On Mar 23, 2009, at 10:46 AM, Kristofer Munsterhjelm wrote:
Dave Ketchum wrote:
On Mar 22, 2009, at 4:24 PM, Kristofer Munsterhjelm wrote:
As stated, it's not summable. But note that the second round,
which is determined by the Plurality count, consists of a pairwise
comparison. Thus, one can make the method summable by simply
storing the information required to simulate any one-on-one runoff
-- in other words, by having a Condorcet matrix. Since Condorcet
is not mutually exclusive with summability, we know Condorcet
matrices can be summed - so that part is summable. We also know
that Plurality counts are summable - if A gets X votes in district
1 and Y votes in district 2, A got X+Y votes in these districts
combined.
Agreed that Condorcet and Plurality, and even Approval, are summable.
But suppose voters say A>B>C and A>C>B:
Condorcet will count A>B, A>B, A>C, A>C, B>C, and C>B into the
N*N matrix.
IRV will only see A>? and A>? until and unless A losing exposes
what remains (B>? or C>?).
True that ballot images could be forwarded, but that does not
really make summable claimable.
Agreed (in turn) that forwarding ballot images doesn't make a method
summable, since otherwise, any method that doesn't care about the
order of the ballots would be "summable".
Also, IRV, in the general case, is not summable. However, what we're
talking about is the contingent vote, an "instant top-two runoff",
which is what the IRV proponents figured out how to make precinct
summable (or thought they had figured out how to make precinct
summable). It agrees with IRV if the number of candidates <= 3.
The contingent vote first counts plurality votes for the various
candidates, as top-two runoff does. Then, again as in top-two
runoff, the two "winners", Plurality wise, go to the next round. The
difference is that the contingent vote uses the same rank ballots
for the second round as for the first, only with all non-winners
eliminated, whereas true TTR has a separate second round.
Let's see:
Plurality and Condorcet look at the ballots ONE time, and never
go back. Does summable require this - never going back to the
ballots, or to the voters, more times?
TTR needs to go back only if the top two were near to a tie -
IRV could do the same.
For TTR the second round presumably always finishes it; get
near enough to a tie and IRV could need more rounds - but they do not
mean extra effort from the voters.
Let's have a concrete example of how the contingent vote works, and
why my approach to it is summable.
District 1: 100: A > B > C
98: B > C > A
27: C > A > B
District 2: 104: C > B > A
121: C > A > B
50: A > B > C
25: B > A > C
Combined: 104: C > B > A
150: A > B > C
148: C > A > B
98: B > C > A
25: B > A > C
For the combined ballot, first do a Plurality count to see who
advances. Note that this Plurality count is what makes contingent
vote equal to IRV for number of candidates = 3, since preserving the
top two is equal to eliminating the last candidate.
Combined, plurality: 104 C, 150 A, 148 C, 98 B, 25 B
hence: 252 C, 150 A, 123 B
So C and A move to the second round.
Eliminating B, we get
104: C > A
150: A > C
148: C > A
98: C > A
25: A > C
summing up,
350: C > A
175: A > C
So C wins.
Let's look at the combined Condorcet matrix. It is
A B C beats
A --- 227 350
B 298 --- 252
C 175 273 ---
Here you can see that the data we're looking for is "C beats A" and
"A beats C". Since there are only two candidates remaining from the
first round, the second round will be an one-on-one, which is the
kind of contest the Condorcet matrix stores information about. If C
beats A more often than A beats C, C is the winner. Incidentally,
this shows that the contingent vote passes Condorcet loser.
But let's do it again, with only the summable information about each
district (that is, the plurality count and the Condorcet matrix).
District 1:
Plurality count: 100 A, 98 B, 27 C
Condorcet matrix: A B C beats
A --- 98 125
B 127 --- 27
C 100 198 ---
District 2:
Plurality count: 50 A, 25 B, 225 C
Condorcet matrix: A B C beats
A --- 129 225
B 171 --- 225
C 75 75 ---
Let's sum this all up:
Plurality count: 150 A, 123 B, 252 C
Condorcet matrix: A B C beats
A --- 227 350
B 298 --- 252
C 175 273 ---
And run the election method again:
First "round": greatest two are A (150) and C (252)
So A and C go to the second "round".
Second "round": A>C by 175, C>A by 350, so C wins.
There you go, the contingent vote is summable.
Not clear why the two districts were even mentioned.
Since Condorcet was mentioned, might make sense to include a cycle and
see how much this complicates life.
As I say above, what qualifies as summable?
I'm not sure about IRV - has anyone devised an STV variant that
handles equal rank? If not, then you're right - again, I'm not sure.
Brian claims, and I cannot disprove, that IRV can be stretched to
tolerate equal rank - questionable whether it would be worth the
expense for real elections.
There are two ways to handle equal rank, in theory, for a weighted
positional method. Plurality is just a weighted positional method
with the weights (1, 0, 0, ..., 0). The first is "whole", which
means that if you rank A = B > C, A and B has the same score, which
is the same as A in A > C. For plurality, that would turn it into
Approval. The second is "fractional", which means that the sum of
the score for all ranked candidates in a certain rank is the same,
no matter how many you ranked. For instance, for Plurality, ranking
A = B > C would give half a point to A and B (so that the sum is 1),
and none to C, whereas ranking A > B > C would give a full point to
A and none to B or C.
Elimination would work the same way however equal rank would be
treated. If you vote A = B > C and A is eliminated, then for the
next round, your vote is B > C.
Assuming the voting machines can handle the input, where would the
expense lie in adding this support? It seems to be more a question
of whether the resulting system would be "IRV" or not... unless the
expense would be in "handling the input", but if you have a machine
that can handle A > B > C > D > E .. > Z, upgrading it to handle A =
B = C > D > E ... doesn't seem to be that expensive a change.
Huh? Noticing whether equal ranks exist; including fractions in doing
sums; etc.
Note that for the current discussion Condorcet is simpler - for each
pair of candidates count A>B or B>A if they exist.
From what I've seen of voting equipment, most limitations seem to
be in the name of expediency. For instance, SF's RCV three-rank
method keeps voters from ranking more than three candidates -
probably to accomodate existing equipment.
I am suspicious as to this relating to existing equipment, but:
Some ways of providing for more ranks significantly burden
equipment design.
Providing for 2 ranks is essential to deserve claim as to ranks
existing; 3 helps some; more than 3 helps real voters little.
One may then ask, how many ranks are required to break Duverger's
Law? Unfortunately, I don't know the answer.
What limitations may exist (such as your IRV example) may be
handled by having a voting machine that permits all ranking types
(full, truncated, equal rank), then having parameters that limit
according to what kind of voting system is being used in the back
end (e.g no equal rank).
Sounds like building in expensive complications.
Doing the specialization in software could be affordable.
There's a tradeoff at this point. Having a generalized machine lets
you build many that are all the same, so that you gain benefits of
scale. However, the generalized machine is more expensive because
you can't cut away what you don't need.
It is in theory possible to make it summable - see above. The
method they did use seems not to be, though - as far as I could
see, they checked, for all possible virtual runoffs (set by
enforcing A and B as winners in the first round), whether A or B
won. Such a binary check is summable only if the results are the
same in both districts - but when they're different, one runs into
trouble. Consider this, for instance:
District 1 X>Y: 1000, Y>X: 990 X beats Y
District 2 X>Y: 1, Y>X: 2 Y beats X
-------------------------------------------------
Summed result X beats Y
but also
District 1 X>Y: 1000, Y>X: 990 X beats Y
District 2 X>Y: 1000, Y>X: 2000 Y beats X
-------------------------------------------------
Summed result Y beats X
In both instances, X beats Y in the first district, and Y beats X
in the second district, but the summed result is different for the
two cases. Thus I think that they would have to store the entire
Condorcet matrix (numbers of voters, not just who won) in order to
be summable. If they did, then they're summable, but if they
didn't, they aren't.
Condorcet cares not as to number of voters - for it simply sum the
matrices.
If nobody equal-ranks, then (A beats B) + (B beats A) = number of
voters. Apart from that, you're right, Condorcet doesn't care. What
I showed was that if they (the IRV proponents) tried to use only
binary arrays instead of integer arrays for their kinda-Condorcet
matrices, they would fail, because there's not enough information
there. A Condorcet matrix has to be integer (or even more fine
grained, e.g for CWP), even when that matrix is only to be used for
determining the winner of the contingent vote.
What do you do when some voters vote for neither A nor B?
Not clear to me what a binary array would be.
I mean freedom as a data format. A rated vote data format can
emulate a ranked vote format, as well as an approval-style data
format.
Saying freedom reminds me of something we sometimes ignore - how
much complication do we burden voters with.
Voting is already irrational from a utilitarian point of view - your
chance of affecting the outcome is way too small for it to be worth
bothering to vote, let alone consider the issues to make an informed
decision. Yet we vote anyway.
That muddies the waters, because we can't use standard utilitarian/
economic theory to find out how much complication is too much.
Perhaps people wouldn't bother with anything more than Approval, but
that seems wrong (since people rate and rank things all the time).
So, how much is too much? I don't know.
Imposing ratings for score is a noticeable complication.
Condorcet can claim a bit of simplification:
Voting as in Plurality should be encouraged whenever that meets
a voter's desires - in many races many voters need nothing more.
Voting as in Approval - ditto.
More complex ranking is really a simplification for those voters
who desire to use that ability, rather than being forced to live with
what Plurality offers.
Of course, the user interface should be good, but that's a separate
issue. I don't particularly like general purpose direct electronic
machines, so the "user interface" may be entirely transparent - put
the appropriate number in the box next to the candidate (for each
candidate), then at some later time, OCR reads off the numbers to
parse the ballot.
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