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.