Contemporary Mathematics: Contemporary Mathematics at Nebraska

Recall the preference table and result from Example 7.2.3:

Using Borda Count, the winner was Scarlet & Cream (S). However, you might have reason to feel uneasy with that result. In the election, there were 11 total voters, and 6 of them voted for Walnut Fudge (F) as their first choice. That’s about 54.5%—a clear majority. Yet Walnut Fudge lost to Scarlet & Cream! This motivates our first fairness criterion.

Example 7.2.3 shows that the Borda Count violates the majority criterion: there is an election in which a candidate (Walnut Fudge) had a majority of first choice votes, but another candidate (Scarlet & Cream) won the election. But, does this always happen with the Borda Count? Let’s consider another example.

We can see that there are really two kinds of question we could ask: whether or not a voting method satisfies a fairness criterion in one specific election, and whether or not a voting method satisfies a fairness criterion in general, i.e., in every election. What we’re ultimately interested in is the second of these questions. Let’s make it a little more precise.

Therefore, Example 7.3.2 does not show that the Borda Count satisfies the majority criterion, because it is one example rather than a general argument. Instead, Example 7.2.3 shows that the Borda Count violates the majority criterion. As soon as we find one example of a violation, we know that the method in question violates the criterion we’re looking at.

It sounds like a strong condition to ask that a voting method satisfy a fairness criterion in any possible election, but hopefully it makes sense why. We don’t want a system that just happens to be fair in some elections; we want a voting method that is guaranteed to be fair in any possible election that will be held in the future.

This does mean, however, that it can take some work to show that a voting method satisfies a certain fairness criterion. We can’t just look at one election, or even many elections. Instead, to show that a voting method satisfies a fairness criterion, we must construct an argument explaining why the criterion will be satisfied in any possible election using that method. Let’s see how this works by looking at the plurality method.

Does the plurality method satisfy or violate the majority criterion? Let’s first look again at the election from Example 7.2.1:

The plurality winner of this election was A, with 6 votes out of 17 (about 35%). This is one of many examples we’ve seen in which a plurality winner did not have a majority. However, this does not represent a violation of the majority criterion. Remember that if no majority candidate exists, then the criterion does not apply: it is neither satisfied nor violated. This election does not tell us anything about whether or not the plurality method satisfies the majority criterion.

What if a majority candidate does exist? By definition, that candidate must have over 50% of the vote. Therefore, all other candidates combined must have less than 50%. No matter what, no other candidate can have as many votes as the majority candidate. Hence, any candidate with a majority must necessarily be the plurality winner, so the plurality method satisfies the majority criterion. This is the kind of explanation we use to show that a method satisfies a criterion: we’ve demonstrated why it must be true in any election.

So, in spite of its problems with electing candidates without majority support, the plurality method satisfies the majority criterion, because when a candidate does have majority support, they will be elected. In contrast, the Borda Count does not satisfy the majority criterion. This relates to the Borda Count’s character, mentioned previously, as a consensus-based method. For instance, in Example 7.2.3, Black Walnut Fudge had a majority of first choice votes, but everyone else put it in last place. The Borda Count rejected it in favor of Scarlet & Cream, which had less first choice votes but was not the last choice of anyone. Some people see this as a positive feature of the Borda Count, since it rejects polarizing candidates. Others see its violation of the majority criterion as a drawback.

Recall the Vermont mayoral election from Example 7.2.12 between Kiss (K), Wright (W), and Montroll (M):

When using the method of pairwise comparisons, we saw that Montroll was the pairwise winner because he won the most pairs of any candidate. We can say more, however: not only did Montroll win the most pairs, he actually won every pair in which he was one of the options. In other words, Montroll would never lose a one-on-one contest against any of his opponents. This is could be considered a strong argument for his superiority over other candidates: if some other candidate wins, the majority of voters will prefer Montroll to have won, but if Montroll wins there will be no such majority for any other candidate. A candidate that satisfies this property is called a Condorcet candidate, named after Nicolas, Marquis de Condorcet.

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Condorcet was a French philosopher and mathematician who lived during the 18th century. His ideas were influencial during the earlier part of the French Revolution, and he was a major author of a proposed constitution. However, his ideas were not adopted, and he became a target during the Reign of Terror. After unsuccessfully attempting to flee, Condorcet was arrested and died in prison in 1794. It is still unknown whether his death was suicide or murder. You can read more about his interesting life at the MacTutor History of Mathematics and, of course, Wikipedia.

A Condorcet candidate would seem to be strongly supported by voters; this leads to our next fairness criterion:

Notice that the Condorcet criterion is very similar in form to the majority criterion. Both stipulate that, if a special kind of candidate exists in an election (a majority or Condorcet winner), then that candidate should win. If not, then they do not specify that any particular candidate should win. The main difference is that determining if a Condorcet candidate exists is more involved than determining if a majority winner exists. We may need to run every pair of candidates against each other to see if one of them wins against every other candidate.

Our next fairness criterion is different from the majority and Condorcet criteria, in that it considers two elections: an initial election, and then another in which the original election has been changed in a way that probably shouldn’t change the outcome. Since this is more involved, let’s first look at an example that demonstrates what can go wrong when this new criterion is violated. We will consider a hypothetical situation based on the election from Example 7.2.8.

This is a really strange outcome! It is so unusual that it motivates our next fairness criterion.

Violation of the monotonicity criterion is one drawback of IRV in particular among the voting methods we’ve studied. In contrast, the plurality, Borda count, and pairwise comparison methods all satisfy monotonicity. Why is this?

Suppose some candidate, A, wins a plurality election. Recall that plurality only looks at voters’ first choice votes. If a second election occurs in which some voters rank A higher than before, then the number of first choice votes for A will never decrease; it will only increase or stay the same. (It could stay the same if, for instance, some voters move A from 3rd to 2nd place.) Thus, A will still win the second election. Since this is true for any election, the plurality method satisfies the monotonicity criterion.

Suppose that some candidate, B, is the winner of an election using the Borda count. If a second election occurs and some voters rank B higher, the number of Borda points awarded to B will only increase. This is because higher rankings always correspond to higher point values in the Borda count method. Thus, B will still win the second election. This reasoning applies to any Borda count election, so the Borda count satisfies monotonicity.

Finally, suppose that some candidate, C, is the winner of an election using the method of pairwise comparisons. If a second election occurs and some voters rank C higher, the number of one-on-one contests that C wins can only increase. This is because moving C higher can never reduce the number of votes C gets in a one-on-one contest: if C was already preferred over the other candidate by some voters, C will retain those votes when moved higher on some preference lists. Therefore, C will earn at least as many points as in the original election. Since no other preferences have been changed, every other pair will have the same result as in the original election, so no other candidate will earn any more points than they did originally. Hence, C will still win the second election. Since this is true for any election, the method of pairwise comparisons satisfies the monotonicity criterion.

Like monotonicity, our final fairness criterion considers two elections: an initial election, and then another in which the original election has been changed in a way that shouldn’t change the outcome. In this case, the second election is one in which a losing candidate drops out.

A losing candidate whose participation in the election affects the outcome is known as a spoiler. In plurality elections, a spoiler is often an independent candidate who has little chance of winning, but is similar enough to one of the major candidates to "steal" some of that candidate’s votes, preventing the major candidate from winning. It would be nice if we could find a method that satisfies IIA, and therefore prevents spoilers. Unfortunately, all of the voting methods we’ve studied violate IIA and allow for spoilers.

Based on the results from this example, you may be hopeful that the method of pairwise comparisons satisfies IIA. Unfortunately, this is not the case; there can be violations in elections where there is no Condorcet candidate.