Elections in which there are only two candidates present no problem. Majority rule is, as we have seen, an eminently successful voting system in both theory and practice. If there are three or more candidates, however, the situation changes quite dramatically. While several voting systems suggest themselves (plurality, the Borda count, sequential pairwise voting, and the Hare system), each fails to satisfy one or more desired properties (the CWC, IIA, the Pareto condition, and monotonicity). Manipulability is an ever-present problem, as we’ll see in the next chapter. Moreover, when all is said and done, Arrow’s impossibility theorem says that any search for an ideal voting system of the kind that we have discussed is doomed to failure.
Where does this leave us? More than intellectual issues are at stake here: More than 550,000 elected officials serve in approximately 80,000 governments in the United States. Whether it is a small academic department voting on the best senior thesis or a democratic country electing a new leader, multicandidate elections will be contested in one way or another. If there is no perfect voting system—and perhaps not even a best voting system (whatever that may mean; i.e., best in what way?)—what can we do?
Perhaps the answer is that different situations lend themselves to different voting systems, and what is required is a judicious blend of common sense with an awareness of what the mathematical theory has to say. For example, while both the Hare system and the Borda count are subject to manipulability, it seems easier to manipulate the latter. Thus, people may tend to vote more sincerely, rather than strategically, if the Hare system is used instead of the Borda count. This may be a consideration when choosing a voting system for a faculty governance system, for example.
For national political elections, there are also practical considerations. The kind of ballot that we are considering (a preference list ballot) is certainly more complicated than the ballots we now employ. There is, however, a voting system that avoids the practical difficulties caused by the type of ballot being used that has much else to commend it. It is called approval voting.
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Description of Approval Voting PROCEDURE
Under approval voting, each voter is allowed to give one vote to as many of the candidates as he or she finds acceptable. No limit is set on the number of candidates for whom an individual can vote. Voters show disapproval of other candidates simply by not voting for them. The winner under approval voting is the candidate who receives the largest number of approval votes. This approach is also appropriate in situations where more than one candidate can win, for example, in electing new members to an exclusive society such as the National Academy of Sciences or the Baseball Hall of Fame.
EXAMPLE 12 Approval Voting
To illustrate approval voting, suppose that we have nine members of a mathematics department who are trying to choose among five finalists for an open faculty position. They decide to use approval voting—the ballots are indicated in the following table. An X indicates an approval vote. For example, Voter 1, in the first column, approves of Candidates , and .
| Voter (department member) | |||||||||
|---|---|---|---|---|---|---|---|---|---|
| Candidate | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| X | X | X | X | X | X | ||||
| X | X | X | X | X | X | X | |||
| X | X | X | X | ||||||
| X | X | X | X | X | |||||
| X | X | X | X | X | |||||
Counting the Xs in each row shows that six department members (1, 2, 4, 6, 7, and 8) approved of Candidate , seven approved of Candidate , four approved of Candidate , and five approved of and . Thus, Candidate wins, with seven approval votes.
Approval voting was proposed independently by several analysts in the 1970s. Probably the best-known official elected by approval voting today is the secretary- general of the United Nations. In the 1980s, several academic and professional societies initiated the use of approval voting. Examples include the Institute of Electrical and Electronics Engineers (IEEE), with about 400,000 members, and the National Academy of Sciences. In Eastern Europe and some former Soviet republics, approval voting has been used in the form wherein one disapproves of (instead of approving of) as many candidates as one wishes.
Is approval voting the perfect voting system? Certainly not. For example, the type of ballot that is used limits the extent to which voter preferences can be expressed. However, it is certainly a voting system with much potential, and the reader wishing to explore it in more detail can start with Brams and Fishburn’s 1983 monograph, listed in the Suggested Readings on page 438.