Fractional Transfer by Hand
New Democracy
donald at mich.com
Sun Jun 8 14:26:06 PDT 1997
Dear List members,
When I asked if anyone had any file material dealing with the
procedure for working a preference election by hand, I was hoping to rely
on more than my experience in writing some text on the subject.
Anyway the following is what I have come up with so far. Anyone with
more experience will be able to detect any flaws or omissions by me.
Yours sincerely,
Don,
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Fractional Transfer by Hand
Fractional transfer by hand can be a bit more tedious - but it has to be
done - we must be exact. Fractional transfer will draw an exact line that
will divide the candidates that should win from the candidates that should
lose - and if you were to do the numbers over a hundred times you would
come up with the same results every time. The people doing the math must
know how to do it and then do it correctly like clockwork - it can be done.
Clockwork Example of Fractional Transfer by Hand
In this example I am going to talk about manual transfer of the ballots
because I want to show that manual transfer can be done for any size
election - besides I do not have software to show you. Computers would be
better but if we do not have the use of a computer we still must do what
has to be done. This example will have twenty candidates running for ten
seats.
Example of First Vote Count
A ==============================2270==============
B ===========================2100============
C ==========================2050===========
D ========================1800========= |
E ======================1730======== |
F =====================1690======== |
G ====================1600======== |
H ================1400======== |
I ============1240======== |
J ===========1200======== |
K =====570=== |
L ====460== |
M ==290= |
N =170 |
O 130 Share = 1898
P 100 |
Q 60 |
R 50 |
S 40 |
T 30 18980 Total votes
* Divide all the paper ballots into separate piles according to the
first preference on each ballot.
* Place a work sheet in front of each pile.
* Count the ballots in each pile and record total on work sheet.
* Add up all the vote counts of all the piles - you should get 18980.
Divide by ten, the number of seats to be filled. This number 1898, the
share amount, is also recorded on each work sheet as a reference.
* Check if any candidate has more votes than this share amount. There
are three candidates that have a surplus of votes, A and B and C.
* Record on their work sheets that this candidate is elected. We need to
record this because all their ballots are going to be changed into a
fractional amount and transferred to the piles of other candidates. This is
how we transfer surplus votes.
* Have three teams work on the ballots of these three candidates. All
three piles of ballots can be worked at the same time - there is no order
needed.
* While the three teams are working on the ballots of candidates A and B
and C, other workers can start preparing the piles of ballots of the other
candidates for when candidates are to be dropped one by one. They prepare
by sorting the ballots according to the next selections. The workers should
start first with the ballot piles of the lowest candidates.
* Each of the three teams is to mark all the paper ballots of their pile
with only the number that is the fractional part of each vote that is going
to be transferred away to some other candidate. This will always be the
rule - whenever a ballot is to be marked with a fraction it must be the
fractional number of the part that is being transferred to another
candidate.
* The A team is to mark the ballots in their pile with the number .1639
- which comes from (2270-1898)/2270. The B team is to mark the ballots in
their pile with the number .0962 - which comes from (2100-1898)/2100. The C
team is to mark the ballots in their pile with the number .0741 - which
comes from(2050-1898)/2050. We need at least four decimal places. I have
seen candidates lose by less than 0.0500 of a vote - a very close
separation between winning and losing. The value of the fourth decimal
place when multiplied by hundreds of ballots can have an influence in the
first or second decimal place.
* All the teams will now divide their pile of ballots according to the
next preferred candidate - except any candidate that is already elected -
in this example candidates A and B and C are already elected - no ballots
are to go to them - instead the ballots are to go to the second next
candidate. Each member of a team can take all the ballots that are to go
to another candidate - count the ballots - calculate the total value using
the fraction marked on ballots - move the ballots to the main pile of this
other candidate - but keep the transferred ballots together and available
in case they are needed to be fractioned a second time - record the
addition of votes - and record the new sum total. When this is done for all
the ballots of candidate A and B and C the transfer of all surplus votes
has been completed.
* Add up all the piles again as a control check - you should still get 18980.
* Check and see if any other candidate has reached quota during the
transfer of ballots.
This will lead us to another operation in which fractional division is to
be used. When votes are transferred to a candidate that is near the share,
the transfer may put this candidate over the share amount needed. In these
cases all the votes being transferred at this one time to this one
candidate are to be divided using fractional division - leaving the
candidate with the correct size part per ballot that will give him enough
votes to reach the share amount. The other parts of these votes go to the
next preferences on the ballots. This will happen when votes of dropped
candidates are being transferred but it can also happen when surplus votes
are being transferred.
If a candidate goes over the share value during the transfer of surplus
votes then all the votes this candidate received from every transfer of
surplus votes are to be divided again a second time. For example: Suppose
that candidate D went over the share value during the transfer of surplus
votes from the three candidates, A and B and C, then all these parts of a
vote from these three candidates to candidate D are to be divided again so
that the sum of one of the new parts will give candidate D the correct
number of votes needed to reach the share value. The other part will go to
the next candidate on each part of a ballot. Because there are three
candidates transferring surplus votes we can have three different values of
vote parts being divided a second time. The first step here is to record on
the work sheet that candidate D has received enough votes to reach share
value - because all these ballots that came to candidate D from candidates
A and B and C are going to be changed and moved to the piles of other
candidates.
In the case of transferred votes from a candidate that has been dropped we
can also have different value ballots being transferred. We will have some
ballots that are worth a whole vote each - some ballots that are only worth
.0741 each - some only worth .0962 each - and some worth only .1639 each.
Suppose a candidate only needed 38 votes from the next transfer of a
dropped candidate in order to reach the share value of 1898 - but the
candidate receives more than 38 - the candidate receives 52 votes, the sum
of 30 whole votes and 40 parts of votes. Some of the ballots are already
marked with a fraction - those ballots get marked with a value that is the
old fraction times the current fraction being used on the current votes
being transferred to this one candidate. They are to be divided into a
.7308(38/52) part for the candidate and a .2692 part to go to the next
candidate and recorded on each of the 70 ballots. All the .7308 parts
should total 38 votes and give the candidate enough votes to reach a share
and be elected.
While the whole ballots will be divided .7308 to the candidate and .2692
marked on the ballot to go to the next candidate - the ballots that are
already fractional parts will be reduced some more - the old fraction times
.7308 to go to the candidate and the old fraction times .2692 marked on the
ballot to go to the next candidate.
The ballots with .0741 parts times .2692 will now all equal .0199
The ballots with .0962 parts times .2692 will now all equal .0259
The ballots with .1639 parts times .2692 will now all equal .0441
* After we have taken care of any candidate that reached quota during
the transfer of surplus votes we start dropping candidates. The candidate
with the lowest vote tally at this point is to be dropped and all his votes
transferred to the next preference on the label of each of his ballots. All
candidates that are either already elected or dropped are not to receive
any more votes but instead the votes pass over these candidates and go to
the next preference on the label of each vote.
* Add up all the piles again as a control check - you should still get 18980.
* Check for any candidate that may have gone past the share amount -
this will happen more as more candidates are dropped.
* We now have a new lowest candidate which is now dropped and his votes
are also transferred.
* We keep repeating this routine until there are only ten candidates
left - these ten are the new elected council.
Rest assured that this not as bad as it seems to be - only a minority of
the ballots need to be fractionalized - but I am sure that you see the help
of computers to handle an election that has a large number of voters and/or
large numbers of candidates. But whether we use computers or do it by hand
we still must do what has to be done - if you use a Hare/STV/Preference
Voting method then you must use fractional transfer. What is important with
all these numbers is that we are constructing a very accurate mathematical
line that will divide the losing candidates from the winning candidates. If
you do not want to use fractional transfer then you may as well use Single
Non-Transferable Vote.
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Donald Eric Davison of New Democracy at http://www.mich.com/~donald
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