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.

35 thoughts on “Some amateur political theory

  1. p.s. For what it is worth I don’t think IRV gets it wrong often enough to bother with reforming it. Our biggest problem in terms of creating a body that is representative of the people is that we have single member electorates such that parliament nicely represent everybody in the centre of politics and hardly anybody else. The result is mediocre policy challenged by mediocre counter arguments. The New Zealand system does a better job and essentially uses plurality voting within an MMP framework.

  2. Terje “For what it is worth I don’t think IRV gets it wrong often enough to bother with reforming it.”

    My feeling too. Still, the availability of voluntary withdrawal could only improve it.

    Yes, parliament composed of single member electorates is basically a failure. A rule of thumb (Duverger’s rule) says it makes for two-party polarised politics and most political scientist would agree that that is the tendency.

    As I pointed out above (#11), left to itself it doesn’t work (breeds civil war) and needs to be supplemented by a second chamber.

    Yet there is ongoing activism in NZ for a return to majoritarian elections. Stupidity or ignorance in my view.

    Given a second chamber, debate in the single-member chamber is worthless – just theatre. Its sittings could be discontinued and the clerk issued with a rubber stamp saying PASSED. That would save a lot of money and the MPs could go back to their electorates and play social worker – which for most of them is what they prefer anyway.

  3. Imagine an election using John Quiggins IRV method in which the Liberal candidate is ahead of the Labor candidate at the end of the count but not if the third party independent, lets call her Helen, voluntarily withdraws. If Helen decides to withdraw then she signals to the world that she favours the Labor party and many of her followers who lean toward the Liberals will be disillusioned. If however she decides not to withdraw then she signals to the world that she favours the Liberal party and all her followers that lean towards the Labour party will be disillusioned. Either way her claim to being an independent third party alternative is compromised. As such it seems to me that the reform proposed by JQ simply leads to a more deeply polarised political culture. As such I would regard it as a backward step.

    The point above is not merely theoretical. I was a candidate for the LDP during the last federal election. We had to give preferences to other parties as part of the senate preference process for each state. Whilst we tried hard to be reasonably neutral towards the various parties the system essentially forced us to rank our choice between them and also signal our position regarding the major parties (Labor and Liberal) in each state. By forcing us into this game the system essentially guaranteed that we would annoy a significant number of our followers, and fuel our critics, one way or another. And in fact this is exactly what happened. Forcing parties to choose between other parties creates a very compromising process that simply reinforces the old left / right mentality and helps to futher undermine conversations centred on policies.

  4. “Imagine an election using John Quiggins IRV method in which the Liberal candidate is ahead of the Labor candidate at the end of the count but not if the third party independent, lets call her Helen, voluntarily withdraws.”

    So Helen came in second on the 1st preference tally and Labor came third. (Is that what you meant? I am not sure it was, but if she came third then her withdrawal would be involuntary and obviously she did not come first.) The people who voted for Helen put second preference to Labor (mostly). The people who voted for Labor, however, must put their second preferences to Liberal – otherwise Helen would win the election. Pressing on…

    Helen has no big decision to make. She will withdraw. For if she did not then she would allow the Lib to win contrary to the wishes of her supporters. Yes, that minority of her supporters who second preferenced Liberal will be disappointed but with their fellow Helen-voters, not with her. Indeed, they would agree she must withdraw if they think the result should reflect the voters’ wishes.

    So your objection falls away completely. If I am missing something, do demonstrate with some percentage voting figures.

  5. Yes I think you are missing something. However given that the something is apparent even in your own example and you still don’t see it then I’m not sure I can help.

    Perhaps another way to look at this is to ask why Helen has a choice in the matter. If as you say she has no decision to make then why make her decision voluntary at all? Why not insist that she must effectively be withdraw as soon as it is apparant that she hasn’t won at position one or two and if most of her supporters favour candidate number two (but not if they mostly favour candidate number one) and then recount accordingly? This would preserve her independance and achieve the same electoral result you seem to advocate.

  6. Still no numbers, Terje? The electoral result I “seem to advocate” is the one which best reflects the will of the voters.

    Presumably it would be her own supporters who would insist that she withdraw. But yes, why make her decision voluntary? Why not change the rules of elimination of candidates to make it compulsory in order to obtain the optimum result?

    I suspect the rules cannot be consistently worked out. With three candidates it seems plain but where there were four or more you probably get paradoxes. Voluntary withdrawal would leave it up to the candidates to sort out in any instance. Even with three candidates, it may not be plain for that is the situation people set up to illustrate the “Condorcet paradox”.

    That brings up something I have been wondering about. With multiple candidates, would there be race to withdraw? Or would there deals done between candidates for a certain sequence of withdrawing? Declaration of the outcome would have to be delayed to allow time for negotiation.

  7. Mike numbers won’t help. The solution may be perfect in terms of finding the condorcet winner but so what. If you think finding the condorcet winner is the only important aspect of elections then I think you are missing an awful lot. You seem to be locked into an assumption that we are trying to optimise for one election only. The solution on offer undermines for many minor candidates the perception of being independent third party alternatives. At the next election Helen will be a less viable candidate because of the damage done at the previous election. Ultimately the reform works to further narrow the focus of elections to the two major parties. It is hard for me to see any real benefit in this except in some esoteric mathematical sense. At best it is an interesting academic exercise.

  8. So you admit the solution may be perfect. (Did you do a bit of doodling with the numbers?) These comments are up to #34 but it seems the matter rests with my (or JQ’s) resolution at #11 to BobT’s numbers at #3.

    You are concerned for future elections? An election is an understanding that the person elected will be the person most congenial to the electorate (or least uncongenial if you wish). If anyone else is “elected” then the rightful winner has been cheated and not only future elections, but the credibility of the system is undermined. Just look at BobT’s complaint. An electoral system should seek that Condorcet winner, even if it is not always achievable.

    It seems to me that the main reason for standing as a third, fourth, etc candidate (who has no chance of winning) is to dicker over preferences with the two or three main candidates (one of whom will win). Thus the Australian preference system allows smaller groups some influence thereby mitigating the two-party polarisation.

  9. Mike – the solution can’t be perfect in reflecting the will of the voters even within your narrow terms of reference simply because it depends on the voluntary decision of an imperfect candidate. However in the real world context that I’m primarily interested in it is definitely not an improvement. If you could turn voluntary withdrawal into a mandatory rules based withdrawal then it may be a real world improvment but you have indicated that beyond three candidates you have not yet figured out how you would do that. And leaving it to candidates to decide who wins is not reflecting the will of the voters. Witness the senate preference process in Australia and the constant complaints about the way Steve Fielding was elected.

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