[EM] Sequential Best Assigment (multiwinner method)

Richard Lung voting at ukscientists.com
Sun Jul 30 00:39:13 PDT 2017

BTV is what I call Binomial Transferable Vote
V/(S + 1/2) is what I call the Harmonic Mean quota. It is a "compromise" 
but it is a principled compromise.
Both are described in my book, Scientific Method of Elections.

Richard Lung.

On 25/07/2017 16:58, Jameson Quinn 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 at jenningsstory.com 
> <mailto: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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Richard Lung.
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