In a message about STV that appeared in
Election-Methods Digest on 29-Aug-2009, I said
the following: "1 vote [per ballot] is a
special case. And, unfortunately, mentioning
only that special case gives opponents of
transferable-vote systems a politically
effective counter-argument. Instead, what
people favoring transferable-vote elections
should say is that (1) every valid ballot will
cast the same number of votes, and (2) the
same outcome will emerge regardless of how
many votes each valid ballot casts, whether
that number is (a) 1 (which simplifies
calculation), (b) the number of open seats
(which might be, say, 3), or even (c) merely a
letter, say, v."
Two people posted comments.
One wrote, "This statement is wrong . In
STV-PR each voter has one vote and one vote
only throughout the entire process. It is
extremely important to refer to STV as the
SINGLE Transferable Vote, because each voter
must have only one vote to ensure PR. PR
cannot be obtained (except by chance) if that
single vote is not transferable ."
The other comment was, "Ahh I see what you are
aiming for. Effectively, each voter gets to
submit v ranked ballots. This could make
counting pretty hard."
Both writers missed the point.
On the other hand, our leading election
theorist, that is, Nicolaus Tideman,
understood. He saw that, as I had asserted,
changing the number of votes cast by each
ballot in a transferable-vote election might
not affect the outcome of the election. And,
remarkably, he tested and illustrated my
assertion.
To do so, Professor Tideman referred to a file
he had containing 460 rankings actually
submitted in a ranked-voting election with 4
winners and 10 candidates. He selected three
different numbers of votes that each ballot
might have cast in that election, namely, 1
(the number actually cast per ballot), 2, and
4 (the number of seats being filled). For
each of these three possibilities, he
determined which candidates would have won the
election.
Professor Tideman used three different
versions of STV. He used the two
most-sophisticated versions currently
available, namely, Meek's method and Warren's
method, and also the Newland-Britton method.
The latter is much advanced over early
versions of STV (for example, over the
primitive version still used in Massachusetts)
but, like them, does not transfer a vote from
a candidate being eliminated to the voter's
next choice if that next choice was elected
earlier in the calculations (as a result, that
voter's following choice receives a larger
transfer directly instead of a smaller
transfer indirectly).
Professor Tideman recently sent me his
results, along with a copy of voters'
rankings. Appendix 1, below, reproduces the
calculations with Meek's method, and the
results with Warren and Newland-Britton are
available on request.
To understand the calculations, you need to
notice which variables changed and which
variables did not change when the number of
votes cast by each ballot changed.
With each of the three versions of STV, three
variables changed as the number of votes per
ballot changed from 1 to 2 to 4. The
variables that changed were the values at each
stage of (a) the quota (that is, the number of
votes that a candidate currently needs to be
elected), (b) the candidates' tallies (that
is, the number of votes that, after transfers
of votes that occurred at previous stages,
each candidate currently has), and (c) the
"excess" (that is, the number of votes that,
because some ballots have incomplete rankings,
cannot be transferred from elected or
eliminated candidates to active candidates).
When the number of votes cast by each ballot
changed, each of those three variables changed
in proportion. Specifically, when each ballot
cast 2 votes, the quota, the tallies, and the
excess at every stage were 2 times the level
they had when each ballot cast 1 vote.
Similarly, when each ballot cast 4 votes, the
values at every stage were 4 times the level
that those variables had when each ballot cast
1 vote.
To see the re-scaling, compare three columns
in Appendix 1, namely, (a) the third column
from the left (that is, the column with the
heading "1"), which contains the quota,
tallies, and excess that emerged at each stage
when each ballot cast one vote, (b) the fourth
column from the left (that is, the column
headed "2"), which contains the quota,
tallies, and excess that emerged at each stage
when each ballot cast 2 votes, and (c) the
fifth column from the left (that is, the
column headed "4"), which contains the quota,
tallies, and excess that emerged at each stage
when each ballot cast 4 votes.
Equally important is what did NOT change as
the number of votes cast by each ballot
changed from 1 to 2 to 4. There was no change
at any stage in the "retention proportion" of
any candidate. (If a candidate's current tally
exceeds the current quota, a
portion--specifically, 1 minus the retention
proportion--of each whole or fractional vote
in that tally will transfer to the next choice
on the ballot where that whole or fractional
vote originated, provided that next-choice
candidate is still "active").
The retention proportions did not change
because they were determined by the algorithm
being used, that is, by Meek's, Warren's, or
N-B's method. Hence, Appendix 1 shows the
retention proportions, not in three columns,
but rather in one, namely, the second column,
which is headed "retain."
Because changing the number of votes cast by
each ballot caused a proportionate change in
the quota, the tallies, and the excess at
every stage, but did not change the retention
proportion of any candidate at any stage, the
outcome was--as predicted--the same whether a
ballot cast 1, 2, or 4 votes. In particular,
there was no change in (a) the candidates who
won and who lost the election, (b) the number
of stages (namely, nine) needed to determine
which candidates won and lost, or (c) the
stage at which each candidate was elected or
eliminated. The column in Appendix 1 labeled
"Status" shows the latter.
On the other hand, because the three versions
of STV use different retention proportions,
they yield different outcomes. The winners
are (a) candidates #10, #1, #7, and #2 with
N-B, (b) #10, #1, #7, and #4 (instead of #2)
with Meek, and (c) #10, #1, #7, and #6
(instead of #2 or #4) with Warren (which
Professor Tideman prefers). Starting in stage
4, Meek's retention proportions differ from
Warren's.
A question probably has come to mind: If
changing the number of votes cast by each
ballot would not change the outcome, then why
not continue to have each ballot cast one vote
and continue to refer to SINGLE transferable
vote? The answer is the second point made in
my message of 29-Aug-2009, namely, that making
each ballot cast as many votes as there are
open seats would help win public support for a
transferable-vote system.
I say that after having a bad experience. My
city, like many others, elects either 2 or 3
members of a 5-member city council every 2
years, using the multi-vote plurality system.
With that system, a voter may vote for as many
candidates are there are open seats, and the
leading vote-getters win those seats.
Recently, there was a popular vote on whether
to substitute STV.
STV lost, and I think a major reason was that,
over and over, opponents asserted that
introducing STV would deprive voters of their
2nd and 3rd votes. Meanwhile, our side
implicitly conceded that point whenever we
mentioned SINGLE transferable vote or tried to
explain how a single vote could support more
than one candidate.
Conversely, I do not see that anything would
be lost by making a ranking ballot cast as
many votes as there are openings. In
particular:
(1) As with 1 vote per ballot, every vote cast
would automatically be apportioned among the
candidates in a way that reflects both the
voter's preferences and how others have voted.
(2) As with 1 vote per ballot, voters would no
longer need to choose between (a) helping
their 1st choice beat their 2nd choice, and
(b) helping their 2nd choice beat candidates
liked even less (of the ballots submitted in
my city's last seven 2-seat elections, 29%
cast only one vote, thereby giving priority to
helping the 1st choice).
(3) As with 1 vote per ballot, whether
spoilers are nominated would become less
important.
(4) As with 1 vote per ballot, the proportion
of open seats filled by a faction's favorite
candidates would, at times, become more like
the proportion of votes cast by that faction.
Moreover, with multiple transferable votes, it
probably would be easier for the public to
understand and appreciate these advantages.
Stressing benefit (4), some advocates of STV
call the system "proportional representation,"
not STV. For them, especially, it should be
interesting to compare the STV outcomes
described above with the outcome that
multi-vote plurality would have produced in
the same election.
If multi-vote plurality had been used in that
election, then (a) each voter would have been
invited to vote for up to 4 candidates, (b) a
candidate would have received one vote for
each voter who had voted for that candidate,
and (c) the winners would have been the 4
candidates receiving the most votes.
How the 460 voters would have voted is not
obvious. Because of the dilemma mentioned
above (see benefit (2)), a voter might have
voted for 4 candidates or for 3, 2, or 1.
Accordingly, I learned which candidates would
have won if every voter voted only for his or
her (a) 1st choice, (b) 1st and 2nd choices,
(c) top 3 choices, and (d) top 4 choices.
Appendix 2 contains the figures.
In all four cases, the outcome with multi-vote
plurality was, to my surprise, reasonable and,
indeed, arguably better than with N-B.
Specifically, the winners in cases (a), (b),
and (c) were, as with Meek's method,
candidates #1, #4, #7, and #10 and, in case
(d), were the same as with Warren's method,
that is, #1, #6, #7, and #10. Hence, changing
from multi-vote plurality to STV--at least to
either of those versions of STV--would not
have made the proportion of open seats filled
by a faction's favorite candidates more like
the proportion of votes cast by that faction.
On the other hand, factions were either absent
or invisible. While 4 seats were open, STV
elected only 1 candidate, namely, #10, before
votes were transferred from an eliminated
candidate. The next-most-frequent 1st choice,
namely, candidate #1, was top-ranked by merely
81 of the 460 voters, and surplus transferred
from #10 was not large enough to carry #1 (or
any other candidate) over the threshold. As a
result, no outcome could have made the
proportion of open seats filled by factions'
favorite candidates resemble the proportion of
votes cast by those factions.
But there are cases, at least hypothetical
cases, where STV probably would--and
multi-vote plurality probably would
not--produce proportional representation, in
the sense that STV would make the proportion
of open seats filled by each faction's
favorite candidates as close to the proportion
of votes cast by the faction as is possible
when the proportion of open seats filled by a
faction's favorites must be a multiple of
(1/number of open seats).
For example, suppose that (a) 2 seats are
open, (b) 6 candidates, namely, A, B, C, D, E,
and F, are running; (c) 100 people vote; (d)
35 voters rank the candidates A > B > C > D >
E > F, 34 voters think C > D > E > F > B > A,
and 31 voters think F > E > D > C > B > A; (e)
with multi-vote plurality, voters will vote
for both their 1st choice and their 2nd
choice; and (f) with a transferable-vote
system, voters will report both their 1st
choice and their 2nd choice.
With multi-vote plurality, candidates (A, B,
C, D, E, F) would receive, respectively, (35,
35, 34, 34, 31, 31) votes. Hence, candidates
A and B would win. As a result, candidates
favored by merely 35% of the voters (and
disfavored by the other 65%) would fill 100%
of the open seats.
With a transferable-vote system, in contrast,
candidate A and candidate C would immediately
receive more than fraction 1/(2+1) of the
votes cast and therefore (in that order) would
quickly be elected. As a result, a candidate
who is the 1st choice of 35% of the voters
would fill 50% of the open seats, and a
candidate who is the 1st choice of 34% of the
voters would fill the other 50%.
Proportionality!
I conclude that (a) a transferable-vote system
will produce the same outcome regardless of
the number of votes cast by each ballot; (b)
to win greater public support for a
transferable-vote system, each ballot should
cast as many votes as there are open
positions; (c) when the electorate is not
polarized, multi-vote plurality may produce
the same outcome as a transferable-vote
system; and (d) even when a transferable-vote
system does not increase proportionality, it
will have other benefits.
APPENDIX 1:
OUTCOME WITH MEEK'S METHOD
Votes per ballot= 1 2 4
Stage 1 Quota= 92 184 368
Cand. Retain Tally Tally Tally Status
#1 1.000 81 162 324
#2 1.000 40 80 160
#3 1.000 15 30 50
#4 1.000 42 84 168
#5 1.000 27 54 108
#6 1.000 41 82 164
#7 1.000 70 140 280
#8 1.000 12 24 48
#9 1.000 14 28 56
#10 1.000 118 236 472 Newly elected
Subtotal= 460 920 1830
Excess= 0 0 0
Total= 460 920 1830
Stage 2 Quota= 91.87 183.73 367.5
Cand. Retain Tally Tally Tally Status
#1 1.000 88.53 177.06 354.1
#2 1.000 41.55 83.10 166.2
#3 1.000 15.89 31.77 63.5
#4 1.000 43.77 87.54 175.1
#5 1.000 28.77 57.54 115.1
#6 1.000 43.21 86.43 172.9
#7 1.000 73.10 146.20 292.4
#8 1.000 15.54 31.09 62.2 To be excluded
#9 1.000 17.10 34.20 68.4
#10 0.779 91.87 183.73 367.5 Elected
Subtotal= 459.33 918.66 1837.4
Excess= 0.66 1.33 2.7
Total= 460.00 920.00 1840.0
Stage 3 Quota= 91.84 183.69 367.4
Cand. Retain Tally Tally Tally Status
#1 1.000 93.63 187.26 374.5 Newly elected
#2 1.000 42.59 85.19 170.4
#3 1.000 18.30 36.59 73.2
#4 1.000 44.59 89.19 178.4
#5 1.000 31.37 62.74 125.5
#6 1.000 44.11 88.22 176.4
#7 1.000 74.89 149.78 299.6
#8 0 0 0 0 Excluded
#9 1.000 17.89 35.78 71.6
#10 0.741 91.84 183.69 367.4 Elected
Subtotal= 459.21 918.44 1837.0
Excess= 0.78 1.56 3.1
Total= 460.00 920.00 1840.0
Stage 4 Quota= 91.84 183.68 367.4
Cand. Retain Tally Tally Tally Status
#1 0.980 91.84 183.68 367.4 Elected
#2 1.000 42.95 85.89 171.8
#3 1.000 18.52 37.04 74.1
#4 1.000 44.94 89.89 179.8
#5 1.000 31.51 63.02 126.0
#6 1.000 44.48 88.98 177.9
#7 1.000 75.15 150.29 300.6
#8 0 0 0 0 Excluded
#9 1.000 17.97 35.95 71.9 To be excluded
#10 0.738 91.84 183.68 367.4 Elected
Subtotal= 459.20 918.42 1836.9
Excess= 0.80 1.59 3.2
Total= 460.00 920.00 1840.0
Stage 5 Quota= 91.83 183.66 367.3
Cand. Retain Tally Tally Tally Status
#1 0.952 91.83 183.66 367.3 Elected
#2 1.000 44.62 89.24 178.5
#3 1.000 21.16 42.32 84.6 To be excluded
#4 1.000 47.12 94.23 188.5
#5 1.000 34.92 69.84 139.7
#6 1.000 48.37 96.74 193.5
#7 1.000 79.30 158.59 317.2
#8 0 0 0 0 Excluded
#9 0 0 0 0 Excluded
#10 0.723 91.83 183.66 367.3 Elected
Subtotal= 459.15 918.28 1836.6
Excess= 0.86 1.72 3.4
Total= 460.00 920.00 1840.0
Stage 6 Quota= 91.77 183.54 367.1
Cand. Retain Tally Tally Tally Status
#1 0.899 91.77 183.54 367.1 Elected
#2 1.000 50.27 100.54 201.1
#3 0 0 0 0 Excluded
#4 1.000 51.78 103.56 207.1
#5 1.000 38.95 77.91 155.8 To be excluded
#6 1.000 49.81 99.63 199.3
#7 1.000 84.49 168.93 338.0
#8 0 0 0 0 Excluded
#9 0 0 0 0 Excluded
#10 0.699 91.77 183.54 367.1 Elected
Subtotal= 458.84 917.65 1835.5
Excess= 1.15 2.31 4.6
Total= 460.00 920.00 1840.0
Stage 7 Quota= 91.04 182.08 364.2
Cand. Retain Tally Tally Tally Status
#1 0.804 91.04 182.08 364.2 Elected
#2 1.000 57.68 115.36 230.7
#3 0 0 0 0 Excluded
#4 1.000 59.62 119.23 238.5
#5 0 0.00 0.00 0.0 Excluded
#6 1.000 58.93 117.86 235.7
#7 1.000 96.90 193.79 387.6 Newly elected
#8 0 0 0 0 Excluded
#9 0 0 0 0 Excluded
#10 0.644 91.04 182.08 364.2 Elected
Subtotal= 455.21 910.40 1820.9
Excess= 4.79 9.59 19.2
Total= 460.00 920.00 1840.0
Stage 8 Quota= 90.85 181.70 363.4
Cand. Retain Tally Tally Tally Status
#1 0.791 90.85 181.70 363.4 Elected
#2 1.000 59.02 118.04 236.1 To be excluded
#3 0 0 0 0 Excluded
#4 1.000 61.92 123.84 247.7
#5 0 0.00 0.00 0.0 Excluded
#6 1.000 60.76 121.51 243.0
#7 0.932 90.85 181.70 363.4 Elected
#8 0 0 0 0 Excluded
#9 0 0 0 0 Excluded
#10 0.635 90.85 181.70 363.4 Elected
Subtotal= 454.25 908.49 1817.0
Excess= 5.76 11.52 23.0
Total= 460.00 920.00 1840.0
Stage 9 Quota= 87.86 175.71 351.4
Cand. Retain Tally Tally Tally Status
#1 0.625 87.86 175.71 351.4 Elected
#2 0 0.00 0.00 0.0 Excluded
#3 0 0 0 0 Excluded
#4 1.000 88.33 176.67 353.3 Newly elected
#5 0 0 0 0 Excluded
#6 1.000 87.38 174.76 349.5 To be excluded
#7 0.726 87.86 175.71 351.4 Elected
#8 0 0 0 0 Excluded
#9 0 0 0 0 Excluded
#10 0.520 87.86 175.71 351.4 Elected
Subtotal= 439.29 878.56 1757.0
Excess= 20.72 41.43 82.9
Total= 460.00 920.00 1840.0
APPPENDIX 2
OUTCOME WITH MULTI-VOTE PLURALITY
Column (1) shows votes received if voters vote only for their 1st choice.
Column (2) shows votes received if voters vote
only for their 1st and 2nd choices.
Column (3) shows votes received if voters vote only for their top 3 choices.
Column (4) shows votes received if voters vote only for their top 4 choices.
Cand. (1) (2) (3) (4)
#10 118 192 239 289
#1 81 163 215 260
#7 70 126 174 214
#4 42 89 133 166
#6 41 86 129 171
#2 40 68 114 159
#5 27 65 111 146
#8 12 42 79 114
#9 14 37 71 113
#3 15 41 74 102
Total 460 909 1,339 1,734
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