The value of mathematical modeling in areas such as voting, apportionment, social welfare, etc. is that one can understand the tradeoffs between desirable properties of a "system" where sometimes one cannot achieve everything one might want simultaneously. Scholars who have done this include K. Arrow, A. Sen, E. Huntington, M. Balinski, H.P. Young, E. Maskin and F. Pukelsheim.

Two relatively accessible books about these matters regarding apportionment, though somewhat "old" are Balinski and Young, Fair Representation, revised edition and H.P. Young Equity In Theory and Practice.

Regards,

Joe

——————————————
Joseph Malkevitch

Email:
jmalkevitch@york.cuny.edu
Web page:
http://york.cuny.edu/~malk/

From: Election-Methods election-methods-bounces@lists.electorama.com on behalf of Kristofer Munsterhjelm km-elmet@munsterhjelm.no
Sent: Sunday, September 15, 2024 12:47 PM
To: Toby Pereira tdp201b@yahoo.co.uk; EM election-methods@lists.electorama.com
Subject: Re: [EM] Proportionality vs utility: redoing 2008 with better units

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On 2024-09-12 14:33, Toby Pereira wrote:

Just an update on this: I had the arguments to the Sainte-Laguë index
function the wrong way around. (Unlike the Euclidean distance, order
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3
instead of 0, and Harmonic's at 0.1 instead of 0.

As a side effect, a lot of the negative proportionality results
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods
all register as having some proportionality relative to random
candidate. And both worst Plurality and worst Antiplurality (electing
the losers of the respective methods) score badly on proportionality
now. So the results seem to be more sensible.

I wrote another implementation from scratch for Harmonic and the
proportionality peaked below delta=0.5 there too, so I'm leaning towards
the problem either being inherent to the model or a result of Harmonic
depending too much on the rating, though it could also be an improper
generalization of the disproportionality index.[1]

The latter argument would go like: Suppose that with some overlapping
opinions, the ratings come out the same was as if there were fewer
issues and the voters were less fragmented, and they were rating the
candidates on quality instead. Then delta=1/2 would choose a balanced
outcome for this lower dimension case, but it would be too large-issue
biased in the higher dimension case.

Such an ambiguity might even be fundamental: that no method can tell
them apart. If the model is unrealistic, that's not a problem, but if it
is, that would mean that the space-unbiased parameter depends on the
complexity of the issue space itself, which would be a bummer.

In a sense, even simpler settings have this, e.g. Warren's "0.5 is not
the optimum divisor" (https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Furldefense.com%2Fv3%2F__https%3A%2F%2Fwww.rangevoting.org%2FNewAppo.html__%3B!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUL4Pdrhw%24&data=05%7C02%7Cmalkevitch%40york.cuny.edu%7Cac96a951253849aa438d08dcd5a6259b%7C6f60f0b35f064e099715989dba8cc7d8%7C0%7C0%7C638620156810778867%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=cLEmx5bdcEe3d12kRfauafEaVkbRE42OE3%2F3GHaplRc%3D&reserved=0 ). But
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs
0.5, or 0.1 vs 0.5 is much more of a big deal.

(Then again, Droop proportionality spanning such a large space might
suggest otherwise. It's hard to tell.)

I'll probably backport some of the reimplementation to clean up my code,
and drop the "candidates also vote" aspect of the model (since in real
elections, the number of voters is so much larger than the number of
candidates that the latter effectively vanishes), and then post some new
plots. Eventually. I'm not going to risk burning myself out.

-km

[1] To test the "improper generalization" hypothesis, I tried to cluster
opinion space into mutually exclusive regions and then report the
fractions of voters/elected candidates falling into each region, instead
of the proportion holding a true opinion on each issue dimension. The
chi-squared test that the Sainte-Laguë index looks like requires mutual
exclusion. But that didn't change the outcome much, and it didn't push
the optimal argument closer to 0.5.

Election-Methods mailing list - see https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Furldefense.com%2Fv3%2F__https%3A%2F%2Felectorama.com%2Fem__%3B!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUtpfOMHQ%24&data=05%7C02%7Cmalkevitch%40york.cuny.edu%7Cac96a951253849aa438d08dcd5a6259b%7C6f60f0b35f064e099715989dba8cc7d8%7C0%7C0%7C638620156810788945%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C0%7C%7C%7C&sdata=nQpkVDE%2FQyaT%2FjUi8E05a5p2jkS3NrRIUq0DZFeH0V8%3D&reserved=0https://urldefense.com/v3/__https://electorama.com/em__;!!NFZiyfBF5EK0!j5CBYEj2AS9TWOuZpZyVOwsLGPAogTBn6ktdX5Y9pMh7It5J4Qzvr4ZSQwGNVcBH7SCfZxcJTnlrDUH_0VlHlAwUtpfOMHQ$  for list info

On 2024-09-15 19:22, Joseph Malkevitch wrote:

That's ultimately what I'm trying to do; that, and determining the
"qualities" of multiwinner methods to guide my design of them, and to
get some idea of how well existing methods fare.

-km

I was just thinking that if I was doing a total score (what we're calling utility here) versus proportionality graph, for proportionality I might just use the var-Phragmen measure + KPT off the voter's utility scores, rather than taking the further step of looking at what the elected candidates would do once elected. I generally think that var-Phragmen gives the best measure of proportionality (and it reduces to Sainte-Laguë).
Toby
On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:

On 2024-09-12 14:33, Toby Pereira wrote:

Just an update on this: I had the arguments to the Sainte-Laguë index
function the wrong way around. (Unlike the Euclidean distance, order
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3
instead of 0, and Harmonic's at 0.1 instead of 0.

As a side effect, a lot of the negative proportionality results
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods
all register as having some proportionality relative to random
candidate. And both worst Plurality and worst Antiplurality (electing
the losers of the respective methods) score badly on proportionality
now. So the results seem to be more sensible.

I wrote another implementation from scratch for Harmonic and the
proportionality peaked below delta=0.5 there too, so I'm leaning towards
the problem either being inherent to the model or a result of Harmonic
depending too much on the rating, though it could also be an improper
generalization of the disproportionality index.[1]

The latter argument would go like: Suppose that with some overlapping
opinions, the ratings come out the same was as if there were fewer
issues and the voters were less fragmented, and they were rating the
candidates on quality instead. Then delta=1/2 would choose a balanced
outcome for this lower dimension case, but it would be too large-issue
biased in the higher dimension case.

Such an ambiguity might even be fundamental: that no method can tell
them apart. If the model is unrealistic, that's not a problem, but if it
is, that would mean that the space-unbiased parameter depends on the
complexity of the issue space itself, which would be a bummer.

In a sense, even simpler settings have this, e.g. Warren's "0.5 is not
the optimum divisor" (https://www.rangevoting.org/NewAppo.html). But
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs
0.5, or 0.1 vs 0.5 is much more of a big deal.

(Then again, Droop proportionality spanning such a large space might
suggest otherwise. It's hard to tell.)

I'll probably backport some of the reimplementation to clean up my code,
and drop the "candidates also vote" aspect of the model (since in real
elections, the number of voters is so much larger than the number of
candidates that the latter effectively vanishes), and then post some new
plots. Eventually. I'm not going to risk burning myself out.

-km

[1] To test the "improper generalization" hypothesis, I tried to cluster
opinion space into mutually exclusive regions and then report the
fractions of voters/elected candidates falling into each region, instead
of the proportion holding a true opinion on each issue dimension. The
chi-squared test that the Sainte-Laguë index looks like requires mutual
exclusion. But that didn't change the outcome much, and it didn't push
the optimal argument closer to 0.5.

Dear Toby,

How do tell or measure how one measure of proportionality is better than another?

Regards,

Joe

——————————————
Joseph Malkevitch

Email:
jmalkevitch@york.cuny.edu
Web page:
http://york.cuny.edu/~malk/

From: Election-Methods election-methods-bounces@lists.electorama.com on behalf of Toby Pereira tdp201b@yahoo.co.uk
Sent: Monday, September 16, 2024 1:08 PM
To: EM election-methods@lists.electorama.com; Kristofer Munsterhjelm km-elmet@munsterhjelm.no
Subject: Re: [EM] Proportionality vs utility: redoing 2008 with better units

• This email originates from a sender outside of CUNY. Verify the sender before replying or clicking on links and attachments. *

This email originated from election-methods-bounces@lists.electorama.com, a sender outside of CUNY. Never send login credentials, financial information, or sensitive information by email. Report suspicious email to reportspam@york.cuny.edu
I was just thinking that if I was doing a total score (what we're calling utility here) versus proportionality graph, for proportionality I might just use the var-Phragmen measure + KPT off the voter's utility scores, rather than taking the further step of looking at what the elected candidates would do once elected. I generally think that var-Phragmen gives the best measure of proportionality (and it reduces to Sainte-Laguë).

Toby

On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:

On 2024-09-12 14:33, Toby Pereira wrote:

Just an update on this: I had the arguments to the Sainte-Laguë index
function the wrong way around. (Unlike the Euclidean distance, order
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3
instead of 0, and Harmonic's at 0.1 instead of 0.

As a side effect, a lot of the negative proportionality results
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods
all register as having some proportionality relative to random
candidate. And both worst Plurality and worst Antiplurality (electing
the losers of the respective methods) score badly on proportionality
now. So the results seem to be more sensible.

I wrote another implementation from scratch for Harmonic and the
proportionality peaked below delta=0.5 there too, so I'm leaning towards
the problem either being inherent to the model or a result of Harmonic
depending too much on the rating, though it could also be an improper
generalization of the disproportionality index.[1]

The latter argument would go like: Suppose that with some overlapping
opinions, the ratings come out the same was as if there were fewer
issues and the voters were less fragmented, and they were rating the
candidates on quality instead. Then delta=1/2 would choose a balanced
outcome for this lower dimension case, but it would be too large-issue
biased in the higher dimension case.

Such an ambiguity might even be fundamental: that no method can tell
them apart. If the model is unrealistic, that's not a problem, but if it
is, that would mean that the space-unbiased parameter depends on the
complexity of the issue space itself, which would be a bummer.

In a sense, even simpler settings have this, e.g. Warren's "0.5 is not
the optimum divisor" (https://www.rangevoting.org/NewAppo.htmlhttps://urldefense.com/v3/__https://www.rangevoting.org/NewAppo.html__;!!NFZiyfBF5EK0!gCmnOKKoIhGZjZz3yP9CaFzsieXzbnVqDZCNEve2z_aAmhoNdjYJkrvGCdjSqwOjf_eVUjU-EAV-Kc_xZQe4NnoXUg$). But
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs
0.5, or 0.1 vs 0.5 is much more of a big deal.

(Then again, Droop proportionality spanning such a large space might
suggest otherwise. It's hard to tell.)

I'll probably backport some of the reimplementation to clean up my code,
and drop the "candidates also vote" aspect of the model (since in real
elections, the number of voters is so much larger than the number of
candidates that the latter effectively vanishes), and then post some new
plots. Eventually. I'm not going to risk burning myself out.

-km

[1] To test the "improper generalization" hypothesis, I tried to cluster
opinion space into mutually exclusive regions and then report the
fractions of voters/elected candidates falling into each region, instead
of the proportion holding a true opinion on each issue dimension. The
chi-squared test that the Sainte-Laguë index looks like requires mutual
exclusion. But that didn't change the outcome much, and it didn't push
the optimal argument closer to 0.5.

Well, I've previously discussed on this mailing list why I think Sainte-Laguë is the most logical measure of proportionality for party-list or apportionment, so for an approval/score method, I'd want it to reduce to that in the case of party voting. var-Phragmen does that in probably the neatest way, and using the KP-transformation for scores doesn't break anything.
https://electowiki.org/wiki/Phragmen%27s_voting_rules
https://electowiki.org/wiki/Kotze-Pereira_transformation
Toby

On Monday 16 September 2024 at 19:36:33 BST, Joseph Malkevitch <jmalkevitch@york.cuny.edu> wrote:  

Dear Toby,
How do tell or measure how one measure of proportionality is better than another?
Regards,
Joe
——————————————Joseph Malkevitch
Email:jmalkevitch@york.cuny.eduWeb page:http://york.cuny.edu/~malk/From: Election-Methods election-methods-bounces@lists.electorama.com on behalf of Toby Pereira tdp201b@yahoo.co.uk
Sent: Monday, September 16, 2024 1:08 PM
To: EM election-methods@lists.electorama.com; Kristofer Munsterhjelm km-elmet@munsterhjelm.no
Subject: Re: [EM] Proportionality vs utility: redoing 2008 with better units 

• This email originates from a sender outside of CUNY. Verify the sender before replying or clicking on links and attachments. *

| This email originated from election-methods-bounces@lists.electorama.com, a sender outside of CUNY. Never send login credentials, financial information, or sensitive information by email. Report suspicious email to reportspam@york.cuny.edu |

I was just thinking that if I was doing a total score (what we're calling utility here) versus proportionality graph, for proportionality I might just use the var-Phragmen measure + KPT off the voter's utility scores, rather than taking the further step of looking at what the elected candidates would do once elected. I generally think that var-Phragmen gives the best measure of proportionality (and it reduces toSainte-Laguë).
Toby
On Sunday 15 September 2024 at 17:47:47 BST, Kristofer Munsterhjelm km-elmet@munsterhjelm.no wrote:

On 2024-09-12 14:33, Toby Pereira wrote:

Just an update on this: I had the arguments to the Sainte-Laguë index
function the wrong way around. (Unlike the Euclidean distance, order
matters.) I fixed it and now QPQ's proportionality optimum is at 0.3
instead of 0, and Harmonic's at 0.1 instead of 0.

As a side effect, a lot of the negative proportionality results
(Antiplurality etc.) vanished. The reasonable single-winner bloc methods
all register as having some proportionality relative to random
candidate. And both worst Plurality and worst Antiplurality (electing
the losers of the respective methods) score badly on proportionality
now. So the results seem to be more sensible.

I wrote another implementation from scratch for Harmonic and the
proportionality peaked below delta=0.5 there too, so I'm leaning towards
the problem either being inherent to the model or a result of Harmonic
depending too much on the rating, though it could also be an improper
generalization of the disproportionality index.[1]

The latter argument would go like: Suppose that with some overlapping
opinions, the ratings come out the same was as if there were fewer
issues and the voters were less fragmented, and they were rating the
candidates on quality instead. Then delta=1/2 would choose a balanced
outcome for this lower dimension case, but it would be too large-issue
biased in the higher dimension case.

Such an ambiguity might even be fundamental: that no method can tell
them apart. If the model is unrealistic, that's not a problem, but if it
is, that would mean that the space-unbiased parameter depends on the
complexity of the issue space itself, which would be a bummer.

In a sense, even simpler settings have this, e.g. Warren's "0.5 is not
the optimum divisor" (https://www.rangevoting.org/NewAppo.html). But
it's not a big deal there: 0.495 vs 0.5 is a very small change. 0.3 vs
0.5, or 0.1 vs 0.5 is much more of a big deal.

(Then again, Droop proportionality spanning such a large space might
suggest otherwise. It's hard to tell.)

I'll probably backport some of the reimplementation to clean up my code,
and drop the "candidates also vote" aspect of the model (since in real
elections, the number of voters is so much larger than the number of
candidates that the latter effectively vanishes), and then post some new
plots. Eventually. I'm not going to risk burning myself out.

-km

[1] To test the "improper generalization" hypothesis, I tried to cluster
opinion space into mutually exclusive regions and then report the
fractions of voters/elected candidates falling into each region, instead
of the proportion holding a true opinion on each issue dimension. The
chi-squared test that the Sainte-Laguë index looks like requires mutual
exclusion. But that didn't change the outcome much, and it didn't push
the optimal argument closer to 0.5.

Posting again because EM was having trouble yesterday:

On 2024-09-16 19:08, Toby Pereira wrote:

I am a bit wary of doing so because I prefer using hidden information to
determine the methods' quality. Similar to how Borda's high VSE isn't
obvious (particularly given its center bias), it may not be obvious what
method is best at being proportional in issue space.

Linking the optimum to a particular method carries the risk of blinding
us to whether some other method is better at it. We wouldn't want to,
for instance, define methods' utilitarian quality as their outcomes'
Kendall tau distance from Borda.

If the strange apparent small-opinion bias persists with different
models, I may have to reconsider. But I think I'll try other models
before I link the optimum to a concrete method.

-km