The electoral system in Australia depends on this form of voting, where voters rank candidates in order of preference.
Proportional representation - the main methods - a document outlining the three major methods of achieving proportional representation
Lower House Voting
The voting for the lower house(House of Representatives) is the simpler. Voters must rank all candidates in order of preference, leaving no candidate unranked. Votes are counted by voters highest ranking candidate(First preference). Candidates are then ranked based on number of votes received, and the candidate with the least votes received is eliminated. Votes currently counting towards that candidate are then distributed to that votes next highest preference. Once these votes have been distributed, the candidates are ranked again, and the process repeated until only two candidates remain, and the winner is the candidate who has a majority of the votes. The final numbers of votes the last two remaining candidates have is referred to as a Two Party Preferred result.
Example: A seat in the lower house has five candidates contesting on behalf of these parties: Workers, Conservative, Environmentalist, Family and Independent.
First preferences are tallied with the following result: Workers 30%, Conservative 25%, Family 20%, Environmentalist 15%, Independent: 10%.
In Round 1, the Independent candidate is eliminated, and his votes are redistributed. Let us assume that 75% of the Independent candidate's votes went to the Environmentalist party, and 25% of their vote went to the Workers party. The new tallies are now: Workers: 32.5%(+2.5%), Conservative 25%(-), Environmentalist: 22.5%(+7.5%), Family: 20%(-).
In Round 2, the Family candidate is eliminated. Any of the votes which next preference the Independent candidate, skip that preference as that candidate is no longer a valid candidate in the race, and will go to the next candidate. Following these rules, 90% of Family's votes go next to the Conservative party, and 10% go to the Workers party. The tallies now read: Conservative: 43%(+18%), Workers: 34.5%(+2%), Environmentalist 22.5%(-).
Finally, the Environmentalist party votes are redistributed as they are eliminated, 80% going to the Workers party, and 20% going to the Conservative party. The final Two Party Preferred numbers now read: Workers: 52.5%(+18%), Conservatives: 47.5%(+4.5%), making the Workers party the winner of the election.
Upper House Voting
Voting in the upper house also uses preferences, but is more complicated. In the upper house(senate), each state is electing multiple senators at any given time, and the state is voting as a whole. This leads to a system where instead of reaching a majority, candidates are attempting to reach a quota of votes, equal to the number of people who voted formally, divided by the number of candidates to be elected plus one(six at a time for each state, so seven), plus one vote .
As with the lower house, the first step is to tally first preferences. When this is done candidates are again ranked in order. Two steps are then gone through and repeated throughout the process of tallying the votes.
Step one: Checking for quotas: If a candidate currently has votes equal to or in excess of a quota, they are successfully elected to one of the senate seats for this electorate. They likely received more votes than exactly enough, in which case some amount of votes is redistributed to other candidates. In order to prevent these votes being able to count for more than one vote, and because each vote elected the successful candidate equally, they must be proportionally reduced. If for example, a candidate received twice the number of votes needed for a quota, then half of the votes they received were part of the quota, and the remaining half should be redistributed. This means that all the votes counted for that elector are redistributed, but they now only count as half of one vote.
Step two: Eliminating unsuccessful candidates: This step is carried out if no candidates currently have a quota. It works exactly as it does in the lower house, the candidate with the lowest current total of votes is eliminated, and their votes are redistributed. Step one and two are now repeated until all seats have been filled.
Example: A senate election is electing Three senators, and there are 100 formal votes lodged thus the quota needed for election is 25% + 1 vote, or 26 votes.
The current candidates are running: WorkerA, WorkerB, ConservativeA, ConservativeB, Environmentalist, Family, Independent.
First preferences are counted as such: ConservativeA: 34,WorkerA: 28, Environmentalist: 14, Independent:10, ConservativeB: 6,WorkerB: 5, Family: 3
Step 1) ConservativeA has 34 votes, which gives them a full quota, they are elected to one of the seats. They have an excess of 8 votes, so their votes will be redistributed at a ratio of 8/34, or 23.5%. However, it is not difficult to imagine that all of the people preferencing a party's first candidate will direct preferences directly to that parties second candidate, so they will flow at a rate of 100%, 8/34 * 34 = 8, so 8 votes flow into ConservativeB, giving them a new total of 14.
Current Tallies: ConservativeA(Elected), WorkerA: 28, Environmentalist: 14, ConservativeB: 14(+8), Independent: 10, WorkerB:5, Family: 3.
Step 1) WorkerA has enough votes for a quota, electing them with an excess of 2 votes, making the same assumption as the Conservative vote, Let us assume however, that this vote is more divided than the conservative vote, with Half of the people voting first for WorkerA giving their second preference to WorkerB, and the other half giving it to the Environmentalist party, after diluting the votes, this then adds 1 votes to WorkerB, and 1 votes to Environmentalist.
Current Tallies: ConservativeA(Elected), WorkerA(Elected), Environmentalist:15(+1), ConservativeB:14, Independent: 10, WorkerB: 6(+1), Family: 3.
Step 1) Noone has a quota, so on to step two for the first time.
Step 2) Family has the least current votes, so their 3 votes get sent to their next valid preference, 2 of those votes went to ConservativeB, and one of those votes went to Independent.
Current Tallies: ConservativeA(Elected), WorkerA(Elected), ConservativeB: 16(+2), Environmentalist: 15, Independent: 11(+1), WorkerB: 6
Step 1) Noone has a quota, on to step two.
Step 2) The second Worker candidate is eliminated, with a split of 5 votes going to the Indepenedent candidate, and 1 vote going to the Conservative candidate
Current Tallies: ConservativeA(Elected), WorkerA(Elected), ConservativeB: 17(+1), Independent: 16(+5), Environmentalist: 15
Step 1) No quotas.
Step 2) Environmentalist is eliminated, with preferences being directed 10 votes to the Independent, and 5 votes to the Conservative.
Current Tallies: ConservativeA(Elected), WorkerA(Elected), Independent: 26(+10), ConservativeB: 22(+5)
Step 1) The Independent now has a quota, and receives the third and final seat.
Result: Candidates elected: ConservativeA, WorkerA, Independent.