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Section5.3Instant Runoff Voting and the Monotonicity Criterion

SubsectionThe Method of Instant Runoff Voting (IRV)

We've seen that plurality voting is not an ideal method. However, the Borda count can violate the majority criterion, which some would argue makes it an unsatisfying solution. Our next method attempts to solve this problem by ensuring that a majority candidate will always win.

Instant Runoff Voting (IRV)

The method of Instant Runoff Voting (IRV) uses several rounds. Each round simulates a runoff, but they are all part of a single election. The rules are as follows:

  1. For the first round, count how many first choice votes each candidate has received. Check to see if a candidate has a majority of 1st place votes. If there is such a candidate, then this candidate wins the election. If there is not a majority winner, continue to the next step.
  2. Eliminate the candidate with the fewest first choice votes.
  3. Use the preference lists of the voters who favored the eliminated candidate to transfer their votes to their next choice among the remaining candidates.
  4. For the next round, count how many votes each remaining candidate has (including newly tranferred votes as well as first choice votes). If a candidate has a majority, they are elected. If not, eliminate the candidate with the fewest votes in this round, and proceed to the next round.
  5. Repeat the above steps until either a candidate wins with a majority or until every candidate but one has been eliminated.

Notice that, because of step 1, any candidate with a majority of first choice votes will automatically win the election. Therefore, IRV satisfies the majority criterion.

What else can we say about IRV? Let's see how it works in practice.

Example5.20

The preference table below shows the same election we considered in Example5.13. We found that Candidate B wins this election when using the Borda count. Let's see who is the winner using IRV.

Number of voters
Rankings 6 5 4 2
1st choice A D C B
2nd choice B B D C
3rd choice D C B D
4th choice C A A A
Table5.21

There are three rounds of IRV, shown in the table below. As we can see, the winner of this election is different with IRV than with the Borda count.

Round 1: Votes for A: 6 D: 5 C: 4 B: 2
B is eliminated, and 2 votes are transferred to those voters' second choice, C
Round 2: Votes for A: 6 D: 5 C: 6
D is eliminated, and 5 votes are transferred to those voters' third choice, C
(since their second choice, B, was already eliminated)
Round 3: Votes for A: 6 C: 11
Candidate C wins the election.

In Example5.20, each candidate was the first choice of a single group of voters. This made the process of transferring votes in each round fairly easy. In the following example, this is not the case.

Example5.22The 2009 mayoral election in Burlington, VT

In 2009, the city of Burlington, Vermont used IRV to elect their mayor. The results of this election are shown below, in a simplified form to include only the three main candidates. 3More complete details can be found at the Wikipedia article. These three candidates were Bob Kiss, Kurt Wright, and Andy Montroll.

Percent of voters
Rankings 34 37 15 9 5
1st choice K W M M M
2nd choice M M K W
3rd choice W K W K
Table5.23

Using this information, we can find the winner of this election using IRV as follows.

Round 1: Percent of votes for K: 34 W: 37 M: 15+9+5=29
M is elimated, and votes are allocated to their different second choices.
Round 2: K: 34+15=49 W: 37+9=46
K wins the election.

Thus, Bob Kiss won this election using instant runoff voting.

Notice that, in this example, the voters who ranked Montroll first had a variety of second choice candidates. When Montroll was eliminated, these votes were transferred to each of these different second choices, rather than all to one candidate.

Additionally, notice that the 5% of voters who did not rank any candidates other than Montroll did not have their votes transferred to any other candidate, since they did not indicate any second choice. Essentially they are saying that, if Montroll does not win, they have no preference between the other candidates. 4Such votes are referred to as exhausted ballots. It may seem strange that their votes were not part of the final tally. However, their votes did count; they just counted only for a candidate that happened to lose.

This example shows how IRV can give a different winner than plurality. The plurality winner in this election was Kurt Wright. However, more votes were transferred from Andy Montroll to Bob Kiss when Montroll was eliminated, so Kiss, rather than Wright, was the winner.

As mayor, Kiss became involved in a scandal and, partially due to his unpopularity, IRV was repealed by a narrow majority of voters in 2010. However, they replaced it with a form of ordinary runoff voting that would still have elected Kiss as mayor had it been used in 2009. This election and its aftermath is discussed on Wikipedia.

Of all the voting systems in this chapter, IRV is the second-most common in real elections (after plurality). In 2016, the state of Maine adopted IRV for most statewide elections. It is also used for local elections in various parts of the US. Internationally, IRV has been used for about a century in Australia (to elect their House of Representatives) and Ireland (to elect their president).

You may encounter different names for IRV in the news. Many in the US call it simply ranked-choice voting, even though it is only one of several voting methods in which voters rank their choices. In Australia it is called preferential voting, while in Britain it is referred to as the alternative vote.

SubsectionThe Monotonicity Criterion

While IRV satisfies the majority criterion, there are other criteria which it fails. To describe one of them, we will consider a hypothetical situation based on the election from Example5.22.

Example5.24A hypothetical mayoral election in Burlington, VT

Suppose that this election had been held as in Example5.22, except that 10% of the voters cast their ballots slightly differently according to the table below.

Percent of voters
Rankings 34 27 10 15 9 5
1st choice K W K M M M
2nd choice M M W K W
3rd choice W K M W K
Table5.25

Before calculating the winner of this election, let's think about who we think should win. The only difference between these preferences and the preferences expressed in the real election is that 10% of voters who originally ranked Wright first now rank Kiss first. Since Kiss won the original election, and the only difference is that now some people like Kiss even better, it stands to reason that Kiss should still win. Right?

Round 1: Percent of votes for K: 44 W: 27 M: 15+9+5=29
W is elimated, and votes are allocated to their different second choice.
Round 2: K: 44 M: 29+27=56
M wins the election.

In the first election, Kiss was the winner. In the second election, when nothing changed except that some voters ranked Kiss higher, the winner changed to Montroll. Do you think this is fair?

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

The Monotonicity Criterion

The monotonicity criterion states that moving a candidate higher on one's preference list should not hurt that candidate.

More precisely, this means that if two elections are held, and the only difference is that the winner of the first election is ranked higher by some voters in the second election, then that candidate should still win the second election.

Exploration5.3
  1. Find the winner of the following election using IRV.
    Number of voters
    Rankings 37 22 12 29
    1st choice Adams Brown Brown Carter
    2nd choice Brown Carter Adams Adams
    3rd choice Carter Adams Carter Brown
    Table5.26Election 1
  2. Find the winner of the following election using IRV. Note that it is similar to Election 1 in the first part of this problem.
    Number of voters
    Rankings 47 22 2 29
    1st choice Adams Brown Brown Carter
    2nd choice Brown Carter Adams Adams
    3rd choice Carter Adams Carter Brown
    Table5.27Election 2
  3. Does a monotonicity violation occur between Elections 1 and 2? Explain why or why not.
  4. Find the winner of the following election using IRV. Note that it is similar to Election 1.
    Number of voters
    Rankings 37 22 2 29
    1st choice Adams Brown Brown Carter
    2nd choice Brown Carter Adams Brown
    3rd choice Carter Adams Carter Adams
    Table5.28Election 3
  5. Compare Election 3 to Election 1. Does this pair of elections demonstrate a violation of the monotonicity criterion? Explain why or why not.
Solution

Election 1: Carter is eliminated in the first round. The 29 votes for Carter are transferred to Adams. In the second round, Adams beats Brown, 66 votes to 34.

Election 2: Brown is eliminated in the first round. 22 of the votes for Brown are transferred to Carter, and 2 are transferred to Adams. In the second round, Carter beats Adams, 51 votes to 49.

Yes, Elections 1 and 2 show a monotonicity violation. The only difference between Election 1 and Election 2 is that 10 voters ranked Adams higher in Election 2 than they did in Election 1. However, Adams won Election 1 and Carter won in Election 2, showing that these 10 voters hurt Adams by ranking that candidate higher.

Election 3: Carter is eliminated in the first round. The 29 votes for Carter are transferred to Brown. In the second round, Brown beats Adams, 63 votes to 37.

No, Elections 1 and 3 do not show a monotonicity violation. Some voters changed their preferences, and the winner did change. However, the difference between Election 1 and Election 3 is that some voters ranked Brown higher in Election 3. The result was that the winner switched from Adams in Election 1 to Brown in Election 3. Thus, the voters who ranked Brown higher helped that candidate by doing so, and therefore the monotonicity criterion was not violated.

Failure of the monotonicity criterion is one drawback of IRV. In contrast, the plurality and Borda count methods both satisfy monotonicity. Why is this?

Suppose some candidate, A, wins a plurality election. Plurality, remember 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.

We have now found that IRV satisfies the majority criterion but violates the monotonicity criterion. The Borda count satisfies the monotonicity criterion but fails the majority criterion. The plurality method satisfies both criteria, but leads to insincere voting and can produce winners with low levels of support. In the next section, we will try to find a method that can solve all these problems at once.