Fun with election maths! N.B. fun<0.

Just a quick point about the following video from the official No2AV campaign.

We’re shown a bar chart representing each candidate’s proportion of first-preference votes in an imaginary five-candidate election. The fifth-placed candidate’s votes are then transferred, pushing the previously second-placed candidate over the 50% threshold – he wins the election. Could this really happen?

Let’s rank the percentages of first-preference votes for the five candidates as follows: a>b>c>d>e, where a+b+c+d+e=100.

If b+e>50, as the video suggests, then necessarily a+c+d<50. Therefore b+e>a+c+d.

Subtracting e and a from both sides gives us b-a>c+d-e.

But a>b, so subtracting a from both sides gives us 0>b-a.

If 0>b-a>c+d-e, then 0>c+d-e.

Adding e to both sides gives us e>c+d.

We know that c>e and d>e. Adding the two equations together gives us c+d>2e.

If e>c+d>2e, then e>2e.

Subtracting e from both sides gives us 0>e.

Given that e is the percentage of the vote achieved by the fifth-placed party – in their examples, the BNP – the scenario they illustrate is only mathematically possible if the BNP can achieve a percentage of the vote that is less than zero. Now, I’ve heard of negative campaigning, but that would be quite something.

“If you break the law, you cannot make the law”.

The words above were spoken by Conservative MP David Davis last week in justification of the current ban on voting rights for most prisoners, and Parliament’s reluctance to accede to the European Court of Human Rights’ ruling that such a ban was illegal. I don’t intend to go into the merits of the ECtHR and the Convention it upholds in this blog, but I would like to respond – however belatedly – to the sound-bite, which Davis and many others clearly feel expresses a compelling logic, with a simple question: why not?

To begin with, it is clearly not the case in fact that “if you break the law, you cannot make the law” – assuming that by “make the law”, Davis means “participate in the election of legislators”. Firstly, civil wrongs, which constitute the vast majority of breaches of the law, do not attract any such ban. Secondly, nor does every criminal conviction – many do not lead to a prison sentence. Thirdly, even in those cases where a prison sentence is served, the right to vote is resumed after the completion of the sentence, although the person has still broken the law. Nobody is suggesting that any of these situations ought to change, so it’s pretty clear that there is no black-and-white rule that breaking the law prohibits an individual from making the law. However you look at it, a line has been drawn according to a mutable judgement.

Of course, there is a clear implication in the above quote that if you break the law, you shouldn’t be able to participate in the legislative process – obviously Davis is justifying something, and not merely stating a fact, however inaccurate. I can see two reasons why one might make such an argument and legislate accordingly. Firstly, in order to protect society from the consequences of not doing so. Secondly, regardless of the consequences, in order to impose personal moral beliefs shared by a majority in Parliament – and, quite possibly, by a majority in society – on those who do not share them, even though these moral differences do not pose any substantial danger to society. Hopefully I don’t need to convince anybody that the latter is an appallingly illiberal approach to government, and shouldn’t be entertained. What remains is the legitimate question of protecting society.

But while it’s a legitimate question, is it a justified fear? What exactly are the dangers of allowing prisoners to vote, and what is the purpose of denying them the opportunity?

One of the principal arguments for criminal remedies such as imprisonment – justified or not – is deterrence through punishment. But does anyone seriously argue that the prospect of a lost vote could deter an individual who was unconcerned by the prospect of lost liberty? I’d be surprised. Davis has expressed a belief that “giving prisoners the vote would not stop one crime in this country”, but surely it’s at least equally pertinent to point out that the reverse is also likely to be true: denying them the vote will not stop one crime either.

It seems to me that central to this point is the base position from which one determines the extent of the right to vote. Specifically: is the right assumed, and only then removed with good reason; or is no right assumed, and the vote granted selectively to those who “deserve” it? In arguing for the former, I will simply say that I would always wish for any denial of basic rights by any authority to require the strongest possible justification, and I don’t think it unreasonable to expect a British government in 2011 to be liberal enough to treat voting as a basic right.

A more practical issue is that of exactly where prisoners vote, and whether prison populations could amount to voting blocks which could lead to skewed and undesirable election results in certain constituencies. Although I have more sympathy with the premise of this argument, in reality, it’s a bit of a swizz. Firstly, it is surely not beyond the bounds of practicality, for example, to allow prisoners to vote by post in the constituency in which they were resident prior to their sentence. Secondly, even discounting this, the largest prison populations in the UK number around 1500. The average constituency size is around 70000. Even the most effectively mobilised prison population would have no chance of unilaterally electing a candidate who was generally thought objectionable by law-abiding citizens.

A final point on the nature of crime itself. Our definitions of crime do not emanate from the earth, and we don’t derive them from some supernatural authority. They are human constructs, borne of our society and its representatives’ consensual judgements about what acts and omissions are inherently dangerous to that society – ideally, at least. The briefest recourse to history will draw your attention to many formerly imprisonable offences which are now broadly considered to be far less serious and, in some cases, not actually objectionable at all; by way of an example, homosexual activity between men was decriminalised in this country as recently as 1967. Now, it may be, after hundreds of years of constantly changing perspectives on what ought and what ought not to be criminal, that we have now “nailed it” (as some no doubt felt that they’d “nailed it” a hundred years ago, and two hundred years ago, and so on), and that none of the laws of today will ever be looked back on ruefully. It is possible, if you insist. But it is undeniably possible that we haven’t “nailed it” at all, and that our perspectives on crime will continue to evolve, as they always have. That understood, it is essential that we put into place mechanisms to ensure that those acts and omissions which are criminalised to the point of demanding incarceration are, at the very least, genuine fringe behaviours, practised and justified by few. It will surely never be an acceptable remedy to incarcerate great swathes of the population for the benefit of the remainder – that approach would drastically misunderstand the variety and difference which is inherent to any human population, and fundamental to the very notion of individual human rights. While the offences for which we imprison are fringe behaviours, practised and justified by few, then we have nothing to fear from granting prisoners the vote – their electoral voice will be diluted to the point of silence. For that voice to be heard would require us to imprison and marginalise in response to behaviours which are, quite simply, not marginal; and that is precisely the point at which that voice must be heard.

A suggested amendment for the Alternative Vote. An alternative, you might say, if this were your blog. But it isn’t.

Before I start, I should say that I intend to vote in favour of changing our electoral system to AV, when the alternative is to stick with the First Past the Post system. I have my reasons, and I may well try to explain those reasons at a later date, but suffice it to say for now that, although AV is far from the perfect system, I would view its adoption as progress. That said, the purpose of this post is to illuminate what I see as a problem in the vote-counting process, and to suggest an improvement, purely for the sake of argument. If it’s already been suggested elsewhere, that wouldn’t surprise me at all, but I think it’s worthwhile regardless.

Being reductive, a victory under FPTP signifies that the victor has the support of the largest group; but of course, this can be achieved with the support of the largest minority, even if the majority consider him or her to be abhorrent. Under AV, the significance of a victory is quite different: that more voters have expressed a clear preference for the winner over any remaining candidates than vice versa. I’m pretty comfortable with that aspect of AV.

What I’m less comfortable with is the process of elimination that reduces the field down to the final set of candidates in the first place. For those who don’t know the system, a very quick summary:

• Voters choose candidates in order of preference, numbering them “1”, “2”, “3” and so on. Once they are no longer interested in expressing a preference between the remaining candidates, they stop – so, for instance, you could mark a single candidate as your first preference and leave it at that, as in FPTP, or you could rank every single candidate, or anything inbetween.
• If one candidate has over 50% of all first preferences, they are declared the winner.
• If no candidate has over 50% of all first preferences, the candidate with the fewest first preference votes is eliminated, and their votes are transferred to the relevant voters’ second preferences.
• If a candidate now has 50% of the vote, they are declared the winner. If not, the candidate who now has the fewest votes is eliminated, and their votes transferred.
• If the candidates are reduced to a field of two without either one gathering over 50% of the vote, the winner is the candidate who is left with the largest number of votes. Bear in mind that it is possible for no candidate to achieve a majority even in a final field of two, since some voters will have had all of their preferences eliminated before the final round of counting. Their votes will therefore not count for any of the remaining candidates.

The problem is that the elimination of a candidate after each inconclusive round of counting serves only to resolve the impasse; the possibility of the eliminated candidate winning would otherwise remain. Because I have a feeling that this is not very clear, I’ll condescend (with apologies) to illustrate it with an example involving three candidates whom – in the spirit of wild creativity – I’ll call Mr Ay, Ms Bee, and Lady Cee of Epping Forest. They are standing for election under AV in a constituency with population 100. It’s an island group, obvs.

In the first round of counting, Mr Ay picks up 35 first-preference votes. Ms Bee also picks up 35 first-preference votes. Lady Cee of Epping Forest picks up just 30 first-preference votes, and is eliminated, and her votes transferred. Of her 30 voters, 10 had ranked Mr Ay as their second preference, and 20 had ranked Ms Bee as their second preference. Therefore, in the second round of counting, Ms Bee has 35 + 20 = 55 votes (to Mr Ay’s 35 + 10 = 45), and is elected as the new MP for Unnamed Island Group Inexplicably Located Somewhere in Epping Forest! Hurrah! Although democracy is, of course, the real winner.

BUT remember the premise: Ms Bee was elected because more electors preferred her to Mr Ay (55) than preferred Mr Ay to Ms Bee (45). The question then is: did more voters prefer Ms Bee to Lady Cee than preferred Lady Cee to Ms Bee? Well the thing is that, in the event, every single person who ranked Mr Ay as their first preference ranked Lady Cee as their second choice. So, in fact, not only did those 30 electors who ranked Lady Cee first prefer her to Ms Bee, but so did the 35 voters who gave their first preferences to Mr Ay – 65 electors in all. Only 35 voters preferred Ms Bee to Lady Cee, meaning that more voters preferred Lady Cee to Ms Bee than preferred Ms Bee to Lady Cee. And actually, every single person who ranked Ms Bee as their first preference had ranked Lady Cee as their second choice too. So, again, not only did the 30 electors who ranked Lady Cee first prefer her to Mr Ay, but so did the 35 voters who gave their first-choice votes to Ms Bee; more voters preferred Lady Cee to Mr Ay (65) than preferred Mr Ay to Lady Cee (35). Should Lady Cee not therefore have won?

The issue is the process of elimination. The only way to discover the above result would have been to eliminate one or other of Mr Ay and Ms Bee after the first round of voting – but this would have seemed grossly inappropriate when, at that stage, they lay in joint first place. Indeed, if you did so and then just continued the process to completion as above, it would be completely inappropriate. But what I am suggesting is that there should be not one counting process, but many – one for each possible final pairing of candidates, with everyone else’s votes transferred in each case. Only that way can we discover which candidate truly has the broadest support by comparison with each of the rest. In an election with three candidates, that means three separate counts – not too unwieldy. In an election with six candidates though, there are 15 possible pairings, and it’s starting to get a bit more arduous (if you’re bothered, the formula to calculate the number of pairings for n candidates is n × (n-1) ÷ 2). But how arduous, really? Given the technology that we have access to, this shouldn’t be any kind of issue.

Unfortunately, this is where my mathematical abilities fail me, because although it seems intuitive that there would always be one candidate who wins each of their counts, I can’t demonstrate that conclusively. If anyone who reads this can either confirm or refute this assumption, I’d be very grateful. To be honest, if anyone reads this at all, I’d be very grateful, and even if you can’t read, I’m grateful to you for casting your tearful gaze on my blog rather than someone else’s. But if you can read and can do that little bit of maths, I will be especially grateful.

That’s it, really. Haven’t we had fun? Oh come on, you don’t mean that.

Blog the first, in which the scrip is blotted, and not much else.

It would appear that I’ve got hold of a “weblog”, and that you’re looking at it. Not to worry – these things happen.

I don’t plan for this to be a tightly themed “weblog”, but I do hope for it not to be too deeply awful, so I’m setting myself a few guidelines from the off. Firstly, no mind-bendingly self-absorbed posts about mundane events in my life – I have a plaster on my finger right now but, man, we all got problems. Secondly, a bit less of the old swearing. Thirdly, no posts with the primary aim of being funny, because I’m not, especially. And lastly for now, I’m not too keen to get into bad-tempered arguments with commenters, should I be lucky enough to attract any; so feel free to disagree by all means, but please also be aware that I will feel free to ignore any particularly brutal displays of keyboardsmanship.

Is that enough of an introduction, you needy git? Then brace yourself for CONTENT. Please. Thanks.