I just noticed this and it seemed worth writing down. It may be a well known result, but 30 seconds of Googling doesn’t turn up everything obvious.

It turns out, that under a certain (highly unrealistic) model of voter behaviour, if you use Approval Voting then you get the Borda Count winner.

In Approval Voting you get to cast a vote for as many candidates as you like, and then the candidate with the most total votes wins.

The question is: How many votes do you cast? Do you cast a vote for only your top candidate (in a close race you probably should)? Do you vote for everyone except the candidate you think is literally Hitler?

I’m actually not sure what the generic tactical answer to this is. For the sake of this post it doesn’t matter, because I’m going to be looking at the following model: What happens if everyone just picks the answer at random? i.e. you have \(N\) candidates, they rank the candidates in order of preference, and then they pick a number c between \(1\) and \(N – 1\) and vote for their top c candidates.

And the answer is that what happens is that you elect the Borda winner.

The Borda Count works as follows:

Everyone ranks every candidate. For each voter, a candidate is given a score: The top candidate gets N, the second candidate gets N – 1, and so on. The candidate with the highest total score wins.

The reason why approval voting results in the Borda winner is quite straightforward: The expected number of votes a voter gives a candidate is just an increasing linear function of the Borda score they would give that candidate.

If you rank a candidate in position t, then you vote for them unless you choose \(c < t\), which happens with probability \(\frac{t – 1}{N – 1}\). So the expected number of votes is \(0\) if this happens and \(1\) otherwise, i.e. \(1 – \frac{t – 1}{N – 1} = \frac{N – t – 2}{N – 1} = \frac{1}{N – 1} B – \frac{1}{N – 1}\), where \(B\) is the Borda score.

Because the the Borda winer is just determined by adding up scores, the scaling doesn’t affect who wins. So assuming that the populace is large enough, the randomization more or less averages out and you elect the Borda winner.

Now, as mentioned, this is a highly unlikely model of voter behaviour – it’s probably just about valid if you have a very small number of candidates (3 or 4), but for larger number of candidates I would expect the number of votes actually cast to be bunched up around either \(0\) or \(N\) with fairly sharp drop-offs moving away from that. I don’t actually know if that’s how it works in practice, but whatever happens in practice it’s very unlikely to actually be uniformly at random.

So I present this mostly as a curiousity rather than as making any point about the actual merits of either system, but I thought it was interesting.

I went to a Lean Coffee event last night. I really enjoyed it.

But naturally I’m constitutionally incapable of participating in a democratic process without thinking about whether it could be improved by randomization, and I think this one can.

Lean Coffee as we practised it had slightly more structure than described in the link:

1. First we voted as described (though actually we had three dot votes for some reason), and then we went discussed the topics in order of decreasing vote.
2. A discussion lasted 8 minutes. A vote was then conducted as to whether people wanted to continue – everyone could vote yes/no/maybe, and you continued for another 4 minutes if there were more yes votes than no votes (maybes are ignored). In theory at the end of that 4 we would vote again, but we actually never voted to continue.

There are two voting mechanisms here, and one is much better than the other: The continuation vote is a good system because you’ve got a simple yes/no question and thus majority rule is a straightforward choice (arguably you can improve it with randomization, but it’s a hard sell).

The voting on topics thing on the other hand didn’t work very well for me. There was a lot of obvious tactical voting caused by both the system and the public nature of the voting. e.g. people were very reluctant to cast their last vote for something that didn’t already have votes.

So I’d like to propose the following alteration to the system:

1. People write cards and just put them into a central pile. Once everyone has finished, the pile is shuffled.
2. You now go through the pile in the shuffled order. You draw a card, the person who wrote it has a short opportunity to explain what it means (normally this happens before the voting phase), and then everyone votes using the majority rule of yes/no/maybe whether they would like to discuss it. Break ties by treating them as no votes.
3. If the conclusion is yes, you discuss it as above. If it’s no, you move on to the next card.

I think this has a number of advantages over the existing system:

1. It removes the element of tactical voting (for the most part. You get an element of time tactical voting where you vote something down because there’s something later you want to discuss more, but that seems quite reasonable to me)
2. What you end up discussing has a much stronger mandate because you’ve had an actual vote on the specific question of whether you want to discuss it.
3. We got a lot of ties which had to be broken arbitrarily in the normal voting for what to talk about process. This intrinsically doesn’t have that problem.
4. It has the usual debiasing advantages of randomization which help offset the tyranny of the majority.
5. It’s actually a more streamlined process – you go straight from writing to discussion with no intervening steps.

As you might recall, there was a referendum four years ago that would have significantly affected the results of the upcoming general election.

Entrenched power very successfully persuaded the British public that electoral form was dangerous and scary and might cause the wrong person to win, and they correspondingly voted against it.

I was, and am, very angry about this.

What I didn’t notice at the time was that the entire referendum was a massive con, that every option was a bad one, and AV would quite possibly have been even worse than the status quo.

Viewed as an alternative to FPTP in the abstract, AV is undoubtedly the better system. It’s not a great system but say we were to want to e.g. elect a mayor then AV is unambiguously preferable to FPTP (indeed, the Mayor of London is elected with a slightly weird and limited form of AV – it’s AV where you can only choose a first and second preference).

The problem is that electing a parliamentary body like the house of commons is not like electing a mayor, because you’re performing the same class of vote many times. This is what gives rise to things like Gerrymandering, but it also means that choosing apparently better electoral systems can give you globally worse results.

AV was sold as reducing tactical voting by allowing you to vote for your actual first choice preference first and not “waste” your vote. The problem is that in reality your first choice vote ends up being a red herring if it’s not for one of the two majority parties, because your candidate will drop out of the race anyway. Sure, you nicely signal your support, but ultimately the only thing about your vote that gets counted is whether you prefer Labour or Conservative.

The problem is that although AV raises the number of votes a minority party will get, it also raises the number of votes they need to get by an even greater margin.

Under FPTP, a minority party can and often will get a seat in a divided area because parties frequently win with a relatively small fraction of the vote, and because voters can “punish” the majority party they favour by defecting to a smaller party. Under AV, this doesn’t happen – they still vote for a major party, and the result is that rather than benefiting smaller parties as was claimed, AV reinforces the two party system.

Does this make AV the “worse” system? No. But I think it’s fair to say that it makes implementing AV at the constituency level a really bad idea. Fundamentally this is a problem with constituencies more than it is a problem with AV, but that doesn’t make implementing AV at the constituency level a good idea just because it’s “not it’s fault” that it would make things strictly worse.

I still think it’s clear that some form of electoral reform is required, ideally some flavour of proportional representation, and if the referendum came up again I would probably still vote yes simply to be able to signal that electoral reform was needed, but I think it’s important to realise what a massive scam that entire referendum was, and reflect on the complete lack of honesty and/or competence that that entails in the people proposing it to us.