Some amateur political theory
As I mentioned, I’m at a conference on Logic, Game Theory and Social Choice. Attending a session on experiments in voting theory (some very interesting ones for which I will try to find links) I started thinking about a case for Instant Runoff/Single Transferable/Preferential systems (like many Australians I’m a big fan of this system which works well for us, with none of the disasters we’ve seen produced in the US and UK by plurality voting). For those interested, an outline of an idea is over the fold. It’s not my field, so I’m quite prepared to be told my argument is wrong, well-known or both.
Update 29/8 The original claim I made was wrong, but now I have one that, I think, works better.
Think about an IRV election where, after the votes have been cast, any candidate has the option to withdraw (there are some potential complications about the order in which this option becomes available to candidates, but I don’t think they matter in the end). Suppose that a candidate will only withdraw if by doing so, they will ensure the election of a candidate preferred by the majority of their voters to the candidate actually elected. I claim that this procedure is a Condorcet method. That is, it always selects the Condorcet winner, the candidate who would beat each of the other candidates in a run-off election, if such a candidate exists.
To see this think about the case of three candidates. IRV elects the Condorcet winner unless she finishes last in the first preference count. For example, there might be three candidates, with the Left and the Right candidate each preferred by 40 per cent of voters and the Centre candidate preferred by 30 per cent. The Centre candidate is preferred by both Left and Right voters to the candidate of opposite orientation, so is the Condorcet winner. The majority of Centre voters prefer the Right candidate. Then the Centre candidate is the Condorcet winner, but, under IRV the Centre candidate will be eliminated, and her transferred votes (second preferences) will elect the Right candidate.
But, if the option of withdrawal is available, the Left candidate, who can’t win, will best serve the preferences of Left voters by withdrawing. This ensures the election of the Centre candidate. So, with three candidates IRV+withdrawal option is a Condorcet method.
To extend to the case of four candidates, we can argue as follows. If the Condorcet winner finishes in the first three places, we have the same case as before. The last-placed candidate is eliminated (or withdraws, it doesn’t matter) and we have a three-candidate race. Suppose that the Condorcet winner finishes fourth. Then (since she’s the Condorcet winner) there must be at least one other candidate whose voters prefer her to the winner under standard IRV. If that candidate withdraws, we are again in a three-candidate race and the previous analysis applies. And so on, recursively, for arbitrary numbers of candidates.
Next observe that if candidates can anticipate votes correctly, and stand only if by doing so they would advance the interests of their own voters, standard IRV will produce the same result, since candidates who would ultimately choose to withdraw will simply not run. We see something like this in the Australian two-party system in seats where there is a strong third-party or independent candidate, but one of the major parties will clearly beat the other in a pairwise choice. The other major party often chooses to run dead, or (mostly in the case of an independent incumbent) not at all, so as to ensure that the other major party is kept out.
If this analysis is correct it seems to me to make a pretty strong case for IRV + withdrawal option and therefore (if decisions not to run roughly match ex post wish to withdraw) for IRV itself. It’s simpler than any other Condorcet method, has actually been used on a large scale, and seems, in practice to work much as claimed in this post.