[EM] Meek style STV - Part One of Two
Donald E Davison
donald at mich.com
Wed Oct 13 04:43:08 PDT 1999
Greetings,
Have you heard about Meek-style STV?
If so, and you understand Meek, what are your comments?
If not, read on. What follows is some text I have just received from
Steve Todd, plus my comments.
Donald
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Steve: If, for the purposes of this exercise, you can put aside your
advocacy of the Hare quota, I think you will agree that these [Meek] rules
are the very best yet devised.
Donald: Why should it be necessary for me to put aside my advocary of
the Hare Quota? If your Meek rules are, as you say `the very best yet
devised', then they should work better with the Hare Quota, or at least,
equally well.
Steve: I can send more information if you become interested.
Donald: I have just become interested. Please send me the basic rules of
your Meek-Style STV via email. One page should do it.
Steve: Please let me have your thoughts in due course. Regards, Steve
Donald: Yes, I will let you have my thoughts about your Meek-Style STV,
if and when I understand its rules. Regards, Donald
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From: TODD Stephen
To: "'Donald E Davison'" <donald at mich.com>
Subject: Meek-style STV
Date: Thu, 7 Oct 1999 17:29:56 +1300
Hi Donald--
This will take more than a page, I'm afraid. I'm sure it will come out
a mess, but here goes. There are Excel objects which probably will not
make it through.
It starts off with my paper explaining the system, followed by extracts
from the bill I wrote for an MP who is promoting STV by means of what we
call a member's bill (as opposed to a government bill). Parliament
rises tomorrow (Fri) for the election campaign, so there will be nothing
happening on that front now before February 2000.
Regards
Steve
Meek-style STV [using the Droop quota]
A simple introduction
With the now widespread availability and use of computers, it is no
longer necessary to perform STV elections using hand-counting rules such
as the Newland and Britton (ERS 1972) rules (which, with slight
variations, are used in Northern Ireland), and the Hare-Clark rules
(which are used in Tasmania and the Australian Capital Territory).
When STV was developed in the nineteenth century, it was thought right
to observe certain rules such as ignoring preferences given for
candidates who were already elected, and transferring only the last
parcel of votes received when distributing a candidate's consequential
surplus, rather than using all the voting papers credited to the
candidate. Even if those features had not been thought right, they
would have been necessary in those days to reduce as much as possible
the number of voting papers that needed to be re-examined for later
preferences, and to speed up the conduct of a count generally. With the
arrival of computers, and the consequent development of the Meek
algorithm, however, arbitrary rules such as these are not only
unnecessary, but can in fact be seen to be undesirable.
It should be emphasised at this point that the purpose of this paper is
not to denigrate hand-counted STV. Where counting by hand is essential,
it does a remarkably good job within its necessary limitations of
practicability. Even the crudest version of STV is very much preferable
to most other electoral systems, but the Meek version is much better
still and is the obvious method of choice where counting by computer is
intended.
The Meek algorithm was devised by Brian Meek (1934-1997), and first
proposed by him in two papers originally published in French in 1969 and
1970 when he was at Queen Elizabeth College, London. The system retains
all the essential features and aims of STV, but uses the power of modern
computers to get a closer realisation of the voters' wishes, better
meeting all the traditional STV virtues.
The Meek rules differ from traditional methods of counting votes, such
as those mentioned above, in four main respects -
(a) Votes are transferred to the next preference in the exact order
indicated by the voter on the voting paper, unless the candidate has
already been excluded:
Donald: This is giving notice to the reader about the Meek change that
votes will also be transferred to candidates that already have a quota.
The word `excluded' as used in this text means eliminated during the
run-off routine.
(b) The total value of the surplus is shared equally across both
transferable and non-transferable voting papers:
Donald: The rules of some STV systems will share the surplus across only
the transferable voting papers. This rule makes Meek STV much better than,
for example, STV under the Northern Ireland rules.
(c) If a candidate is elected later in the count, or an elected
candidate receives further votes, the surplus to be transferred is
shared across all voting papers credited to that candidate in
appropriate proportions, not just across the voting papers which gave
immediate rise to the surplus:
(d) As votes are credited to the non-transferable total, the quota
is recalculated to reflect the smaller total of votes remaining active.
The processing of votes exactly in accordance with the wishes of the
voters means that in a Meek count votes are, in effect, being recounted
again and again. The iterative nature of a Meek count is derived from
two simply-stated principles-
Principle 1. If a candidate is excluded from the election, all
voting papers are treated as if that candidate had never stood.
Principle 2. If a candidate has attained the quota, that
candidate retains a calculated proportion of every vote received, and
transfers the remainder to the next non-excluded candidate, the retained
total equalling the quota.
When interpreted literally, these principles are stating that votes are
transferred to non-excluded candidates, including candidates who are
already elected, giving such candidates new surpluses which must then be
transferred away again in due proportion from all relevant votes. In
addition, non-transferable votes not only drop out of the count, but are
treated as informal or as if they were never cast, which means the quota
for election is recalculated each time the number of such votes
increases. The operation of these principles ensures that-
(a) The number of wasted votes in an election, i.e. votes which do
not contribute to the election of any candidate, is kept to a minimum;
Donald: Meek Droop STV will have more wasted votes than Normal Droop STV,
because the last Droop Quota will be larger. It is larger because Meek will
balance up all the final vote totals including the vote total of the last
Droop Quota, thereby making the last Droop Quota larger than it would
otherwise be if normal Droop STV were used. Because the last Droop quota is
larger, Meek will have more wasted votes when the last Droop Quota is
wasted. This is not the fault of Meek, it is the fault of Droop.
If Meek really wanted to keep wasted votes down to a minimum, it would
use Hare Quota in place of Droop. Meek would work better with Hare, then it
could truthfully say that it is keeping wasted votes down to a minimum.
(b) As far as possible, the opinions of each voter are taken equally
into account; and
Donald: Except for the voters in the last Droop Quota. Steve does not like
my harping on the Droop Quota, but I think it's silly for anyone to claim
the "superiority of the Meek rules" while at the same time the system
contains this big flaw, this flaw of disqualifying a quota of votes, which
is the same as not allowing these people the right to vote.
(c) There is no incentive for voters to vote in any way other than
according to their actual preference.
Whenever the quota is recalculated, based on the reducing number of
valid votes remaining in the election, a reduced value for the quota is
obtained. This means candidates who are already elected, again have new
surpluses which must also be transferred. To complicate matters
further, surpluses of elected candidates will often be transferred in
part to other elected candidates, involving the transfer of
progressively smaller surpluses between two candidates in a seemingly
infinite regression. Furthermore, some of the transfer will usually go
to non-transferable, thus causing a further quota reduction and new
surpluses.
Although these processes seem to be impossibly convoluted, as indeed
they would be for elections where the counting of votes is carried out
manually, they can be handled very easily by computers. The
never-ending transfer of surpluses can even be carried out to
completion, or at least to a point where the surpluses remaining to be
transferred are less than, say, 1/10000 of a vote.
The procedure required to conduct a Meek count is based on each
candidate being assigned a scaling factor representing the proportion of
each vote that will actually be credited to each candidate, out of the
votes that are potentially available to each candidate. The
calculations necessary to produce these scaling factors, called "keep
values", would be very time-consuming for election staff to perform by
hand, but computers can carry them out rapidly. A detailed description
of a Meek count, in which the concept of keep values is expanded upon,
is set out below.
Clearly, in a large public election, the processes set out in the
preceding paragraphs would be far too tedious to carry out by hand, and
almost impossible to complete within an acceptable timeframe. It is for
these reasons that, realistically, a Meek count could only be undertaken
using computers.
Compared with traditional STV, what is the same?
1. Each voter votes by listing some or all of the candidates in
order of preference.
2. Each voter is treated as having one vote, which is assigned
initially to that voter's first-preference candidate.
3. A quota is calculated, as the minimum number of votes needed by
a candidate to secure election.
Donald: In other words, all members shall be elected by the same number of
votes - a quota.
4. If a candidate receives a quota of votes or more, then that
candidate is elected, and any surplus votes (above the quota) are
transferred to other candidates in accordance with the later preferences
expressed by the relevant voters.
5. If, at any stage of the count, no surplus remains to be
transferred, but not all seats are yet filled, then the candidate who
currently has fewest votes is excluded. Votes assigned to that
candidate are then transferred to other candidates in accordance with
the later preferences of the relevant voters.
What is different?
6. All surpluses are transferred simultaneously, instead of in a
particular order.
Donald: This is a good policy, even for normal STV. I have long advocated
that all surplus votes should be treated equally.
7. Surpluses are taken, in due proportion, from all relevant votes,
not only from those most recently received.
8. To make that work properly, it is necessary to give votes to
already-elected candidates and not "leap-frog" over them. This does not
waste votes as the same number are transferred away again, but now in
due proportion to all relevant votes.
9. Whenever a candidate is excluded, the count behaves as if that
candidate had never existed (except that anyone previously excluded
cannot be reinstated).
10. Whenever any votes become non-transferable, the quota is
recalculated, based on active votes only. This lower quota then applies
not only for future election of candidates, but also to already-elected
candidates, giving them all new surpluses.
11. No candidate is ever elected without reaching the current quota.
12. For surpluses, every relevant vote goes to the voter's next
choice, at fractional value. If there is no next choice, the fraction
becomes non-transferable.
13. At an exclusion, all the relevant votes are dealt with at once.
There is no doing one little bit at a time.
14. The only disadvantage is, the count is too tedious to do by
hand, but has to be performed by computer.
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Dear Steve,
Donald: This is as far as I understand your text, but I think I have a fair
understanding of Meek. Besides, I only asked for one page.
My current comments will be on points I understand. I may comment on
other points and parts in the future.
My current comments on Meek Droop STV:
Based on my current understanding of the Meek Droop STV, I would say
that one feature of the method is that it requires all members to be
elected by the same number of votes - a quota. In a normal STV election, we
can expect that half the members will be elected by less than a quota.
One of my requirements of the `Ideal-PR' election is that all members
are to be elected by the same number of votes, but I regard this as a goal
that may never be reached. Meek does reach this goal, but this raises the
question: Is it necessary?
Meek goes through all this repeating fractional mathematics to get
these exact results. Has the election been improved?
We must consider the fact that the results of a Meek conducted
election can be different from an election conducted by normal STV. Both
cannot be correct - one is wrong. If you accept the policy that all the
votes held by a candidate are to have an influence on the transfer of any
surplus votes, then Meek is necessary. The election has been improved.
Fractional transfer has been a great improvement in STV because it
draws an exact line between the candidates that should be elected and the
candidates that should not be elected. Does Meek gives us a better placed
line, a more honest line? If so, then this would be the main good feature
of Meek over normal STV.
If Meek is trying to draw a better line between candidates, then it
should include all the voters. Meek-STV brags that it gets "a closer
realization of the voters' wishes". If this is the true intent of Meek,
then it should also go through its fancy mathematical gyrations for the
voters in the last Droop quota. If it cannot, then the Droop should be
eliminated - `Drop the Droop'.
The Good in Meek-STV:
I do see some good in Meek. First I approve of every member being
elected by the same number of votes. Then, if Meek does draw a better line
between the candidates, I would approve of Meek Hare STV, not Meek Droop
STV.
But, I also see a silver-lining in this Meek method.
Meek may expose to the public, that Droop is the flaw it is, and
something not to be embraced - not to be used.
The mathematics of Meek will make the votes of all the elected members
equal at the end. The last Droop quota will also be included in this
equality. The vote difference of the last quota will only be less in votes
by a number that is equal to the number of members, plus one.
Suppose an election of three members and the final quota is 120,004
votes. This will make the final vote totals as follows:
120,004 A 120,004 B 120,004 C 120,000 D
A person like me would have a field day with these numbers. I will be
able to point to these results and make charges, like, why is it the voters
of A, B, and C are able to elect a member, while the voters of D, who have
99.9967 percent as many votes, are not allowed to elect a member. Those
numbers are close enough to claim equality.
I say good for Meek, if it causes the Droop to be Dropped, then we
really will have a superior STV system.
Donald
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Note: To whom it may concern: I have read the text below, but do not ask me
any questions about anything in there, because I was unable to follow.
Donald
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Detailed description of the count
1. At each stage in the count, each candidate has an associated
"keep value", which indicates the proportion of every vote, or part of a
vote, received by that candidate which is kept, the remainder being
transferred. Every candidate's keep value is initially set to 100%, or
1.000, and it does not change until that candidate is either elected
(when it is reduced below 100%) or excluded (when it is permanently
reset to 0%, or 0.000).
2. Each time the votes are counted, it is done in the following
way. Suppose that candidate A's keep value is 80%, or 0.800, candidate
B's is 50%, or 0.500, candidate C's is 100%, or 1.000, and candidate D's
is 0%, or 0.000. Then a voting paper listing DCAB (in that order) would
be counted as-
nothing to D,
100% of a vote to C,
nothing to A or B (because C has taken the lot).
A voting paper listing ABC (in that order) would be counted as-
80% of a vote to A,
10% of a vote to B (i.e. 50% of the remaining 20%),
10% of a vote to C (i.e. 100% of the remainder).
A voting paper listing BDA (in that order) would be counted as-
50% of a vote to B,
nothing to D,
40% of a vote to A (i.e. 80% of the remaining 50%),
10% of a vote regarded as non-transferable (because this
remaining 10% has run off the end of the list).
3. After each count of the votes, the current quota is calculated
as-
(number of votes currently assigned to candidates)
divided by (number of vacancies + 1)
where the number of votes currently assigned to candidates is
the total number of valid votes cast, minus the current number regarded
as non-transferable.
4. Any candidate whose votes equal or exceed the current quota is
elected (if not already elected earlier) and given a new keep value,
calculated as-
(candidate's current keep value) times (current quota)
divided by (candidate's current votes).
Thus, for example, a candidate who has 4/3 times the number of
votes necessary for election needs to keep only 3/4, or 75%, of what
that candidate previously kept.
5. After every such change, to one or more candidates, the votes
are recounted using the new keep values. This has the effect of
transferring the surplus votes of all the elected candidates in
accordance with the voters' later preferences. However, it does not
necessarily remove all surpluses in a single step, since some of A's
votes may go to B, but some of B's votes may go to A simultaneously.
This will leave each of them with a surplus, though the total surplus
will be smaller than before. It is necessary to repeat steps 3, 4 and 5
until, for all practical purposes, no surplus remains. In the present
implementation, this is taken to be when the total remaining surplus is
less than 1/10000 of a vote.
6. If, at the end of any count of the votes, no surplus remains,
but the number of candidates elected so far falls short of the number of
vacancies to be filled, then the candidate who currently has fewest
votes is excluded, and that candidate's keep value is reset to 0%. The
votes are then recounted. (If an exclusion is necessary and two or more
candidates have equal fewest votes, then the one who had fewest votes at
the earliest point at which they had unequal votes is excluded, but if
they have always been equal, then one of the tied candidates is chosen
by a random (or pseudo-random) process for exclusion.)
7. It is usually clear before all surpluses are transferred that an
exclusion will be required, and which candidate it must be. In such
cases the exclusion may be made at once, giving a short cut which cannot
change the final result of the election.
Examples
1. A very simple, though artificial, example of the superiority of
the Meek rules is seen in 4 candidates for 3 vacancies. If there are
only 5 voters and the votes are-
2 ABC, 2 ABD, 1 BC
it is obvious to anyone, whether knowing anything of STV or not,
that the correct solution must be to elect A, B and C, as the Meek rules
do. Yet traditional hand-counting rules elect A and B, but declare the
third vacancy to be a tie between C and D. (See result sheets and
associated information, attached.)
With the quota for this election being 1.2500, it is clear that
under Meek rules A keeps 0.3125 (1.2500/4.0000) of the first four votes,
and B 0.2291 (0.3333 * 0.6875). B's keep value is 0.3333 because, after
the transfer of A's surplus, B has 3.75 votes, but needs only one-third
of them. The remaining 0.4584 of the two ABC votes go to C (0.9167) and
the remaining 0.4584 of the two ABD votes go to D (0.9167). B retains
0.3333 of the BC vote, to keep B at the quota, and the remaining 0.6667
goes to C, taking C over the quota to 1.5833 votes.
Under traditional hand-counting rules, however, such as the ERS
1976 rules, A's four first-preference votes transfer to B at a transfer
value of 0.68 (surplus of 2.75/4, with a non-transferable remainder of
0.03), which enable B to attain the quota. These same four votes are
then transferred two each to C and D at a transfer value of 0.61
(surplus of 2.47/4, with a further non-transferable remainder of 0.03),
leaving them tied for the third vacancy on 1.22 votes each.
2. The following illustration provides further evidence of the
superiority of the Meek rules. In a real election there were many
candidates, of whom we shall call four, A, B, C and D. At the last
stage, A and B had each been elected with a surplus, C had been excluded
and D was still continuing, to be either the last elected or the
runner-up. Four of the votes gave preferences starting as ABCD, ACBD,
CABD and ABD. As C had been excluded, these became identical votes,
each now having A as first preference, B as second and D as third. The
Meek method would have treated them identically, but the hand-counting
rules actually used gave D wildly different portions of these votes, as
follows-
Vote Rules as used Meek
rules
Portion of vote assigned to
Portion of vote assigned to
A B C D A B C
D
ABCD 0.72 0.28 - - 0.471 0.285 -
0.244
ACBD 0.72 - - 0.28 0.471 0.285 -
0.244
CABD - - - 1.00 0.471 0.285 -
0.244
ABD 0.72 0.28 - - 0.471 0.285 -
0.244
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correction by donald of the above mixed data:
Vote Rules as used Meek rules
Portion of vote assigned to Portion of vote assigned to
A B C D A B C D
ABCD 0.72 0.28 - - 0.471 0.285 - 0.244
ACBD 0.72 - - 0.28 0.471 0.285 - 0.244
CABD - - - 1.00 0.471 0.285 - 0.244
ABD 0.72 0.28 - - 0.471 0.285 - 0.244
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Under the rules as used, the first and fourth votes helped to
elect A and were then transferred to B at a transfer value of 0.28. The
second vote helped to elect A and then transferred to C at a transfer
value of 0.28. When C was excluded, it passed over B, who was already
elected, and transferred to D. The third vote transferred to D at full
value upon the exclusion of C, because A and B were already elected.
The variation between all of the vote going to D, and none of it doing
so, is really startling.
Under the Meek rules, however, the four votes would have
benefited the three non-excluded candidates exactly as the relevant
voters intended. Upon exclusion, C would have been treated as never
having stood. At this point in the count, A's keep value was 0.471,
leaving 0.529 of each vote available to benefit candidates B and D. B's
keep value was 0.539, giving B 0.285 (0.539 * 0.529) of a vote in each
case. As D's keep value was still 1.000, the remaining 0.244 of each of
the votes went to that candidate.
How was my vote used?
If an election has been conducted by STV using the Meek rules, and the
final keep values have been published (as they should be), any voters
who remember their preference orders can work out how their votes were
used, as follows.
Suppose you voted for Bodkins as first preference, for Edkins as second
preference, etc. and their final keep values were published as 0.310,
0.772, etc. as shown in the table below. The first thing to do is to
make such a table with the order of preference that you actually used
for the real candidates and fill in their published final keep values in
column 3.
Always start with 1.000 as the first item, one line above your first
candidate, in column 6, and then in each row in turn, fill in columns 4,
5 and 6, using the rules shown.
<<...>>
When an excluded candidate appears, such as Atkins above, the keep value
is 0.000, so no part of the vote is kept. When a candidate is either
the runner-up or the last to be elected, such as Firkins, the keep value
is 1.000, so that candidate keeps everything received and later
preferences get nothing.
Column 5 shows how the vote was used. 0.310 of it went to help elect
Bodkins, 0.533 of it went to help elect Edkins, 0.110 of it went to help
elect Dawkins and the remaining 0.047 went to Firkins and, if Firkins
was the runner-up, was unused.
Notes
1. In the above description, and in the computer output that
follows, keep values and numbers of votes are rounded to just 3 or 4
figures after the decimal point, but it should be noted that the actual
calculations, within the computer, are made to a much greater accuracy.
2. It should not be thought that the number of votes that an
elected candidate has at the end, gives any indication of merit. Thus
in example 1, candidates A and B finish with 1.2500 votes each, but
candidate C finishes with 1.5833 votes. This does not mean that C has
done better than A and B. They finished with 1.2500, as votes had been
transferred away from them to get them down to the quota, whereas when C
was elected the whole election was over, so there was no point in making
any further transfers.
Stephen Todd I. D. Hill
Secretary, Wellington Branch Immediate Past Chairman,
Electoral Reform Coalition Technical Committee
Electoral Reform Society (UK)
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| Q U O T A T I O N |
| "Democracy is a beautiful thing, |
| except that part about letting just any old yokel vote." |
| - Age 10 |
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