Evolution of STV PR

Specific STV methods vary in how the quota is calculated, how surplus votes are transfered, and how ties are treated (ties can arise in Find Winners and Elimination). We’ll discuss STV variations in a historical context, looking at how increasingly sophisticated methods have been developed. Our historical approach to the evolution of STV methods follows Nicolaus Tideman’s Collective Decisions and Voting Chapter 15 [Tideman 2006].

1819: Thomas Wright Hill

In 1819, Thomas Wright Hill published a system for electing a committee of The Birmingham (England) Society for Literacy and Scientific Improvement. Each voter cast a ballot for one candidate, with the ballot also identifying the voter. The votes were counted, and any candidate who received five or more votes was elected. If a candidate received more than five votes, excess ballots cast for that candidate were selected at random, and the voter was invited to cast his vote for another (as yet unelected) candidate. When no surplus votes remained, voters whose ballots were cast for unelected candidates were invited to vote again. The process repeated until all the empty seats were filled, or until there were fewer than five left-over ballots.

Wright’s son Rowland Hill used a somewhat similar method in an Australian municipal election. A list of candidates was presented, and each voter in turn wrote his name in a candidate’s column until a candidate had the number of votes required for election. Subsequent voters would, of course, vote for another (as yet unelected) candidate.

One problem with Rowland Hill’s system is that later voters have an advantage over earlier voters, in that they can use their vote to elect a second or third choice if, by the time they vote, their earlier choices have already been elected. In Thomas Hill’s system, voters who cast their votes in later stages have a similar advantage. We call this advantage “free riding”.

1856: Carl Andrae

In 1856, Danish mathematician and politician Carl Andrae (who was unaware of the Hills’ systems) introduced the idea of a ranked ballot. The ballots were counted in random order for the first choice on each ballot that had not yet been elected. Unlike Hill’s system (and later STV systems), Andrae’s system did not allow for the reallocation of ballots originally counted for losing candidates. The system saw limited use in Denmark until 1915.

Ranked ballots solve the voting-order problem, in that voters make their selection without knowing how others have voted.

1865: Thomas Hare

Thomas Hare, a London barrister, unaware of both Hill’s and Andrae’s work, described a system similar to Andrae’s in his 1859Treatise on the Election of Representatives: Parliamentary and Municipal. In The election of representatives parliamentary and municipal : a treatise (1865), he added the idea of successively eliminating the candidate with the fewest votes and transferring those ballots to the next (unelected and uneliminated) candidate on the ballot. This was the first system to include the fundamental elements of all future STV systems: ranked ballots with transfers of surplus votes and transfers from eliminated candidates.

Hare’s system, like Andrae’s, transfers surplus ballots at random. This works reasonably well for large numbers of ballots, but there is of course no absolute guarantee that the ballots chosen randomly are completely representative. This is a problem with all STV systems that transfer only whole votes.

1868: Henry Droop

In the previous systems, the threshold of election (quota) was the number of voters divided by the number of seats to be filled, with any remainder ignored. In STV systems that deal exclusively with whole (as opposed to fractional) votes, this is the largest number such that there are enough quotas to fill all the seats, and is known as the “Hare quota”, after Thomas Hare. Henry Droop argued that a lower threshold, approximately the number of voters divided by one more than the number of seats to be filled (with rounding that depends on the counting method) had several advantages. We describe these advantages elsewhere, but they were and are sufficiently compelling that virtually all STV systems today use some version of the Droop threshold.

Hare’s system, with Droop’s threshold, is essentially the system used today by Cambridge, Massachusetts.

1880: J B Gregory

The methods we’ve discussed so far choose surplus ballots at random to distribute to lower-choice candidates. With a large enough electorate, this isn’t a serious problem, but especially in smaller elections it raises a question of fairness, since those ballots chosen for surplus distribution have a measure of elective power that the ballots not so chosen do not. J B Gregory proposed an improvement in which, instead of choosing a number of ballots at random, we instead distribute all the ballots to their next choices, but count those ballots at a reduced weight. (Gregory actually gave each ballot an initial weight of 100 and then used whole-vote counting, but the effect is the same as transferring hundredths of votes.) In this way, every voter contributing to a surplus has a voice in how the surplus is distributed. There are variations on the Gregory method that differ in which ballots are deemed to be surplus, and how exhausted ballots are treated; we describe them elsewhere.

The variation knows as the “Weighted Inclusive Gregory Method” (WIGM) is the most widespread of STV systems today, and is used with minor variations by Ireland and Scotland, and in the US for internal Green Party elections. It was also the system recommended, but not approved, for use in British Columbia.

1973: Robert Newland & Frank Britton

For integer-based STV methods, the Droop threshold is calculated as the integer portion of 1+ballots/(seats+1).
When Robert Newland and Frank Britton produced a detailed set of STV counting rules for the UK’s Electoral Reform Society, based on a variation of the Gregory method, they observed that the exact Droop threshold of ballots/(seats+1) produced better results. This threshold is sometimes called the Newland-Britton (or NB) threshold (or quota) to distinguish it from the integral Droop threshold, but is more often treated simply as a variation on the Droop threshold.

1969: Brian Meek

In the STV methods described so far, the details of surplus vote transfers depend on the order in which candidates are elected or eliminated. This is a problem because it can motivate voters to vote strategically—that is, in a manner other than their actual preferences. In 1969, Brian Meek published a pair of papers in which he described a system that observes two principles:

Principle 1. If a candidate is eliminated, all ballots are treated as if that candidate had never stood.

Principle 2. If a candidate has achieved the quota, he retains a fixed proportion of every vote received, and transfers the remainder to the next non-eliminated candidate, the retained total equalling the quota.

The resulting method, known as “Meek’s method”, is impractical to count without a computer, but it avoids nearly all the shortcomings of previous methods, including a form of free riding to which the Gregory family of methods is vulnerable.

In 1983, C H E Warren proposed a variation on Meek’s method that differs in a detail of surplus transfer. There is no consensus on which method is superior, but fortunately they tend not to differ much in their results.

Meek’s method is used for some local elections in New Zealand.

Composite STV methods

STV methods have a property known as “later no harm”. That is, voters can rank candidates lower (later) than their favorite without any concern that so doing will make the election of their favorite less likely. A consequence of such a guarantee, though, is that when there are no immediate winners, STV can fail to find “everybody’s second choice”. A variety of approaches have been suggested to try to avoid the problem of “sure losers” preventing the election of other candidates with broad but not first-choice support while minimizing the violation of conventional STV’s later-no-harm property.

Three of these proposed approaches are composite methods. That is, they produce their results by using a conventional STV method more than once, typically many times.

Sequential STV

David Hill first proposed Sequential STV in 1994, refining it with Simon Gazeley in 2002 and again in 2005. In an election to fill nseats, the general idea is to find a set of n candidates such that those n are elected from each set of n+1 consisting of those n and one other. Thus every rejected candidate has been fairly tested without interference from others who are not elected.

Such a set is not guaranteed to exist, and when it does not, Sequential STV tries to find the set of n candidates that comes closest to doing so. On the other hand, it is possible for there to be more than one such set of n; in such a case, Sequential STV will find one of them.

In the case of a single seat to be filled, Sequential STV finds the Condorcet winner if there is one.


In 1995, Nicolaus Tideman proposed a composite method called CPO-STV (CPO for “comparison of pairs of outcomes”). Tideman writes:

The comparison of two outcomes in CPO-STV proceeds somewhat like plain vanilla STV, beginning with the calculations of a quota. However, it is then necessary to list every set of potential winners and then compare these sets two at a time. In each such comparison of two sets of potential winners, each vote is allocated to the first candidate in that voter’s ranking that is in at least one of the two sets being compared. However, votes are transferred only from those candidates (if any) that have more than quota and are in both sets. The count is finished as soon as all such surpluses have been transferred. The result is given by the difference between the sum of the votes of the candidates in one set and the sum of the votes of candidates in the other set. When all pairs of outcomes have been compared in this fashion, the winning set is the set, if there is one, that beats all other sets in these head-to-head comparisons. If there is none, the winner is the set whose worst loss is least bad.

As Tideman points out, CPO-STV may not be computationally practical in the general case.

Schulze STV

Schulze STV addresses, at least in part, the issues of vote management and free riding. See Wikipedia Schulze STV article for details.


10 Responses to Evolution of STV PR

  1. Richard Lung says:

    Dear PR Foundation,
    STV Evolves.

    I have evolved a transferable voting method, descriptively called (abstentions-inclusive keep-value averaged) Binomial STV.
    And a new quota: the Harmonic Mean quota.
    An explanation (which will later be in my e-books) is here:

    Binomial STV and the Harmonic Mean quota.

    Binomial STV.
    Traditional single transferable vote is uninomial: it is just a preference vote count. Binomial STV also conducts a reverse preference count. The latter is an exclusion count to do away with the critique, “premature exclusion” of a trailing candidate when the transferable surplus votes have run out.
    Unfilled preferences or abstentions must also be counted to preserve the structure of the relative importance of greater to lesser preferences. (The abstentions may achieve a quota that leaves a seat vacant.)

    A preference election and an unpreference exlusion is a first order Binomial STV count. The result is obtained by inverting the exclusion keep values and averaging them with the election keep values (using the geometric mean).
    Keep values were introduced by the computer-counted Meek method STV. However, I extended their use from candidates who are elected with a surplus of votes, to candidates still in deficit of an elective quota. This extra information is useful in Binomial STV, because winners are those who do best on average.

    Binomial STV can be taken to indefinitely higher orders of election and exclusion counts.
    If preference, p, plus unpreference, u, count is given in binomial theorem form, (p+u), then the second order count is given by (p+u)^2 = pp + up + pu + uu. This is the formula for a second order truth table of four logical possibilities.

    The algebra is non-commutative because up and pu represent two different operations.
    The second order count qualifies the two first order counts with four counts: “pp” means that the most prefered candidate has votes re-distributed to next preferences; “up” means the most unprefered candidate has votes transfered to next preferences. The candidates keep values for these two counts are averaged for an election count. The process is repeated, with pu and uu, for an average exclusion count, which is inverted, and averaged with the average election count, for an over-all average result.
    In turn, a third order count (p+u)^3 may qualify the second order count. And so on.

    Binomial STV can be used on any STV count but would require a computer program and its level of analysis would only make sense in conjunction with Meek method, but without the quota reductions to help fill all the seats. Since Binomial STV includes preference abstentions in the count, it makes no sense to worry about any seats left vacant when the preferences run out.

    Now you dont need Binomial STV for ordinary elections. Traditional (uninomial or zero order) STV is by far the best method compared to the anarchy of rotten voting methods in politics.
    For systematic analysis of preferential data, Binomial STV might interest the data-mining community.

    Harmonic Mean quota.
    The Hare quota (votes/seats) and the Droop quota (votes/(seats plus one) are respectively maximum and minimum range limit quotas. Since they are harmonic series, the average of their range is the harmonic mean, which works out at: votes/(seats plus one-half). Hence, this is the Harmonic Mean quota.
    The HM quota is justified because the Droop quota may deliver statisticly insignificant margins of victory, and the Hare quota may require levels of support only possible with deferential voting.

    Ive just published the first of two free e-books on electoral reform and research:

    “Peace-making Power-sharing” can be obtained free from Smashwords here in epub format:

    It is also available from Amazon, which makes the author charge at least $0.99 until they find out it is free elsewhere.
    The Amazon kindle (mobi) format is available here:

    Richard Lung.

    • Richard Lung says:

      Three averages Binomial STV.

      Recently, I realised that second order or higher order Binomial STV could be subjected to another error-minimising averaging of recounts. Formerly, I assumed that recounts from redistributing the vote could be done on a plurality basis. The most prefered or the most unprefered candidate would have their votes redistributed to modify candidates keep values (according to the binomial theorem patterns of preference and unpreference counts).
      I modify this method by redistributing the votes of each candidate, in turn, and then taking the arithmetic mean of the candidates keep values for each redistribution count.

      This means that Binomial STV can be subject to three error-minimising averages: the geometric mean of election and exclusion keep values; the arithmetic mean of possible redistribution recounts; as well as the Harmonic Mean quota to neutralise the respective undemocratic shortcomings of the Hare and Droop quotas.

  2. Richard Lung says:

    John Stuart Mill identity of Proportioal Representation with “Personal Representation” (1861). (“Mirage” demonstration).

    I call my demonstration of this identity, the mirage demonstration.
    A mirage is a scene of deceptive proximity (caused by atmospheric lensing).
    In the context of elections, I define a mirage as a deceptive (ap)proximation.
    The idea that there is a proportional count of parties or groups is a mirage that progressively recedes the more proportional the count is made, as more parties win less seats between them.
    The more proportional the count, the more particular the extra parties become.
    Taken to its extreme, every individual would be their own party. This is the reduction to the absurd (reductio ad absurdum) of party proportional counting for representative democracy.
    Even a mirage has a basis in reality. Electorally, that reality is freedom of individual choice. That is the moral of the mirage. As John Stuart Mill said: “Proportional Representation” is “Personal Representation” (by “Mr Hare’s system”). His posthumous Autobiography explicitly accepts both terms.

  3. Richard Lung says:

    The election referendum paradox,
    and “HG Wells principle” (1916).

    If there is no true election method, then no election method can truly choose an election method.
    If there is a true election method, then, the truth being one, there is no choice of true election methods.

    Without knowing the right election method for electing an election, there is no way of knowing how to elect it. If you do know, the election of an election is superfluous.

    The widespread assertion that there is no such thing as right and wrong, in election methods, is a paradox.
    If this statement were right, it would contradict itself, and is therefore wrong.
    If this statement were wrong, then there is such a thing as right and wrong, in election methods.

    This demonstrates (in The Elements of Reconstruction) “HG Wells principle” (1916) of scientific treatment – in this case – of voting method: voting method is “not a matter of opinion but a matter of demonstration.”

  4. Richard Lung says:

    HG Wells law (1918): Law of electoral entropy.
    The HG Wells formula.

    “The problem that has confronted modern democracy since its beginning has not really been the representation of organised minorities – they are very well able to look after themselves – but the protection of the unorganised masses of busily occupied, fairly intelligent men from the tricks of the specialists who work the party machines.”
    HG Wells, 1918: In The Fourth Year.
    Quoted by George Hallett with Clarence Hoag: Proportional Representation. The key to democracy. (1937 ed.)

    The law of electoral entropy proposes that the organised few (as in parties) forestall the organisation of the many (as for government) by disorganising the electoral system.
    First and most decisive degrading or disorganising (of the Andrae and Hare system) was to omit the preference vote from most proportional elections, leaving “The Vote” as a stub vote, a mere X vote, or one-preference vote to count only for the organised few, the parties, rather than the disorganised many, or all the people.

    By 1918, HG Wells already was having to avoid misunderstanding, by defining the organised voting system, as opposed to its relentless disorganising by the meanest interests (so vividly described by Hallett and Hoag).
    “The HG Wells formula” is proportional representation by the single transferable vote in large constituencies.
    For, confining the vote to one preference, an X vote, no longer freely transferable in a proportional count, and confining the choice of representation, to one or few seats per constituency, enfeeble the information value of a general election.

  5. Please see:
    The Angels Weep: H.G. Wells on Electoral Reform.
    (Edited with a post-script by Richard Lung.)

    FAB STV: Four Averages Binomial Single Transferable Vote.

    Also available from Amazon.

  6. Robert Trad says:


    I decided to tackle the issue of elections in my country (Lebanon) back in 2015 and decided to write an STV algorithm that considers the allocation of seats according to sects. This created edge cases not normally seen in classic STV.

    I also designed the algorithm to be as “pure” as possible. I choose the Hare quota and the Warren method. The result is a system that can easily be visualized and understood by the normal population.

    During the quarantine, I decided to build a website to explain and promote my system.
    There’s a page explaining in detail the algorithm with flowcharts and videos.
    There’s also a page where you can test the algorithm and visually see every step of it with animated progress bars.


    I would really like to hear your feedback.

  7. Tom Monto says:

    Here’s a re-write or clarification of what the timeline shows:
    1857 Thomas Hare (1806-1891) published his original work on PR Machinery of Representation in 1857. His concept was based on Single Voting in a multi-member at-large district (one covering all of the UK) with a method of transferring votes to prevent waste of votes through surplus votes being left with successful candidates. Soon, preferential voting (ballots bearing back-up preferences) became a part of his system. This has proved itself as being the basis of a family of fair proportional representation systems – although the systems as used today vary in some ways from Hare’s concept.
    For elections of government bodies larger than cities (and even in some city elections) usually separate multiple-member districts are now used, to allow representation of local areas.
    Mostly the Hare quota (one-ninth of the electorate in a nine-seat district, for example) has been replaced by the equally-effective Droop quota (slightly more than one-tenth of the electorate in a nine-seat district, for example).
    The transfers conducted under STV may be done in a strictly random manner – by taking a number of ballots equal to the surplus at random from the ballots of the successful candidate and transferring the votes to the next available preference shown on the ballot (that is, to candidates who have not already been elected or excluded).
    But even in Hare’s time, mathematical calculation of the transfers was envisioned, using just the next usable preferences marked on all or some of the votes held by the successful candidate. (Later, a more exact method using all the preferences marked on some of the ballots held by the successful candidate was invented – see 1880.)

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