Discussion:
[EM] Multiwinner methods with weighted votes
steve bosworth
2017-06-24 23:51:11 UTC
Permalink
Re: Multiwinner methods with weighted votes

To Kristofer and everyone:

________________________________

From: Kristofer Munsterhjelm <***@t-online.de>
Sent: Sunday, April 2, 2017 1:00 PM>

To: steve bosworth; election-***@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 a multiwinner
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”?

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 they value 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.



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).



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.



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).



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.



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 435ths of the nation’s registered voters, 3 if at least three 435ths, 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.

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?

Steve
Kristofer Munsterhjelm
2017-07-04 23:40:32 UTC
Permalink
Post by steve bosworth
Re: Multiwinner methods with weighted votes
------------------------------------------------------------------------
*Sent:* Sunday, April 2, 2017 1:00 PM>
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.
Post by steve bosworth
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.
Post by steve bosworth
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
Post by steve bosworth
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.
Post by steve bosworth
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.
Post by steve bosworth
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.
Post by steve bosworth
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.
Post by steve bosworth
*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.
Post by steve bosworth
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.)
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