[EM] IRV proponents figure out how to make IRV precinct-summable

[Kristofer Munsterhjelm]

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.

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.

>>
>>
>> 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.

> 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.

> Not clear to me what a binary array would be.

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).