[EM] Three ideas

[Forest Simmons]

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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