Here's a multiwinner system that's so simple that it should have a name,
but I don't think it does.  Let me know if it does.

It uses rated ballots.  The goal is to repeatedly find the candidate whose
top quota's-worth of grades are highest and elect that candidate, then
de-weight a quota's-worth of voters.  Some names worth considering:

Sequential Best Assignment
Sequential Constituent Matching
Sequential Quota Allocation

The method:

N = Number of voters
S = Number of seats

1. Every voter grades every candidate.  (I'd say 4 or 6 grades.)

2. Each voter starts with weight 1.

3. Choose quota Q = N / S. (*)

4. For each candidate, calculate the minimum of their top Q grades.  Let G
   be the highest minimum.  Elect the candidate with that minimum.  (Break
   ties as in GMJ: calculate for each candidate what fraction of their G
   grades are in their top Q grades, and elect the candidate with the smallest
   such fraction.  Break further ties by choosing the candidate with the least
   number of G grades in their top Q grades.)

5. Deweight some voters to decrease the total voter weight by Q, in this
   manner:
   a) any voter who gave the minimum grade to all remaining candidates is
   deweighted to 0.
   b) for the voters not deweighted in (a) who gave this candidate a grade
   of G or above, find the deweighting D such that when the deweighting
   formula:

W_new = max(W_old - D, 0)

is applied, the total voter weight in this round is decreased by Q. (**)

6. Repeat steps 4 and 5, applying voter weights when calculating the top Q
   grades, until S seats are filled.

(*) With this quota, when you are filling say, 4 seats, then 25% of the
voting weight gets used up with each seat filled.  25% of the voting weight
will remain when choosing the last seat.  That last seat will be determined
by the tie-breaker rule, so it is essentially equivalent to approval
voting, with any above-bottom grade counting as approval.

The other common choice of quota, Q = N / (S + 1), could also be
considered.  When filling 4 seats, then, 20% of the voting weight gets used
up with each seat filled.  40% of the voting weight remains to choose the
last seat, so the last seat is essentially filled with a median-based
method (GMJ).  20% of the voters' opinions are, by design, left without a
representative.

(**) I thought about another step (a') where anyone who gave a grade
strictly above G was deweighted completely, but I think it gives the voters
too much incentive to down-weight candidates who they think can get elected
without their help.

I also considered another step (a'') where anyone who graded the chosen
candidates strictly above all other candidates was deweighted completely,
but I don't think there's much benefit for the added complexity.

Any thoughts on which quota is better or on the right name?

~ Andy Jennings

Election-Methods mailing list - see http://electorama.com/em for list info

This is a good idea.

But on thinking about it further, I'm not sure whether it's not the
same as BTV.

BTV, like Bucklin, works by gradually lowering a "pseudo-approval
threshold", and electing and deweighting candidates as they reach a
quota of "pseudo-approvals". Andy's proposal, like MJ, works by
directly looking at the "quota-th" highest rating, and electing and
deweighting the candidate who's highest by that measure.

But of course, we know that, aside from tiebreakers, MJ and Bucklin
are the same thing. So the more I think about it, the more I think
that (aside from quota choice, tiebreaker, and deweighting scheme;
none of which are really specified by the label "BTV") Andy's proposal
and BTV are the same thing.

I could be wrong about this... can anybody else check my logic here?

Still. Even if this is just a new name for BTV, it's a good excuse to
discuss that system.

We could talk about how good it is. Pretty excellent! I like that it
avoids the horrible center-squeeze breakage of STV. Even though the
problems with center squeeze are much less in a multiwinner setting
than in IRV, it's still ugly.

When designing GOLD
http://wiki.electorama.com/wiki/Geographic_Open_List/Delegated_%28GOLD%29_voting,
I chose STV rather than BTV as a substrate. That wasn't because I
prefer STV theoretically; it's just because of its longer track record.

Also, we could talk about the ancillary design decisions: quota
choice, tiebreaker, and deweighting scheme.

Quota choice: I tend to prefer Droop, or a compromise V/(S+.5), over
Hare. Basically, when you're assigning the last seat, you're left with
the voters who are most atypical; the "crumbs" of the party system. If
you use a Hare quota, then at best you'll find a candidate with some
appeal to a full quota; but realistically, you might just find the
biggest of a group of crumbs, who could easily have support from just
35-40% of a quota (based on 1/e, my SWAG for this kind of situation).
If you go with a Droop quota, on the other hand, the entire pool is 2
quotas; and 2/e is 70-80% of a quota, much closer to fair.

Andy's suggested deweighting scheme might help encourage bigger
crumbs, but I'm not sure about that.

Tiebreaker: I don't have a lot to say about this. GMJ-style seems like
a good choice.

Deweighting: This is where things get interesting. You don't want to
have too much of a free-riding incentive, but you do want to deweight
the votes which are "more satisfied" with the winners and not-deweight
those which are "less satisfied" with the future potential winners.

I like Andy's concept of subtractive, rather than multiplicative,
deweighting. It makes things a little bit harder to describe, but it
does mean that somebody who is "halfway decisive" twice will be fully
deweighted, rather than keeping 1/4 of their voting weight; that seems
fair to me.

I think that Andy's rejected idea of "for those who only gave the new
winner the threshold rating, deweight them last" was doing it wrong,
so I'm not surprised that he decided it led to too big of a free rider
incentive. If you're doing a GMJ tiebreaker anyway, then from a BTV
point of view, those voters are essentially giving a fraction of an
approval to the new winner. I think that only that fraction of their
ballot should be at risk for deweighting; so their subtractive
deweighting should be the minimum of their GMJ fraction and the
overall deweighting.

The other way to do things is to try to avoid deweighting voters
insofar as they still have useful opinions about the remaining
candidates. That's what Andy's proposed "completely deweight those who
rate all remaining candidates at 0" rule would do. But this could
still leave a very "crumbly" remainder at the end; imagine if the 100
candidates for the last seat each had 1% of the remainder giving them
a top-rating.

So I can imagine more complicated schemes to do this. For instance:

1. Find the R candidates with the highest quota-th ratings, where R
   is the remaining number of seats. In other words, the prospective
   winners if you proceeded from here on without any deweighting.
2. Of the deweight-able votes (counting only the GMJ subtractive
   portion ot threshold votes), find the Q which have the lowest max
   rating for those R candidates. Deweight these completely.

Note that the incentive of the above is not so much to downvote early
winners, as with traditional free riding (though of course that is
still possible if you downvote them below their winning threshold),
but rather to up-vote late winners. That creates a couter-free-riding
incentive; a possibility I'd never considered before.
....

But all-in-all, I think that Andy's suggested deweighting scheme is
pretty good, and I'd rather go for "simple" than "theoretically
awesome" here.

Jameson

2017-07-24 21:58 GMT-07:00 Andy Jennings <elections@jenningsstory.com
mailto:elections@jenningsstory.com>:

 Here's a multiwinner system that's so simple that it should have a
 name, but I don't think it does.  Let me know if it does.

 It uses rated ballots.  The goal is to repeatedly find the
 candidate whose top quota's-worth of grades are highest and elect
 that candidate, then de-weight a quota's-worth of voters.  Some
 names worth considering:

 Sequential Best Assignment
 Sequential Constituent Matching
 Sequential Quota Allocation

 The method:

 N = Number of voters
 S = Number of seats

 1. Every voter grades every candidate.  (I'd say 4 or 6 grades.)

 2. Each voter starts with weight 1.

 3. Choose quota Q = N / S. (*)

 4. For each candidate, calculate the minimum of their top Q
 grades.  Let G be the highest minimum.  Elect the candidate with
 that minimum.  (Break ties as in GMJ: calculate for each candidate
 what fraction of their G grades are in their top Q grades, and
 elect the candidate with the smallest such fraction.  Break
 further ties by choosing the candidate with the least number of G
 grades in their top Q grades.)

 5. Deweight some voters to decrease the total voter weight by Q,
 in this manner:
   a) any voter who gave the minimum grade to all remaining
 candidates is deweighted to 0.
   b) for the voters not deweighted in (a) who gave this candidate
 a grade of G or above, find the deweighting D such that when the
 deweighting formula:

   W_new = max(W_old - D, 0)

 is applied, the total voter weight in this round is decreased by
 Q. (**)

 6. Repeat steps 4 and 5, applying voter weights when calculating
 the top Q grades, until S seats are filled.


 (*) With this quota, when you are filling say, 4 seats, then 25%
 of the voting weight gets used up with each seat filled.  25% of
 the voting weight will remain when choosing the last seat.  That
 last seat will be determined by the tie-breaker rule, so it is
 essentially equivalent to approval voting, with any above-bottom
 grade counting as approval.

 The other common choice of quota, Q = N / (S + 1), could also be
 considered.  When filling 4 seats, then, 20% of the voting weight
 gets used up with each seat filled.  40% of the voting weight
 remains to choose the last seat, so the last seat is essentially
 filled with a median-based method (GMJ).  20% of the voters'
 opinions are, by design, left without a representative.

 (**) I thought about another step (a') where anyone who gave a
 grade strictly above G was deweighted completely, but I think it
 gives the voters too much incentive to down-weight candidates who
 they think can get elected without their help.

 I also considered another step (a'') where anyone who graded the
 chosen candidates strictly above all other candidates was
 deweighted completely, but I don't think there's much benefit for
 the added complexity.


 Any thoughts on which quota is better or on the right name?

 ~ Andy Jennings


 ----
 Election-Methods mailing list - see http://electorama.com/em for
 list info

Election-Methods mailing list - see http://electorama.com/em for list info

I think you're right that this matches BTV in the major details.  Is
http://rangevoting.org/BucklinTV.html the best reference for BTV?  It
doesn't have a page on Electowiki, yet, right?  We should add one.

There aren't many good, simple, rated-ballot, multiwinner systems, so
it deserves to get talked about more, no?

I wasn't suggesting to change the name, but if you are...

Regarding the quota:  I see what you're saying about the crumbs.  I
just have reservations about a quota that's not even going to try to
represent 1/(s+1) of the population.  I realize that STV does it and
STV has a track record.  But STV does it because it might come down to
a one-on-one at the end and you want to say that a majority in the
final round is the same as the quota for all the other rounds.  A
rated system doesn't have that restriction.  I kind of like approval
voting in the last round, even if the winner only gets 30-40%.

If we choose one quota for the default, I'd hope we could add a
footnote that the other one was a possible alternative.

Tiebreaker and deweighting:  I don't feel strongly about these, but
it's good that we're considering different options, looking for
simplicity but also looking for corners that cause adverse
incentives.  I think it's better to recommend good defaults than just
including a bunch of options.

Tiebreaker: If we were to use Hare quota (and "approval voting" in the
last round), then the critical grade for the last round is going to be
0(F), so a GMJ tiebreaker or "most votes at or above critical" are not
going to help break ties.  We'd either have to have a different rule
for the last round, or a second tiebreaker.  "Most votes strictly
above critical" would do it.

Deweighting: If we wanted to, we could "assign" voters to the
representatives for whom they were deweighted.  In that case, it would
be advantageous to deweight in chunks as large as possible.  So
subtractive deweighting would be better than multiplicative and rules
that "deweight completely" are good.  But multiplicative is probably
simpler.

~ Andy

On Tue, Jul 25, 2017 at 8:58 AM, Jameson Quinn
<jameson.quinn@gmail.com mailto:jameson.quinn@gmail.com> wrote:

 This is a good idea.

 But on thinking about it further, I'm not sure whether it's not
 the same as BTV.

 BTV, like Bucklin, works by gradually lowering a "pseudo-approval
 threshold", and electing and deweighting candidates as they reach
 a quota of "pseudo-approvals". Andy's proposal, like MJ, works by
 directly looking at the "quota-th" highest rating, and electing
 and deweighting the candidate who's highest by that measure.

 But of course, we know that, aside from tiebreakers, MJ and
 Bucklin are the same thing. So the more I think about it, the more
 I think that (aside from quota choice, tiebreaker, and deweighting
 scheme; none of which are really specified by the label "BTV")
 Andy's proposal and BTV are the same thing.

 I could be wrong about this... can anybody else check my logic here?

 Still. Even if this is just a new name for BTV, it's a good excuse
 to discuss that system.

 We could talk about how good it is. Pretty excellent! I like that
 it avoids the horrible center-squeeze breakage of STV. Even though
 the problems with center squeeze are much less in a multiwinner
 setting than in IRV, it's still ugly.

 When designing GOLD
 <http://wiki.electorama.com/wiki/Geographic_Open_List/Delegated_%28GOLD%29_voting>,
 I chose STV rather than BTV as a substrate. That wasn't because I
 prefer STV theoretically; it's just because of its longer track
 record.

 Also, we could talk about the ancillary design decisions: quota
 choice, tiebreaker, and deweighting scheme.

 Quota choice: I tend to prefer Droop, or a compromise V/(S+.5),
 over Hare. Basically, when you're assigning the last seat, you're
 left with the voters who are most atypical; the "crumbs" of the
 party system. If you use a Hare quota, then at best you'll find a
 candidate with some appeal to a full quota; but realistically, you
 might just find the biggest of a group of crumbs, who could easily
 have support from just 35-40% of a quota (based on 1/e, my SWAG
 for this kind of situation).  If you go with a Droop quota, on the
 other hand, the entire pool is 2 quotas; and 2/e is 70-80% of a
 quota, much closer to fair.

 Andy's suggested deweighting scheme might help encourage bigger
 crumbs, but I'm not sure about that.

 Tiebreaker: I don't have a lot to say about this. GMJ-style seems
 like a good choice.

 Deweighting: This is where things get interesting. You don't want
 to have too much of a free-riding incentive, but you do want to
 deweight the votes which are "more satisfied" with the winners and
 not-deweight those which are "less satisfied" with the future
 potential winners.

 I like Andy's concept of subtractive, rather than multiplicative,
 deweighting. It makes things a little bit harder to describe, but
 it does mean that somebody who is "halfway decisive" twice will be
 fully deweighted, rather than keeping 1/4 of their voting weight;
 that seems fair to me.

 I think that Andy's rejected idea of "for those who only gave the
 new winner the threshold rating, deweight them last" was doing it
 wrong, so I'm not surprised that he decided it led to too big of a
 free rider incentive. If you're doing a GMJ tiebreaker anyway,
 then from a BTV point of view, those voters are essentially giving
 a fraction of an approval to the new winner. I think that only
 that fraction of their ballot should be at risk for deweighting;
 so their subtractive deweighting should be the minimum of their
 GMJ fraction and the overall deweighting.

 The other way to do things is to try to avoid deweighting voters
 insofar as they still have useful opinions about the remaining
 candidates. That's what Andy's proposed "completely deweight those
 who rate all remaining candidates at 0" rule would do. But this
 could still leave a very "crumbly" remainder at the end; imagine
 if the 100 candidates for the last seat each had 1% of the
 remainder giving them a top-rating.

 So I can imagine more complicated schemes to do this. For instance:

  1. Find the R candidates with the highest quota-th ratings, where
     R is the remaining number of seats. In other words, the
     prospective winners if you proceeded from here on without any
     deweighting.
  2. Of the deweight-able votes (counting only the GMJ subtractive
     portion ot threshold votes), find the Q which have the lowest
     max rating for those R candidates. Deweight these completely.

 Note that the incentive of the above is not so much to downvote
 early winners, as with traditional free riding (though of course
 that is still possible if you downvote them below their winning
 threshold), but rather to up-vote late winners. That creates a
 couter-free-riding incentive; a possibility I'd never considered
 before.
 ....

 But all-in-all, I think that Andy's suggested deweighting scheme
 is pretty good, and I'd rather go for "simple" than "theoretically
 awesome" here.

 Jameson


 2017-07-24 21:58 GMT-07:00 Andy Jennings
 <elections@jenningsstory.com <mailto:elections@jenningsstory.com>>:

     Here's a multiwinner system that's so simple that it should
     have a name, but I don't think it does.  Let me know if it does.

     It uses rated ballots.  The goal is to repeatedly find the
     candidate whose top quota's-worth of grades are highest and
     elect that candidate, then de-weight a quota's-worth of
     voters.  Some names worth considering:

     Sequential Best Assignment
     Sequential Constituent Matching
     Sequential Quota Allocation

     The method:

     N = Number of voters
     S = Number of seats

     1. Every voter grades every candidate.  (I'd say 4 or 6 grades.)

     2. Each voter starts with weight 1.

     3. Choose quota Q = N / S. (*)

     4. For each candidate, calculate the minimum of their top Q
     grades.  Let G be the highest minimum.  Elect the candidate
     with that minimum.  (Break ties as in GMJ: calculate for each
     candidate what fraction of their G grades are in their top Q
     grades, and elect the candidate with the smallest such
     fraction.  Break further ties by choosing the candidate with
     the least number of G grades in their top Q grades.)

     5. Deweight some voters to decrease the total voter weight by
     Q, in this manner:
       a) any voter who gave the minimum grade to all remaining
     candidates is deweighted to 0.
       b) for the voters not deweighted in (a) who gave this
     candidate a grade of G or above, find the deweighting D such
     that when the deweighting formula:

       W_new = max(W_old - D, 0)

     is applied, the total voter weight in this round is decreased
     by Q. (**)

     6. Repeat steps 4 and 5, applying voter weights when
     calculating the top Q grades, until S seats are filled.


     (*) With this quota, when you are filling say, 4 seats, then
     25% of the voting weight gets used up with each seat filled. 
     25% of the voting weight will remain when choosing the last
     seat.  That last seat will be determined by the tie-breaker
     rule, so it is essentially equivalent to approval voting, with
     any above-bottom grade counting as approval.

     The other common choice of quota, Q = N / (S + 1), could also
     be considered.  When filling 4 seats, then, 20% of the voting
     weight gets used up with each seat filled.  40% of the voting
     weight remains to choose the last seat, so the last seat is
     essentially filled with a median-based method (GMJ).  20% of
     the voters' opinions are, by design, left without a
     representative.

     (**) I thought about another step (a') where anyone who gave a
     grade strictly above G was deweighted completely, but I think
     it gives the voters too much incentive to down-weight
     candidates who they think can get elected without their help.

     I also considered another step (a'') where anyone who graded
     the chosen candidates strictly above all other candidates was
     deweighted completely, but I don't think there's much benefit
     for the added complexity.


     Any thoughts on which quota is better or on the right name?

     ~ Andy Jennings


     ----
     Election-Methods mailing list - see http://electorama.com/em
     for list info

Election-Methods mailing list - see http://electorama.com/em for list info

BTV is what I call Binomial Transferable Vote