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Section 11.3 Instant Runoff Voting

Instant Runoff Voting (IRV), also called Plurality with Elimination, is a modification of the plurality method that attempts to address the issue of insincere voting.

Subsection 11.3.1 Instant Runoff Voting (IRV)

In IRV, voting is done with preference ballots, and a preference schedule is generated. The choice with the least first-place votes is then eliminated from the election, and any votes for that candidate are redistributed to the voters’ next choice. This continues until a choice has a majority (over 50%).

This is similar to the idea of holding runoff elections, but since every voter’s order of preference is recorded on the ballot, the runoff can be computed without requiring a second costly election.

This voting method is used in several political elections around the world, including election of members of the Australian House of Representatives, and was used for county positions in Pierce County, Washington until it was eliminated by voters in 2009. A version of IRV is used by the International Olympic Committee to select host nations.

Example 11.3.1.

Consider the preference schedule below, in which a company’s advertising team is voting on five different advertising slogans, called A, B, C, D, and E here for simplicity.

Table 11.3.2.
3 4 4 6 2 1
1st Choice B C B D B E
2nd Choice C A D C E A
3rd Choice A D C A A D
4th Choice D B A E C B
5th Choice E E E B D C

If this was a plurality election, note that B would be the winner with 9 first-choice votes, compared to 6 for D, 4 for C, and 1 for E.

There are total of \(3+4+4+6+2+1 = 20\) votes. A majority would be \(11\) votes. No one yet has a majority, so we proceed to elimination rounds.

Round 1: We make our first elimination. Choice A has the fewest first-place votes, so we remove that choice.

Table 11.3.3.
3 4 4 6 2 1
1st Choice B C B D B E
2nd Choice C D C E
3rd Choice D C D
4th Choice D B E C B
5th Choice E E E B D C

We then shift everyone’s choices up to fill the gaps. There is still no choice with a majority, so we eliminate again.

Table 11.3.4.
3 4 4 6 2 1
1st Choice B C B D B E
2nd Choice C D D C E D
3rd Choice D B C E C B
4th Choice E E E B D C

Round 2: We make our second elimination. Choice E has the fewest first-place votes, so we remove that choice, shifting everyone’s options to fill the gaps.

Table 11.3.5.
3 4 4 6 2 1
1st Choice B C B D B D
2nd Choice C D D C C B
3rd Choice D B C B D C

Notice that the first and fifth columns have the same preferences now, we can condense those down to one column.

Table 11.3.6.
5 4 4 6 1
1st Choice B C B D D
2nd Choice C D D C B
3rd Choice D B C B C

Now B has 9 first-choice votes, C has 4 votes, and D has 7 votes. Still no majority, so we eliminate again.

Round 3: We make our third elimination. C has the fewest votes.

Table 11.3.7.
3 4 4 6 2 1
1st Choice B D B D B D
2nd Choice D B D B D B

Condensing this down:

Table 11.3.8.
9 11
1st Choice B D
2nd Choice D B

D has now gained a majority, and is declared the winner under IRV.

Problem 11.3.9. Try It Now.

Consider again the election,

Table 11.3.10.
14 44 20 70 22 80 39
1st choice G G G M M B B
2nd choice M B G B M
3rd choice B M B G G

Find the winner using IRV.

Answer.

G has the fewest first-choice votes, so is eliminated first. The 20 voters who did not list a second choice do not get transferred - they simply get eliminated.

Table 11.3.11.
136 133
1st Choice M B
2nd Choice B M

McCarthy (M) now has a majority, and is declared the winner.

Subsection 11.3.2 What's Wrong with IRV?

Example 11.3.12.

Let’s return to our City Council Election. Who wins using the IRV method?

Table 11.3.13.
342 214 298
1st Choice Elle Don Key
2nd Choice Don Key Don
3rd Choice Key Elle Elle
Solution.

In this election, Don has the smallest number of first place votes, so Don is eliminated in the first round. The 214 people who voted for Don have their votes transferred to their second choice, Key.

Table 11.3.14.
342 512
1st Choice Elle Key
2nd Choice Key Elle

So Key is the winner under the IRV method.

We can immediately notice that in this election, IRV violates the Condorcet Criterion, since we determined earlier that Don was the Condorcet winner. On the other hand, the temptation has been removed for Don’s supporters to vote for Key; they now know their vote will be transferred to Key, not simply discarded.

Example 11.3.15.

Consider the voting system below.

Table 11.3.16.
37 22 12 29
1st Choice Adams Brown Brown Carter
2nd Choice Brown Carter Adams Adams
3rd Choice Carter Adams Carter Brown

In this election, Carter would be eliminated in the first round, and Adams would be the winner with 66 votes to 34 for Brown.

Now suppose that the results were announced, but election officials accidentally destroyed the ballots before they could be certified, and the votes had to be recast. Wanting to “jump on the bandwagon”, 10 of the voters who had originally voted in the order Brown, Adams, Carter change their vote to favor the presumed winner, changing those votes to Adams, Brown, Carter.

Table 11.3.17.
47 22 2 29
1st Choice Adams Brown Brown Carter
2nd Choice Brown Carter Adams Adams
3rd Choice Carter Adams Carter Brown

In this re-vote, Brown will be eliminated in the first round, having the fewest first-place votes. After transferring votes, we find that Carter will win this election with 51 votes to Adams’ 49 votes! Even though the only vote changes made favored Adams, the change ended up costing Adams the election. This doesn’t seem right, and introduces our second fairness criterion.

Definition 11.3.18. Monotonicity Criterion.

If voters change their votes to increase the preference for a candidate, it should not harm that candidate’s chances of winning.

This criterion is violated by the election in the previous example. Note that even though the criterion is violated in this particular election, it does not mean that IRV always violates the criterion; just that IRV has the potential to violate the criterion in certain elections.