Can we count on your vote?
Key text
This topic is sponsored by Australian university mathematical sciences departments and the Australian Government's National Innovation Awareness Strategy.
There are almost as many different voting systems in the world as there are elected assemblies. The one thing they all have in common is their reliance on mathematics to calculate the results.
In Australia, we elect a group of people, a Parliament, to make decisions and laws on behalf of the whole country. We elect a new Parliament approximately every 3 years. At this time each adult citizen votes for the candidates in their electorate who they think will make the best decisions on their behalf. Candidates are then elected according to votes received, in accordance with the Australian electoral system.
While there are many different electoral systems around the world, all are based on mathematical criteria of fairness. However, electoral systems are generally chosen or refined for political rather than mathematical reasons.
The system used in Australia was based partly on the British Westminster system and partly on the American system. Over time some distinctly Australian characteristics have developed, such as the principle of 'one person, one vote', and a candidate needing to receive a majority of votes to be elected.
Many countries still use the 'first past the post' system, where the candidate with the highest number of votes wins. The problem with this system is that the person with the most votes may still have less than half of all the votes.
For example, let's say that Anna, Brett, Christine and Daniel are candidates in an electorate where 100 votes are cast as follows: Anna 36, Brett 30, Christine 23, and Daniel 11. You can see that even though Anna got the most votes (36), there were almost twice as many people who didn't vote for her (64).
An option used in some countries is to have several rounds of elections - the candidates with the least number of votes drop out, and the whole election is held again with fewer candidates, until someone gets more than half of all votes.
Rather than have several rounds of elections, which requires asking voters to vote more than once, it is mathematically possible to achieve the same result by asking voters to mark their preferences for all candidates. This is called a preferential system and it is used in Australia to elect our Parliament.
What is preferential voting?
In a preferential voting system you vote for each candidate in order of preference, rather than voting for just one candidate. This tells us every voter's first preference, second preference, third preference, and so on. After first votes are allocated, the candidate with the least number of votes drops out and their votes are given to the others according to preferences. This process is repeated until someone gets more than half the votes, as this example shows:
Anna, Brett, Christine, and Daniel are again candidates in an election with 100 voters. The count starts with the first preferences, cast as follows:
| Anna Brett Christine Daniel Total |
36 30 23 11 100 |
Daniel, with the least number of votes, drops out. His 11 votes are now given to their second preference: 3 votes had Anna at number 2, 6 had Brett at number 2, and 2 had Christine at number 2:
| Anna Brett Christine Total |
36 30 23 89 |
+ + + + |
3 6 2 11 |
= = = = |
39 36 25 100 |
Because no-one has more than half the number of votes, another candidate is dropped. This time Christine has the least number of votes so her votes are distributed according to their second preferences: 8 to Anna, 13 to Brett, and 2 to Daniel. But Daniel has already dropped out, so those last 2 go to the third preference, one each to Anna and Brett. The two votes that previously went from Daniel to Christine are now given to their third preference, this time both go to Brett. So now we have:
| Anna Brett Total |
36 30 66 |
+ + + |
3 6 9 |
+ + + |
8 13 21 |
+ + + |
1 1 2 |
+ + |
2 2 |
= = = |
48 52 100 |
Brett now has more than half the total vote so he gets elected.
(Incidentally, according to the Australian Electoral Commission there were seven seats at the October 1998 election where the leader on first preferences lost after the full distribution of preferences.)
The Australian Parliament
The Australian Parliament has two chambers – the House of Representatives and the Senate.
How is the House of Representatives elected?
Australia has 148 electorates that each elect a single member to the House of Representatives, using the preferential voting system.
Most electorates have between 80,000 and 90,000 voters. Electorates have similar numbers of voters because each electorate gets exactly one vote in Parliament. It means that all Australians have roughly equal representation. But it does mean that more than half of all seats come from New South Wales and Victoria because more people live there.
How is the Senate elected?
There are only eight electorates for the Senate – the six States and two Territories. These electorates are much bigger than those for the House of Representatives but each State elects twelve Senators and each Territory elects two. Notice that New South Wales with nearly 3,900,000 voters has as many senators as Tasmania with 320,000 voters (less than one-tenth as many as New South Wales).
One of the reasons for creating the Senate was to give people in the less populated States more of a say in Parliament.
The Senate also uses preferential voting, but because more than one person is elected from each electorate the situation is more complex. Usually, a large number of candidates stand for Senate seats and ordering them all in order of preference can take a long time. As a shortcut, the different political groups work out their preferences before the election and by voting once for a group your preferences are distributed according to the wishes of the group you voted for.
Another complication is that the State senators usually sit through two Parliaments – six Senate seats are filled at one election and the other half at the next election. (There are rare situations when all twelve Senate seats are up for election.) The Territories elect two Senators at each election.
The number of votes needed to win a Senate seat is called a quota:
- the quota needed when there are two senators being elected is one vote more than one-third of all votes;
- the quota needed when there are six senators being elected is one vote more than one-seventh of all votes;
- the quota needed when there are twelve senators being elected is one vote more than one-thirteenth of all votes (Box 1: Counting the Senate vote).
Box
1. Counting the Senate vote
Posted August 1999.