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Maths of AV: a reading list

On the recent Math/Maths Podcast, among other things, we discussed the upcoming referendum on the UK voting system. Since then, I’ve become aware of a few more articles and blog posts that may be of interest. The referendum asks for a “Yes” to change the method of running the election of MPs to Westminster to the Alternative Vote (AV) system, or “No” to keep the existing “First-Past-The-Post (FPTP) system. The BBC have a Q&A that covers the basics. The referendum is this Thursday, 5th May. If you are undecided, or interested in the issues, here is a reading list. I am focusing on those articles which deal with the topic from a mathematical point of view. (Alright, a few of them are more ‘economics’, but we all enjoy a bit of subject-line-blurring, don’t we?) There is much writing purely on the politics of the debate, but you can find that elsewhere (although be warned: the campaign has been called “a new low in the quality of British political argument“).

A piece by Jacob Aron in the New Scientist, ‘Mathematicians weigh in on UK voting debate‘, looks at, and provides some commentary over, two blog posts: ‘Two cheers for AV‘ by economist Dennis Leech and ‘Is AV better than FPTP?‘ by mathematician Tim Gowers. Both look at some misconceptions of the whole debate and, while giving a fairly impassioned and to some extent balanced look, both are nevertheless pro-AV. In particular, Gowers’ list of claims made by the “No” campaign is well worth a read.

A typically mathematician approach is to reach for proven results and several articles highlight theorems in voting theory, most notably Arrow’s theorem, which gives conditions in which no voting system can produce a fair result. David Broomhead, writing in the Guardian under the heading ‘A formula for fair voting‘ and sub-heading “The AV debate so far has been riddled with false assertions. Here’s the mathematics to prove it”, touches on Arrow’s work and also explains the Gibbard-Satterthwaite theorem, on tactical voting. Tony Crilly goes into Arrow’s work in some depth in Plus under the title ‘Which voting system is best?‘ and also gives an entertaining voting scenario in which twenty people are voting to elect one of three candidates and, depending how votes are counted, any of the three candidates can win.

This morning, Tony Crilly has a piece in the Independent, ‘The maths of AV: A small step towards a fairer vote‘, in which he explains a few quirks of the two systems and gives a little history.

Tim Harford wrote on Twitter today to highlight two blog posts he wrote: ‘Vital, yet unrepresentative. That’s democracy for you‘ in the run up to the 2010 General Election, on the proportion of the vote needed by each party to win a majority and, yes, Arrow’s theorem; and, ‘Why small parties can punch above their weight‘ on the morning after that election, about the game theory of forming coalitions.

There is also some interest in the claim by the “No” campaign that AV is too complicated for people to understand. This is covered by Gowers in his piece and also by Johann Hari in his pro-AV piece: ‘If you get the X-Factor, you can get AV‘, which contains provocative talk about “a campaign that thinks you are too thick to count to three”.

A fan of a good visual representation, I note the “Yes to AV (and beer)” graphic posted by Adam Ramsay in a piece entitled ‘11 reasons to vote Yes on Thursday‘.

There has been a lot of “Yes”-leaning writing listed here. I haven’t seen anyone arguing strongly that FPTP is mathematically a better system than AV. Most arguments against seem to be those listed by Gowers – or, as I have seen it called, “scaremongering and peddling untruths“. The closest I’ve heard is something like ‘vote no because AV doesn’t go far enough’. I’m happy to be corrected.

Further contributions are most welcome in the comments or via Twitter.

4 Responses to “Maths of AV: a reading list”

  1. Avatar Dave Gale

    This is a nice article Peter. Like you, I’ve been trying to look for balanced arguments but finding it hard to find impartial ones.

    I like the links you’ve given and I do like the beer graphic but am a little concerned that it blurs the reasons why people were voting for each pub in particular.

    I am voting yes to AV but has anyone got a convincing way of showing it is one vote each? I’ve heard people saying that if you vote for someone unpopular, you get to have another go. I know this isn’t quite right but can’t find a way to point out that that isn’t exactly how it works.

  2. Avatar Peter Rowlett

    Hi Dave,

    I think Tim Gowers deals with this well in his post, under the heading “3. Under AV, some people get more votes than others.”. Particularly:
    The idea that it is unfair for some people to have their vote counted more often than others is — in so far as it means anything at all — just plain wrong. The NO2AV campaigners are saying that supporters of unpopular parties get more votes. What they actually get is more opportunities to change their vote. Since each change is from a higher preference to a lower preference, changing one’s vote is not something one wants to do.

    Someone I met said it’s not fair since because he votes for, let’s say, Party A, which is a very popular party, he will only get one vote while people who vote for Party D, a minority party, get several votes. First, I said that was nonsense and he was just parroting Party A’s anti-AV rhetoric. Second, I said: “Imagine you lived in a constituency that Party B and Party C contest and Party A doesn’t get a look in. In that case, when your Party A vote is cast out in the first round, how will you feel? Will you think: ‘Hurrah, I get to have another go!’; or, ‘Oh no, Party A aren’t going to win.'” I don’t think AV is perfect but the idea that your first choice losing the election is a bonus is laughable.

    You’re right about the beer graphic, but I think it highlights well the problem of vote splitting allowing a third party to win. If I want to go to one of the pubs and I have to compromise, I’m more likely to settle for a different pub than I am the coffee shop.


  3. Avatar Dave Gale


    I think ‘nonsense’ is a bit risky as it really does feel that voting for an unpopular party does mean you’re likely to get another go (although I realise it’ll be for a party you are less in favour of).

    Colin Graham suggested you accept that it isn’t one vote, but when you recount in round 2, _everybody’s_ vote gets recounted. You all get recounted an equal number of times but some people may be changing their mind along the way.

    As for the beer, it assume that choosing pub = choosing beer. One of the pubs may have a particularly attractive member of bar staff. Another may have a darts board that the others don’t have. In both these cases, if you don’t get your choice of pub it may not be true that ‘any other pub’ is your 2nd choice.

    I’m still yes to AV. ;-)

  4. Avatar Peter Rowlett


    Re. multiple preferences, what Colin suggests is sensible but I wonder if someone who felt passionately about the problem might find the idea that their vote is recounted in each round just, erm, ‘nonsense’ ;)

    Re. beer, You’re right, of course. The More or Less pudding skit was more nuanced.



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