The Gregory methods, used to transfer surplus votes in some STV systems, is super-precise. It often involves fractions, and when it does, it makes the vote count difficult. not so difficult that it can't be done but un-necessarily difficult.
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 Mantioba by Harold John Jansen
https://www.collectionscanada.gc.ca/obj/s4/f2/dsk2/tape15/PQDD_0004/NQ29051.pdf points 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.)
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.
So here is a brief description of the various Gregory methods.
the systems can be catgorized according to two criterion:
relevant ballot papers when making surplus transfers:
GM last parcel only, IGM, WIGM all votes held by elected candidates are relevant.
how vote tallies are reported: fractions or whole vote
(whole vote is where 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)
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Here is a quick run-down on the various types of Gregory Method and how they transfer surplus votes differently.
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 are recorded each time.
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.
Simple GM (last parcel) whole vote only* (Tasmania)
votes (only whole votes) 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.)
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)
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.
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.
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)
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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
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* see the "Gregory integer whole-vote method" below
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Two categories of "Whole vote" surplus transfers
When looking at the transfer of surplus votes, the term "whole vote" is used in two different ways.
Whole votes and no fractions are used in the non-Gregory Method systems
there are two types of non-Gregory Method systems:
Random methods
(there are actually varying systems within the random category:
some 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.)
some systems draw the surplus equally from each polling place or precinct (this method was used in Ashtabula, Kalamazoo, Sacramento and Cleveland city elections in 1920s (Hoag and Hallett, PR (1926), pgs. 346, 391)
Cincinnati method -- ballots are numbered and the numbers used to set the transfer. For example, ballots divisible by four are transferred if one quarter of votes have to be transferred. (currently used in Cambridge, Mass. city elections ) https://www.opavote.com/methods/cambridge-stv-rules) (Hoag and Hallett, PR 1926), p.
"Exact methods"
(not random at all as to next usable preference but random if the
back-up preferences piggybacked with the vote transfers have to be used later)
two types of "exact method" systems:
the British-Irish-Canadian method (the term used in Hoag and Hallett, pgs. 395)
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.
U.S.-style "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.
In summary, the above methods - random and British-Irish-Canadian method and U.S. exact method - 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.
see Montopedia blog "Timeline of electoral reform" (footnote) for more info.
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."
"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.
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)
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More on Gregory methods
[following are some salient points taken from the essay "Developing STV rules...", by James Gilmour, followed by a question or two]
[advantage of integer whole vote transfers]
"With integer vote totals, candidates will either be separated by at least one vote or have the same number of votes.
[What to name the fractional votes?]
Of course, the fractional parts of the vote totals that are not transferred to the candidates cannot be ignored; they must be accounted for properly. These fractional parts are shown separately on the Australian integer result sheets as ‘Lost by fraction’. I prefer the term ‘Vote fraction not transferred’ because it is more correctly descriptive and does not convey the idea that any votes can be “lost”" (from "Developing STV rules..." by Gilmour)
[How are whole vote integer transfers done?]
This truncation to an integer value is applied only to the total value of all the parcels and sub-parcels being transferred to any one candidate; it is not applied to the values of the individual parcels and subparcels before the candidate’s transferable total is calculated. There is only one truncation for each candidate to whom votes are transferred in any one stage. That way the ‘Vote fraction not transferred’ is minimised.
...
[sub-stages]
STV counting rules that use the Gregory Method of transferring surpluses usually provide for sub-stages during exclusions, in which the transfer of a parcel of ballot papers of the same value constitutes a substage. The transfer of first preference ballot papers before the transfer of other ballot papers of value 1 vote also constitutes a separate sub-stage in the Northern Ireland rules [3]. If any candidate attains the quota at the end of a sub-stage, that candidate is ‘deemed elected’ and no further transfers are made to that candidate.
This is consistent with the ‘exclusive approach’ to STV that seeks to keep the voters in discrete, ‘exclusive’ groups so far as possible. Although it is clearly not directly related to WIGM, the sub-stage approach to handling exclusions seems incompatible with the ‘inclusive’ approach that underlies WIGM.
I have, therefore, made no provision for sub-stages during exclusions.
(from "Developing STV rules..." by Gilmour)
[BC STV 2007 technical report also did not make any allowance for sub-stages during transfers of surplus votes] -- see next.
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How BCSTV (2004) described Gregory
BCSTV technical report ("Appendix: glossary") says this:
Gregory (method)
In counting votes under a single transferable vote system, if a candidate has more than the minimum number of votes needed to be elected (see Droop quota), a procedure is needed to allocate the surplus votes to other candidates. The may be done by taking a number of ballots equal to the surplus at random from the ballots of the successful candidate and assigning votes to the next available preference shown on the ballot (that is, to candidates who have not already been elected or excluded).
[no mention here of a successor to the random method -- the "exact method" AKA the British-Irish-Canadian whole-vote method. This is the method used in Edmonton, Calgary and Winnipeg to elect MLAs from the 1920s to the 1950s. see my Montopedia blog "Timeline of electoral reform" (footnote) for info. on this method.]
In 1880, J.B. Gregory contended that this process of random selection could produce varying results depending on the choice of the randomly selected ballots used for making the transfers to other candidates. He suggested that all the relevant ballots should be recounted, assigned to other candidates according to the preferences of the voters, but at a reduced value called the transfer value. The transfer value is calculated by dividing the surplus votes by the total number of relevant votes.
There are three variations of the Gregory method which differ as to the definition of ‘relevant votes’ for calculating the transfer value.
[1] Gregory’s original suggestion was that only the ballots that last contributed to the creation of the surplus votes should be counted (the Gregory last parcel method).
[2] Some Australian elections (such as Aus. Senate) use a second method, the Inclusive Gregory method, where relevant votes are defined as all the votes that contributed to a candidate’s surplus.
[3] The BC-STV system recommended by the Citizen’s Assembly uses the Weighted Inclusive Gregory method under which all votes are counted and assigned to other candidates still in the count according to the voters’ preferences, but the ballots are given separate transfer values depending on their origin (that is, whether they are first preferences, or transfers from one or more other candidates).
The Citizens’ Assembly decided that the Weighted Inclusive Gregory method was most in keeping with the goals of proportional representation by the single transferable vote, was fairer to the voters than the other options, and did not add significantly to the task of counting (or recounting) ballots."
That seems pretty good description of the three sorts of Gregory methods,
but here (and everywhere else in the technical report) there is no mention of the number of decimal points to be used or the use or not of whole-vote reporting of vote tallies, and perhaps many other technical details ]
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Here's more categorization and description of the Gregory methods
the distinction of whether or not they use whole-vote vote reporting is not mentioned. Above, that distinction is used to make up five or more Gregory methods.)
Three types
Gregory where only last parcel is used to determine transfer
Inclusive Gregory Method where all votes held by successful candidate are used to determine transfer
Weighted Inclusive Gregory Method
where all votes held by successful candidate are used to determine transfer and the transfer value of the ballot when it came to the candidate is used in calculation of the present transfer value
(IGM and WIGM each have a whole-vote variant)
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WIGM The “weighted inclusive Gregory method” is an evolution of an STV method first described by J B Gregory in 1880. Gregory’s original method, and several of its successors, have significant flaws, and should not be considered for adoption. WIGM is countable by hand, albeit with some effort, which is the principle reason that it might be preferred today to Meek’s method.
here's examples of WIGM
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Description of how to conduct transfers using WIGM :
(even under WIGM, the unweighted method is used if candidate elected with only first preferences (No. 4 below)
Weighted IGM used if candidate was elected with combo of first preferences and votes previously transferred from others. (No. 6 below)
4. If a candidate on the first count gains more than the minimum number of votes needed to be elected, the candidate is declared elected, and the number of votes in excess of the number of votes needed to be elected (the surplus) is recorded. All of the elected candidate's ballots are then re-examined and assigned to candidates not yet elected according to the second preferences marked on the ballots of those who gave a first preference vote to the elected candidate. These votes are allocated according to a "transfer value."
The formula for the transfer value is:
(surplus votes cast for elected candidate) / (total number of votes received by elected candidate)
5. If two or more candidates on the first count gain more than the minimum number of votes needed to be elected, all of those candidates are declared elected. The ballots of the candidate with the largest number of first preference votes will be re-examined first and assigned (at the transfer value) to candidates not yet elected according to the second preferences marked on that candidate's ballots, or the next available preference, if the second preference candidate has already been elected.
The ballots of the other elected candidate(s) will then be re-examined and their surpluses distributed in order according to the number of first preference votes each candidate received.
6. If a candidate reaches more than the minimum number of votes needed to be elected as the consequence of a transfer of votes from an elected candidate, the number of votes in excess of the number of votes needed to be elected (the surplus) will be transferred to other candidates. This transfer will be to the next available preference shown on all of this candidate's ballots. These ballots now include
1) the candidate's first preference ballots, and
2) the parcel(s) of ballots transferred to the candidate from one or more elected candidates.
The transfer value for the candidate's first preference ballots is:
(surplus votes cast for the elected candidate) / (total number of votes received by the elected candidate)
The transfer value for each parcel of ballots transferred to the candidate from one or more elected candidates is:
(transfer value of parcel of ballots received by the candidate) × (surplus votes cast for the candidate) / (total number of votes received by the candidate)
from
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PR Soc of Aus
has submission sub186
available online
on diff. between weighted and unweighted Inclusive Gregory system on one hand versus Meek on other.
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Gregory method in Australia falls into two categories
Most STV systems in use in Australia fall into one of two categories:
exclusive Gregory methods (the original Gregory method) (used, for example, by the ACT Legislative Assembly) (only last parcel of votes received is used),
and inclusive Gregory methods. (used by Aus. Senate)
Within the category of inclusive Gregory methods, systems can be divided into weighted and unweighted inclusive Gregory methods.
The Weighted Inclusive Gregory Method (WIGM) is regarded by the Proportional Representation Society of Australia as one of the options in a gold standard STV system, and is used for WA Legislative Council* and NSW local government elections.
The unweighted inclusive Gregory method is the method currently used to elect members of the Australian Senate (it is flawed as some votes are valued at more than 1 whlie others are valued at less than 1 (see below))
Western Australia uses WIGM
see download
WIGM surpus transfers:
All the elected candidate’s ballot papers are examined and distributed at a transfer value (TV). For ballot papers received from the elected candidate’s first preference votes, the current TV is Number of surplus votes of elected candidate divided by
Total votes of elected candidate [this is current TV]
For ballot papers received from previous surplus distributions, the TV is current transfer value X previous transfer value
Total votes for each of the continuing candidates are calculated by multiplying the number of ballot papers to be transferred to a continuing candidate by the transfer value.
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explanation on why unweighted Inclusive Gregory Method is flawed
shows that under unweighted Inclusive Gregory Method a different person is elected compared to the WGIM and WGIM is more fair as the voting block that has three quota elects three
A particular voting block (B and C parties) is due three seats.
The 1900 B1 voters, 770 B2 voters and 400 C2 voters all prefer each of B1, B2, C1 and C2 to either of A1 or A2.
He says "the 3070 combined B1, B2 and C2 voters are a solid coalition with over 3 Droop quotas of votes."
yes but B1 voters' No. 2 preference was not either B2 nor C2 but was C1
B2 voters' next preferences were B1, then C1 then C2
C2 voters' next preference was C1 not eith B1 or B2.
so you can call it a solid coalition meaning parties B and C but that doensot mean you leave out C1 evenouth ough got no votes at first because voters for B1 and C2 preferred C1 over any other options. Voters did not mark their preferences as if B1, B2, C2 were solid coalition,
in both WIGM and unweighted IGM we see three of B1, B2, C1, C2 elected..
Under unweighted IGM we see A1, A2. B1, C1 elected, so not three of B1, B2, C1, C2 are elected.
Under WIGM he crows that three of B1, B2, C1, C2 are elected. [which, although the analyst did not see it, was also achieved in unweighted IGM, the only difference that one of that group was replaced by another of that group compared to the winners under unweighted IGM.]
the analyst says A1, B1, B2, C1 elected under WIGM.
In unweighted IGM example we see
the analyst says
"In the weighted inclusive Gregory method, the underlying principle is that each of C1's votes should have an equal chance of being transferred in the surplus distribution. The unweighted inclusive Gregory method does away with this principle, and proceeds on the inexplicable basis that each voter should have the same number of votes transferred in the surplus distribution."
(oddly this explanation has given each voter 1000 votes but cast as solid chunk, instead of breaking each vote into as many as 1/1000ths fractions, multiples of which would go are split between two candidates in each transfer. (sorry for this complication)
getting past the 1000 votes each voter is given,
we see In the example this situation:
This whole vote working of the data re=echoes results under WIGM,
(the IGM result is different (includes election of A2, not B2) but seems based on wrong principles )
1st 2nd 3rd 4th 5th
B1 1900 -899* 1001 elected
A1 1010 -9* 1001 elected
A2 920 +2* 922
B2 770 +305* 1075 elected
C2 400 -400 0 eliminated
C1 0 899* 899 +9* 908 1308 -307* 1001 elected
Total 5000
four seats being filled quota 1001
*2nd Count 1900 votes at TV of approx. .47
3rd Count 1010 votes at TV of approx. .009
5th Count
1900 votes at TV of approx. .47 = 899
1010 votes at TV of approx. .009 = 9
400 at TV of 1
all transfered at TV of .235
so 899 at TV of .235 now equals 211
9 at TV of .235 = 2 goes to A2
400 at TV of .235 now equals 94
total 307 (2 to A2, 305 to B2).
I wrote the blogsite to point out where I think he went wrong in Dec 2023:
I think I see what you are saying and I see that WIGM and even IGM is complicated. But I think you missed something.
You say "That this calculation is meaningless should be self-evident. Most obviously, the 1010 voters with a first preference for A1 have each contributed only 9 votes to C1.
How can they now each transfer 92 votes from C1 to further preferences?
If we conducted this process using physical ballot papers, giving each voter 1000 each, the absurdity would become obvious at this point where we must transfer more ballot papers than physically exist."
But A1 voters did not transfer 9 votes but the 1010 voters transferred each their vote at transfer value of .009, so just part of 1010 votes went to C1 equating in total 9 votes (but not being 9 votes).
Actually 1010 separate votes each with piggybacked preferences, at the transfer value.
When you give each vote 1000 votes it complicated the situation while making the math easier. Then because you don't see how 9 votes are transferred to make 92 (which they couldn't if they were 9) or for some other reason, you give A2 92 votes when he shoud get just 2. (.235 X 9, or .235 X .009 X 1010) (perhaps because you did transfers "analogously," instead of on math principles)
so then you complain that A2 is elected when no such thing would have happened under any fair STV. Anyways as C1 did not have any votes as first preferences,
here's some examples of WIGM in use: https://glennsoutreach.com/stv/"
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James Gilmour on how to use manual counting under STV
Gregory but only using last parcel
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