A simple guide to electoral systems-Proportional systems

The second main type of electoral system is proportional systems. Proportional representation systems are designed to provide a close match between votes cast and seats won. Proportional representation systems are only used for legislatures, as a single office cannot be divided proportionally.

There are two types of proportional representation. There are party list systems, which work by dividing seats between parties based on their vote shares and then electing members from lists of candidates, and the single transferable vote. The single transferable vote is slightly harder to explain, but it will be explained below.

Party list systems

There are two basic elements to party list systems. The first is what sort of system is used to divide seats between lists (the method of seat apportionment) , and the second is how the party lists are composed.

Seat apportionment

Largest remainder methods

Largest remainder methods are one of two ways of distributing seats to parties. Under largest remainder methods, voters vote for a party list. The first step in seat allocation is to calculate a quota. There are a number of quotas. The Hare quota, invented by Thomas Hare, is total votes/seats. The Droop quota, invented by Henry Droop, is total votes/(seats+1) or total votes/(seats+1)+1. The Imperiali quota is total votes/(seats+2), and a quota used for Italian elections during the 1950s is total votes/(seats+3).

The votes for each party are divided by this quota. A party receives one seat for each filled quota. If there are still unfilled seats, the party with the highest remainder is given one seat. This process continues, with no party being allocated more than one seat by highest remainders.

The table below shows how the system works. In the election below, 5 seats are at stake, and 4 parties are competing.

As you can see, three seats are allocated on full quotas: one each to parties A, B, and C. This leaves 2 seats unallocated. These are filled by parties A and B, as they have the 2 highest remainders.

This system can be modified to only allocate seats on full quotas. Under this system, known as the remainder-transfer system, seats are allocated to parties for each quota that they win. Unused votes are transferred to a higher tier multi-member constituency, where all seats are allocated using quotas and largest remainders.

Highest averages method

The highest averages method is somewhat different. However, voters still cast votes for a party. These votes are tallied up. Under highest averages, a numerical sequence is decided on before the election. There are a number of different numerical sequences. There is the D'Hondt method, invented by Belgian mathematician Victor D'Hondt. This sequence starts at 1 and increased by 1 at each increase; it goes 1,2,3,4 and so on. The second system is the Sainte-Lague method. Invented by a French mathematician, this sequence consists of all positive odd numbers, starting at 1: it goes 1, 3, 5, 7 and so on. This system can be modified to increase the barrier for a first seat by changing the first divisor to 1.4. There is the Imperiali method, which starts at 2, goes up by 1 each time. There is also the Danish method, which goes 1,4,7,9 and so on.

So, how do these systems translate votes to seats? Well, at the start, all votes are divided by the first number in the sequence. The party with the highest number is then awarded one seat, and their vote is divided by the next number in the sequence. At this point, the party with the highest vote wins a seat, and has their vote divided by the next number. This process repeats until all seats are filled.

The table below shows a sample election, the same sample election as above. The D'Hondt method is used.

As you can see, Party A has the highest vote, and it receives the first seat. Party A's vote is divided by 2 (the next number in the D'Hondt sequence). Because of this division, Party B wins the next seat. Party B's vote is then divided by 2 as well. Party C then wins a seat, and has their vote divided by 2. At this point. Party A has the highest vote, and they win another seat. Their vote is then divided by 3 (the next number in the sequence). This division makes Party B the largest, and they win the fifth and final seat.

Composition of party lists

In order to fill these seats, parties need to create ranked lists of candidates. There are different ways of creating these lists.

Closed list system

Under a closed list system, parties draw up a ranked list before an election. Voters vote only for a party, and the candidates on the list are elected in list order.

Flexible list system

Under a flexible list system, voters may vote for a candidate within a party. This vote counts as a vote for the party. If the vote for a candidate excludes a certain threshold, that candidate is elected. After all candidates with a personal vote over the threshold are elected, remaining seats are filled in party list order. This system is used in Sweden and the Netherlands.

Open list 

Under an open list system, voters may vote for one or more candidates within a party. This vote counts as a vote for the party. Seats are filled by the candidates with the highest votes. This system is used in Finland and Brazil

Free list

Under a free list system, voters have as many votes as there are candidates to be elected. Voters may cast multiple votes for a single candidate, or cast votes for candidates of different parties. Each vote for a candidate counts as a vote for the party that they are running with. Within parties, seats are filled by the candidates with the most votes. This system is used in Switzerland and Luxembourg.

Single Transferable Vote

Australian Capital Territory single transferable vote ballot paper. (Credit:Elections ACT)

The Single Transferable Vote (STV) is one of the rarer voting systems. It is used only in Australia (for upper house elections in New South Wales, Victoria, South and Western Australia, for lower house elections in Tasmania and the Australian Capital Territory, and for the Federal Senate), Ireland (for all elections), Scotland (for local elections), India (for indirect federal upper house elections), and Malta (for all elections).

Under the Single Transferable Vote, voters vote for candidates, not parties. Voters vote by numbering boxes next to candidates: a voter's most preferred candidate gets a 1, second most preferred a 2, and so on. 

At the start of the count, all number 1 votes are counted, and votes are tallied for each candidate. Following this, a quota is calculated. Under the first STV elections, the quota was the Hare quota (votes/seats), but this was abandoned, as this quota tends to increase the value of tactical voting (don't ask).  Usually this quota is the Droop quota (votes/(seats+1)+1).

Any candidate with more than the quota, or the same number of votes as the quota, is elected. If a candidate has the same number of votes as the quota, they are elected, and nothing needs to be done. However, if a candidate has more votes than the quota, then they have a surplus. Different STV methods differ on how to deal with this surplus; the Tasmanian STV system gives each surplus vote a value (number of surplus votes/total votes for elected candidate with further preferences), and distributes the votes at that value, while other systems take a random sample of surplus votes. Under most STV systems in use, all ballot papers are now treated as if that candidate never stood for election.

If no candidate has a surplus, the lowest polling candidate is excluded. Their ballot papers are examined for further preferences: ballot papers are transferred to the next available preferences. If there are no further preferences on a ballot paper, that paper is marked 'exhausted', and is ignored for the remainder of the count. Again, all ballot papers are now treated as if the excluded candidate had never stood for election.

This process repeats until all seats are filled. If, at the final stage, the number of candidates still in the count is equal to the number of seats unfilled, those candidates are elected, even if they have not reached the quota.

Formerly for the Australian senate and for the Victorian, Western Australian, and South Australian upper houses, a modification is added. Parties have the option of lodging 'party voting tickets'. A party voting ticket is a list of candidates in an election ranked by a party. Under group ticket elections, voters either number every box for every candidate (only five in Victoria), or vote for a party ticket. Party tickets have been controversial for taking preference power out of the hands of the voters; this has resulted in anomalous results, like Ricky Muir getting elected to 1 of 6 Victorian Senate seats with 0.5% of the vote.

In 2016, the Turnbull Liberal government made changes to the electoral law for the Senate. Group voting tickets were abolished. Ballot papers look the same, with voters having the option to vote for a parties above the line or individual candidates below the line. However, an above-the-line vote for a party has ceased to become an acceptance of a group voting ticket; it counts as a vote for the party's candidates in order of preference.

For example, a person voting 1 above the line for the Labor Party (3 candidates) and 2 for the Greens (2 candidates) would have their vote go 1 to the first Labor candidate, 2 to the second, 3 to the third, then fourth to the first Green candidate, and fifth to the second. Onc but it would only go to Labor candidates. Voters can number multiple boxes above the line, and are told by the AEC to number six (though one is formal) but a vote exhausts when there is no remaining preference.