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

Thu Apr 11 14:28:09 PDT 2019

[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 at pcc.edu>
> To: EM <election-methods at 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?
>
>
>
>
>
>
>
>
>
>
> ****
> >
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> Message: 2
> Date: Wed, 10 Apr 2019 18:30:26 -0400
> From: Abd ul-Rahman Lomax <abd at lomaxdesign.com>
> To: election-methods at lists.electorama.com
> Subject: Re: [EM] A New Multi-winner (PR) Method
> Message-ID: <be0d97ba-67e4-4982-f916-ba8208cf4b41 at lomaxdesign.com>
> Content-Type: text/plain; charset="utf-8"; Format="flowed"
>
> I continue to be amazed that someone like Forest Simmons, who was an
> early writer about what later was called Asset Voting, as named by
> Warren Smith, and was simply a tweak on STV when invented by Charles
> Dodgson in the 1880s, which can produce *perfect* representation,
> beyond? mere "proportional representation," still works on complex
> single ballot deterministic methods that must compromise and prevent
> true /chosen/ representation in favor of some sort of theory of
> "optimised goodness."
>
> Asset Voting can very simply produce unanimous election of seats, where
> the seat represents a quota of voters who have */unanimously/*
> consented, directly or, probably much more commonly in large-scale
> elections, through chosen "electors," I call them, who becomes a proxy
> for the anonymous voters, for the election of the seat. Asset allows the
> electors --- those who become public voters -- to cooperate and
> collaborate for the selection of seats.
>
> The "Electoral College," fully and accurately, represents *all* the
> voters. But it might be very large, possibly too large to even meet
> directly. But all that is needed is a way for electors to communicate
> and to register their vote assignments for the creation of seats. They
> could use delegable proxy to advise them how to transfer.
>
> Asset was actually tried once, and it produced a result that most would
> have considered impossible. 17 voters, five candidates for a three-seat
> steering committee. A rather sharp division, but the final result was
> the election of the three seats *with every vote represented directly or
> by proxy*. Before the third seat was elected, nevertheless two seats
> were elected, so the steering committee, if needed, could have made any
> decision by unanimous vote (two agreeing) through the representation of
> two-thirds of the electorate. It could, in theory, have decided to elect
> the third seat by using the Droop quota. (the quota had not actually
> been specified, and the one who called the election and created the
> process was not totally sophisticated on Asset, which does not need to
> specify an election deadline, it can, instead, leave that last seat or
> seats open, and consider the rest of the electorate as Robert's rules
> considers unrecognizable ballots: they count for determining "majority"
> but not as a vote for or against any candidate.
>
> So, say, there is a jurisdiction with a million voters. It is decided
> that an optimal assembly would be 49, so the basis for a quota could be
> 50, and thus the quota would be 2% of the votes cast. This allows that
> if all electors assign their votes in exact measure to seats, 50 seats
> could be elected, but that outcome is improbable. (But if it happens,
> whoopee!), so, normally, 49 seats might be elected. Instead of electing
> the last seat by plurality, depending on some deadline for vote
> assignments, I have suggesting leaving the election open. Further, those
> electors could be consulted by the Assembly. This is the power of having
> public voters who, collectively, represent the entire electorate.
>
> Unanimous election of seats. Bayesian regret, zero. No losers. Minimal
> damage to the dregs. No expensive or extensive campaigning necessary. (I
> expect that the tradition would develop rapidly that one would only vote
> for people one could meet face-to-face, because how else can they truly
> represent? But this would be voluntary, not coerced. Basically, anyone
> who registers as an elector may participate further, the only
> requirement being a willingness to vote publicly (which is already
> required of elected representatives!)
>
> The U.S. Electoral College was a brilliant invention, knee-capped by the
> party system. It did not, however, represent the people, but
> jurisdictions, specifically states. It could have been reformed to
> represent the people, but it went, instead, toward representing the
> party majority in each state, in effect, becoming a rubber stamp and
> creating warped results.
>
> Asset is simple, close to tradition, but actually revolutionary, a
> dramatic shift from the entire concept of "elections" as contests. The
> voters would be 100% represented in the College, though voluntary
> choices, no need to consider "electability," hence no need for "voting
> strategy." Choose the available person, from a large universe, you most
> trust. As Dodgson noticed, that was relatively easy for ordinary voters,
> much easier than sensibly ranking many candidates. That complexity is
> completely unnecessary with Asset. No votes are wasted.
>
> The only tweak I see as needed involves making it difficult to coerce
> votes, by making it impossible to know that a specific person did *not*
> vote for one. This would not be needed for small NGO elections. In
> public elections, I'd have a set of known candidates who received
> substantial votes in a prior election, or something like that, and when
> electors register, they would assign their own vote to another
> candidate. If they receive less than N votes, their vote would be
> transferred as directed. If they receive N votes, they become an
> elector. Electors would not vote in the election, so, if they receive N
> votes, they get their own declared vote back and so they have N+1 votes
> to transfer. N should be the minimum size necessary to ensure that they
> cannot know that a specific person did not vote for them. If they do not
> receive N votes, their received votes are privately added to the total
> for the candidate they chose when registering. So N might be two, but if
> it is a bit higher, it could provide increased security. 3 might be
> completely adequate, together with it being very illegal to coerce votes.
>
> *No more original content below.*
>
> On 4/10/2019 5:08 PM, Forest Simmons wrote:
> > 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?
> >
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