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With three or more candidates, we find no shortage of additional procedures that suggest themselves and that seem to represent perfectly reasonable ways to choose a winner. Closer inspection, however, reveals shortcomings with all of these. We illustrate this with a consideration of several well-known procedures. Additional procedures (and additional shortcomings) can be found in the exercises.

Plurality Voting and the Condorcet Winner Criterion

In plurality voting, only first-place votes are considered. Thus, while we will consider plurality voting in the context of preference list ballots, a ballot here might just as well be a single vote for a single candidate. The candidate with the most votes wins, even though this may be considerably fewer than one-half the total votes cast. This is perhaps the most common system in use today. It is how the voters in Florida chose George W. Bush over Al Gore, Ralph Nader, and Patrick J. Buchanan in the presidential election of 2000.

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The CWC is certainly a property that one would like to see satisfied. We record plurality voting’s failure in this respect with the following.

Perhaps a more fundamental drawback of plurality voting is the extent to which the ballots provide no opportunity for a voter to express any preferences except for naming his or her top choice. No use is made, for example, of the fact that a candidate may be no one’s first choice but everyone’s close second choice.

To illustrate this point, we return to the mathematics department’s five-person hiring committee that has interviewed four candidates: Adam, Beth, Carol, and Dan. They have ranked the candidates on their five ballots (reproduced in the table below), but now, let’s suppose that instead of using Condorcet’s method, they decide to use plurality.

Finally, there is yet another shortcoming of plurality voting: There are elections in which it is to a voter’s advantage to submit a ballot that misrepresents his or her true preferences.

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In the presidential election of 2000, many voters who ranked Ralph Nader or Patrick J. Buchanan over George W. Bush and Al Gore chose to vote for Bush or Gore rather than to “throw away” their vote on a candidate they believed had no chance. Condorcet’s method, it turns out, is not manipulable, and this is one of its most desirable properties. We’ll explore this further in the next chapter.

The Borda Count and Independence of Irrelevant Alternatives

In many elections that use preference list ballots, the goal is to arrive at a final group rank ordering of all the contestants that best expresses the desires of the electorate. The purpose is not only to determine the winner—say, the class valedictorian—but also to arrive at who finished second, third, and so on, as in the case of one’s rank in the senior class. In other applications, such as an election to a hall of fame, the first few finishers each win, while the remaining nominees are also-rans.

One common mechanism for achieving this objective is to assign points to each voter’s rankings and then to sum these for all voters to obtain the total points for each candidate. If there are 10 candidates, for example, then we could assign 9 points to each first-place vote for a given candidate, 8 points for each second- place vote, 7 for each third-place vote, and so forth. The candidate with the highest total number of points is the winner. Subsequent positions are assigned to those with the next-highest tallies.

The Borda count is named after Jean-Charles de Borda (1733–1799), who was a contemporary of Condorcet.

Rank methods other than the Borda count are common. For example, a track meet can be thought of as an “election” in which each event is a “voter” and each of the schools competing is a “candidate.” If the order of finish in the 100-meter dash is school , school , school , school , then points are often awarded to each school as follows: 5 points for first place, 3 for second place, 2 for third place, and 1 for fourth place.

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Sports polls often use point assignments that qualify as rank methods according to our definition. The following example provides an illustration of this.

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There is an easy way to calculate the Borda score of a candidate. You can count the number of occurrences of other candidate names that are below this candidate’s name. For example, consider the following ballots:

Because there are three candidates, each first-place vote is , or ; each second-place vote is ; and each third-place vote is . If we were to calculate the Borda score of Candidate algebraically, we would say that has two first-place votes, worth 2 points each (a total of 4 points), and three second-place votes, worth 1 point each (a total of 3 more points). Thus, the Borda score of Candidate is .

But instead of calculating this Borda score algebraically, we can mentally replace each occurrence of a letter below by a box, □, and simply count the boxes.

Notice that there are seven boxes, giving us the correct value of 7 as the Borda score for Candidate . Of course, you don’t actually have to draw any boxes. We are just emphasizing the fact that, in the counting process, it is “spaces” that we are counting, without regard to which letter occurs in the space. A quick glance at the original ballots (without the boxes) reveals that the Borda score of Candidate is 6 and the Borda score of Candidate is 2. When calculating Borda scores this way, be sure that each individual ballot is listed separately, as opposed to using a single list to represent the ballots of several voters (as we often do).

The Borda count certainly seems to be a reasonable way to choose a winner from among several candidates (or to arrive at a group ranking of the candidates). It also has its shortcomings, however, one of which is the failure of a property known as independence of irrelevant alternatives (IIA).

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To describe this property, suppose that an election yields one candidate (call it ) as a winner and another candidate (call it ) as a nonwinner. Suppose that a new election is now held and that, although some of the voters may have changed their preference list ballots, no one who had previously ranked over changed his or her ballot to rank over now.

If this new election were to yield as a winner, the new outcome would seem strange, especially because not one of the relative individual preferences for over had changed in ’s favor. The ballot changes responsible for the new outcome involve candidates other than or . One could argue that these other candidates ought to be “irrelevant” to the question of whether is more desirable than or is more desirable than . This inspires the name “independence of irrelevant alternatives.”

Condorcet’s method satisfies IIA. That is, if we have a sequence of preference list ballots that yield as a Condorcet winner and as a nonwinner, then defeats every other candidate, and in particular, in a one-on-one contest according to these ballots. If no voter reverses the order in which he or she ranked and , then will still defeat one on one, and thus remains a nonwinner.

The following illustration shows that the Borda count, unlike Condorcet’s method, fails to satisfy IIA. Suppose the initial five ballots are as follows:

Our counting procedure shows that the Borda scores are as follows:

The winner is (with 6 points), and is a nonwinner (with 5 points). But now suppose that the two voters on the right change their ballots by moving down between and . The ballots then become:

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Our counting procedure shows that the Borda scores are as follows:

The Borda count therefore now yields as the winner (with 7 points). Thus, has gone from being a nonwinner to being a winner, even though no one changed his or her mind about whether is preferred to , or vice versa. Hence, the fact that wins and loses is not “independent” of where the “irrelevant” alternative is ranked.

The above discussion establishes the following theorem.

Sequential Pairwise Voting and the Pareto Condition

In our voting-theoretic context, an agenda will be understood to be a listing (in some order) of the candidates. This listing is not to be confused with any of the preference list ballots, and to avoid confusion, we will present agendas as horizontal lists and continue to present preference list ballots vertically.

For a given sequence of individual preference list ballots, the particular agenda chosen can greatly affect the outcome of the election, as we’ll show in the next chapter. Nevertheless, we will see later in this chapter that sequential pairwise voting arises naturally in the legislative process. Notice also that because of our assumption that the number of voters is odd, there is always a unique winner with sequential pairwise voting.

There is something very troubling about the outcome of the preceding example, especially if you are Candidate . Everyone prefers to , but ends up winning! This example shows that sequential pairwise voting fails to satisfy what is called the Pareto condition.

Again, the Pareto condition (named after Italian economist Vilfredo Pareto, 1848–1923) is a property we would like to see satisfied. But Example 6 (with Candidate in the role of and Candidate in the role of ) establishes the following.

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The following sequence of three preference list ballots illustrates the Pareto condition further:

Every one of the three voters prefers to . Hence, if we were using a voting rule that satisfies Pareto, we would conclude that is not among the winners. However, we cannot conclude that is among the winners. Indeed, there are very reasonable voting rules, like plurality, that satisfy the Pareto condition but would produce as the unique winner using these ballots.

Runoff Systems and Monotonicity

The voting system known as the Hare system, which was introduced by Thomas Hare in 1861, is also known by names such as the “single transferable vote system.” In 1862, John Stuart Mill described the Hare system as being “among the greatest improvements yet made in the theory and practice of government.” Today, the system is used to elect public officials in Australia, Malta, the Republic of Ireland, and Northern Ireland.

We now give another example of the Hare system. It is a bit more complicated than Example 8, but as with earlier examples in this section, it will reveal a serious shortcoming of a seemingly very reasonable voting system (the Hare system, in this case).

Clearly, this is once again quite counterintuitive. Alternative won the original election, the only change in ballots made was one favorable to (and no one else), and then lost the next election. This example shows that the Hare system fails to satisfy what is called monotonicity.

As with the CWC and the Pareto condition, monotonicity is a property we would like to see satisfied. But Example 9 (with Candidate in the role of ) establishes the following.

The fact that the Hare system does not satisfy monotonicity is considered by many—and with good reason—to be a glaring defect. A 17-voter example in which only a single candidate is eliminated in the first round can also be used to show that the Hare system does not satisfy monotonicity (see Exercise 28 on page 436). For an even more glaring version of this defect, one in which alternative goes from winning to losing because voters move from last place on their ballots to first place on their ballots, see Exercise 29 (page 436).

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In spite of these drawbacks, the Hare system is used in important ways today. For example, it is essentially the method that was used to choose Rio de Janeiro as the site of the 2016 Summer Olympics. Chicago was eliminated in the first round on the basis of fewest first-place votes, then Tokyo in the second round, and Madrid in the final round.

There are other runoff systems, some more frequently used than the Hare system. One such example is the following.