Dave Ketchum wrote:
On Mar 23, 2009, at 4:38 PM, Kristofer Munsterhjelm wrote:
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.
I'm not talking about IRV. I'm talking about the Contingent Vote. IRV is
to exhaustive runoff (do balloting, remove loser, do another balloting,
remove loser, ...) as CV is to top two runoff.
I'm talking about the Contingent Vote because that was what I thought
they were emulating. CV is not a particularly good voting method, in itself.
Again, the steps for the Contingent vote is as follows:
1 - Do a plurality count. If someone has a majority, he wins.
2 - Otherwise, eliminate all but the top two. Reread the ballots with
all but the top two eliminated. Whoever has the greatest Plurality count
(i.e, is voted above the other more often) wins.
Do you see that the first step can be done by a plurality count, which
is summable, to find the top two? Also, do you see that the second can
be checked by a pairwise comparison, no matter who the top two are?
(Although, of course, it's a different pairwise comparison depending on
who the top two are)
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.
I never claimed IRV itself to be summable, just that the Contingent Vote
is. I don't think it's possible to make IRV (the proper method) summable
at all - the chaotic aspect that makes it nonmonotonic and causes those
weird shapes in the Yee diagrams hints at that (small change in votes -
large change in outcome).
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!
I claimed it was summable. Then I showed it for a particular instance by
first showing the result by counting both districts as one (central
count), then showing the "data chunk" for each district, summing them,
and showing that the result using the summed chunks was the same as the
result by central count.
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.
Ah, yes. SF's RCV is sort of summable, though. Because there are only
three ranks, the possible number of unique ballots are:
n * (n-1) * (n-2)
where n is the number of candidates. This increases more slowly than
n^3, hence the data chunk (which simply consists of the ballots
themselves, each prefixed by how many voted this way), is polynomial
with regards to the number of candidates.
With explicit equal rank, the equation is somewhat more complex. If by
"3 ranks", A = B = C count as 3 ranks, then it's still summable. If A =
B = C only counts as a single rank, then it's not.
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.
Truncation is a term that's been used on EM before. It simply means
stopping the ranking before you have ranked everyone. The machine or
method doesn't truncate, the voter does.
See, for instance,
http://lists.electorama.com/pipermail/election-methods-electorama.com/2002-
September/008656.html , which is an EM post (from 2002) where the term
"truncation" is used.
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