Can we count on your vote?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.
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Key textIn 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:
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:
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:
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 first preference votes are:
Emiko has reached the quota and is elected. She received 16,502 votes above (or surplus to) the quota of 66,996. Her surplus votes are now worked out as a proportion of her total vote:
Emiko's 83,498 votes are then given to their second preference, after being multiplied by 0.19763347:
So now we have:
You can see that the number of preferences allocated is exactly Emiko's surplus:
None of the three has reached the quota of 66,996 votes. And even though Ingrid received most of Emiko's surplus, she still gets excluded because she has the least number of votes. Ingrid's 18,237 first preference votes are now distributed as follows:
The situation now is:
These totals don’t include Emiko's 16,502 surplus votes, which have to be redistributed again. They are distributed between Gerard and Harry, since Ingrid is no longer a candidate. This is done by multiplying Emiko's first preference votes (83,498) by the same proportion as before (0.19763347) and allocating them as follows:
The final picture is:
Harry reaches the quota and is elected to the second seat. Related site
About the House September 2006, pages 47-49 Voters for life (by Georgie Oakeshott) Reports on a survey of young Australians about democratic processes and their level of interest in politics.
Australasian Science October 2004, pages 14-18 The perils of electronic voting (by Stephen Luntz) This article examines the electronic alternatives to voting with paper and pen.
New Scientist 28 April 2010, pages 28-31 Electoral dysfunction (by Ian Stewart) Evaluates different democratic voting systems. 12 April 2008, page 30-33 Why first-past-the-post voting is fundamentally flawed (by Phil McKenna)
21 July 2007, page 25 California court voids electronic vote Reports on a court ruling which voided the results from electronic votes.
30 June 2007, page 25 Electronic polling gets vote of no confidence Reports on the problems linked with e-voting machines.
21 October 2006, page 30-31
22 June 2005 US e-voting proponents say no to paper trails (by Celeste Biever) Reports on a debate in the US Senate over whether electronic ballots should be backed by paper copies.
4 November 2004 Success claimed for US e-voting machines (by Celeste Biever) Reports on the e-voting systems used in the US presidential election.
28 October 2004 US boosts e-voting software security (by Celeste Biever) Looks at an encryption method to provide e-voting software with a tamper-proof 'fingerprint'.
16 October 2004 The great American e-voting experiment (by Eugenie Samuel Reich and Celeste Biever) Summarises some of the problems faced when counting votes.
Scientific American December 2006, pages 12-13 Voting with the heart (by Charles Choi) Suggests that people make emotional decisions when voting.
October 2004, pages 60-65 Fixing the vote (by Ted Selker) Looks at electronic voting systems used in the United States.
March 2004, pages 64-69 The fairest vote of all (by Partha Dasgupta and Erick Maskin) Describes a voting system that gives the best reflection of the electorate's views.
This site covers many aspects of voting systems and includes different voting methods and fairness criteria. Effective use of examples.
Voting systems (Electoral Reform Society, UK)
Provides an overview of different voting systems and the advantages or disadvantages of each system.
ACE Project – Administration and cost of elections (Electoral Knowledge Network)
Ask the experts – mathematics (Scientific American)
Three mathematicians give their answers to the question ‘Has there been any progress in developing fairer ways for people to vote in elections?’.
Electronic voting and electronic counting of votes (Electoral Council of Australia)
Defines electronic voting and describes possible options for it in Australia.
Other electoral sites in Australia (Australian Electoral Commission)
Provides links to other electoral organisations.
How e-voting will work (How Stuff Works, USA)
Uses the 2000 US presidential election to explain how electronic polling places and Internet voting would work.
criteria of fairness. See Fairness criteria (The Center for Teaching and Learning, University of Alabama, USA). electorate. The total number of all qualified voters within specified boundaries. The boundaries of an electorate can be changed. parliament. A political assembly of the representatives of a nation. quota. The prescribed number of votes needed to win an election. In the Australian Senate, the quota is dependent upon the number of senators being elected. For example, the quota needed when there are two senators being elected is one vote more than one-third of all votes, and the quota needed when there are twelve senators being elected is one vote more than one-thirteenth of all votes.
External sites are not endorsed by the Australian Academy of Science. Posted August 1999. The Australian Foundation for Science is also a supporter of Nova.
This topic is sponsored by Australian university mathematical sciences departments and the Australian Government's National Innovation Awareness Strategy.
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