All elections are unfair!

Consider the following hypothetical scenario. The prime minister is to be chosen by 15 electors. Each is asked to order their preference – first, second, third choice – from three candidates. The results are as follows.

Six vote for Cameron first, Clegg second, Brown third. Five vote for Brown first, Clegg second, Cameron third. And four vote for Clegg first, Brown second and Cameron third. Who wins?

Under the UK's current system, first-past-the-post, Cameron wins – he was the first choice of more voters (six) than anyone else.

But is this fair? Nine of the fifteen voters not only didn't place him first, they placed him third.

So who else do we choose? Nine of the voters preferred Clegg to Brown, and nine of the voters preferred Clegg to Cameron. So Clegg wins.

But is this fair? Looking at the first choices, Cameron won six votes and Brown won five. Clegg only won four, so let's eliminate him and have a run-off between Cameron and Brown. In the run-off, six voters would choose Cameron (because six of them placed him ahead of Brown) but nine would choose Brown (because they had placed Brown ahead of Cameron). So Brown wins.

So who wins the election depends on the system you choose to run it. You can make anyone be the winner with the "right" system.

There's a great article about this in New Scientist, and a book on the subject, Numbers Rule, has recently been published, which was reviewed in Times Higher Education.

How would you choose a leader?