Some amateur voting theory

by on August 28, 2009

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.

Note 29/8 I initially put up this post with another, related, claim, convinced myself that this claim was wrong, and deleted it, leaving the post as a placeholder until I could do something better. The first few comments refer to this.

Think about an IRV election, and suppose that there is no strategic voting (I’ll argue that it won’t be needed, so voters will always vote sincerely. Now suppose that , 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.

1

MR Bill 08.28.09 at 1:01 pm

Here as in “here at Crooked Timber” or here as in “the link is not present”?

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MR Bill 08.28.09 at 1:03 pm

You mean the previous post?

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Kieran Healy 08.28.09 at 1:11 pm

Are you sure you’re really an economist?

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Marcus 08.29.09 at 12:58 am

I don’t completely understand your argument, so forgive me if the following questions seem obtuse or cavil.

Think about an IRV election, and suppose that there is no strategic voting…

In standard IRV, there is actually a strong incentive for strategic voting because of something called `non-monotonicity’ (also called the `paradox of increased support’). There are profiles where a candidate, say, A, will win. But if A’s support increases slightly amongst the electorate (say, 5% switch A from their third-place candidate to their second-place candidate), then A will then LOSE the IRV. If the voters can anticipate this, they may chose to deliberately falsify their preferences to manipulate the outcome.

Now suppose that , after the votes have been cast, any candidate has the option to withdraw … 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.

Here I think you have an incentive problem. If I am the candidate thinking of withdrawing, then I will withdraw if and only if it ensures the election of a candidate who >I< prefer to the current winner. This may, or may not, be the candidate prefered by the majority of my supporters to the current winner.

To see this think about the case of three candidates. IRV elects the Condorcet winner unless she finishes last in the first preference count.

(Is this always true? Is it a theorem?)

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.

First, I think for your example to work, the Right must have an edge over the Left —say the Right has 41% support and the left has 39% support (not 40% vs 40% as in your example). This is probably what you meant.

Second, in your example, all the voters have `single-peaked preferences’, because the candidates can be arranged on a left-right continuum. This is why Center is the Condorcet winner. However, single-peaked preferences are an extremely well-behaved special case (first studied by Duncan Black in the late 1940’s) where most voting rules are known to be especially well-behaved. Does the logic of your argument work for arbitrary preference profiles? I’m not sure.

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 doesnâ€™t matter) and we have a three-candidate race.

But wait… what if one of the first three candidates withdraws instead? Why can’t this happen?

Suppose that the Condorcet winner finishes fourth. Then (since sheâ€™s the Condorcet winner) there must be at least one other candidate whose voters prefer her to the winner under standard IRV.

Not clear. If Y i’s Condorcet, she is prefered by a majority of the entire electorate to any other candidate. But it may be the case that, for any particular candidate X, 51% of X’s supporters prefer the current winner to Y. (So Y only gets to have a majority over the current winner because of her own solid base of support).

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.

This sort of `rational expectations’ argument is probably attributing way too much foresight to the candidates. After all, elections are rarely that predictable. And if they were that predictable, then all the `no-hope’ candidates wouldn’t both wasting time and money running, so you would often see only two-person races, or even election by acclamation. The whole question of voting rules would become moot.

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. Itâ€™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.

The problem is that Condorcet winners exist only in a minority of situations —arguably the `nicest’ situations. So being a Condorcet rule is a necessary, but not sufficient condition for a voting rule to be acceptable. (Actually, I don’t personally think it’s even necessary, but then I’m a Borda count guy….) The real test of a voting rule is how it behaves when there ISN’T a Condorcet winner. As I’ve already mentioned, IRV can have some unpleasant properties (e.g. non-monotonicity). So the question is whether IRV behaves acceptably `most’ of the time (whatever that means). —or at least, behaves better `on average’ than the alternatives

Sorry if I’m sounding negative or discouraging. If your idea works, then it’s pretty neat.

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Anthony 08.29.09 at 4:29 am

Your example assumes that candidates act as the agents of the people who vote for them, when this is clearly not the case. Look at Nader in 2000 – he knew that there was a significant chance that his candidacy would cause the election of Bush, even though most of his voters would have preferred Gore. The same is true of Perot in 1992.

There is also the problem of selecting candidates – in countries with strong party discipline, the result may be like your Australian example, but in the United States, even if there is no Republican or Democrat with the blessing of the party officials, the requirements for getting on the ballot are fairly low, and there will likely end up being a crackpot candidate who isn’t known as a crackpot, who will draw away some significant number of votes strictly by having the party identification. (There are sometimes internal party political reasons to run even for hopeless seats; the Republicans have had contested primary elections in San Francisco and Berkeley in recent memory.) Someone who has made such a run is not likely to withdraw “for the good of his voters”.

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Duncan 08.29.09 at 6:44 am

I agree with the above comment about spoilers — Nader actually claimed to prefer Bush to the Democratic alternatives both in 2000 and 2004 (he said that someone actively causing harm to the environment etc. would cause more people to become aware) and he clearly thought that he would benefit professionally from a Bush win, although this was perhaps shortsighted given the backlash against him. The interests of his voters, however, would presumably have been better served by Gore.

7

Not clear. If Y iâ€™s Condorcet, she is prefered by a majority of the entire electorate to any other candidate. But it may be the case that, for any particular candidate X, 51% of Xâ€™s supporters prefer the current winner to Y. (So Y only gets to have a majority over the current winner because of her own solid base of support).

It’s still true that there has to be at least one candidate whose voters prefer Y to the IRV winner (W). Since the majority of the electorate prefers Y to W, and W has more first-prefs than Y (Y’s last on first prefs!), we know Y is preferred by more than half of the other two candidates’ voters. But that can’t be the case unless she gets more than half from at least one of them.

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Kenny Easwaran 08.29.09 at 9:05 am

Furthering Anthony’s point, I believe that recent Australian political history indicates that leaving decisions about when and how to move their votes to someone else up to the candidates rather than the voters sometimes leads to disaster. (For those of you who aren’t in the know, I’m talking about how the Family First party got a seat in the Senate, because Labour, along with several other parties, made a deal where they gave preference to Family First over the Greens.)

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John Quiggin 08.29.09 at 9:15 am

@Marcus – thanks, this is exctly what I was hoping for.

Sincere voting

If the mechanism works as claimed, then sincere voting is optimal, as is withdrawal by candidates in the stated situations. This is the kind of equilibrium argument economists and game theorists like – I know it’s a bit tricky but bear with me for the rest.

To see this think about the case of three candidates. IRV elects the Condorcet winner unless she finishes last in the first preference count.

(Is this always true? Is it a theorem?)

Yes. Suppose the Condorcet winner finishes first or second in the initial count. The third candidate is eliminated and we are left with a two-candidate race which, by definition, the Condorcet winner must win.

But waitâ€¦ what if one of the first three candidates withdraws instead? Why canâ€™t this happen?

This can happen, but we still reduce to the three-candidate case, and everything still works, I think. I’ll check on this one.

On the next point I agree with Adrian

This sort of `rational expectationsâ€™ argument is probably attributing way too much foresight to the candidates.
I wouldn’t want to claim this all the time, but there are occasions when this works. But I’ll qualify this point if I get to writing it up.

Non Condorcet cases
I haven’t got to these cases yet. There may be some problems here, depending on the order in which candidates withdraw. But I don’t have a strong intuition as to who should win.

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John Quiggin 08.29.09 at 9:17 am

Kenny, this is a different case to the one I’m considering, but it was indeed disastrous (thanks, Vic Labor!). In my story, candidates are directed by voters, not vice versa. In fact, you can remove the discretionary element altogether, I think.

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anonymous 08.31.09 at 4:16 am

Just friggin use the Schulze method. Then you get a Condorcet winner without having to count on auspicious withdrawals–especially since beyond the simplest cases it won’t always be obvious whether it’s in a particular candidate’s interest to withdraw or not. The Schulze method requires more computation but that’s all completely opaque to voters anyway; the ballot is exactly the same for Schulze as for IRV. The only thing IRV seems to have going for it is that it’s relatively easy to explain to people–but how many people actually understand the details of how apportionment in the US House works or how congressional districts are determined? I think in the end people would be more concerned with whether the system actually produces a result that follows intuitively from voters’ preferences (Condorcet, monotonic, etc.) than whether they understand every detail of it. Especially since even though IRV is “simple” most people still aren’t aware of its critical shortcomings.

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ogmb 08.31.09 at 8:53 am

What do you make of this example?

A: 49 votes; 25 with C>D>B, 24 with D>C>B
B: 18 votes; all with D>C>A
C: 17 votes; all with D>B>A
D: 16 votes; all with C>A>B

D is the Condorcet winner. A and B, with the most primary votes, get paired in the runoff, which A would win. B’s voters prefer C and D over A, so B withdraws. Under “normal” succession rules, C replaces B and beats A in a runoff. Since A’s voters prefer C over D, A would not withdraw in favor of D, and C is declared the winner. A way to solve this would be to allow B to handpick D as her replacement — a dicey proposition under any circumstances but with an added twist here because it was D’s voters who caused B to lose the runoff.

Also, if you flip A’s support groups, B would withdraw in favor of C, and A in favor of D. D would beat C in a runoff, which creates the curious outcome that the last finisher in the primary wins the runoff over the second-to-last primary finisher after the two top finishers have both withdrawn, including A with more than three times as many first-preference votes as D. Even if you get a Condorcet winner you’d have a lot of credibility problems with such an outcome.

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

If you don’t allow for anticipation, IRV+ will create the kind of post-election haggling as in the example above, even if it creates a Condorcet winner. With limited anticipation, I don’t see an argument why IRV+ is a better process than traditional (delayed) runoff. In particular since with a delayed runoff you have the primary results as factual basis on which you can make withdrawal decisions before the runoff is held (unless in IRV+ the election committee is supposed to keep the secondary votes under wraps until the withdrawals have been announced??). And of course with perfect foresight, we don’t need elections because we already know who will win…

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John Quiggin 08.31.09 at 9:51 am

“Bâ€™s voters prefer C and D over A, so B withdraws. Under â€œnormalâ€ succession rules, C replaces B and beats A in a runoff.”

As I am thinking about things, B’s withdrawal means that her second preferences are distributed as they were cast, so B has no capacity to nominate how they should be allocated. In the present case, they all go to D who is then elected regardless of whether C or A withdraws.

“Also, if you flip Aâ€™s support groups, B would withdraw in favor of C, and A in favor of D. D would beat C in a runoff, which creates the curious outcome that the last finisher in the primary wins the runoff over the second-to-last primary finisher after the two top finishers have both withdrawn, including A with more than three times as many first-preference votes as D.”

This would replicate a very common outcome in two-party contests with plurality voting. The Republicans overwhelmingly prefer the unelectable wingnut, but nominate the RINO who has a chance of winning. Meanwhile the Dems would like to put up a genuine liberal, but stand a Blue Dog who can beat the RINO. Desirable or not, this kind of outcome is virtually inevitable, regardless of system, and is the mechanism by which Duverger’s law works.

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ogmb 08.31.09 at 10:10 am

As I am thinking about things, Bâ€™s withdrawal means that her second preferences are distributed as they were cast

Nevermind, I wrote this under the assumption that the runoff pairs the two top primary vote getters, in which case withdrawal creates a need for a clear succession rule (none of which seem to ensure Condorcet). If your model assumes a runoff between all candidates except the last vote getter, you’d still need to stipulate that the last finisher is reinstated if any of the other candidates withdraws. Since D is that last finisher in the primary he wouldn’t be eligible to get B’s votes unless specifically reinstated.

This would replicate a very common outcome in two-party contests with plurality voting. The Republicans overwhelmingly prefer the unelectable wingnut, but nominate the RINO who has a chance of winning.

That’s still a big difference to the kind of strategic maneuvering after the voting results are known (to the candidates? to the public?) that the IRV+ seems to engender.

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Jameson Quinn 08.31.09 at 5:50 pm

This is just way way far from true. Others have provided logical arguments above, but all you have to do is to look at the pictures here (read the first bit to understand what the pictures depict and what a “good” picture should look like, then search for IRV and see the fugly pictures that result) to know that IRV’s problems are deeper than such a simple fix.

Note that even the fugly IRV pictures are arguably better than the pathetic plurality pictures, so I still support IRV as a reform. There are much much better reforms, though, and if you’re looking to find one, supporting an existing proposal is better than making a new one based on IRV (for two reasons: the proposal itself is likely to be better, and you’re likely to find more allies).

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Bob Richard 08.31.09 at 9:28 pm

The last paragraph of the OP seems to imply that IRV + withdrawal option has been used in real world elections. Is that true? Where?

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Bob Richard 09.01.09 at 5:41 pm

John suggests that “if decisions not to run roughly match ex post wish to withdraw” then IRV comes closer to meeting the Condorcet criterion than is otherwise the case. But the decision whether to run or not is greatly affected by the amount of information available about the relative strengths of other potential candidates. So if this conjecture is true at all, it can only be true in elections important enough to generate good polling data (or extremely good hunches) before the decision to run must be made.

This seems to me to be a frequent problem with models based on game theory. Either they ignore the role of missing (and wrong) information, or they make simplistic assumptions about the nature of the information available.

BTW, my previous question is probably based on misreading the paragraph in question. It now seems apparent that “actually used on a large scale” refers to conventional IRV rather than IRV + option to withdraw.

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