Jacob Aron, reporter
On 5 May, the UK will hold a referendum to determine which voting system the country should use in future elections, with voters asked to decide whether they want to adopt the alternative vote (AV) or stick with the first-past-the-post (FPTP) system, which is currently used. Can mathematics tell us which system is the most fair?
AV requires voters to rank candidates in order of preference. If no candidate gains 50 per cent of the vote, the candidate with the least votes is eliminated and their voters' support is switched to their second choice. The process repeats until one candidate has 50 per cent or more.
By contrast, FPTP, also used by the US, is much simpler as it only requires voters to choose a single candidate, and the one with the largest number of votes wins. However, as we explain in a recent feature story, it is often criticised for its ability to "throw up a winner who in a straight fight with the runner-up would not be the majority's choice". This was illustrated by the 2000 US presidential election, won by George W. Bush, where "it is now widely accepted that the Green party candidate Ralph Nader took votes that would otherwise have counted towards Al Gore".
Back in the UK, campaigners on both the "yes" and "no" sides of the AV question are plugging their cause, yet few politicians are talking about the theories behind each voting system. Filling the gap are a number of mathematically-minded individuals who have weighed in on the debate.
Leading the charge is blogger Timothy Gowers, a mathematician at the University of Cambridge. He argues that many of the claims made by "No to AV" supporters are mathematically wrong.
I find it very dispiriting to live in a country where it can benefit a politician to use a provably incorrect argument
He also complains that, despite its name, there is actually no such thing as "the post" in FPTP, whereas the concept does exist under AV: candidates must past "the post" of 50 per cent to win.
AV is also supported by Dennis Leech, an economist at the University of Warwick, UK, who calls FPTP "just about the worst election method ever devised".
However, he also highlights some problems with AV, including the finding that, in some situations, a candidate's chances of winning can actually be reduced if voters rank them higher on the list. He adds however that this may just be a theoretical problem with the system that is unlikely to play out in the real world.
Highlighting a more radical solution is David Maclver, a software engineer with a background in mathematics. In 1963 economist Kenneth Arrow proved that no voting system can satisfy a few reasonable and democratic conditions - in other words, democracy is always unfair. MacIver points out that Arrow was only considering certain types of voting systems, and claims that there is an alternative, "perfect" system.
Maclver's system is identical to FPTP in all but one respect. Voters in each constituency choose a single candidate, but then one voter is picked at random from each constituency and their choice determines which candidate gets elected. The random element means the system isn't covered by Arrow's theorem.
It sounds horribly unfair but it would actually produce results that are more proportional to the views of the country as a whole, argues MacIver, as it is simply a random sampling of the population. So if a party has 20 per cent of the national vote, it should end up with roughly 20 per cent of the seats in parliament.
It turns out Maclver's idea isn't a new one - the system is known as a random ballot. But it isn't one of the choices being offered to the UK public.
Gowers and Leech make good arguments for why AV is mathematically a better system than FPTP, but ultimately the decision people make in the ballot box next week is all about politics, not mathematics. Do you want a system that picks a winner with strong support from a minority of voters (FPTP) or one where the leading candidate is vaguely liked by a majority of people (AV)? No amount of equations can help you reach an answer.