Revisiting an old voting system design

A while ago I designed a voting system which is basically how you apply random ballot to multi-member elections. To recap, here’s how it works:

You are voting to elect N candidates. Each voter selects N of the candidates who are running and lists them in order of preference. You then repeatedly select (with replacement!) a random voter and elect the candidate they most prefer who has not already been elected. Once you have done this N times, you have elected N candidates and your job is done.

I was thinking about this again recently, and I’ve realised that it’s a much better idea than I initially realised. In particular:

  1. It is indeed strategy proof – you have no incentive to vote outside your true order (note: See edit below. This is wrong)
  2. It has a very nice strong proportionality property
  3. It can be used to patch the single biggest objection people have to my towards a more perfect democracy proposal.

That it’s strategy proof is essentially obvious: At each point you are performing a random ballot, which is strategy proof, for the set of people who have not already be elected. So you you should always be voting for your most preferred person who is not already elected. This means that if you prefer A to B then you should rank A higher than B – if A has already been elected, you will vote for B, if not you will vote for A.

Edit: Wrong! You need to take into account the effect of other peoples’ votes. Suppose there are three candidates A, B, C and we’re electing two of them. Suppose everyone other than me is voting in order A, C. Now, suppose that I really really hate C and think A is a little better than B. This means that I strongly want the set of elected candidates to be A and B. Therefore I should vote B, A. Conditioning on my being picked on the first round (if I’m not it doesn’t matter what I vote – my second round selection will always be B, because A was elected in the first round), if I put A first then we will end up with {A, C} with very high probability (whenever I’m not picked for the second round too) . If I put B first then we will end up with {A, B} with probability 1. Therefore I should put B first even though I would prefer A.

This example can be extended further so that I may not even vote for my top N candidates.

The proportionality property it has is kinda interesting. It has the following property:

Suppose some group fields at least N candidates. If x% of the population prefer every member of that group to everyone not in that group, that group will have an expected number of seats at least x% of N.

So, for example, if x% of the population fill out their ballot entirely with members of a single party, that party will expect to have at least x% of the elected seats.

Why at least x%? Because the remaining voters can  mix it up by e.g. preferring some members of that group to people not in it, but some people not in it to some people in it. If no-one outside those core x% of voters vote for the group then the expected proportion is exactly x%.

Finally, the patch to the perfect democracy proposal: One of the biggest things people object to about it is that people with a strong mandate can still fail to be elected – even if you have 90% of the vote, that’s still a one in ten chance of not being elected. This significantly depletes the chances of having an experienced person who people love staying in power, which is a bit of a shame.

The nice thing about this system instead of that it is that it handles this case much better by giving each candidate multiple chances to get in. As a result, the probability of a candidate with a fraction \(p\) of the first votes getting elected is at least \(1 – (1 – p)^N\) (it may be more depending on the distribution of second and higher votes). So if they have 90% of the first votes, with N = 2 they have at least a 99% chance of getting in, with N=3 they have 99.9%, etc. With a more modest 50% of the vote, they have a 99% chance of getting it at \(N \geq 7\) seats.

So this is the fix: Rather than having a large number of single member constituencies, you instead have a smaller number of multi-member constituencies. 10 is probably a good number – it’s on the cusp of what is feasible to get people to vote on, but it means that anyone with at least 40% of the first votes is extremely likely to get a seat, and anyone with 50% of them is virtually certain to get a seat (about every 10-20 general elections you will expect one candidate in the entire house with 50% of the vote in their constituency to fail to get a seat).

It definitely complicates the initial proposal, but I don’t think it’s that difficult to get people to reasonably rank 10 candidates they like. A more difficult thing might be persuading people to go for multi-member constituencies.

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