The other randomized voting system

The title of this post is of course misleading. There are myriad randomized voting systems. But there are two (well, sortof three) which have a specific special property: That of being immune to tactical voting. In “Manipulation of Schemes That Mix Voting with Chanceh” by Allan F. Gibbard in Econometrica (which I’ve never actually read due to the great academic firewall) demonstrated that if you have a ranked voting system such that when there are at least three candidates running and:

  1. There is no dictator
  2. Anyone who receives a unanimous votes wins
  3. There is no incentive for a voter to lie about their preferences

then it is one of the following:

  1. Random ballot. i.e. pick a voter at random, use their first choice
  2. Random pair. Pick two candidates at random, use the one that more people have ranked first
  3. A mixture of the two in the sense that we randomly choose between them with some probability

EMERGENCY EDIT: this version of the theorem, which I originally got from’s reporting of it, turns out to be a complete lie and a much richer class of systems is permitted. I owe you one correct accounting of the actual theorem.

I’ve written quite a lot about random ballot. You might have noticed. I’ve never written about random pair as far as I can remember, despite the fact that I think it’s quite interesting.

Why? Well because it looked like it was impossible to make work in practice. It’s so easy to game – people can’t feasibly rank a large number of candidates, so you have to make do with a small number of candidates, and then it’s easy to win by stuffing the ballots with people you agree with so that people get a choice between two options which you both like.

I realised in the shower this morning (I say that a lot. The shower is where I do some of my best thinking) that this reasoning is in fact completely false. There is a practical voting system which is in some sense equivalent to random ballot (the way it’s run may distort the results, but if people truly have a transitive ordering amongst candidates decided in advance then they have no incentive to vote otherwise in this system) and you can easily run it on as many candidates as you want.

In particular you can run it on a sortition. I suspect for many cases where a sortition is desirable this may be a better system. It will tend to shift things towards the prevailing biases of the country (e.g. it’s possible that it will be more gender biased than a sortition), but it may also bias more heavily towards competence.


Well the key realisation is that the only preferences that matter are between the candidates who are randomly chosen. So you don’t need to ask about the other preferences.

This is what you do: Rather than rank then choose, you choose then vote. You pick two candidates at random from the eligible populace, then you do a run-off vote between them in which people simply select which candidate they prefer and the one with the majority of votes wins. If people vote in line with their transitive preferences, this is equivalent to random ballot.

In order to do this in a way that doesn’t bias towards recognisability, here’s what you do in practice: A year before the election you select the two candidates (people have the opportunity to refuse, in which case you select a new one). You now pay each of them a decent salary for the next year and give them a staff of campaign advisors. They now have the next year to campaign for why they should be elected MP. At the end of that year you do the vote.

What are the characteristics of this? I don’t know. It’s obviously vastly more unstable than random ballot – the chances of the same MP being elected twice in a row are essentially zero. This is a downside. On the other hand, you lose some of the privilege bias where recognition leads to votes.

A good solution might be to adopt a mixture approach – you can either do this with a coin-flip for selecting which one you use or you can do this by electing two MPs per constituency, maybe on alternating cycles.

Potentially you could also use one for the commons and one for the lords. This might be an interesting way to elect the lords if every time a new lord is called for you run this procedure over the whole population.

In all honesty I’m still not sure this is a good idea, but it’s gone from a completely impractical idea to one that we can usefully reason about the consequences of, and that makes it interesting to think about.

This entry was posted in voting on by .

3 thoughts on “The other randomized voting system

  1. Nick

    One problematic aspect of this is that the two formulations are only equivalent if we assume that the public selection of candidates does not effect the likelihood of people to go and vote, which seems unlikely. In particular, the expected utility of your vote is different once you’ve observed the candidate selection.

  2. Pingback: Stating the probabilistic Gibbard-Sattherthwaite Theorem | David R. MacIver

Comments are closed.