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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.

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. 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.



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

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

\rm votes \over \rm seats

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 candiates would be elected with less than a quota as the last remaining.

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

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

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

Some elections use the Imperiali Quota:

\rm votes \over \rm {seats+2}.

For those keeping track, the size of the quota is then generally Hare > Droop > Imperiali.

Counting The Votes

Process A: Top-preference votes are tallied. If one or more candidates have received 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. Some versions use random selection which makes the count non-deterministic; others count each ballot fractionally.

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.

An example

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

16 voters rank the candidates:

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

24 voters rank the candidates:

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

17 voters rank the candidates:

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

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

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 (in a fractional system the 40 second preference votes are transfered with a value of 0.5 (half) a vote each, which the value of the excess (20) divided by the total (40)[[1]]). 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.


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 contrasts with many other proportional voting systems for which proportionality is apportioned as a percentage of single-preference-only votes for each candidate or party. STV provides proportionality by transferring votes to minimise waste. Votes are wasted when they have no affect on the outcome of a vote, because they either go to a losing candidate or go to a winning candidate who does not need them. Under STV each voter has a single (transferable) vote, regardless of whether there is one vacancy or several.

The more winners there are in a single constituency, the fewer votes that may be wasted (assuming voters are willing to vote for enough candidates) and therefore the more proportionate the outcome will be in that constituency. In some countries, such as Australia, voters have to rank every candidate when casting a vote.

Failures to produce exact proportionality in elections can be controversial, and this situation has arisen in elections using STV. 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 provision for the possibility of bonus seats. These bonus seats needed to be used in 1987 and again in 1996. Similarly, 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.

STV also suffers from 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 would occur if much of her support is from transfers rather than first preference votes.

Potential for 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.

However, in the older STV systems used in many countries there is a loophole: 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 parties standing more than one candidate in the election. Standing too few may result in all the candidates 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 among them, and several being eliminated before any are elected and their second-preference votes distributed, if 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.

Filling 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 at least three possible solutions:

  1. Recounting those votes originally cast with first-preferences for the party where the vacancy arose, as if it was an instant-runoff election between the candidates from that party not originally elected; this works in Malta where there are usually many more candidates from a party than seats won by that party, and necessary since people are often elected for more than one constituency, but would not work where parties do not run more candidates than the number of seats they hope to win.
  2. 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.
  3. Allowing someone to nominate a replacement: for the Australian senate this is done by the state Premier at the suggestion of the party concerned; disputes over who had the final choice contributed to the Australian constitutional crisis of 1975.

In practice

Places that use STV for governmental elections include:

All local governments in Scotland will be using STV to elect their councillors. The Local Governance (Scotland) Bill [6] passed on June 23, 2004.

STV enjoyed some popularity in the United States in the first half of the 20th Century. The community school boards of the City of New York [7] used STV until they were abolished in 2002.

This method used for electing the Legislative Assemblies of Tasmania and the Australian Capital Territory and 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 [8] by the Citizens' Assembly on Electoral Reform.

Some non-governmental organisations also use STV. For instance, all National Union of Students of the United Kingdom 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.

Use to determine list order for List PR

STV is sometimes 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 choses 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 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.

Historical assessments

Thomas Hare is generally credited with conceiving of Single transferable voting, in the 1850's although a version of it was independently devised at roughly the same for use in the Danish Folkting.

An early proponent of STV was John Stuart Mill, who 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."

It spread largely through the polities of the former British Empire - leading for it to be occaisionally known as British Proportional Representation .

It has found use in America, and in the 1940's it was used in New York City municipal elections for a short while, but this reverted again to First Past The Post

See also

External links

Last updated: 05-07-2005 12:42:06
Last updated: 05-13-2005 07:56:04