Single Transferable Vote

The Single Transferable Vote, or STV, is a preference voting system designed to minimise wasted votes in multi-candidate elections while ensuring that votes are explicitly for candidates rather than party lists. In its most basic form, it works by allocating an elector's vote to their highest ranked candidate who has not already been removed from contention through either election or elimination.

Names

When promoted as a proportional representation method in multi-party multi-seat elections, it is generally known as Proportional Representation through the Single Transferable Vote or PR-STV; in Australia it in its purest form is known as the Hare-Clark Proportional method, whereas the same system with Parties able to indicate preferences is called STV. When a similar method is applied to single-seat elections it is sometimes called instant-runoff voting or the alternative vote or, simply, preferential voting, and has different proportionality implications for a similar ballot. In both systems of voting the ballot choices represent an ordinal ranking of preferences, but an "instant runoff" for only one position or measure is a simple calculation.

Voting

If a class of children were choosing representatives, say, they could line-up behind the candidate of their choice. Since they would all know that each candidate only needs a certain number of classmates to vote for them to be elected, those arriving last in line for a candidate who already has enough votes would choose to not waste their vote and would instead move to another line to help someone else to win. Likewise, those children whose candidate obviously could not win, would move to another line, and so on, until all the representatives are chosen.

When using an STV ballot, these preferences are set out in advance, as instructions to the counters.

Each voter ranks all candidates in order of preference. For example:

1. Andrea
2. Carter
3. Brad
4. Delilah

Setting the quota

Droop quota

The most common formula for the quota used now is the Droop Quota which is most often given as:

<math>\left({{\rm votes} \over {\rm seats}+1}\right)+1<math>.

Unlike the Hare Quota, this does not require that all preferences must reach a final home. It is only necessary that enough votes be allocated to ensure that no other candidate still in contention could win. This leaves nearly a quota's worth of votes unallocated, but it is held that this quota simplifies voting, and that counting these votes would not alter the eventual outcome.

Hare quota

When Thomas Hare originally conceived his version of Single Transferable Vote, he envisioned using the quota:

<math>\rm votes \over \rm seats<math>

This has thus become known as the Hare Quota. It would require that all votes cast be divided equally between the eventually successful candidates. The only differences, thus, between the votes received for each candidate would be based on the distribution of voters between constituencies (Hare's original proposal was for a single national constituency) and the number of non-continuing votes, i.e. people who did not express a preference for all candidates, meaning that some candidates would be elected with less than a quota as the last remaining.

Imperiali quota

Some elections use the Imperiali Quota:

<math>\rm votes \over \rm {seats+2}<math>.

Choice of quota

The size of the quota is then generally Hare > Droop > Imperiali. The reason that the Droop quota is now usually preferred is that with the Hare quota even if each voter expresses full preferences, at least one candidate is likely to be elected with less than a full quota, while with the Imperiali quota more candidates may obtain a full quota than there are seats to the filled. Under the Droop quota, however, if each voter creates a full list of preferences then it is guaranteed that every candidate elected will win by meeting the quota rather than by being the only one remaining after others are eliminated.

Counting the votes

Process A: Top-preference votes are tallied. If one or more candidates receive at least as many votes as the quota, they are declared elected. After a candidate is elected, she may not receive any more votes (though see below for a modernisation).

The excess votes for the winning candidate are reallocated to the next-highest ranked candidates on the ballots for the elected candidate. There are different methods for determining how to reallocate excess votes. (See below)

Process A is repeated until there are no more candidates who have reached the quota.

Process B: The candidate with the least support is eliminated, and his votes are reallocated to the next-highest ranked candidates on the eliminated ballots. After a candidate is eliminated, he may not receive any more votes.

After each iteration of Process B is completed, Process A starts again, until all candidates have been elected or eliminated.

Surplus re-allocation

Hare's method

With the school children (above) all the votes that arrived after the candidate reached a quota of votes were simply redirected to their next preferences. The same can be done with ballot papers, on the rough assumption that the votes in the surplus are a representative sample of all the ballots cast. The possibility remains, however, that they will not represent a properly mixed sample.

Cincinnati method

The city of Cambridge, Massachusetts, uses a method called the Cincinnati method to increase the likelihood of a representative spread of transfers. The votes from each polling district are counted and merged together into a randomised sequence. Then, taking the formula

<math>\left( \frac{\mathrm{Candidate's\ Total\ Ballot}} \mathrm{Candidate's\ Surplus} \right)= n<math>

every nth ballot is removed from the candidates pile and transferred. If no continuing preference is stated on the paper, then the next available ballot is chosen. If the candidate's votes are not reduced to the quota after removing every nth ballot, every (n+1)th ballot is removed.

Clarke method

In Australia, A.I. Clarke (see below) devised the method whereby all a candidate's next preferences are counted, and their surplus is transferred according to the various ratios of such preferences found. e.g. if a candidate has 100 surplus votes, and 25% of all their votes have the same next preference, 25 will be allocated accordingly.

Senatorial rules

Another method is known variously as the Senatorial rules – after its use for Irish Senate elections, or the Gregory method – after its inventor in 1880 J.B. Gregory of Melbourne. This continues Clarke's method. Instead of dividing the surplus out according to the fractions of next preferences a candidate received, all of their votes are passed on to their next preference, with their new effective value found by:

<math>\left( \frac{\mathrm{Quota}} \mathrm{Candidate's\ votes} \right)<math>

To avoid confusion, the decimal point is sometimes removed, and votes are given a nominal value of 100 instead of 1.

Meek method

All of the above methods assume manual counting of ballot papers. In 1969, B.L. Meek devised a method which is suitable for computer counting.

All candidates are allocated one of three statuses – Hopeful, Elected, or Excluded. Hopeful is the default. Each status has a weighting, or keep value, which is the fraction of the vote a candidate will receive for any preferences allocated to them while holding that status.

The weightings are:

Thus, if a candidate is Hopeful they retain the whole of the remaining preferences allocated to them, and subsequent preferences are worth 0.

If a candidate is Elected they retain the value of their weighting and the remainder of the value of the vote is pass along fractionally to subsequent preferences depending on their weighting, with the formula

<math>\left (1 -{\mathrm{nth Weighting}} \right)<math>

being carried out at each preference.

This results in a fractional excess, which is disposed of by altering the quota, hence Meek's method is the only method to change quota mid-process. The quota is found by

<math>\left({{\rm votes – excess} \over {\rm seats}+1}\right)<math>,

a variation on the droop quota. This has the effect of also altering the weightings for each candidate.

This process continues until all the Elected candidates' vote values almost equal the quota (within a very close range, i.e. between 0.99999 and 1.00001 of a quota). [1]

An example

2 seats to be filled, four candidates: Andrea, Brad, Carter, and Delilah.

The votes are cast with the following preference groupings:

	16 Votes	24 Votes	17 Votes
1st	Andrea	Andrea	Delilah
2nd	Brad	Carter	Andrea
3rd	Carter	Brad	Brad
4th	Delilah	Delilah	Carter

The threshold is: <math>\left({57 \over (2+1)}\right) +1 = 20<math>

In the first round, Andrea receives 40 votes and Delilah 17. Andrea is elected with 20 excess votes. Her 20 excess votes are reallocated to their second preferences. For example, 12 of the reallocated votes go to Carter, 8 to Brad.

As none of the remaining candidates have reached the quota, Brad, the candidate with the fewest votes, is eliminated. All of his votes have Carter as the next-place choice, and are reallocated to Carter. This gives Carter 20 votes and he is elected.

Thus:

	Round 1	Round 2	Round 3
Andrea	40	20	20	Elected
Delilah	17	17	17	Defeated
Brad	0	8	0	Eliminated
Carter	0	12	20	Elected

STV in Australia and other countries

Australia

Australia uses two forms of STV, usually called the Hare-Clark System and STV.

The Hare-Clark System is used to refer to the system used in Tasmania's House of Assembly and the Australian Capital Territory (ACT) Legislative Assembly. This is essentially the system described above using the Droop quota (not the Hare quota), but candidates' placements are randomised by Robson Rotation rather than grouped by party.

STV or 'proportional voting' is the term used (somewhat inaccurately) to refer to the system used in the Australian Senate and the Legislative Councils of New South Wales, Victoria, Western Australia and South Australia. This system is counted in the same way as in Hare-Clark, but group voting tickets are used.

Each form has its pros and cons. The Hare-Clark system with Robson Rotation is advocated on the grounds that voters know who they are voting for as they must fill all their preferences, that each party's candidates compete with each other and the effect of 'donkey voting' is reduced because of the randomised ordering. The alternative system is advocated on the grounds that informal voting is reduced because only one number need be written; on the other hand, it greatly increases the potential for tactical voting by parties.

Ireland

Republic of Ireland, for all elections [2]. (However, presidential elections and most by-elections elect only one candidate and so reduce to the Alternative Vote.) Ireland shows one major advantage of STV — you can elect a president or mayor by the same method as the parliament or the city council.

Northern Ireland

Northern Ireland, for local, Assembly and European elections

Malta

STV applies for all elections in Malta. See [3]

New Zealand

New Zealand [4], where used STV for the first time for district health board and some local authority elections in October 2004New Zealand has chosen STV by Meek's method which overcomes the non-monotonicity problem in STV.

Scotland

All local governments in Scotland will be using STV to elect their councillors. The Local Governance (Scotland) Bill [5] passed on June 23, 2004.

North America

STV enjoyed some popularity in the United States in the first half of the 20th Century. The United States, where the only official governing bodies that use STV to elect representatives are the City Council and School Committee of Cambridge, Massachusetts. The community school boards of the City of New York used STV until they were abolished in 2002. It was used for the elections in the province of Alberta, Canada from 1926 to 1955.

British Columbia will decide by referendum on May 17, 2005 whether to adopt STV to replace its current First Past the Post electoral system, after a recommendation of STV [6] by the Citizens' Assembly on Electoral Reform.

NGOs

Many non-governmental organisations also use STV. Most Australian political parties, unions and peak business organisations use STV. All National Union of Students of the United Kingdom and Oxford Union elections and those of their constituent members are under the system. It is used in several political parties for internal elections such as the British Liberal-Democrats and all the British Green Parties. It is also used to elect members of the General Synod of the Church of England. The UK Royal Statistical Society [7] uses STV with the Meek method to elect their council.

Ordering candidates for List PR

STV can be used used as a social preference function to produce an overall ranking of candidates. For example, closed party-list PR allots seats proportionally to lists and chooses candidates from that list in the order set down on the list. Some parties decide this centrally but others give the choice to the membership. STV has been used in such elections; in such an election the party members mark their ballots in just the same way as in a normal STV election.

When the votes come to be counted the same ballots are used first for an STV election to choose one winner, as an instant runoff election; then the same ballots are used to see which two would win when the votes are counted for a two winner election and so on. The head of the list is the party member who would be elected in the one winner count. The second place is taken by the next party member who succeeds in the two winner race (in the unusual event that the winner of the single winner count did not succeed in the two winner count, then the first to be elected in the two winner count), and so on.

Like most other vote-based ordering systems, in Arrow's impossibility theorem STV fails to achieve independence of irrelevant alternatives; it also fails the monotonicity criterion.

Issues

Proportionality

The outcome of voting under STV is proportional within a single election to the collective preference of voters, assuming voters have ranked their real preferences. However, due to other voting mechanisms usually used in conjunction with STV, such as a district or constituency system, an election using STV may not guarantee proportionality (across all districts put together). For example, in STV elections to the Australian Senate, states with vastly different populations have the same number of seats, and so while the results for individual states are proportional, the nationwide result is not, giving greater voting power to individual voters in less populated states; the lack of proportionality is derived from unequal representation rather than any deficiency in STV.

Because STV is a preference voting system, whereby voting is done by ranking a list of candidates, the type of proportionality obtained by STV contrasts with many other proportional voting systems such as party-lists. Under these systems, seats are apportioned as a percentage of single-preference-only votes for each candidate or party. By contrast, in STV each voter has a single (transferable) vote, regardless of whether there is one vacancy or several. STV provides proportionality by transferring votes to minimise waste, and therefore also minimises the number of unrepresented voters. Votes are said to be wasted when they have no effect on the outcome of an election, either because they go to a losing candidate or to a winning candidate who does not need them. For example, in an STV election using the Droop quota method for 9 seats, at most only 10% of the vote will be wasted. Depending on the method of STV used and the preference distribution of the voters, this share of wasted votes can either be completely localised within 10% of the electorate (who would get no representation) or fractionally spread out amongst a larger share. Importantly, the more winners there are in a single constituency, the fewer votes that may be wasted under STV and therefore the more proportionate the outcome will be.

Failures to produce partisan proportionality exactly analogous to the party affiliations of top candidates in elections can be controversial, and this situation has arisen in elections using STV. For example, the Northern Ireland elections in 1998 led to the Ulster Unionists' winning more seats than the Social Democratic and Labour Party, despite winning a smaller share of first-preference votes. In the Republic of Ireland in 2002, Sinn Fein won 6% of the votes and 3% of the seats, being the fourth most popular party in terms of first preferences, but coming sixth in terms of the number of seats won. The outcome may be particularly controversial in close elections such as the 1981 election in Malta. In this election the Maltese Labour Party won a majority of seats despite the Nationalist Party winning a majority of first-preference votes. This caused a constitutional crisis, leading to a provision for the possibility of bonus partisan seats which were again awarded after elections in 1987 and 1996.

STV also experiences the theoretical curiosity that, unlike proportional party list systems, if a candidate is elected in an n seat constituency, she may not be elected in the same constituency with n + 1 seats even when voters express exactly the same preferences. This is due to the inherent mechanism for compromise built into the STV system – intuitively, a candidate who was elected largely because of transfers from two similar groups may not be elected when the number of winning candidates increases, as both groups would instead get their preferred candidates elected rather than automatically compromising on their mutual second choice when their votes transfer.

Tactical voting

The single transferable vote eliminates much of the reason for tactical voting. Voters are "safe" voting for a candidate they fear will not be elected, because their votes will be reallocated in Process B. They are "safe" voting for a candidate they believe will receive overwhelming support, because their votes will get reallocated in Process A.

Though still theoretically possible, figuring out how to vote tactically in STV systems by exploiting the non-monotonicity of STV is extremely computationally difficult. It is NP-hard to determine whether there exists an insincere ballot preference that will elect a preferred candidate, even in an election for a single seat. This makes tactical voting in STV elections vastly more difficult than with other commonly-used election methods. Importantly, this resistance to manipulation is inherent to STV and does not depend on hopeful extraneous assumptions like the presumed difficulty of learning the preferences of other voters. Furthermore, it is NP-hard to determine when an STV election has violated the monotonicity criterion, greatly reducing the likelihood that the electorate will know if even accidental tactical voting has occurred. As a consequence, the difficulty of tactical voting in STV elections increases sharply as the number of voters, candidates, and winners increase.

Of special note is that voters have a real incentive to list their preferences honestly in STV, as it's the best strategy for securing representation if tactical voting is either impractical or impossible. This is frequently the case, as successful tactical voting (when possible) requires both nearly perfect information about how others are voting and the computation of a virtually unsolvable math problem. This contrasts heavily with non-proportional, plurality-based systems, where there is tremendous incentive and ability to vote tactically in order to avoid the spoiler effect.

However, in the older STV systems still used in many countries there is a loophole that can allow for easier tactical voting: candidates who have already been elected do not receive any more votes, so there is incentive to avoid voting for your top-ranked candidate until after he has already been elected. For example, a voter might make a tactical decision to rank her top-place candidate beneath a candidate she knows will lose. If the voter's true top-place candidate has not been elected by the time her fake top candidate loses, the voter's full vote will count for her true top-place candidate. Otherwise, the voter will have avoided either having had her ballot in the lottery to be "wasted" on their top-ranked candidate or with only a fraction transfered, and will continue on to lower-ranked candidates.

Note that in some more modern STV systems, this loophole has been fixed. A vote receives the same fractional weighting regardless of when it arrives at the successful candidate. This modernisation has not been adopted in all STV systems.

There are also tactical considerations for political parties standing more than one candidate in the election. Standing too few candidates may result in all of them being elected in the early stages, and votes being transferred to candidates of other parties. Standing too many candidates might result in first-preference votes being spread too thinly amongst them, and consequently several potential winners with broad second-preference appeal may be eliminated before others are elected and their second-preference votes distributed. This effect is amplified when voters do not stick tightly to their preferred party's candidates; however, if voters vote for all candidates from a particular party before any other candidates and before stopping expressing preferences, then too many candidates is not an issue. In Malta, where voters tend to stick tightly to party preferences, parties frequently stand more candidates than there are seats to be elected, although this behaviour is also partially a result of the Maltese system's potential for granting parties bonus seats. Similarly, in Australian Senate elections that require a voter to make a complete list of preferences, voters also tend to vote along party lines due to the relative ease of selecting a party's declared preferences rather than individually casting their own.

Vacancies

It is difficult to fill vacancies which occur under STV given the way that results depend on votes given to other candidates in the same party and in other parties. There are several possible solutions:

Countback

The countback method is used in Tasmania, Malta, and Cambridge, Massachusetts. A new representative is selected by using the data from the previous election. The candidate who held the seat is eliminated, and a new election result is counted from the first and later preferences credited to the vacated candidates. Importantly, a clone of the replaced representative would be guaranteed to win.

In 1985 the Tasmanian parliament amended the electoral act to allow true by-elections if no candidates of the same party as the outgoing MP remain. In this circumstance the party may request that a by-election be held; this has not yet happened.

Appointment

The state legislatures appoint replacements members to the Australian senate. This is done at the suggestion of the party of the outgoing senator. Disputes over Senate vacancies contributed to the Australian constitutional crisis of 1975. In 1977 the people amended the Constitution of Australia to provide that the legislature must elect a member of the same party as the outgoing senator. Vacancies in the New South Wales legislative Council are filled in a similar way by a joint sitting of the legislative council and assembly.

IRV By-Election

Holding an instant-runoff by-election, as happens in the Republic of Ireland; this allows the parties to choose new candidates and all voters to participate, but often leads to the most popular party picking up an extra seat. This is because if you voted for a continuing member your vote still counts and you effectively get to vote a second time while the member you elected at the general election remains in office. If you voted for the vacating member you only get one vote.

Subdistricts

If each legislative district is divided into as many sub-districts as there are representatives, each representative can be elected in the single-member sub-district where they perform best. Vacancies can then be filled by an instant runoff ballot within that sub-district.

An important consequence of the Recount and Countback methods for selecting replacements is that they are known before the vacancy actually occurs, potentially influencing the circumstances which create the vacancy in the first place.

History

Thomas Wright Hill first proposed transferable voting in 1821.

In 1855 Carl Andrae proposed a transferable vote system in Denmark. It was used in 1856 to elect the Rigsdag. In 1866 it was adapted for indirect elections to the second chamber, the Landsting, until 1915.

Thomas Hare is generally credited with conceiving of Single Transferable Voting, and he independently developed the idea in the 1857. He proposed a single national constituency. His view was that it should be a means of "making the exercise of the suffrage a step in the elevation of the individual character, whether it be found in the majority or the minority". He proposed that electors should have the opportunity of discovering which candidate their vote had counted for, to improve their personal connexion with voting.

John Stuart Mill was an early proponent of STV and praised it in "On Representation." In the "English Constitution" Walter Bagehot praised the Hare system for allowing everyone, even ideological minorities, to elect an MP, but said that the Hare would create more problems than it solved. "[the Hare system] is inconsistent with the extrinsic independence as well as the inherent moderation of a Parliament – two of the conditions we have seen, are essential to the bare possibility of parliamentary government."

STV spread through the British Empire – leading it to be sometimes known as British Proportional Representation. Andrew Inglis Clark was successful in persuading the Tasmanian House of Assembly to be the first parliament in the world elected by Hare-Clark in 1896. Hare-Clark is named after Thomas Hare and Andrew Inglis Clark.

It was used in Canada in Calgary and Edmonton from 1926 to 1955.

It has found use in America, and from 1936 to 1947 it was used in New York City municipal elections for a short while to break the Democrats' strangehold on the city. This reverted again to First Past The Post, after strenuous attempts by the Democrats to reverse the change, aided by scraes over the election of Communist candidates.

Meek's version contained the innovation that electors could rank preferences equally, but this option has not been used.

See also

• List of democracy and elections-related topics
• Table of voting systems by nation
• Single Non-Transferable Vote
• preference voting
• Instant-runoff voting
• voting systems
• Hagenbach-Bischoff quota
• Hare quota
• Imperiali quota
• Ross quota

External links

• Simulating 2005 BC Election if STV was used
• pSTV--software for computing the single transferable vote
• Electoral Reform Society
• Proportional Representation Society of Australia
• The Single Transferable Vote procedure in Ireland
• Information on BC-STV. British Columbia's Referendum on STV
• Single Transferable Vote Resists Strategic Voting a paper which discusses the NP-completeness of the problem of determining when monotonicity is violated and voting tactically in STV elections
• A New Approach to the Single Transferable Vote a paper by Brian Meek which advocates Meek's method for eliminating the problem of tactical voting and ensuring monotonicity.
• Algorithm 123 —Single Transferable Vote By Meek’s Method
• Single Transferable Vote Animation (BC-STV) detailed flash animation explaining the process of Single Transferable Vote.
• STV in New York