The maths of AV: A small step towards a fairer vote

The Yes and No to AV teams both say their voting system is fairer. But which is? We asked mathematician Tony Crilly to do the sums

Supporters of the "No" campaign, those who reject AV take us to the races and lampoon the winner of the Grand National as the horse that came in third. Those who campaign for "Yes", advocating AV, point to the imagined scenario of a TV talent contest, the winner of which is determined by just 20 per cent of viewers. Much better a contest, they say, where competitors are voted off week by week. With two competitors finally left, one will automatically get more than 50 per cent of the votes.

Mathematicians are used to stripping away inconsequential embroidery, so let's consider the voting question from a mathematical standpoint. In an election contested by three (or more) candidates, how can the winner be judged fairly if none managed to gain the majority of votes? It might be that candidate A gets 40 per cent, B gets 35 per cent and C 25 per cent. What should be done? Is there a voting system to resolve the matter?

As candidate A has most votes, the FPTP camp would say that A is the winner. The obvious drawback to this is that 60 per cent of the voters did not support A. Under FPTP voters are only offered the chance to put down their first preference. Voters do not have the option of indicating support for any other candidate if their first and only preference fails.

The AV camp allows voters to signal second, third... lower order preferences, and these may come into play as "support" for a candidate.

The mathematical problem is how to bring these later preferences into an acceptable calculation. After all, critics will be primed for a retort along the lines that you "can do anything with numbers".

In the AV system the candidate with the least vote is dropped off the list, and the second preferences of their vote reconsidered. In the example C will be eliminated. If 80% of C voters (ie, 20% of ALL voters) put down B as their second preference while the other 20% (5% of all voters) plumped in favour of A, then in the second round B would win the election with 35% + 20% = 55% of the vote with only 40% + 5% = 45% for A.

In AV, instead of the putting down your X, you may now record your preferences, as many or few as you like. A purer form of it has been used in Australia since 1918 and termed the "preferential system", but like AV it is nothing like a true Proportional Representation system (PR) where the number of MPs is directly proportional to the number of people who elected them, that is proportional to the global popular vote.

For 150 years Britain has flirted with different ways of electing its representatives but never took the step of changing. The mathematician Charles Dodgson, otherwise known as Lewis Carroll (the writer of Alice in Wonderland and other classics), was an active campaigner for voting reform, as were such political heavyweights as John Stuart Mill. Thomas Hare, a pioneering Victorian in voter reform, proposed a Proportional Representation system, which took root in Tasmania and was influential in the adoption of the Australian method of preferential voting.

Australian activist for voting reform, Edwin Haber, reminds us that the worth of an electoral voting system is measured by whether it "reflects the nation's mind". With some candidates elected with less than 50 per cent of the vote it is hard to see how the FPTP rates any marks on this criterion. True, PR would do it but it would take a huge reorganisation of the electoral system in Britain. Apart from the widening of the electorate, little change has happened in 150 years and it is difficult to see how this could happen. AV would be but a small step in the right direction.

The Big Questions: Mathematics by Tony Crilly; Quercus (£9.99)