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

[Dave Ketchum]

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.