[EM] Sequential Best Assigment (multiwinner method)

Andy Jennings elections at jenningsstory.com
Wed Jul 26 08:12:32 PDT 2017

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 at gmail.com>

> 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_(GOLD)_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 at 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
>> ----
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