[EM] A Framework for adapting single winner methods to the task of Proportional Representation in multi winner districts

[Forest Simmons]

The Multiwinner Method I have in mind chooses the winners sequentially.
It is based on the idea that ballots have an initial weight of one, and
that as candidates supported by a ballot are added to the winners' circle,
the weight is reduced according to some rule designed to diminish the
influence of the voters who have already achieved some level of
satisfaction.

At each stage in the election the new seat is filled by the candidate
picked by the single winner method applied to the entire ballot set with
the current ballot weights in force.

How, in general, do we diminish the weight of a ballot? Perhaps the
simplest way is to make the current weight 1/(1+S) where S is the current
satisfaction obtained by comparing the ballot preferences (whether ratings
or rankings) with the winners elected so far. As long as the current
satisfaction is zero, the weight remains at one since 1/(1+0) is just one.

Let's use Sequential Proportional Approval voting as a guide for what to
do. In SPAV the current satisfaction is simply the number of winners so far
that are approved on the ballot in question.  So what would we use in place
of that approval?  How could we reliably estimate how many of the current
winners would be approved when no approval cutoff is provided by the ballot?

Here's what I propose: bear with me and the rationale will become clear as
we proceed.

For each ballot we form a square matrix P whose entry in row i and column j
is given as follows

P(i, j) is 100% if candidate i is ranked or rated strictly above candidate
j. If j is ranked or rated strictly above candidate i, then P(i, j) is
zero.  If i and j have the same ranking or rating or are both truncated,
then P(i, j) equals P(i,i) = P(j,j), and the common value is  100% or zero
percent for equal top or equal bottom respectively.  If they are rated
equal and strictly between the two extremes, then P(i, i) is their common
rating.  If they are ranked (without ratings) strictly between  the
extremes, then P(i, i) is taken to be fifty percent.

We can think of P(i, j) as the probability that the voter who marked this
ballot would approve candidate i in an approval election where candidate j
was the main rival of candidate i.

Now let p(i) be the minimum over j of P(i , j).  This value may be thought
of as the minimum probability that the voter of this ballot would approve
candidate i no matter which j turned out to be the main rival candidate.

Now we can define the voter satisfaction for the voter of this ballot: it
is just the sum of the p(i) as i ranges over the current winners.

If the ballots are approval style, then this method reduces to Sequential
Proportional Approval Voting.  But it applies to any method based on
ranking or ratings with equal rankings and truncation allowed.

In particular, it seems (IMHO) to be the simplest, reasonable way to
convert River, BeatPath, RankedPairs, or MAM into a multiwinner PR method.

I admit this is a very terse description of the method, so I expect lots of
questions.  Especially people new to the EM List, I hope you will ask
questions for the benefit of all.  And everybody (new or not) make your
strongest criticisms so we can improve it (or discard it, if hopeless)!

Thanks,

Forest
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20200121/b160547e/attachment.html>