator if a motion will pass if and only if is in favor, and the votes of the other members make no difference.
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If one of the voters in the system is a dictator, what whole number or numbers could be?
Dummy Voter DEFINITION
A participant in a voting system who never has an opportunity to cast a deciding vote is called a dummy voter.
If a voting system has a dictator, all the participants except the dictator are dummy voters, but there are situations where there are dummy voters who may not be aware that they have no power.
EXAMPLE 2 Dummy Voters
In the voting system , the weight-1 voter is a dummy because a motion will pass only if it has the support of the weight-5 and weight-3 voters, and then the additional 1 vote is not needed. For a subtler example, consider a committee with members Andy, Beth, Cathy, and don. They use the weighted voting system . Any two of Andy, Beth, and Cathy have enough votes to pass a motion. don plus one other member have just 51 votes, less than the quota. To pass a motion, don would have to get two of the other members to vote with him. Because the other two members could pass the motion without don, he is a dummy voter.
If exactly one of the voters in the system is a dummy, what whole number or numbers could be? Can be chosen so that no voter is a dummy?
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.)
Veto Power DEFINITION
A voter whose vote is necessary to pass any motion is said to have veto power.
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Allen, the faculty adviser in the Film Selection Committee, has veto power because his vote is required in order to approve a selection. In any weighted voting system, a voter has veto power if the sum of the voting weights of all of the other voters is less than the quota. There are voting systems, such as juries in criminal trials, that require the voters to be unanimous to pass a motion. In these voting systems, each voter has veto power.
EXAMPLE 4 The Film Selection Committee Revisited
Betty has just graduated and is now working on her Master’s degree. To reflect her distinction, the Film Selection Committee now uses the system . Allen, the weight-5 voter, still has veto power because the combined weight of Betty and Cao is only 4, and the quota is 6. No motion can pass if Allen is opposed.
If exactly two of the voters in the system have veto power, what whole number or numbers could be? Would the other voters be dummies?
The voters in Example 4 are not equally powerful: Allen has veto power, while neither Betty nor Cao do, yet Betty and Cao are not dummies. We can’t compare power by comparing the voting weights, either. Betty will be disappointed to find that she has the same voting power as Cao. Together, they can prevent Allen from passing a motion, and either one can combine with Allen to pass a motion.
The Electoral College Spotlight 11.1
In the United States, the Electoral College votes to elect the president and vice president. Each state is represented by electors, appointed in a manner prescribed by its legislature. The number of electors for a state is equal to the size of its delegation in Congress (senators and representatives). in addition, the District of Columbia appoints three electors. in every state, there is a general election in which voters choose a “ticket” consisting of a candidate for president and a candidate for vice president, nominated by a political party. in 2012, the Republican and Democratic parties nominated the Romney-Ryan and the Obama-Biden tickets, respectively. All but two states chose their electors by a statewide contest. For example, in California the Democratic ticket received 60% of the vote and the Republican ticket received 37%, but California’s 55 electors all voted for the Obama-Biden ticket. The two exceptions are Maine and Nebraska, which use the District System to choose their electors. With the District System, two electors (corresponding to senators) go to the ticket that received a plurality in the statewide election. The remaining electors are chosen by congressional district. in 2012, Maine’s four electors voted for the Democratic ticket, and Nebraska’s five electors voted Republican. (in 2008, the Nebraska electors were split: Four voted Republican and one voted Democratic.) if the District System interests you, see Writing Project 3 on page 501 and Exercises 12 and 25 on pages 497 and 498, respectively.
Although the framers may have envisioned the Electoral College as a deliberative body, each elector has to vote his or her party’s ticket—there is no room for negotiation. Effectively, the Electoral College functions as a weighted voting system, in which there are 56 participants: the 50 states, the District of Columbia, and five congressional districts in Maine and Nebraska.