Re: [EM] A New Multi-winner (PR) Method
[pesky text editor sent my previous attempt prematurely]
Correction to an important oversight: as a ballot is added to a
candidate's pile it must be multiplied by that ballot's favorability for
the candidate. Later the normalization step that makes the max possible
goodness equal to one, must take this adjustment into account. I'll
indicate these corrections with inline edits of the original message below.
Before I go to Lomax, let me just mention that other measures of
favorability may turn out to be better, particularly those based on
singular value decompositions (SVD's) like the ones Warren is working on.
Now to Lomax:
I fully agree that Charles Dodgson's solution (that we now call Asset
Voting) is the simplest and best over all method for most multi-winner and
even single winner elections.
And I appreciate your insights relative to the electoral college and other
historical context, as well as your practical suggestions for
implementation of the method or a version of the method, since it seems
that tweaks about the details are irresistible.
And any other method can be combined with Asset Voting, like the
Australians do by allowing the voters to vote the party line or copy
candidate cards.
But mostly, some of us are interested in finding the mathematical
limitations of methods based on Range/Score/Cardinal Rating style ballots.
Thanks for your valuable comments.
[messages being replied to]
1. A New Multi-winner (PR) Method (Forest Simmons)
2. Re: A New Multi-winner (PR) Method (Abd ul-Rahman Lomax)
Message: 1
Date: Wed, 10 Apr 2019 14:08:37 -0700
From: Forest Simmons fsimmons@pcc.edu
To: EM election-methods@lists.electorama.com
Subject: [EM] A New Multi-winner (PR) Method
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.
Before throwing each ballot into its respective pile, scale it by its
favorability (as defined below) to the candidate inot whose pile it is
being thrown.
(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.
Actually, because of the nature of the favorability factor by which each
ballot has already been scaled, no ballot can contribute more than one unit
to the total, so the max total is the number of ballots in the pile, namely
two thousand, not five 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?
Hi Forest, this is in reaction to your new "card dealing" PR multiwinner
voting method.
"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)."
--My belief is, either
(a) a computer can enumerate all binomial(C,W) possible W-winner subsets
of the C candidates, or
(b) too many for the computer.
In case (b), I have arguments that virtually any multiwinner method is
inherently ridiculously
ultra-vulnerable to strategy. Suggesting none should be used in this regime.
"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."
--this measure seems suspicious if there could be many cloned candidates.
If we had ballots of the form "you (the voter) have 100 points to distribute to
the candidates in any way you please" it would be less suspicious. But then
voters would likely be strategically motivated to give one 100 & all others 0.
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.)
--sounds like a freaking major "detail" to me. And if the candidates,
or the voters, produce
low-information ballots (such as plurality style voting) then seems
likely there will just not be enough information in there to allow any
intelligent way of deciding how to assign them.
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.
Actually, because of the nature of the favorability factor by which
each ballot has already
been scaled, no ballot can contribute more than one unit...
--I am not seeing that. Favorability is in [0,1]. Ratings are
nonnegative integers.
The product can be any nonnegative rational, not upperbounded by 1.
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."
--you could have omitted taking the 5th root, and just used the product as
the goodness. And also, you could have avoided multiplying things,
and instead summed the logs of things, and just used that sum-of-logs
as the goodness.
These are equivalent restatements. As I daresay you already knew.
I believe it takes into account "out-of pile" preferences as much as
possible without destroying proportionality.
--that sounds like it could be the beginning of some interesting train
of thought.
But I don't know what it is.
any questions about the method?
--I have not figured out what it is. Pseudocode might help.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)
On 4/11/19, Warren D Smith warren.wds@gmail.com wrote:
Hi Forest, this is in reaction to your new "card dealing" PR multiwinner
voting method.
--another reaction: in the method you described, there are a lot of exact ties.
You say stuff like "choose the next card to deal as the one most
favorable to candidate 3"
but there will be many tied-for-most. I presume you break all such
ties randomly.
In your example with 10000 voters & 5 winners, I think it would be common for
there to be more than 1000 such randomness-assisted events.
That is a lot of randomness. So the "quality function" would actually not
be a function of the ballots and winner-set alone, it also would
be a function of a truckload of random coin tosses.
That is problematic.
--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)