The following is a description of the main methods of trasferring surplus votes under STV. There are four random methods, two "Exact methods" and five Gregory Methods.
The basic premise of STV is:
Principle 1. If a candidate is eliminated, ballots are treated as if that candidate had never stood. Votes held by the candidateare transferred in accordance to the next usable marked preference. Any vote that would be transferred to the eliminated candidate are instead transfered to the next usable marked preference.
Principle 2. If a candidate's vote tally exceeds the quota, he retains a quota worth of votes but the surplus votes are transferred proportionally or randomly to the next usable preference marked on the ballot.
In fact transfers often make little or no difference - and by that I am including both transfers arising from surpluses of elected candidates and also transfers arising from eliminated candidates.
I say that transfers are not as important as they are generally thought to be because most or all winners are determined in the first count before any transfers are conducted anyway. Comparing the final winners to the popularity ranking of candidates in the first count in all instances of STV shows this.
(My statement is backed up by a recent scholarly publication which says that intensive research has shown that under STV only 1 out of 10 on average are elected due to vote transfers, the rest were in winning positions before transfers and are elected irrespective of transfers.
Stephen Quinlan, The Transfers Game
The transfers game: A comparative analysis of the mechanical effect of lower preference votes in STV systems (sagepub.com
the thesis on STV used in Alberta and Manitoba by Harold John Jansen
https://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/tape15/PQDD_0004/NQ29051.pdf ppoints out that each party received at most about half of a quota in vote transfers from other parties. so that also is meaure that shows transfers not as important as might be thought. Each party pretty much had the same number of votes at the end as in the first Count. A party might see someone move up from lower ranks due to transfers mostly from internal party transfers (This happened when a party started with its vote tally spread over three or more candidates but through transfers, concentrated that vote behind just one or two usually to get them over quota or to win a seat through purality at the end.)
Transfers are only done when a candidate is elminated or elected,
and if elected the surpus votes are only transferred if all three of these conditions apply -- there are still unfilled seats
-- the number of remaining candidates exceeds the number of remaining open seats
-- the amount of surplus is enough to make a difference in the rank ordering of the bottom two candidates.
The transfer is done mathematically only if the number of transferable votes exceeds the number of surplus votes.
In all other cases, either no transfer of surplus votes are done or it is done through simple consideration of the next usable preference marked on the few transferable votes. In both those situations the method used to transfer surplus votes is moot.
In the history of Alberta's use of STV, only about a third of the seats were fileld in such a ways that mathematical transfers were necessary. (This was likely a result of using only the "last parcel" received to determine the transfer of surplus votes.) (for more info on this, see the Montopedia blog "Plumping in Alberta...")
In elections to the Irish Dail, it is possible mathematical transfers resulted from a larger portion of the seats filled. This remains to be investigated.
In elections of Scottish local authorities , the proporton is also not yet known.
Despite the limited impact of vote transfers, much energy has been devoted to the invention of a variety of Gregory methods to transfer surplus votes under STV.
I examine these here, as it seems Gregory (especially one of its more complicated versions -- WIGM) is discussed or proposed as the choice to be a new STV system in the Canadian setting.
For example when BC voted on STV in 2005, it voted on the adoption of WIGM, which was the transfer method of choice of the BC Citizens Assembly of that period.
When the Scottish local authorities began to use STV in 2007, the Scottish Assembly chose WIGM.
As mentioned, not all transfers that are done use math.
Math is only ever used to transfer surplus votes, not transfers arising from elmination of a candidate.
Math is not used where there are fewer transferable votes than surplus to be transferred
But where math is used - where there are more transferable votes than the surplus to be transferred, we see a variety of methods used.
The transfer methods (where transferable votes exceed the surplus) can be categorized according to three criterion:
-- relevant ballot papers when making surplus transfers:
GM last parcel only, IGM, WIGM all votes held by elected candidates are relevant.
-- how many preferences on those ballots are considered or at least transferred proportionally
-- how vote tallies are reported: fractions or whole vote
More on these characteristics below.
Two types of surplus vote transfers depending on context
As well, some systems use one method for transfers of surplus votes belonging to candidates elected in the first count and a different method for surplus votes belonging to a candidate elected in a subsequent count.
Those that take place just after the first count relate to candidates that have only first-preference votes
Those that occur after a candidate has received transfers have mixture of first-preference votes and votes that already have a transfer value attached to them.
In the first context, any transfer method of the 11, except No. 1 (simply stopping and re-directing transfers), can be used. Any of the 11 methods can be used for the post-first Count transfers.
but usually the applied method was used in these options:
a random is used all the time,
two different random methods used in the different contexts
Exact method is used all the time,
Exact method for First Count successful candidates and a random method (no. 1) is used for laters,
Gregory of one sort or another is used all the time.
combinations of different Gregory Methods: Inclusive Gregory for first-count winners and then the "original" Gregory Method for subsequent winners.
U.S. cities (the U.S. exact method) used the whole-vote method for the first case and then resorted to a random method for the later transfers. (example -- model PR election rules produced by PR League, reprinted in Hoag and Hallett (1926), p. 347)
British-Irish-Canadian systems used the whole-vote method for both cases, and it seems used just the last parcel of votes received by the successful candidate.
The election of the lower houses in the Australian Capital Territory and Tasmania used the Gregory method. The exact-ness of this method is foiled to a degree for the later transfers, by the systems using only the last parcel of voters that the candidate received to determine the transfer.
(Hoag and Hallett, PR (1926), p. 345-346, 389-394,
Farrell and McAllister, The Australian Electoral Systems, p. 60)
info on Australian elections see https://prsa.org.au/hareclar.htm
The following is a description of the main methods of trasferring surplus votes under STV.
There are four random methods, two "Exact methods" and five Gregory Methods.
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Random systems - four variants
There are a variety of systems within the random category:
To transfer the surplus, some STV systems use simple methods that allow random-ness.
(1) Some systems simply direct that when a candidate achieves quota, no further votes will be moved to the candidate and the vote is moved to the next usable marked preference instead. Depending on how average the re-directed votes are, such a random method could have an impact on who wins later.
These systems simply stop the candidate receiving votes past the quota (this is the method used in STV elections held by private organizations (Hoag and Hallett, PR (1926), p. 390) (This method is so simple that Hare described it in his book Machinery of Representation, written back in 1857 - when a candidate exceeds quota, simply stop giving him more votes, any vote he would get is instead diverted to the next usable marked preference on the ballot - pretty straightforward.)
(2) some systems say to draw the necessary number of votes randomly, at choice of the election official, from the votes. (I don't know of any STV systems that use this but it seems necessary to list it.)
Some STV systems use simple methods that produce less random-ness or guarantee more random-ness actually.
(3) STV systems that were used in Ashtabula, Kalamazoo, Sacramento and Cleveland in 1920s prescribed that the votes to be transferred would be drawn at random but in equal numbers from each polling place. This prevents votes being drawn from only one part of the district, cognizant of how some types of voters congregate in particular areas of the district. (Hoag and Hallett, PR (1926), pgs. 346, 391)
(4) In the STV system used in Cincinnati (1924-1957) and in Cambridge city elections since 1940, votes received by a winning candidate were numbered sequentially, then if the surplus votes made up one quarter of the votes held by the successful candidate, each vote that was numbered a multiple of four was extracted and moved to the next usable marked preference on each of those votes.
so this is a very non-random type of random system!
https://www.opavote.com/methods/cambridge-stv-rules) (Hoag and Hallett, PR 1926), p.
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"Exact methods"
(not random at all as to next usable preference but random if the back-up preferences that are piggybacked with the vote transfers have to be used later)
The "Exact Method" transfers votes in true proportion to the next usable marked preference maked on the relevant ballot papers but does not consider any lower preferences.
Two variants:
-- (5) the British-Irish-Canadian version where exact method used for all surplus transfers;
-- (6) the U.S. version, where Exact Method was used for surplus votes belonging to candidates elected in first count; the random method was used for surplus votes of candidates elected after the first count.
No. 5 The British-Irish-Canadian method (the term used in Hoag and Hallett, pgs. 395)
used in British, Irish and Canadian uses of STV (excluding Calgary city elections and Irish Senate and perhaps others in Britain, Ireland or Canada)
ballots held by elected candidate are consulted and sorted by next usable preference. Part of each pile is transferred in accordance to portion of surplus over votes held by candidate.
(The "exact method" (B-I-C variant) is described fully in John D. Hunt's 1923 publication A Key to P.R.)
No. 6 The U.S. "exact method" (described in Hoag and Hallett, pgs. 392-3)
Surplus votes of those elected in the first count are transferred as per the British-Irish-Canadian method,
but when candidate is elected in any count after the first count, the candidate just does not receive any more incoming votes after achieving quota, same as the random system used in STV elections held by private organizations.
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The whole-vote "Exact method" of transferring surplus votes
This method uses a simple formula, b/c X s = B rounded down, to formulate the transfers of surplus votes.
Where that does not neatly compose the various transfers to the other candidates, largest remainder is used. The largest remainder system is used to allocate the last few votes, to give accurate results. (A version of this is used today in Denmark to calculate party seat counts and is reputed to be very proportional.)
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The relevant ballot papers depends on circumstance and the variant system of "exact method" in use, whether it is all the votes held by the elected candidate or only the last parcel.
(Because the exact method does not consider secondary preferences when making transfers, it does have a potential element of chance in its mechanics.
However as all or most of the candidates in the winning position in the first count are elected in the end, transfers overall have only small effect, the surplus transfers in particular have less, and the effect of chance in transfers of surplus votes even less than that, so the effect of chance under the "exact method" seems in many cases to be only theoretical.)
The "Exact method" might have been used in Tasmania's first STV elections for members of its state assembly in the 1890s (or a whole-vote form of Gregory might have been used.
The Exact Method was used in Sligo's STV 1919 election, the first STV election in U.K., and/or maybe to fill university seats in U.K. HofC starting in 1918. By 1918 the exact method was likely being used in several cities in BC.
The exact method was used in 1920 in Winnipeg to elect both MLAs and city councillors, and in Malta in national elections, starting in 1921, and then Ireland and Northern Ireland, for sure, starting in 1922
(but not it seems used anywhere in Australia after 1907, and it seems it was last used in North America when Winnipeg last used STV in the late 1960s. (Cambridge, Mass. uses a random method to transfer surplus votes in its city elections.)
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Who invented exact method?
It is not known when someone first described the "exact method" in writing...
(Buckalew in his 1872 book P.R. does not describe how transfers would work.
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The "exact method" was applied in two different systems --
the "American exact method" system (No. 6 above)
and the "British-Irish-Canadian method." system (No. 5 above)
(Hoag and Hallett, PR (1926) p. 395).
(for more info, see Montopedia blog "Gregory Method" and the footnote in the Montopedia blog "Timeline of Electoral Reform".)
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In summary, the above methods - random and British-Irish-Canadian method and U.S. exact method package - use only whole votes and no fractional votes.
All ballot papers are always transferred at a value of ‘one vote’. In such systems (Republic of Ireland and Malta) the surplus, a fractional proportion of the candidate’s total vote, is transferred by selecting a proportion of the “whole vote” ballot papers with a total value equal to the surplus to be transferred.
That is, a fraction (portion) of all the votes are transferred,
Under the Gregory method, discussed next, a fraction of each vote is transferred.
But that is not to say fractions are always in use even under the Gregory Method.
Sometimes under a Gregory Methods, whole votes are recorded, with fractions not recorded. Thus when looking at the transfer of surplus votes, the term "whole vote" is used in two different ways, as we see in three methods described below.
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Various Gregory methods
The Gregory Method prevents the element of chance that is produced when votes are transferred just according to the next usable preference but then piggybacked lower preferences are brought into play later.
The Gregory Method does this by transferring a portion of each vote, leaving none behind.
There are various Gregory Methods, distinguished by
-- relevant ballot-papers consulted to determine transfers
-- whether transfer values are carried forward or ignored
-- whether fractional votes or whole votes (composed of totals of fractional votes) are recorded each time.
There are three Gregory methods. But for each, two different ways are used (or may be used) to record those transfers.
The three Gregory methods - Gregory Method (GM), Inclusive Gregory Method (IGM) and Weighted Inclusive Gregory Method (WIGM) - are all fractional transfer methods where all ballot papers held by the candidate, or all the ballots just in the last parcel that came to the candidate, are distributed but at a fractional value, the Transfer Value, based on ratio of surplus to total votes held by candidate.
But different ways are used to record those transfers.
the transfers are recorded as an integer number with or without a fraction for any fractional left above the integer.
with the fraction it looks like this 24.33 for example;
without fraction left over, it looks like this -- 24, for example.
In the latter example, any fractional remainders were disregarded, but their total was recorded on the result sheet as “Votes lost by omitting Fractional Remainders”.
(from Review of some aspects the Single Transferable Voting system for local elections in Wales
Review-of-some-aspects-the-Single-Transferable-Voting-system-for-local-elections-in-Wales (2).pdf by James Gilmour 2021)
A Gregory Method (fractional transfer based on transfer value) of one sort or another is/was used In some Australia elections, for the Ireland Senate, and in old-time Calgary city elections.
A random method or an Exact Method, described above, are or were used in most Britain, Ireland and Canadian uses of STV.
There is no temporal dividor between the uses of the groups of systems.
One of the earliest applications of STV in the world used a Gregory Method (Tasmania, Hare-Clark, 1909).
U.S. cities adopted a random method in the late 1910s.
Exact Method was adopted around 1920 for provincial (Alberta and Manitoba) and national applications (Ireland and Malta).
But then the Ireland Senate was elected through a Gregory Method in 1922.
Cambridge city elections held using a random method starting in 1940, and still does to this day.
Australia Senate elected through a Gregory Method starting in 1949.
Scottish local authorities began to use a Gregory Method in 2005?
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Here are the various types of the Gregory Method.
ACT and Tasmania lower houses use last parcel Gregory Method.
Tasmania is No. 8 below
ACT's specific method unknown
(7) Simple GM (last parcel)
uses only “last parcel” transfers when transferring surpluses.
these implementations of the Gregory Method use decimal calculations credited and report fractional parts of votes, to different numbers of decimal places.
======================
(8) Simple GM (last parcel) whole vote only* (Tasmania)
votes (whole votes arrived at by adding together fractional votes transferred) are credited to continuing candidates following the calculation of fractional transfer values.
uses only “last parcel” transfers when transferring surpluses.
Importantly such a practice means a candidate who wins on the basis of transferred ballots never will have those ballots on which she was the first choice be transferred. Back-up preferences marked on ballots only in the most recent incoming parcel are used to determine the vote transfer.
(The original implementation of the Gregory Method used proper fractions to designate the values of the ballot papers being transferred to transfer a surplus. The calculations were done (divisions and multiplications) and the “whole votes” (integer part) credited to the receiving candidate.)
(Tasmania uses what it calls Hare-Clark, which is a whole-vote version of GM*)
(Although some state the election of a “wrong” candidate—meaning a candidate who is less popular than a candidate who is not elected — is unlikely, others say that such unfortunate occurence is statistically possible.
For that reason, the Inclusive Gregory Method and the Weighted IGM was invented.)
(9) Inclusive GM (whole vote)* (Aus. Senate)
(whole vote even after adopted decimal arithmetic)
votes (only whole votes) are credited to continuing candidates following the calculation of fractional transfer values.
For those STV elections the precision is not limited at all [19], but this has no consequences because of the ‘value averaging’ method that is used in those rules to calculate transfer values de novo for each surplus. (from "Developing STV rules..." by Gilmour)
see this blog for info on the weakness of IGM:
(IGM described here
(but note in No. 4 at the end of the article there is a mistake:
it says "So, if Winner A received 20% over the threshold, the votes transferred to the second-ranked candidates are only 20% of a whole vote."
but actually transfer value is determined by surplus divided by total votes so in this case 120/20 so 16.7 percent, not 20 percent as stated.)
pitch for IGM
(for info. on why unweighted IGM is flawed, see below)
(10) WIGM (West Aus.) (decimal math) (whole vote)*
WIGM in Western Australia uses decimal arithmetic for its calculations but transfers and reports only “whole votes” (integer values) to the receiving candidates. In WA this “whole vote” approach is applied separately to the calculations for each sub-parcel of differently valued ballot papers, i.e. a fractional part is likely to the “lost” for each differently valued parcel.
see these online articles for more info on West Aus.'s adoption of WIGM:
Curiosities in drafting for the weighted inclusive Gregory method of STV (yingtongli.me)
adoption of WGIM/STV in West. Aus.
Proportional_Rep_WA.pdf (elections.wa.gov.au)
When Western Australia adopted WIGM, they also implemented a similar integer (whole vote) transfer, applying the ‘integer only’ calculation separately to each parcel of ballot papers of different transfer value.
(from "Developing STV rules...", by James Gilmour)
(11) WIGM (Scottish) (decimal math) (fractions recorded, not whole vote)
WIGM (Scottish) (five decimal places)
WIGM in Scotland uses decimal arithmetic to five decimal places, and transfers and reports candidates’ votes to five decimal places.
Scotland WGIM:
surplus transfers - close the door when a receiving candidate achieves quota. and any later sub-parcels stil to be transferred are routed to next marked preference.
exclusion transfers -- all votes are transferred before looking at accumulated vote to see if receiving candidate achieved quota.
The fundamental difference between the Gregory Method and WIGM is that there is no ‘last parcel’ restriction in WIGM when a transfer has to be made. This is the “Inclusive” part of the name WIGM.
That applies both to transfers of surpluses and to transfers on exclusion.
Surplus
So when transferring a surplus the RO takes all of the elected candidate’s ballot papers and transfers them all. When the candidate’s pile of ballot papers is made up of parcels of ballot papers of different current values the transfer values are calculated separately for each parcel to ensure that each ballot paper retains the correct overall value of ‘one vote’. This is the “Weighted” bit of WIGM.
Exclusion/Elimination
The corresponding no ‘last parcel’ in an exclusion is that there are no sub-stages, i.e. all of the excluded candidate’s ballot papers are transferred to whatever ‘next available preference’ is marked. The ballot papers will be transferred in parcels by current value, but no account is taken of any receiving candidate’s vote until all the ballot papers have been transferred.
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Unsorted instances of use of Gregory
WIGM of some sort is used in the local government council elections in New South Wales.
some sort of Gregory other than WIGM is (or was in 2007)) used in District Council elections in Northern Ireland. likely simple Gregory
simple Gregory is used to elect the No. Ireland Assembly.
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*footnote -- Whole-vote Gregory methods "Gregory integer whole-vote method"
The Gregory Method version used may or may not record fractions.
Some use the whole-vote method by simply marking fractional votes as "lost" or "not transferred."
Under whole-vote Gregory Methods, fractions are not recorded but are indicated by number of votes and the transfer value.
say for instance 123 votes are transferred at Transfer Value of .33 this would be recorded at 40 votes with .59 being logged in under the heading "vote fraction not transferred" (or "lost by fraction")
(the former term is recommended by James Gilmour in his proposed whole-vote WGIM
voting rules accessible here : https://www.votingmatters.org.uk/RES/gilmour7.pdf)
now say those votes have to be transferred at .2 Transfer Value and the back-up preferences carried by these 1230 votes have to be considered and say 80 are marked for C and 40 for D (as next usable back-up preference).
we see this math:
(800 X.33) X .2 = 52.8
recorded as 52 votes transferred to C
.8 recorded as "vote fraction not transferred"
(430 X .33) X .2 = 28.4
recorded as 28 votes transferred to C
.4 recorded as "vote fraction not transferred"
(the "vote fraction not transferred" previously noted may or may not be resurrected in later vote transfers)
"Gregory integer whole-vote method"
votes (only whole votes) are credited to continuing candidates following the calculation of fractional transfer values.
This first occurred in the Tasmanian use of the Gregory Method, in 1907 (although possibly Tasmania used the whole-vote Gregory system also in the 1890s when its two two main cities used STV.
This “whole vote” (integer part) transfer was adopted when the Australian Senate IGM was devised. The averaged transfer values are calculated to an indeterminate number of decimal places, but only the integer (whole votes) part is credited to the receiving candidate.
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New South Wales legislative council uses Inclusive Gregory for first count winners, and uses only the last parcel (the Gregory method) to determine transfer of surplus votes held by winners who won due to incoming transfer of votes.
(The WGIM method was on the ballot in BC's 2018 referendum)
Victoria upper house uses the Weighted Inclusive Gregory method (a blog presents the weakness of Victoria's WIGM: https://antonygreen.com.au/inclusive-gregory-another-serious-problem-with-the-victorian-legislative-councils-electoral-system/#comments)
On the other hand,
the New South Wales upper house uses whole-vote GM of some sort.
(for more on Australia's adoption of STV see below)
====================
[note: the method used for transfer of surplus votes is unclear.
Farrell and McAllister present conflicting info -- p. 62 does not mesh with table on p, 60/61)
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Surplus vote transfers occur in three parts of the vote count:
-- A. arising from 1st Count victories
-- B. arising from people being elected when 1st Count surpluses are transferred
-- C. arising from elections after elimination of candidates.
Table showing which surplus transfer method is in use in each part
(as described above)
(this table may need some editing later but as it is, it presents useful information in easily read-able format.)
WV = whole vote
LP last parcel only
EM exact method (whole vote, only next usable pref consulted on relevant ballot papers)
RT Re-direct transfer (No. 1)
BR Balanced random - used in AKSC (no. 3 above)
NR Numbered random (Cincinnati method) (No. 4 above)
AKSC Ashtabula, Kalamazoo, Sacramento and Cleveland
A B C
Cambridge* RT RT RT
Cincinnati NR NR NR
AKSC BR BR BR
Brit-Irish-Can. exact method inclusive EM LP EM LP EM
U.S. exact method inclusive EM RT RT
Gregory methods
(7) GM last parcel inclusive GM LP GM LP GM
NSW upper house inclusive GM WV LP GM (WV?) LP GM (WV?)
(8) Tasmania lower house inclusive GM WV LP GM WV LP GM WV
(9) IGM whole vote Aus Senate inclusive GM WV inclusive GM WV
(10) WIGM whole vote inclusive GM WV weighted inclusive GM WV
Scottish Assembly
(11) WIGM (dec. math) inclusive GM weighted inclusive GM
*Cambridge -- a website gives conflicting info. on method used to transfer surplus votes. in one place it says NR used, in another place it says RT (Random method No. 1).
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(NSW lower house, Tasmania upper house IRV)
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Why transfers?
Here's an explanation of Irish elections that says it well:
Only one of the preferences in your vote is active at a time. The vote
stays with your first preference candidate unless and until he/she does
not need it any more (either because the candidate has been elected and
your vote has become one of his or her surplus votes -see section 3.3 -
or the candidate has been excluded from the count). If your vote is
transferred, it passes to your next highest preference for a candidate still
in the running. Your vote could transfer a number of times at the same
election to your lower preference candidates [successively, one at a time]....
3.3 Overview of the surplus distribution procedure
An elected candidate's surplus is distributed based on the next available preferences for continuing candidates (i.e. candidates not elected or excluded) contained in the last parcel of votes that brought the elected candidate over the quota. Where the ballot papers of the elected candidate whose surplus is to be transferred consist ONLY of ballot papers with first preferences for that candidate, all of that candidate’s ballot papers are examined to ascertain the next available preferences. This is always the case where a surplus is being transferred at the second count.
Where the number of transferable papers is greater than the surplus, only a proportion of them can be included in the surplus distribution. This proportion is calculated by working out the ratio of the surplus to the total number of transferable papers and applying that ratio consecutively to the total number of next preferences for each candidate still in the running. This calculation gives the number of next preferences for each candidate that should be included in the surplus distribution. The resultant number of next preferences for each continuing candidate to be transferred as part of the surplus distribution is taken from the top of his/her sub-parcel of next preferences made up from the last parcel of votes received by the elected candidate.
from assts.gov.ie "A Guide to Ireland's STV voting system"
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