An earlier article in this series showed that the spoiler effect can happen in a ranked-choice election with five candidates. A later article showed this can even happen in a ranked-choice election with
only three candidates. There is no reason to think that these are the only examples, of course, and in this article, we provide a third example. After three strikes in baseball, you are out, but ranked-choice voting somehow seems fixed in many minds as the only available alternative to plurality voting.
This example may seem simpler and less contrived than the previous ones. As with the last example, it involves only three candidates, A, B, and C. Each of the three candidates is opposed by roughly, though not exactly, a third of the voters. And each pair of candidates has the support of roughly a third of the voters and those voters find either of the two equally acceptable. Specifically, the 10,000 voters are divided as follows:
- A is opposed by 3600 voters who support B and C equally.
- B is opposed by 3300 voters who support A and C equally.
- C is opposed by 3100 voters who support A and B equally.
It happens that the candidates are listed on the ballot in alphabetical order and when these voters are forced to decide arbitrarily between two candidates, they more often choose the one that appears earlier on the ballot. In this example, sixty percent choose the one that appears first and only 40% choose the later one.
In summary then, in the election the ranked lists on ballots are distributed as follows:
- A,B 1860 votes
- B,A 1240 votes
- A,C 1980 votes
- C,A 1320 votes
- B,C 2160 votes
- C,B 1440 votes
It is worthwhile pausing now to think about which candidate deserves to win the election. It seems clear C is opposed by the fewest voters and that might be a hint that C should win. A has the most opposition and that at least suggests A should not be the winner.
In a BAV election, C is in fact the winner with support and opposition distributed as follows:
- A has a net vote of 2800 = {6400-3600)
- B has a net vote of 3400 = (6700-3300)
- C has a net vote of 3800 = (6900-3100)
This shows C winning with B next in line and A coming in last. It seems especially appropriate for C to win since, in addition to having the smallest opposition vote, C also has the largest number of supporters. In contrast, a win by A would seem especially inappropriate since A has the least support and the most opposition. But this inappropriate outcome is exactly what happens with the ranked-choice election.
With only three candidates, this is easy to see. In the first round of counting, the tallies for the three candidates are:
- A gets 3840 votes
- B gets 3400 votes
- C gets 2760 votes and so is eliminated
With C eliminated, the votes in the second round are:
- A gets 5160 votes
- B gets 4840 votes and so B is eliminated
A is declared the winner of the ranked-choice election with B in second place and then C. The three candidates come in with exactly the reverse order from the BAV election.
It seems very hard to justify a claim that this is a satisfactory outcome given the attitudes of the voters toward the candidates.
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