[EM] Multiwinner methods with weighted votes

[Kristofer Munsterhjelm]

On 06/25/2017 01:51 AM, steve bosworth wrote:
>
>
> Re:  Multiwinner methods with weighted votes
>
> To Kristofer and everyone:
>
> ------------------------------------------------------------------------
>
> *From:*Kristofer Munsterhjelm <km_elmet at t-online.de>
> *Sent:* Sunday, April 2, 2017 1:00 PM>
>
> *To:*steve bosworth; election-methods at lists.electorama.com;
>
>
>
>>    K:  [….]  I'd like to note a few things.
>>>
>>>      - You could use MJ instead of IRV. Since APR prior to weighting is
>>>      basically "elect candidates by IRV …”, and you've stated
>>>      that MJ is better than IRV, APR should be better if you replace it with
>>>      "elect candidates by MJ a bunch of times". All you need is another way
>>>     of counting which voters contributed to which candidates' election. My
>>>      party list-type Bucklin method will do the job, for instance.

>> K:  As I mentioned in the previous mail, I have constructed amultiwinner
>> generalization of Bucklin/MJ.It can be used both for STV-type methods
>> and weighted vote-type methods. See
>> https://github.com/kristomu/voting-scripts/tree/master/new_methods/bucklin_range.

>> In essence, you count the number of Excellent votes alone first. If any
>> candidate has more than a threshold's worth of Excellent votes alone, he
>> is elected with weight equal to the number of votes he got, and the
>> votes are removed from consideration.
>
>
>
> You count the number of Excellent and Very good votes, and repeat the
> logic. You choose the
> threshold to be the lowest number so that the number of elected
> candidates comes out right (to 435 or whatnot).  [….]
>
>
> S:  Thank you for suggesting that I consider using a form of Bucklin/MJ
> rather than a form of IRV for my ARP.  Please correct me if I have not
> fully understood your brief description of your own “multiwinner
> generalization of Bucklin/MJ”. Using the USA as an example, below I
> will try to modify my APR proposal so that it is in line with my
> understanding of your Bucklin/MJ. I will call it Evaluative
> Associational Proportional Representation (EAPR).  Contrary to my first
> expectations, I see now how EARP, like ARP would allow the electorate to
> be represented most clearly as being composed of up to 435 groups, each
> enthusiastically holding a somewhat different worldview.  How does this
> compare with your “party list-type Bucklin method”?

There are two significant differences between weighted methods and 
party-list methods. The first is that, in party list, the effective 
weight of each party is quantized. A party can have one seat out of 435 
or two seats, but not 1.5 seats. The second is that the total number of 
seats depends not on the number of parties elected, but on their total 
weight. If a party gets five seats out of 435 in a party list method, 
there are only 430 seats left, which is the same as if two parties got 
five seats in total. But in a weighted system, the total weight a party 
holds is in principle unrelated to the number of seats they hold.

It seems like you're leaning in the party list direction for both, which 
I think is a good choice, but I'd like to verify that I've understood 
your method correctly.

> Similar to APR, EAPR would also enable each citizen to guarantee that
> their one vote would be added to the “weighted vote” of the one
> congressperson theyvalue most highly.  If no candidate she had evaluated
> at least as Acceptable, the EAPR ballot would allow her to require the
> candidate she had evaluated most highly to transfer her one vote to the
> congressperson he sees as the one most “fit” for the office.  If she had
> given more than one candidate her highest evaluation, the one that would
> pass her vote on in this way would be determined by lot.

You could also just give 1/n of a vote to each candidate at highest 
evaluation. E.g. if the voter had given more than two candidates the 
highest evaluation, each would get half the vote, as if each voter was a 
very large number of microvoters and you picked randomly among them.

With a very large electorate, the results are about the same as by using 
lot. The advantage of spreading it evenly is that you get the same 
result if you have 10 voters as if you have 10 million, whereas chance 
would matter more when there are 10 voters than when there are 10 million.

> Well before the general election, EAPR’s “primary election” would help
> to provide EAPR’s benefits. In this primary, citizens would be asked to
> use their “evaluations” rather than their “preferences” to help society
> discover the group of nationwide applicant organizations (e.g. political
> parties, interest groups, electoral districts, etc.) who are most valued
> by the electorate, i.e. the “associations” through which all voters will
> elect the 425 congresspersons later in the general election.  Each
> citizen would be asked to give one of the following “grades” to as many
> of the applicant organizations as they might wish:  Excellent, Very
> Good, Good, Acceptable, Poor, or Reject.  In this way, each citizen
> would record the extent to which they see each applicant organization as
> likely to offer a list of attractive candidate for any US voter to
> “grade” in the later general election.  Each organization not marked by
> a voter would be counted as Rejected by that voter.
>
>
> Each organization that receives at least 1/435 of all the registered
> voters in the country as giving it their highest available evaluations
> would become an “association”.  At the same time, each citizen who had
> given their evaluation to one of these associations would become a
> registered voter through that association during the later general
> election.  If none of the organizations given a “grade” of Acceptable or
> above by a citizen becomes an “association” in this way, that citizen
> would instead automatically become a registered voter through the
> geographically defined “association” in which they reside (i.e. their
> local electoral district).

I assume 425 was a typo and you meant 435 congresspeople :-)

As far as thresholds go, there are two varieties of my party-list 
Bucklin. The first is fixed quota (threshold), and the second is 
floating. Very briefly, fixed quota is to largest remainders party list 
methods as floating quota is to highest averages methods.

It seems you're using a fixed quota method here, where you set the 
threshold to some function of the number of candidates. The most common 
is the Droop quota, and I think that's what makes most sense for fixed 
quota, because it's a natural generalization of the concept of a majority.

In single-winner MJ, we're interested in the candidate that gets the 
best grade from a majority of the voters, which we could take as "more 
than 50% of the voters". So the threshold in MJ is "at least 50% + 1" or 
"more than 50%".

The Droop quota generalizes this by setting the threshold to "more than 
1/(s+1) of the total number of voters", where s is the number of seats. 
If s is 1, we get "more than 1/2 of the total number of voters", which 
matches the concept of a majority.

By that reasoning, the threshold above shouldn't be 1/435, but 1/436 for 
the same reason that the threshold for MJ shouldn't be 100% of the 
votes, but 50% of them. So the rule:

> Each organization that receives at least 1/435 of all the registered
> voters in the country as giving it their highest available evaluations
> would become an “association”.

becomes

"Each organization that receives more than 1/436 of all the registered 
voters in the country as giving it their highest available evaluations 
would become an 'association'."

Only 435 associations can all receive more than 1/436, for the same 
reason that only a single candidate can receive more than a majority in 
a single-winner election.

> Again, these EAPR “associations” would be composed of all the most
> valued organizations that had received at least the above threshold of
> voters giving  them their highest available evaluations. The primary’s
> discovering of these associations would start by counting only the
> number of Excellents received by each applicant organization.   Any
> organization receiving the largest number of Excellents above the
> threshold number would be the first association to be discovered.  Each
> citizen who had evaluated this organization as Excellent would become a
> registered voter through this association during the later general
> election at their local polling station.  The total number of
> evaluations received by each association would help to determine the
> number of congresspersons which must be elected later in the general
> election to represent it in the House of Representatives, e.g. an
> association receiving 1/435 would elect one congressperson, an
> association receiving 2/435 would elect two.  See the Endnote for more
> details.

That's right, except it should be "more than 1/436", "more than 2/436" 
and so on.

> For the discovery of each succeeding association, one by one, all the
> remaining evaluations given to all the remaining applicant organizations
> by each citizen who had already become a registered voter through one of
> the previously discovered associations would not be used to help
> discover any other association.  This guarantees that each citizen’s one
> vote in the primary will count only once, i.e. for the association in
> which they become a registered voter.  Consequently, the second
> association to be discovered would be the one, if any, which had
> received the next highest number of Excellents above the threshold.
> Similarly, any other organizations who had received at least the
> threshold of the remaining Excellents would also be one of this first
> group of associations to be discovered.  The second group of
> associations to be discovered would be composed of all those remaining
> organizations that had received at least the threshold but only by also
> adding all the remaining Very Goods they had received.  Similarly, the
> third group would be composed of all other organizations that had
> received at least the threshold but only by also adding all the
> remaining Goods they had received.  Similarly, the fourth group would be
> composed of all the other organizations that had received at least the
> threshold but only by also adding all the remaining Acceptables they had
> received.  Any citizens who had not yet become a registered voter for
> the general election through one of these associations as a result of
> the above counts would nevertheless now be added to one of these if they
> had given it at least an evaluation of Acceptable.  However, if none of
> their evaluations would allow them to become a registered voter in this
> way, they would instead automatically become a registered voter through
> their local electoral district (the geographically defined “association”
> in which they reside).

I would probably add some kind of tiebreak here (and to the general 
election as well). Suppose that there are two parties or associations: 
one that's center left and one that's center right, so they share a lot 
of voters. Suppose that both parties/associations get above the 
threshold at the same time (e.g. when Very Goods are added in). Then it 
might be the case that if you register all the voters for the 
center-left party, the center-right no longer has enough voters left to 
pass the threshold, and vice versa.

A reasonable tiebreaker, I think, would be to check who clears the 
threshold by the most voters. If there are multiple candidates (parties, 
associations), check who had the highest count at the previous grade 
level, and so on.

E.g. suppose parties X and Y have 100 voters giving them Good or better. 
If X and Y have a different number of voters giving them Very Good or 
better, the party with the most "Very Good or better" votes wins the 
tiebreak. If they're still equal, the party with the most voters giving 
them Excellent wins.

> During the later general election of congresspersons, each citizen would
> be asked to evaluate as many of the candidates in the country as they
> might wish, i.e. giving each one of the above “grades”.  Any candidate
> not marked by a voter would be counted as Rejected by that voter.
> Similar to the count in EARP’s primary, the first candidate in the
> country to be elected to the House would be the one, if any, who had
> received the highest number of Excellents above the threshold, i.e. at
> least 1/435 of all citizens in the country who have voted.  Again, to
> honor the principle of one-person-one-vote, all the evaluations given to
> other candidates by voters who have elected an earlier candidate would
> play no part in electing any later candidate.   Also, the next
> candidates to be chosen, one by one, would be the one, if any, who had
> received the next highest number of remaining evaluations above the
> threshold.  For example, the second candidate to be elected would be the
> one, if any, who had received the next highest number of remaining
> Excellents above the threshold.  All such winners would constitute the
> first group of congresspersons elected.
>
>
>
> Similarly, the second group would be those that had received at least
> the threshold but only by also adding all the remaining Very Goods they
> had received.  The third group would be those that had received at least
> the threshold but only by also adding all the remaining Goods they had
> received.  The fourth group would be those that had received at least
> the threshold but only by also adding all the remaining Acceptables they
> had received.  The fifth group would be composed of the most highly
> evaluated candidates but those who had not received even the above
> threshold number of evaluations, i.e. only the number of congressperson
> who are still needed to complete the exact number to be elected to
> represent each association as previously determined by EAPR’s primary.
> In any case, each congressperson would have a “weighted vote” in the
> House exactly equal to the total number of evaluations from citizens
> used to elect them.
>
>
>
> Next, the vote of any citizen whose vote had not yet been counted toward
> the “weighted vote” of an elected candidate as a result of any of the
> above counts would now be added if possible to the “weighted vote” of
> the congressperson to whom she had given at least an evaluation of
> Acceptable.  However, if a citizen’s vote still could not be added in
> this way to the “weighted vote” of one of the candidates who has already
> been elected, EAPR’s ballot allows a citizen to require the non-elected
> candidate to whom she had given her highest evaluation to transfer her
> one vote to the ”weighted vote” of the congressperson he believes is the
> one most qualified for the office.  In this way, each citizen guarantees
> that their vote will continue fully to count within the deliberations of
> the House.
>
>
>
> Finally, all the above counts must be interpreted so as to guarantee the
> election of the exact number of congresspersons to represent each
> association in the House as determined by the results of EAPR’s primary.

I'm not quite sure how your primary and general elections relate to each 
other. Is this correct?

- In the primary, voters vote for associations. They become registered 
voters of the association that they helped elect, and the number of 
seats each association gets is based on the MJ/Bucklin party list method 
(here, with fixed quotas).

- In the general, each voter votes for candidates within his assigned 
association. A candidate who gets more than 1/436 of the total general 
election turnout, according to the MJ/Bucklin procedure, gets a seat at 
full weight, and the remaining candidates to fill out the seats get 
reduced weight.


>
>
>
> *Endnote*
>
> When initially discovered, each “association” would immediately know the
> minimum number of congresspersons it would be allowed to elect, e.g. 2
> if it had at least two 435^ths of the nation’s registered voters, 3 if
> at least three 435^ths , etc.  However, if together all these
> associations had not yet been authorized to elect all 435
> representatives, the remaining number needed to complete the 435 would
> be distributed between these associations as follows: One by one, the
> right to elect an additional representative would be given sequentially
> to the association that currently has the “highest remainder’’.  A
> ‘remainder’ here is the number of electors beyond the minimum required
> to allow an association to elect one, two, three, or x number of
> representatives, as previously explained.  The second additional
> representative would be added to the association with the second largest
> remainder, and so forth.  This adding process would continue until the
> exact number of representatives that each association would elect as its
> contribution to the 435 had been discovered.

As I understand the primary, if you're using party list Bucklin with the 
Droop quota, you should always get seat allocations whose numbers sum up 
to 435. This is similar to how Bucklin always eventually finds a 
majority in a single-winner election, and MJ always finds a majority of 
the ballots who have graded the winner at the winning grade (say 
Acceptable) or above.

So you shouldn't ever get a shortage of allocated seats - if you use the 
Droop quota. (Nor should you get an excess, for that matter.)

The floating quota method would be more fair and still preserve that 
property that the seat allocation numbers sum up to 435. It would also 
avoid some population-pair monotonicity problems that largest-remainder 
party list methods have and highest-averages ones don't, but all of that 
comes at the cost of greater complexity.

> I now see EARP as superior to ARP because:
>
>  1.
>
>     It is easier and more informative for citizens to grade candidates
>     rather than rank them.
>
>  2.
>
>     It does not eliminate any candidate from consideration in the count
>     until all the winners have been discovered.
>
>  3.
>
>     It entirely eliminates the possible occurrence of either the
>     Condorcet or Arrow paradox.
>
> What do you think?

I think I need to understand the details a little better :-) But it 
seems to be, in general, good.

However, I'd like to point out that 3. only holds for single-winner MJ 
in the common grade language context that B&L use. If the voters rank 
rather than grade, 3. fails.

Furthermore, the multiwinner nature of the method might induce 
additional strategy, like vote management, or may lead to IIA violations 
even given the common grade context used by B&L. I can't say for sure 
that the multiwinner method fails IIA, but I do know that it is not 
impervious to vote management. (I did write a post some time back about 
a constraints-based version of Bucklin that would be much less 
vulnerable to vote management, but unfortunately the method is much too 
complex to be of practical use.)