[EM] A New Multi-winner (PR) Method

[Forest Simmons]

As near as I know the following PR method based on Range/Score style
ballots is new.

This method is based on maximizing a measure of "goodness" of
representation to be specified later.  Slates of candidates are nominated
individually for consideration, because in general there are too many
possible slates to consider every one of them (due to combinatorial
explosion).  Among the nominated slates, the one with the best measure of
"goodness" of PR is elected.

To reduce the abstraction, suppose that there are only 100 candidates and
that only five vacancies to be filled.   Suppose further, that there are
ten thousand ballots (one for each of ten thousand voters).

Given a subset S of five candidates, we decide how good it is as follows:

Order the set S according to their Range totals, so that the highest to
lowest score order is c1, c2, ...c5.  This order only comes into play to
determine the cyclic order of play as the candidates "choose up teams" so
to speak.

Ballots are assigned to each of the candidates cyclically so that the
ballot most favorable to c1 goes to c1's pile, of the remaining the one
most favorable to c2, goes to c2's pile, etc. like the way we used to
choose teams when we were in grade school.

(Eventually we'll get to how to automate judgment of favorability.  Be
patient)

After 2000 times around the circle, each pile will contain exactly 2000
ballots. (Thanks for your patience.)

For our purposes the relative favorability of ballot V for candidate C is
the probability that V would elect C if it were drawn in a lottery; i.e.
V's rating of C divided by the sum of all of V's ratings for the candidates
in S including C.

What happens when one of more of the candidates is not shown any
favorability by any of the remaining ballots?  The other candidates
continue augmenting their piles until they reach their quotas (two thousand
each in this case), and the remaining ballots are assigned by comparing
them to the official public ballots of the candidates whose piles are not
yet complete. (We won't worry about the details of that for now.)

For each candidate C in S add up all of the ratings over all of the ballots
in the pile, but not the ratings for candidates outside of S.  Divide this
number by the total possible, which in this case is two thousand times five
or ten thousand.

We now have five quotients, one for each candidate.  Multiply these five
numbers together and take the fifth root.  This geometric mean is the
"goodness" score for the slate.

Among the nominated slates, elect the "best" one, i.e. the one with the
highest "goodness."

It is easy to show that this method satisfies proportionality requirements.
And (I believe) it takes into account "out-of pile" preferences as much as
possible without destroying proportionality.

No time for proofs or examples right now, but first, any questions about
the method?










****
>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20190410/36201b26/attachment.html>