Sunday 24 April 2011

I'm Backing AV 110%

(Sorry to go on about it, but I'm just fascinated by mathematics.)

One of the headline reasons being put forward in favour of AV is that it means that no one can become an MP without winning more than 50% of the vote (assuming every voter uses all of their votes, which is a detail I'm not too fussed about). I'm not sure this is such a great thing, since some of that support might be, to say the least, grudging, coming from voters who could actually think of six or seven candidates that they would rather have had than the winner for whom their vote was finally counted.

However 50% is still 50%. And what's really great about AV is it's not only the winner who gets more than 50% of the vote, so might some of the losers.

Let's return to the constituency I mentioned in my last post, but give it a more definite result. First-choice votes go:

Labour: 15,000
Liberal: 11,000
Tory: 9,000

The Tory drops out and let's suppose 7000 of his second-choice votes were for the Liberal. The Liberal now has 18000, which is more than 50%, and wins. First-choice votes of one candidate plus second choice votes of another gives greater than 50%.

The trouble is, the Labour candidate also got more than 50% if you add the first choice votes of one candidate and the second choice of another. The Labour first-choices plus the Liberal second-choices (assuming a reasonable split) also come to greater than 50% . (It's actually mathematically possible that the Tory could get more than 50% too if the Labour candidate's second choice votes were largely for the Tory - but that's unlikely).

So 50% is a necessary but not sufficient condition to win an AV election. How then is the winner decided? We could go back to a FPTP-style approach, where it's the candidate with the highest number of votes (first and second choice combined), but in this example, that would probably be Labour. Here the Liberal wins with more than 50% of a particular set of first and second choice votes, but still fewer than the combination of votes that Labour got, but didn't get counted.

So the total number of votes cast is more than 100%?

Let's consider the definition of percentage. The percentage of votes for a candidate is:

(n / T) * 100

where n is the number of votes received by the candidate and T is the total number of votes overall.

But that term T could actually have several meanings. It could be the total number of voters, or the total number of votes cast or the total number of votes counted. (It could also be the total number of eligible voters, but turnout is a problem under any system.) Under FPTP those three definitions of T are all the same thing, because FPTP is one person-one vote.

But AV gives several votes to each voter. (You can argue that it's a good thing, but you can't deny it - the voter gets to indicate support for more than one candidate. It may not be that multiple votes are counted, but multiple votes are cast.) If you go with T being the number of voters, then it's clearly true - the winning candidate has more than 50% of the vote. In this case, 51%.

But if T is the total number of votes cast, with three votes per voter (assuming that there were no other candidates and that every voter used all their votes) that gives a total of 105,000 votes - and the winning Liberal candidate gets (18,000 / 105,000) * 100 = 17%.

On the other hand, if T is the number of votes counted, things get more complicated. In the first round, 35,000 votes were counted. The Tory dropped out and so in the second round a further 9000 votes were counted, giving a T of 44,000. So now the winning Liberal candidate's percentage is (18,000 / 44,000) * 100 = 41%. Not a bad result for a winner under FPTP, but this is AV, which supposedly guarantees the winner gets more than 50%. You may think I'm being unfair, counting those second-choice Tory votes into T with the same weight as first-choice votes, but if you do, then you must surely also object to them being counted into n (the votes for the winning candidate) with equal weight - that's one of the main objections that many people have to AV.

So the question that we must all answer, whether we are pro or anti or could not care less, is:

What is your definition of T?


  1. T is the number of voters, as each voter has (at most) one vote that counts at any given point in the process. Once my preferred candidate is out my first vote is null and no longer exists.

  2. But if that logic is valid when your first vote is for the third place candidate, why is it invalid if your first vote is for the second place candidate?

    When the second place candidate loses, first-choice votes for them become, as you say, null, but none of your other votes count. By that calculation, T becomes the number of votes for the winning candidate, and the percentage becomes 100%, which I can see is an argument for AV.

  3. I should add that even without the issue I just mentioned, your argument does still lead to a slightly ambiguous definition of T.

    If your first-choice candidate drops out and that vote becomes null, and if you didn't vote for a second-choice candidate, then your entire vote is lost. Thus your definition of T is the total number of votes counted in the final round, which may be different from the total number of voters.

    It's a bit of an unfair point for me to make, since I did originally set aside the question of uncompleted papers, but it does highlight the issue that there is yet another possible definition of T.

  4. A greater concern for me has always been the low percentage of votes cast. The 50% finishing pole is nice for those who feel that a simple majority is a clear mandate, but it only means 50% of surviving ballot slips from an active (or perhaps with AV, a hyperactive) minority of the eligible population. Whichever system we adopt, we have to get more people voting, otherwise we will always get governments whose majority is illusory, whether because of the overcounting that bothers you, or the low percentage of votes that can trigger an electoral majority under FPTP. At least with AV you need the assent of a majority of votes cast, whereas with FPTP a candidate can be elected despite the possibility of over 60% of actual voters not being prepared under any circumstances to support that candidate.

  5. I heartily agree on the idea that increasing turn-out should be one of the highest goals of any electoral reform.

    I'm not sure it's helpful to say 'At least with AV you need the assent of a majority of votes cast' in reply to a blog which fundamentally asks 'what do we mean by votes cast?'

    Also many proponents of AV suggest that it works best when all voters use all their choices (as is compulsory in Australia). Thus the idea of 'voters not being prepared under any circumstances to support that candidate' is meaningless. In AV you can only be less prepared to support one candidate than another.

    And even in practice (with the possibility of not voting for certain candidates at all), I doubt whether under any reasonable electoral system there would be 60% of voters who were not 'prepared under any circumstances to support [the winning] candidate'. In a constituency where the winner, on 40%, was Labour, Liberal, Tory or Green, you'd still find 95% were happier with that than the proverbial BNP winning. Thus there are circumstances in which they are happy with the result.

  6. I should point out that this is not just a matter of a few decimal places. Consider the results of the last French presidential election. (This works on a kind of abbreviated AV called called 'two round run-off voting'. All candidates stand for the first round and then the two highest placed stand for a second round, thus ensuring inescapably, as in AV, that someone gets more than 50%).

    In 2007 Sarkozy won the final round against Royal with 53% of the vote, which provides the pleasing but inevitable headline that the president is support by more than half the electorate.

    But if you look at the total number of votes cast over the two elections, and the total cast for Sarkozy over the two elections, it only comes to 42%.

  7. (Ah, lovely paragraphs ... back from Vienna and no longer crippled by iPad!)

    You claim that "the idea of 'voters not being prepared under any circumstances to support that candidate' is meaningless". Well, the idea does not become actually "meaningless" even if we adopt what you describe as the Australian system (where one is forced to use all ranks of vote). It would just become inapplicable.

    But *are* we adopting that system? As far as I can see, we don't know ... we will be asked as follows: "At present, the UK uses the “first past the post” system to elect MPs to the House of Commons. Should the “alternative vote” system be used instead?" We aren't told *which* alternative vote system.

    I think it highly unlikely that we would adopt a voting system where placing all votes was compulsory, for the simple fact that it would mean that every voter might conceivably contribute a vote to the election of the BNP, which I believe most UK politicians would strive to avoid.

    In short, my idea, far from being meaningless, is direcly applicable. It is highly likely that the AV system will enable voters to vote only for parties they support, and will not compel them to vote for parties that they do not, which means that the only party that can be elected is one that commands the assent (if not the wholehearted support) of 50% of voters.

    Under the current system, a party that is absolutely hated by 64% of the constituent voters can be elected by the remaining 36%, which is oligarchy, not democracy. (Whether AV qualifies as democracy is also up for grabs, but it seems more like it to me.)

    Oh, and I didn't comment on your math because math interests me little and should interest you less than it does. This is definitely an issue on which you should strive to see the wood rather than counting the leaves on the branches of the trees.

  8. You are right - inapplicable is the mot juste. Happily my term was close enough for you to unerringly hit on what I meant.

    I have to say, I'm going to find it difficult to reply to you in a form you like, since you quote percentages at me and then suggest I avoid using maths. What's a boy to do?

    (And I know you're only trying to annoy me by saying 'math' not 'maths'.)

    However, your point that the ability not to use all your votes in is significant is very true, and I shouldn't have ignored it. It leads to a definition with T that I don't think many will disagree with. It's actually essentially Katie's definition, with the inclusion of the possibility of not using all your votes.

    T = the number of people who can stomach at least one of the final two candidates

    But that leads to the fact that the key claim were discussing her is wrong - even wronger than I had thought.

    As you put it: 'the only party that can be elected is one that commands the assent (if not the wholehearted support) of 50% of voters.'

    But suppose (an extreme example, but saves dazzling you with maths) the only two candidates in an election were a fascist and a communist. Most people can't stomach either of them, so don't vote at all (just an extreme extension of not using all your possible votes in AV). One of them still wins and gets more than 50%, but you can't say that most people are happy with that result.

    Now that's not an argument against AV. (How could it be, since the same is true under FPTP?) It's simply to refute one of the dubious claims made in favour of AV (which is by and large all I've ever been doing).

  9. Exactly. (You can take that as congratulation for several points if you like.)

    This is why voter turnout is the real battle. People speak of low turnout as being due to "voter apathy", but it is also attributable in part to the fact that no party represents the political aspirations of the electorate. Governments speak of having a "mandate" to govern, but no British government has had such a mandate in my lifetime, except perhaps for the Thatcher government after the khaki election.

    We always end up speaking of "50% of voters" because it sounds bad to say that we will be electing a government on the basis of a finishing line drawn at, what?, 24%?, of electors. As you pointed out previously, AV provides contituency elections with a fixed finishing line, and that finishing line is not pretty!

    As your fascist/communist example shows, AV is just as susceptible as FPTP to electing people who are hated by a majority of the population, but it cannot elect someone who is hated by 50% of the electors. In real terms, the mathematical consequences of this are not enormous, because we are looking for an example where a party that would win under FPTP with 36% cannot get to 50% after recounts of eliminated candidates. I'm not sure that is going to happen very often. (AV will likely still produce a duopoly of national government, but one in which second votes from minority parties put the leading parties over the top.)

    But, as I acknowledge, the AV system is only a very small improvement.

  10. I hate to sour this tone of agreement, but you're wrong on one point. AV can easily elect someone who is hated by more than 50% of the actual voters.

    Unfortunately, since you're impervious to mathematics, I cannot explain why this is the case.

    For anyone other than Rob who's still reading however, here's an example:

    Suppose we have a constituency with 100% turnout (thus percentage of eligible voters and percentage of those who vote are the same thing). Suppose there are three candidates (or we're down to three) and the first-choice votes go:

    Tory: 15,000
    Liberal: 11,000
    Labour: 9,000

    So, the Labour candidate drops out and his votes are reallocated. Labour voters hate the Tories (because they're Tories) so have not ranked the Tory at all. But Labour voters also hate the Liberals (because they formed a coalition with the Tories and because you can't trust a word they say in their manifesto). So actually there few or no votes to be transferred from the Labour candidate (actually, there could be up to 4000 Labour supporters who don't hate the liberals and it still wouldn't change things).

    If we look at the Liberal second-choices (even though they'll never be counted) none of them are Tory either, since the Liberals hate the Tories because of the referendum and because of continually dropping them in it during the coalition.

    Thus in the final round the Tory still gets 15,000. The Liberal gets some votes transferred from Labour, but not enough and the Tory wins with 15,000. There were as many as 20,000 voters who listed the Tory nowhere on their papers. They hated him, but they still got him.

    It's less likely than under FPTP, for sure, but it's still a claim for AV that is demonstrably false.

  11. It seems to me that you have said that if no party has reached the 50% finishing post after the third placed candidate was eliminated, then the first placed candidate won the constituency. That's the relevant rule that I did not know about and it doesn't require numeracy to understand. Are you sure that this provision will be made though?

  12. Not quite.

    It may often amount to the same thing, but the rule is that the winner is the candidate who gets more than 50% of the votes counted in the final round (or in fact of the votes counted in whatever round they pass 50%). That's what the whole question of 'T' was about. If that were not the case then what would they do? Declare that there was no winner? (More of that tomorrow...)

    This issue has been fairly widely discussed in the media.

  13. Ah, okay, got it.

    Having to understand distinctions such as that is precisely the thing that put me off having to get into the figures in the first place ...