[EM] Three ideas (Forest Simmons)

[Forest Simmons]

It looks like it turned out to be four over-lapping ideas.

I promised to elaborate on the third idea.

First I want to talk about what we mean by a weighted ballot, for example,
how do we count some ballots with a weight of 1/2 ?

The easiest way is to duplicate all of the ballots, and only keep one copy
of the ballots that are supposed to be weighted 1/2.

What if (in a certain round) some ballots are to have a weight of 1, some
1/2, and some 1/3 ?

Make six copies of every ballot.  Keep all of the full strength ballots,
keep half of the half weight ballots, and keep one third of the 1/3 weight
ballots.  Use the resulting set of ballots in this round.

Etc.

Next, by using score ballots we were able to come up with a natural system
of assigning weights to the various ballots at the various stages.  The
greater the satisfaction with the previously chosen winners, the smaller
the weight.  Jere are some possible alternatives to using the sum of the
ballot scores of the previous winners as the measure of satisfaction for
that ballot:

(1)  Use the number of winners so far that the ballot preferred over the
second place candidates in their respective rounds. [There are various
natural ways of defining second place.  Any of them will do.]

(2)  For each member X of the winner's circle, add a one to the
satisfaction score of  ballot B if ballot B rated X higher than the average
rating of X in its winning round.

(3)  Same as (2) only use median ratings instead of average ratings.

(4)  Use the number of winners so far with implicit approval (above zero
score).

(5) Same as (4) only allow an explicit approval cutoff.

Etc.

Finally, note that some of the above ideas do not require score style
ballots, in particular (1), (4), and (5) can be done with ordinal ballots
(rankings) as long as truncations and equal top rankings are allowed.

In this connection it is important to note that almost all methods
discussed on the EM list, including all versions of Condocet that allow
truncations and equal top rankings, still reduce to Approval when voters
use only those two extremes.  That's the crucial feature that makes all of
these methods equivalent to sequential PAV in the proportionality tests,
i.e when each faction gives top rank to each of their guys and bottom
position to all others.




>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Thu, 19 Nov 2015 15:26:48 -0800
> From: Forest Simmons <fsimmons at pcc.edu>
> The first idea was sparked by a feature that Warren Smith has incorporated
> into Range Voting, namely that instead of counting a non-rating (i.e.
> blank) score for a candidate as zero, it is considered to be a willingness
> to not influence the average score of that candidate.
>
> Idea I:  Interval Voting
>
> For each candidate C, each voter V  places the score for candidate C
> between two limits  min score for C and max score for C. Voter V's core for
> candidate C is the point of the interval specified by V that is nearest to
> the average score of the other voters. If that average is inside the
> interval, then V's score will be that average, otherwise it is given by the
> endpoint closest to that average.
>
> So Warren's blank rule is equivalent to taking the entire range as the
> interval.
>
> If the voter makes the upper and lower limits coincide, then the other
> voters' average score for C has no effect on V's score for C.
>
>
> Idea II:  Use interval voting with Majority Judgment or any other median
> based, Bucklin-like method.
>
> Interval voting would be particularly easy to implement in Bucklin-like
> methods like Majority Judgment:  Each voter submits two scores for each
> candidate.  If the median score of the other voters' submissions is between
> V's two submissions, then automatically V's submissions have no influence
> on the outcome.
>
> Idea III.  A new family of sequential PR methods.
>
> Suppose we have a method that (1) uses score style ballots, and (2) reduces
> to Approval when all voters use only the extreme scores. Any such method
> can adapted in a standard way to yield Proportional Representation.
>
> Here's how: First use the method unmodified to pick the first member of the
> winning set.  If the method is Beatpath (based on score ballots), then the
> beatpath winner W1 will be in the winning set.
>
> Next, run the method again with the same ballots, but where each ballot B
> is counted only 1/B(W1) of a normal ballot.  Let's call the winner of this
> round W2.
>
> In the third round, ballot B is counted only 1/(B(W1)+B(W2) + B(W3)) .
>
> Etc.
>
> Since the test for proportionality involves only factions that vote 100
> percent for their own candidates, and zero for all others, that puts us
> squarely in sequential PAV, which is well known to be proportional.
>
> That said, it is possible to use different weights adapted to the special
> features of each method.   More about that in a future post.
>
> Idea IV:  Andrew Jennings has shown how any sequential method for PR can be
> used to generate single winner lottery methods that satisfy a  proportional
> probability requirement.  More about that in a future post, ,also.
>
> Forest
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