Hello from Denis Falvey in Nova Scotia

Hello!
I have been analyzing data and jurisprudence for Canadian electoral reform for over eight years now. My applicable backgrounds are in mathematics, data analysis, and computer programming, since 1970.
This approach has led to an essay on Constitutional Electoral Reform, which I would like members of the forum to critique for logical errors.
I don't particularly care if anyone likes SMDPR and I am not interested in delving into the minutia of anyone's favourite alternative electoral system.
What matters to me is the Canadian Constitution as it applies to elections, the logic of my objective measures of democracy and local representation, and factbased common sense. But if there are logical or factual errors in my work I would like to know.
This appears to be the appropriate place to post such an essay, so I will seek to post it in draft form.

@denisfalvey Welcome.
the logic of my objective measures of democracy and local representation,
Do you have documentation on this in something succinct. You may have trouble getting people interested in reading a longer form essay specific to a Country. If you have short summaries of these objective measure please provide a link.
If you do not have such documentation, electowiki.org/ would be a great place to put it. There already exists a page for the concept of Proportionate Representation which is important in the Canadian Constitution.
https://electowiki.org/wiki/Proportionate_representation
Is this what your measure of local representation is quantifying? If not there are several others
https://electowiki.org/wiki/Category:Types_of_representation 
Hi,
I am happy to proofread a work for mathematical error & style. My background is in math though, not writing.
Feel free to either post it here or you can send it privately.

@keithedmonds Contrary to what might seem obvious, I have not stopped interest in getting what I have done 'out there'.
I have in fact prepared four articles, three of which are published in preprint form, and have published a book.
How do I get this across to an audience that can understand? The measures are idiotically simple, but were over the heads of some seven editors I approached. Innumeracy seems endemic!
If you are interested to help I will send the DOI for these, or pdfs.
Wiki may be a solution, but most of these are already published; does that matter?

Thanks for your offer Andy.
I became engrossed in learning to write and in presenting my arguments in digestible chunks. The result has been publishing three articles and a book.
Now I am trying to get them know about.
I approached seven editors, who helped with language but were frankly innumerate.
I am fine with writing the math, but it seems to me it may have been a mistake to remove Euclid from the school curriculum!

@andydienes PS: Here is the DOI for the measures article
Measuring Democratic Voice, Falvey, D. Doi: 10.20944/preprints202211.0180.v1What is your opinion? Hard to understand?

@denisfalvey I read the above preprint. Your Legislative Empowerment Measure (LEM) seemed clear and logical to me, but I didn't scrutinize every detail. I had almost no prior familiarity with Gini coefficients or Lorenz curves before, but good explanations were easy to look up. I could see how the same concept could be used for malapportionment; but with no axes labels on your Lorenz curve graphs I still couldn't follow what you had done.
The description of SMDPR (single member district proportional representation) in the above preprint lacked details necessary to assess if the numbers you extrapolated were even plausible. I did, however, find a detailed description at APSA Preprints. The negligible surplus vote looks suspicious, but given the use of Hare quota and proportion of vote that year to very small parties, it does seem plausible.
The description of ALRM (accountable local representation measure) has a typographical omission. It is clear that ALRM was calculated as LEM per representative per district not LEM per representative. The bigger issue is that, as a function of districts per representative, ALRM is only useful for comparing election systems for the same size country with the same number of representatives. A universal measure could be a function of districts per citizen, but that would make it an accountable personal representation measure, a very small number. Or it could simply be a function of districts, or districts per thousandths of the country's citizens, to bring it back to a similarly scaled number.
I also question the utility of a combined measure when the subject matter could be conveyed with greater clarity using independent threshold and limit measures, that would answer the following questions.
If all qualified citizens cast valid ballots, what would be the minimum number of votes needed by an individual candidate to clinch a seat? What percentage of votes in the riding would this be? What percentage of votes in the allocation division would this be? What percentage of votes in the country would this be? If the choice of riding would yield different results express the answer as a range.
What would be the maximum number of votes that could be combined toward selecting an individual candidate?(same conditions and supplemental questions for this and the following)
What would be the minimum number of votes needed to clinch a seat for a party if their supporters are optimally concentrated?
What would be the minimum number of votes needed to clinch a seat for a party if their supporters are evenly distributed throughout the country?
What would be the maximum number of votes that can be combined toward securing seats for a party?

Oh good. The description of the logic of LEM and ALRM was not too opaque.
Any detail of how things were calculated is in the data at
D. Falvey, "SMDPR 2022," 28 Oct 2022. [Online]. Available: https://data.mendeley.com/datasets/tn373nrphc/3 .
This really needs to be consulted to understand the details.
ALRM is LEM per average matching division representative, i.e., the number of seats in the nation divided by the number of matching divisions. It is inherently a local metric.
Technically, I suppose, the Lorenz curve is the equality line; Gini was the economist/mathematician that plotted the curved lines first. Or, given what I know about 'discovery', he was probably recognized as having put into words the ideas that are associated with his name, but came by them from existing ideas.
The axis values of a Gini/Lorenz graph are both always percents of the latent variable. For instance, the latent variable in a malapportionment Gini chart is the number of electors in each riding. What each point of the curve tells you is what percent of ridings has a value of the latent variable that is that percent of the latent variable value. This is the same as with wealth disparity, where people will say, " 90% of the worlds wealth is concentrated in the hands of 10% of the worlds people", or something similar.
The GINI coefficient is a single number that characterizes the disparity of the latent value being measured. It is in the same descriptive category as standard deviation and significance measures, but has the utility of providing a single coefficient as descriptor, which is much more useful in constructing LEM and ALRM.
This paper focused on measures, of which I have provided three: ballot split by type, LEM, and ALRM. These are only possible with the existence of a clear definition of democracy.
My definition is based on voice. This is in accord with both a thousand years of parliamentary evolution toward democracy and the enjoinders of the Supreme Court of Canada cases
https://scccsc.lexum.com/scccsc/scccsc/en/item/766/index.do
and
https://www.canlii.org/en/bc/bcsc/doc/1989/1989canlii248/1989canlii248.html
I recommend you read at lest these.Choosing voice allows all systems to be compared, regardless of whether they are considered democracies or not. LEM is a continuous scale from democracy to dictatorship.
The difference between a measure and an index is very important in what I have done.
Choosing a measure does not have any effect on the information accessible concerning the latent variable; choosing an index does. For instance, what you learn about mass from the various experiments designed to confirm the inverse square law or gravity is invariant of whether you use Imperial or SI units. What you learn about democracy changes with each index used to measure. This is because the pertinent academic literature has left democracy undefined.
Adhering to a broad, generous, but purposive definition of democracy allows direct measurement of what differentiates all governance systems, viz, voice and who has it.
Your other observations show a keen grasp, but this article was about the measures and how they can be used. So, I needed countries to compare and another system for Canada to compare to SMP. But SMDPR was not the focus.
I have published several things, in addition to the underlying data referenced above
• For a Euclidean development of voice expression theory, with examples, see The Theory of Democratic Voice Expression, Falvey, D. Doi: 10.33774/apsa2023rm17k
• For a mathematical treatment of political measurement theory and practice, see Measuring Democratic Voice, Falvey, D. Doi: 10.20944/preprints202211.0180.v1 (you've read this)
• For a more indepth look at SMDPR, see Single Member District Proportional Representation, Falvey, D. Doi: 10.33774/apsa2023pglpl
• For a grounding of the above material in historical and constitutional context, and for complementing SMDPR with Demeny voting and Constituency Assemblies, as well as a more indepth presentation of the idea of an ERC, see, Democratic Voice: A Brief, Falvey, D., Amazon, 2023I don't expect to live many more months, so I wanted to put my work over the last 10 years in the hands of the general public.
I really don't see how electoral reform can reasonably proceed without a standing Electoral Reform Commission (ERC). There will never be agreement on what system to use, especially with political parties chivying and lying to protect SMP. Plus, there will never be a perfect system. A Commission acting in accord with constitutional principles, like the Electoral Boundaries Commission, seems the obvious solution.
I can demonstrate that the SMDPRtype systems provide the three important aspects of any electoral system: voice, accountable local representation, and stable governance. These three must be provided in unison, and comparison of systems must address all three for fairness.
For instance, MMPR systems provide voice almost as high as partylist systems, using LEM as measure. But MMPR, and all other systems, are comparatively bad at providing accountable local representation, as shown with ALRM.
Stable governance begins with an unbiased threshold, which the allocation/matching division structure of SMDPR provides. You will see much more about this in the SMDPR article referenced above.
I am still working on a stability measure, which I think can be constructed from the Gini curves of partisan disparity. It is difficult to see mathematically why Canada's 338 matching divisions results in three or four federal parties. The reason it happens is that it takes a lot of organization to create a federal party; underlying this ability is the matter of sufficient population. But how to turn that into a formula?
Canada gets four federal parties with 338 ridings and no threshold. Israel and Sweden have the same partylist mechanics, but the number of parties in Sweden is far lower than in Israel, apparently because of matching divisions, and even though both of these countries have bad accountable local representation. Sweden's ALRM is five time better than Israel's, but how do I turn that info into a measure?
Anyway, I will be spending less time on electoral reform during this part of my life. Ten years of intensive data analysis, case analysis, and extensive reading will have to be enough.
The only other mathematical device I can add to the arsenal is how to combine measures and indices into a single unbiased number. This is high school vector math.
Make a vector of n measures/indices, each normalized to [0,1]; it will live in ndimensional space. Use its norm or length to identify an equivalence class of vectors that are all the same length. Now choose the vector in the class that has equal component values; use that component value as a canonical representation of the class, and therefore of the original set of index/measure values.
There is no guarantee that the components of the vector are independent, which is necessary for the above calculations to be valid; otherwise, the choice of indices/measures could affect the canonical component value.
To address this, look at the historic or other values of each index/measure. Create a vector of these values for each index. Then used the inner product function to determine the angle between these historic vectors in pairs and use these angles to orthonormalize the components of the original vector of indices/measures. This gets rid of bias.
Anyway, I like to think I will engage fully in a discussion of my ideas, but that may not be possible much longer. Hopefully someone will understand what I was trying to get at and run with it.
SMDPR systems are the best I've found. I would have been happy to present the concept to an Electoral Reform Commission. But presenting it to people who don't really know much about electoral reform, the case law, and mathematics at the same time requires more than I have left.
Cheers

@denisfalvey Welcome! Thanks for sharing and discussing your work here!