On Mar 23, 2009, at 4:38 PM, Kristofer Munsterhjelm wrote:
Dave Ketchum wrote:
On Mar 23, 2009, at 10:46 AM, Kristofer Munsterhjelm wrote:
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.
To be clear here, we're dealing with two sorts of election methods.
There are one-round methods, like Plurality, Condorcet, contingent
vote, etc.; and then there are two-or-more methods, like TTR,
exhaustive ballot, eliminate-one runoff, etc.
It's possible to turn a multiple-round method into a single-round
method by assuming the voters would never change their ballots.
Doing so with eliminate-one runoff produces IRV, and doing so with
TTR produces the Contingent Vote.
Let's try it slowly for IRV, assuming multiple districts to avoid
shortcut temptations:
1 Count ala Plurality. If leader has a majority, that is winner.
2 Sum vote counts, starting with weakest count and ending before doing
the next candidate that would make a majority. None of those counted
could win, so mark them all as losers and go back to step 1.
Never needing step 2 is single round. In IRV voters do not have
opportunity to change ballots - but step 2 to decide on losers and
recounts is not avoidable. Note that with three candidates step 2 is
trivial for there is only one candidate for it to find.
But, with four candidates, such as A 29 , B 28, C 27 , D 5, only
D can be discarded for round 2 - but for A 29 , B 28, C 6 , D 5, C and
D can be discarded for second (final) round.
Summability is only properly defined for one-round methods. A method
is summable if it's possible to process any group of ballots into a
certain data chunk, where running the method on this data chunk
produces the same result as if it was run on that subset itself, and
where two chunks can be combined so that the same is true, and that
a chunk is of size determined by a function that increases no faster
than some polynomial of the number of candidates in the election.
Less formally, the method is summable if you can "count in
precincts" to produce managable data chunks that can then be
combined to get the result for all precincts or districts involved,
no matter the size of each district.
Not clear how this helps. You have to get the totals for round 1 to
decide how to proceed - matters not how many chunks.
Let's have a concrete example of how the contingent vote works,
and why my approach to it is summable.
[SNIP]
There you go, the contingent vote is summable.
Not clear why the two districts were even mentioned.
The two districts were mentioned so as to show that using only the
plurality counts and Condorcet matrices for each district, one could
get the same result as by counting all the ballots combined. That
is, that the Contingent Vote (the method) is summable.
Huh!
Since Condorcet was mentioned, might make sense to include a cycle
and see how much this complicates life.
Although my "summable CV" uses a Condorcet matrix, it's not a
Condorcet method. It passes Condorcet loser (like IRV), which is
simple to see: assume that the method eliminates all but the
Condorcet loser and some other candidate in the first round. Then
the Condorcet loser will lose the second round. Thus, the Condorcet
loser can't win.
However, it is not Condorcet. A simple example shows this:
11: A1 > B > A2 > C
10: A2 > B > A1 > C
9: B > C > A1 > A2
8: C > B > A2 > A1
A1 and A2 go to the second round, but B is the CW.
As I say above, what qualifies as summable?
It's summable if you can merge managable-size data chunks into
larger data chunks and find the result by referring to the data
chunk alone, so that you don't have to forward the (potentially
unmanagably large) ballot data to a central location.
("Managable size" being polynomial wrt the number of candidates)
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?
That implies either explicit equal-ranking, or truncation, which in
some sense is equal ranking last.
We have IRV ballots permitting only 3 ranks - with more than three
candidates.
Not clear to me what a binary array would be.
Clearly not useful here.
Take a Condorcet matrix like this:
A B C beats
A --- 98 125
B 127 --- 27
C 100 198 ---
"A beats B" is true for the binary matrix iff more people voted A >
B than B > A, so
A B C beats
A - F T
B T - F
C F T -
It's not really important, though; especially not given that Kathy
has said the IRV proponents weren't doing Contingent Vote or using
binary Condorcet matrices after all.
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.
I think this implies that any ranked vote system should deal with
less fine-grained ballots. That is: voters should be able to bullet-
vote or vote Approval style. That, in turn, means that the ballot
system should both support explicit equal ranking (for Approval
style) as well as truncation (for Plurality type counts). Supporting
truncation makes sense in any case, because otherwise you get
Australian conditions (that degrade into a form of external party
list PR through how-to-vote cards).
Your use of "truncation" bothers - I think of it as the system
discarding what it sees as excess data rather than the voter choosing
to say less than the method's limits.
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