[EM] Approximating Non-Summable Methods with Summable Methods

[Kathy Dopp]

> Date: Sat, 03 Apr 2010 22:52:08 +0000 (GMT)
> From: fsimmons at pcc.edu
> To: election-methods at lists.electorama.com
> Subject: [EM] Approximating Non-Summable Methods with Summable Methods
> Message-ID: <e4f0cf4ca370.4bb7c698 at pcc.edu>
> Content-Type: text/plain; charset=us-ascii
>
> Recently Kathy Dopp initiated a thread rerquesting the best summable, monotone, PR method ideas.
>
> In general non-summable methods can be approximated by summable methods if range ballots are used
> in the following way:
>
> At each precinct, for each candidate C take a weighted average of the ballots that rate candidate C at
> the top where the weight of a ballot is one over the number of candidates rated at the top on that ballot.
> Then the weighted averages (together with the total weights for each average) are passed along to the
> central location where they can be averaged together using their respective weights, to get one weighted
> ballot for each candidate.  The method you had in mind can be applied to these ballots.
>
> In particular, if you have a monotone PR method based on Range Ballots, you can apply that method.  If
> you don't have a monotone PR method based on Range Ballots, you can go to Warren Smith and get
> one; he collects them.
>
> Of course, after the averaging, you can convert the range ballots into ordinal rankings, and then use your
> favorite method based on rankings if you don't like range ballots.
>
> Note that if the original ratings are zero/one only, i.e. approval ballots, then the method still works.
>
> In summary, you can get a summable version of your favorite method.  And if you want, no matter what
> kind of ballots you method uses, you can get the required information via approval ballots.
>
>

I apologize for not having enough time to really reply or to spend
studying the great responses to this question.  Just briefly, I think
James that your method above at least at the precinct stage is exactly
equivalent to a method I proposed earlier that has the feature of
ensuring that each voters' vote is worth 1 no matter how many
candidates they rank or rate, the same value as every other voter.
This may or may not be considered a desirable feature, but some US
courts have declared it to be a vital feature for an electoral method.

As far as reweighting the ballots from the precinct, I think you've
only got weights for each individual voter, 1 divided by the number of
candidates they ranked or 1 divided by the total number of points they
assigned candidates, so what process are you using to obtain weights
for a precinct, the number of voters voted in that precinct?

As long as the method used to combine the tallies from each precinct
at the central office was also monotonic (not IRV/STV or other
nonmonotonic methods), I see no problem at all with this approach and
would probably support it as far as I can tell.

Thanks for thinking about it and coming up with an idea.

Regards,


Kathy Dopp
http://electionmathematics.org
Town of Colonie, NY 12304
"One of the best ways to keep any conversation civil is to support the
discussion with true facts."

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