This is mainly a link dump post of some interesting things I’ve read about voting, mostly recently.

Michael Sproul recently wrote an independent Rust implementation of the Australian Senate voting rules. His post about it on medium and the source code. Basically: Australian Senate uses a fairly vanilla STV variant with some weird rules about ballots, but it’s counted electronically on proprietary software that is not made publicly available. He wanted to be able to reproduce the results, so read through a bunch of legalese and turned it into code. The ballots cast are made available as CSV files. Fortunately either the election wasn’t rigged or the people rigging the election were smart enough to rig the voter data too.

But they probably didn’t, because ranked ballots are trivially deanonymizable using the ballot-as-signature attack. So as much as I love weird voting systems and love open data, you’re not really supposed to combine the two like Australia is.

Another reason I’m glad to have read the article and the rust implementation is that it cleared up something I’d been wondering about with how rounding is handled in fractions when implementing STV. The answer is simply that the legislation specifies a number of decimal places of precision.

I ran into this again in this nice piece arguing that Scotland should have used Meek STV instead of the Weighted Inclusive Gregory Method (if I’m honest, I’m not sure what the difference between WIGM and what I think of as “normal” STV is), which mentions that New Zealand specifies nine decimal places of accuracy for calculations.

Both of these clear up some of my concerns with the Meek method. At least people are thinking about this.

On a non-STV proportional relation subject, this one page explanation of why D’Hondt voting works the way it does is excellent and basically cleared up up any confusion I had on the subject.

I’ve also been reading about Borda-STV and Dummet’s Quota-Borda system. I’ve yet to be super convinced by the quota Borda system (mostly because of doubts about how it handles overlapping coalitions that I’ve yet to resolve), but Borda-STV on the other hand just looks like a solidly good idea: It’s more or less the thing I proposed as generalized STV, but in a way that can be regarded as the minimal change you can make to STV to improve it: Instead of having the candidate with the fewest first choice preferences in the current round drop out, you calculate the Borda count off the initial votes and the candidate who drops out in each round is always the unelected candidate with the lowest ranking in that. This has a number of appealing features for reducing the chaos of STV and ensuring a broadly more popular parliament. I think this might fall into the category of “Given this system exists there’s not really a reason to use the more common but worse variant” that I put Approval voting into.

On the other hand, Borda count is not without its problems The ultimate of chaos resulting from weighted voting systems, Donald Saari proves a result that really is spectacular enough to justify that title: Given N sufficiently different weighting systems (i.e. values \(w_1, \ldots, w_n\) such that each vote gives the candidate it ranks in position \(i\) a score of \(w_i\), which are linearly independent when viewed as vectors), and \(n\) different permutations of the candidates \(\sigma_1, \ldots, \sigma_n\), there is some voter profile such that when run with weighting \(i\) the outcome ranking is \(\sigma_i\). i.e. different weighting systems produce arbitrarily different results under some voter profiles! I confess I have only skimmed the methods section of this paper, but the result is really interesting and the contents look pretty approachable if you have some linear algebra background, I just haven’t got around to it.

This isn’t necessarily an obstacle to using Borda, you just have to be very clear up front about what your choices of weightings are designed to promote (e.g. how much more do you really want to weight first preferences than second). Of course you could just let voters decide how much they want to weight those votes using Range Voting but uh it turns out that in the presence of tactical voting, Range Voting is more or less just equivalent to Approval Voting (in large electorates).

That paper also contains a fairly spectacular result (which is an extension of a result in an earlier paper that I don’t have access to): Under any set of preference profiles, and for any candidate that is not the Condorcet loser for the true rankings under those profiles, there is a trembling hand perfect equilibrium (My game theory is a bit weak, so I do not entirely understand what this means, but the result is interesting even if you replace trembling hand perfect equilibrium with Nash equilibrium) where that candidate is the plurality voting winner. The extension in this paper is that this also works for Cumulative Voting (which in large electorates it shows is tactically equivalent to plurality voting). This whole paper is lovely and I strongly recommend reading it.

Anyway, so if you’re using Range Voting you might as well use Approval Voting instead (this has always been my suspicion). It’s a good thing Approval Voting is so tactically simple and nothing complex can happen in it, right? Oh, uh, about that. This paper has some pretty bad examples of what tactical voting under Approval voting can look like. In particular the third example shows a case where there’s another trembling hand perfect equilibrium where one of the voter’s strategy is to cast an insincere vote: They prefer the first three out of four candidates, but in this equilibrium they only vote for the first and the third. Again, my game theory is a bit weak, so I don’t understand all of this paper, but the examples themselves and some of the features of them are very easy to follow. This paper also has an example of an equilibrium where the Condorcet winner is elected with very low probability, but the only stable (also not entirely clear on the significance of “stable” here) equilibrium for plurality voting elects the Condorcet winner. This somewhat calls into question my “There is no point in using Plurality voting when Approval voting exists” stance, but I think I’m going to stick with it until I’ve thought about it a bit more.

In conclusion, voting is hard. Maybe we should just embrace the chaos, go back to our classical roots, and use the ten round system of alternating lotteries and approval votes that they used to elect the doge of Venice.

On the less link dumpy side, I’m currently parallel reading both of the following books which I can recommend reasonably strongly:

• Principles of Electoral Reform is a nice little book (which I recommend buying second hand rather than for the rather steep first hand price) which is a good primer on both some of the basics of voting theory and more generally about how to think about electoral reform and its role in society.
• Electoral Systems: A Comparative Introduction is a great account of what electoral systems look like in actual practice rather than in theory. I realised I was quite light on that, and I’m finding the extra context very helpful.

No, this isn’t about the recent US presidential elections.

In one of those “Yay I get to combine all my weird obsessions” moments, I’ve been playing around with using linear programming to construct provably minimal examples of specific election results. It actually works pretty well – it’s easy to to do for things like plurality voting, borda count and Condorcet winners, and it turns out to be almost possible to do for instant runoff voting (I haven’t figured out how to specify the instant runoff winner in a single linear problem, but you can specify an exact instant runoff drop-off order and then brute force by just considering all the relevant permutations).

Anyway, in the course of doing this I came up with the following really nasty small election example:

• 6 votes of A, C, B, D
• 3 votes of B, D, C, A
• 3 votes of D, B, C, A
• 1 vote of C, D, B, A

Why is this election so nasty? Basically the electorate is divided into two groups: A near majority who think A is great, and a bare majority who can’t agree on much of anything except that they hate A. If you want an interpretation of how these results came about, you could imagine that A is a right wing candidate in a slightly majority left-wing constituency, C is a centrist candidate, and B and D are two left wing candidates who are running against each other and have split the vote.

But regardless of interpretation, what results is an election that is very sensitive to which voting system you use. In particular four fairly common choices of system each elect a different one of the four candidates:

• The plurality winner is A, with 6 votes.
• The Borda count winner is B: The Borda scores are A at 31,  B at 35, C at 34, and D at 30.
• The Condorcet winner is C:  7 voters (the first and last group) prefer it to B and D and 7 voters (everyone but the first group) prefer it to A.
• The Instant Runoff Voting winner is D: In the first round C drops out with only 1 vs 3, 3 and 6 first choice preferences. This transfers a vote to D, causing B to drop out in the next round at 3 vs 4 and 6 first choice preferences. B then transfers their first choice votes to D as well, and finally D has a majority at 7 out of 13 votes, so D wins.

This is the smallest election with these winners and this specific drop-off order, in the sense that it has the fewest number of distinct votes cast (caveat: Where no individual vote has more than 100 voters casting it. This probably doesn’t matter), and amongst those elections with four distinct votes having this property it also has the smallest number of total votes (it also does some minimizing amongst those, but that’s more for aesthetics and doesn’t correspond to an obviously interesting property).

I originally thought all of the voting systems were doing badly on this election, but actually on further reflection I think they’re all behaving in an extremely typical fashion for them.

To frame this in terms of the interpretation above:

• Plurality voting elects the right wing candidate because the left wing vote got split.
• IRV collapses the two left wing candidates together and elects the one with slightly larger support due to the centrist tie breaker vote.
• Borda vote does similarly but breaks the tie in the opposite direction because it counts the fact that the A voters prefer B over D for something.
• The Condorcet candidate is a centrist compromise between the two sides that everybody can tolerate but nobody is thrilled about.

So which candidate you think is correct really depends on what you’re going for – there’s a legitimate question to be asked about how centrist you really want your elected leader to be, and about how justified that is in this case. If you decide that because you’ve got a bare majority the you should elect a more left-wing leader, the question of B vs D seems to come down to whether you want to go for a candidate the other side hates or one the other side really hates.

I don’t know about you, but right now having leaders the other side only hates sound pretty good to be honest. So although I didn’t intend this to be an argument for any of these systems, maybe Borda count comes out of this looking pretty good.

Given that, I feel the title is rather over promising matters. So here’s an even worse election:

• 6 votes A, B, D, C
• 5 votes C, A, B, D
• 5 votes D, C, B, A
• 2 votes B, C, A, D
• 2 votes B, D, C, A

This has more or less the same properties as the above one: A is the Plurality winner, B is the Borda winner, D is the IRV winner.

However, C is no longer the Condorcet winner because there no longer is a Condorcet winner. Instead C is merely the Kemeny-Young winner, a system which always elects the Condorcet winner when there is one and has some claim to being the “best” such electoral system (in the sense of being the closest to capturing the same spirit as the Condorcet criterion), but is horribly impractical for any significantly large number of candidates.

To see why this happens, lets look at the majority preferences amongst the candidates:

• 11 voters prefer A to B
• 13 voters prefer A to D
• B and C are tied with 10 votes each
• 15 voters prefer B to D
• 14 voters prefer C to A
• 13 voters prefer D to C

So we have a Condorcet cycle where A is strictly preferred to B, B is strictly preferred to D, D is strictly preferred to C, and C is strictly preferred to A.

The Kemeny Young rule works by looking at all rankings of the candidates and giving them a cost associated by how much they disagree with this ordering. For each voter and each pair of candidates, the ordering incurs a cost of one if the voter disagrees with that ordering of that pair.

There are better algorithms for this than just considering all orderings of the candidates, but fortunately we only have four candidates so we can just brute force it to see why this works out this way.

The full list isn’t very informative, but here’s a partial list:

This gives us the following scores:

• “C, A, B, D” scores 50
• “A, B, D, C”, “B, C, A, D”, “B, D, C, A” and “C, B, A, D” each score 52
• “B, A, D, C “scores 54
• Everything else scores 58 or higher
• “D, B, A, C” scores 70

(The example was specifically constructed to give “C, A, B, D” as the unique Kemeny-Young best ordering: In much the same way as with IRV, it’s easy to specify the Kemeny-Young ordering but hard to specify just the Kemeny-Young winner)

Because it’s the leader of the unique minimum cost ordering, C wins under the Kemeny-Young rule.

As you can see it’s quite a close fight – there isn’t a huge amount of difference between the lowest cost and the runner up cost, and in the second lowest cost orderings don’t really agree on who should win – the only thing that seems to be consistent is that it shouldn’t be D. This sort of close race tends to be why Kemeny-Young calculations become difficult very quickly.

I’m finding it hard to come up with an interpretation of this election, so lets run the instant runoff vote and see if it’s enlightening (I’m ambivalent about IRV as a system, but watching how votes transfer can help elaborate on coalitions).

Initially B drops out, having only four first choice votes. These split equally between C and D, leaving us with the following configuration:

• 6 votes A, D, C
• 7 votes C, A, D
• 7 votes D, C, A

Now A, the previous plurality winner, has the lowest first choice votes so drops out and transfers votes to D:

• 7 votes C, D
• 13 votes D, C

So D wins the runoff over C.

So I think maybe the way to look at this is that A is an independent candidate of some sort that is distracting from the main stream set of candidates B, C and D. Lets see what happens to the election if we start by taking A out:

• 8 votes B, D, C
• 5 votes C, B, D
• 5 votes D, C, B
• 2 votes B, C, D

So B becomes the plurality candidate, with fully half the first choice vote. The majority preferences remain as they were (because they are not affected by the removal of A): C is still majority tied with B, B is still strictly preferred to D, D is still strictly preferred to C. So there’s still no Condorcet winner.

Removing has made instant runoff voting result in ambiguity though: C and D are tied as to who will drop out. If C drops out then it transfers votes to B, which wins. If D drops out, it transfers votes to C, which is now tied with B. In neither case does D, the original IRV winner, win! This is particularly interesting because the voters who put A first all strictly prefer B to D, and C to D, so removing their first choice candidate arguably results in a better result for them (if you break ties by flipping a coin, they get a better option 3/4 of the time).

The Borda winner (which can change by dropping a candidate) is still B.

I don’t know how enlightening that was, but it seems to reinforce the interpretation that B, C and D are an extremely fractious lot who are probably not going to get along, and unless we elect RON instead (the only candidate who can promise change!) we’re probably going to see the continuation of voting through other means.

I’m not a huge fan of IRV (Also known as AV or Hare voting), despite its status as being probably the most widely used single-winner ranked voting system in practice. It seems unclear whether it’s actually better than plurality voting, but it’s certainly not a lot better than plurality voting. About its only real redeeming value is that Single Transferable Vote is built out of it (and my generalized single transferable vote is still basically a form of IRV in disguise).

Still, people use it, which makes it worth thinking about and studying.

However, some examples I ran into yesterday have caused me to start to think that there are simple variations that are strictly better than IRV in the same way that Approval voting is strictly better than Plurality voting: It’s not that these systems are intrinsically amazing, it’s just that they are (or in this case may be) in all ways better than the systems they replace.

The variation is this: Instead of running potentially as many rounds as candidates, you only ever run two rounds. In the first round, everyone scores as per their plurality (first choice) votes. In the second round, the top two candidates from the first round stay in and everyone else drops out. You then elect whichever of those two candidates the majority prefers.

I’m not yet wholly convinced this is a strict improvement on IRV, but it leads to some things that I think in many cases will be.

The major reason I prefer it to IRV is that it acts as a rather pleasant blend of plurality and Condorcet method: As long as the Condorcet winner is one of the top two plurality winners, this will always elect the Condorcet winner.

Naturally the Condorcet winner is not always one of the top two candidates, but this is still a much stronger claim than IRV can make. The following election is one where the Condorcet winner is also the Plurality winner, but loses in IRV!

23 voters are electing a winner from 5 candidates: A, B, C, D and E.

• 7 votes of A, B, C, D, E
• 6 votes of D, A, B, C, E
• 5 votes of B, A, C, D, E
• 3 votes of C, B, A, D, E
• 2 votes of E, D, A, B, C

The Plurality and Condorcet winner is A, but the IRV winner is B. The two round IRV winner is of course A because it is both plurality and Condorcet winner.

It can of course also go the other way when the Condorcet winner is not in the top two Plurality winners:

 

14 voters are electing a winner from 4 candidates: A, B, C and D.

• 5 votes of D, B, A, C
• 4 votes of A, B, C, D
• 3 votes of B, A, C, D
• 2 votes of C, B, A, D

The IRV and Condorcet winner is B, but the two-round IRV winner is A (the Plurality winner is D here): In this case the Condorcet winner is the third place candidate with the Plurality votes. In full IRV when C drops out it transfers enough votes to B to keep it in the race so A drops out instead of it and then it beats D, but in two-round IRV only A and D make it to the second round, and then A beats out D on majority preferences.

In many ways Two Round IRV is out-IRV-ing IRV here: The way in which IRV tends to fail to elect Condorcet winners is that it gives too strong a weight to first choice preferences. So this is a failure mode that is already present, it just happens that IRV manages to avoid it in this example and two-round IRV does not. I do think that B would be a much more solid choice of winner here (as well as being the Condorcet winner, it’s also the Borda winner), but I’m not very surprised that a system would get it wrong.

So my current feeling from these examples is that two-round IRV is not obviously worse than IRV and that the promise of “often” giving the Condorcet winner is a modestly strong recommendation for it over IRV.

I haven’t actually done the simulations to check (I couldn’t get the original code to compile and haven’t yet put in the effort to write my own), but I would also expect two-round IRV to look a lot smoother than full IRV because of this criterion – IRV has a lot of weird spiky edges in the geometry of voting that I would expect to be damped down by the fact that it usually agrees with the Condorcet winner where that’s in the same rough region as the Plurality winner.

So that’s my evidence for two-round IRV being surprisingly competitive with full IRV and possibly better. Additionally, it’s easier to explain and simpler to follow the reasoning of an election. IRV is not itself that hard to explain or follow the reasoning of, but it’s nice when a system can both be better and simpler.

In and of itself these differences aren’t very interesting and I suspect there’s likely not much in it, but it leads to additiona interesting variations. In particular:

1. If we’re only doing two rounds, why on earth are we using such a bad voting system for selecting who makes it into the second round? Why not e.g. use Borda count and then do a majority runoff between the top two?
2. If we’re only doing two rounds, is it really worth making it an instant runoff?

Both of these questions stem from a single root question: Why are we making voters go to all the work of providing a full ranking and then using so little of it?

Full rankings are quite a lot of work (even partial rankings are a fair amount of work), and while IRV in theory uses most of your ranking, two round IRV uses very little of it – it only looks at your first choice preferences and then the order in which you put two pairs. Why not just split this out into a normal non-instant runoff?

And, while you’re at it, if you’re using a normal non-instant runoff why are you bothering to use plurality voting for the first round? As I mentioned at the top: Plurality voting is a sign you should be using Approval voting.

So, if you’re using IRV why not consider the following non instant variation? My suspicion is that it will be outright better:

1. in the first round, vote for candidates with Approval voting.
2. In the second round, pick the top two candidates from the first round and do a simple majority vote between them.

It does require people to go out to the polls twice, which may be a reason to prefer IRV and variants for large scale elections (though there are plenty of examples of people doing it anyway). However for a lot of use cases it’s very practical to just run two rounds. e.g. I discussed the best democratic solutions for picking a lunch venue recently, where the polls are literally just people raising their hands. It’s not substantially harder to just pick the top two options from the previous election and hold another election for lunch venue.

So, for many and possibly most use cases if you’re tempted to use IRV I would recommend using the above delayed runoff system instead.

This may also be an improvement on straight Approval voting too: One of the problems I have with Approval voting is that it lacks the ability to express a nuanced distinction between two candidates: You can say “I’d be OK with either of these candidates” but you can’t say “But given a choice between them I’d definitely pick this one”. Adding a follow up runoff election adds some of that nuance back in.

One question is whether you always need the runoff election. Suppose only one choice makes it through the first election with a majority of approvals. Should you still run a runoff between that and the second choice?

My suspicion is yes, not because I expect it to reverse the decision in that case but because I think it’s likely to reduce tactical voting in the first stage: If there’s the possibility of no runoff election, you’ll want to very carefully choose your approvals so that your favourite candidate makes it to majority and any other candidate you favour doesn’t.

I don’t know how big a factor that is, and always holding the runoff certainly doesn’t eliminate tactical voting (nothing will except ignoring peoples’ votes in the first round and replacing it with a pure lottery over all the candidates. For many cases this might be a reasonable thing to do, but people seem not to like it very much).

This piece will end without a resounding conclusion, because I haven’t done enough work or reading to decide for certain, but I do feel fairly confident in the claim that a delayed two-round runoff with approval voting in the first round will often be a better choice than IRV and you should consider using it.

Also, if you enjoy these small election examples, I’m putting together a small ebook collating some of them. Let me know if you’d like to be a reader of an early draft.

Finally (you’re going to be seeing messages like this a lot over the next year), if you like reading about this sort of thing and haven’t already, do consider voting with your wallet and sending some money to my Patreon for supporting my blogging here.

I was interested in how range voting, majority judgement, and Condorcet winners interacted, so I thought I’d put together some small elections to demonstrate this. Perhaps unsurprisingly the answer is they’re fairly orthogonal.

The setup is that we’re considering 5 point scores, 0 through 4 (or if you prefer you can interpret these as “Terrible, bad, OK, good, great” for the majority judgement case).

Here is an election where the range winner is the Condorcet winner. We have three voters voting between two candidates, and they assign scores as follows:

• 3, 4
• 3, 2
• 0, 1

The first candidate has a range voting score of 6 and the second has a range voting score of 7, so the second wins the range vote. But the median score for the first is 3, while the median for the second is 2, so the first candidate wins the majority judgement vote. However, two out of three voters (the first and the third voter) prefer the second candidate to the first, so the second is also the Condorcet winner.

Conversely, in the following election the majority judgement winner is the Condorcet winner:

• 0, 1
• 0, 1
• 3, 0

The first candidate is the range voting winner because it totals 3 rather than 2, but it’s the majority judgement loser because it has a median score of 0 rather than 1. The Condorcet winner is also the second candidate, because the first two voters prefer it.

Sometimes range voting and majority judgement agree but they agree on someone who isn’t the Condorcet winner:

• 0, 1
• 1, 2
• 2, 0
• 2, 0
• 2, 3

The first candidate scores 7 vs the second’s 6, so it’s the range winner. It also has a median of 2 vs the second candidate’s 1, so it’s the majority judgement winner. But the first two and the last voters strictly prefer the second candidate, so the second candidate is the Condorcet winner.

Finally, here’s an example where all three disagree:

• 1, 2, 0
• 2, 0, 3
• 3, 4, 3
• 3, 4, 3
• 4, 0, 3

The first candidate is the range voting winner (13 vs 10 and 12), the second candidate is the Condorcet winner (3 out of 5 voters strictly prefer it) and the third candidate is the majority judgement winner (median 3 vs 2 for the second. It ties on the first median with the first candidate, but on removing one 3 score from each candidate the first candidate breaks downwards to 2 while the third stays at 3).

I don’t have strong opinions on what the right answers for any of these elections are, other than “it’s complicated”. In particular I don’t think any of the three options (Condorcet, Range and Majority Judgement) obviously produce the best outcome in all of these examples.

My rough intuitive judgement (formed without explicitly reminding myself of which systems won which, though I’m probably still biased by that knowledge) goes:

• There’s not much in it but the second is slightly better (range vote, which is also Condorcet)
• This election sucks, but the second candidate sucks less (majority judgement, which is also Condorcet)
• First candidate seems a much better compromise (majority judgement and range voting)
• Third candidate  seems pretty universally popular other than the first voter who kinda hates everyone, so seems the best option (majority judgement only).

So on this blatantly biased sampling of four elections I like the range voting winner on three of them, the majority judgement winner on three of them, and the Condorcet winner on two of them. I don’t think that significantly updates my opinions about any of the three mechanisms, but it’s still interesting to see the comparison.