[EM] Sequential Best Assigment (multiwinner method)

Jameson Quinn jameson.quinn at gmail.com
Tue Jul 25 08:58:11 PDT 2017

This is a good idea.

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

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
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.


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