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CSIRO Education, Australian Mathematical Sciences Institute and the Australian Government Department of Education, Employment and Workplace Relations distribute a useful and interesting newsletter. The newsletter consists usually of a main article and several small problems. This time, the main article dealt with Preferential voting. Towards the end there appeared a claim to the effect that "The Australian system of preferential voting is much more complicated than other forms of voting, but it makes sure that the best candidate wins."

The authors of the article are obviously within their rights to be satisfied with the manner in which the choose their government. I wonder whether their satisfaction slipped a little too far. They conclude the article with

This way, a little bit of maths makes sure that we get the best possible government.

Below I include the whole article that in fact does a very good job describing the preferential voting system.

A few weeks ago, the Prime Minister called an election. On August 21, all Australian adults will be voting to elect a member to represent them in parliament. But how do we work out who wins?

In many countries, the system is very simple. Each voter gets one vote. They write on a paper, known as a ballot paper, which person they want to vote for and then put it in a sealed box. After all the votes have been cast, they are counted and the person with the most votes wins the seat.

This system is easy to understand, but it is not always fair. For example, imagine an election where three "similar candidates" (they have similar policies) run for the same seat against another candidate. Each of the "similar candidates" gets 20% of the votes in the election, and the other candidate gets 40%. The other candidate wins, even though the "similar candidates", between them, got more than half the votes.

In Australia, we have a more complicated system, called preferential voting. With preferential voting, a voter ranks every single candidate, from the best to the worst, by numbering them 1, 2, 3 and so on, on the ballot paper. When the votes are counted, each candidate gets the votes that they were ranked first on.

If one candidate has more than half the votes, then they are elected. They are definitely the most popular person. If not, then the least popular person (the one with the least votes) is eliminated and the ballot papers are redistributed. Each vote is now given to the person ranked second on each voting form.

If there still isn't a candidate with more than half the votes, then the next most unpopular person will be eliminated. These votes are given to the next ranked person on each voting form. This process of eliminating the least popular person and giving out their votes continues until one candidate has more than half of all the votes. Once a person has more than half the votes, they cannot be beaten, so they are declared the winner.

The Australian system of preferential voting is much more complicated than other forms of voting, but it makes sure that the best candidate wins. This way, a little bit of maths makes sure that we get the best possible government.