As is often the case, I think the importance of Arrow's Theorem is overstated in that article. Arrow's Theorem essentially says "With a few reasonable background assumptions, no ranked-ballot method passes Independence of Irrelevant Alternatives." But this was already known for centuries from the Condorcet Paradox. I don't really know why it's gained so much traction over the years, as it was nothing like the paradigm shift people credit it as.
Toby
On Tuesday, 7 November 2023 at 04:29:31 GMT, Forest Simmons forest.simmons21@gmail.com wrote:

Rob,
Thanks for clearing up a lot  of the confusion... and for putting the current status in perspective.
I like the comparison of the "impossibilities of voting" with the impossibilities of faster than light travel, etc.  The 2nd law of thermodynamics is especially relevant... because as Prigogene showed in the 70's, the impossibility of decreasing entropy in closed systems still allows for local pockets of possibility ... that make life possible .... until the "heat death" of our island space-time big bang remnant ... while miriads of new "inflationary bubbles" appear from random virtual quantum fluctuations.
We used to "know" that the event horizon was a boundary of no return .... nut now evaporation of black holes through quantum tunneling is taken for granted.
In the early 1800's Gauss proved the impossibility of trisecting an arbitrarily given angle .... inside the rules of classical geometric ruler and compass constructions.
But it turns out that (as any first year topology student can show) any angle can be transformed into atrisectable one by an arbitrarily small perturbation.
I'm fact, once you learn the binary point expansion of 1/3 ..., you can get within a relative error tolerance of 1/2^n precision with n bisections... bisections being the first constructions you learn in geometty.
Pockets of possibility like these .... adequate "For All Practical Purposes" pervade mathematics ... including the mathematics of voting systems.
Sometimes you have to discover new tools not included in the classical tool kit. In  the case of angle trisections, if you are allowed to make a few marks on the ruler... hen the general ruler and compass trisection suddenly resolves itself.
Thanks,
Forest
On Sun, Nov 5, 2023, 11:34 PM Rob Lanphier roblan@gmail.com wrote:

Hi folks,
I just wrote a letter to the editor(s) of Scientific American, which I've included below.  My letter was in a response to the following article that was recently published on their website:
https://www.scientificamerican.com/article/see-how-math-could-design-the-perfect-electoral-system/
Y'all may have other thoughts on the article.

Rob
---------- Forwarded message ---------
From: Rob Lanphier roblan@gmail.com
Date: Sun, Nov 5, 2023 at 11:22 PM
Subject: Regarding using math to create a "Perfect Electoral System"
To: Scientific American Editors editors@sciam.com

To whom it may concern:
I appreciate your article "Could Math Design the Perfect Electoral System?", since I agree that math is important for understanding electoral reform, and there's a lot of good information and great diagrams in your article:https://www.scientificamerican.com/article/see-how-math-could-design-the-perfect-electoral-system/
There's some things that the article gets wrong, but the good news is that the article title and its relation to Betteridge's law.  This law states "Any headline that ends in a question mark can be answered by the word 'no'."  The bad news: the URL slug ("see-how-math-could-design-the-perfect-electoral-system") implies the answer is "yes".  The answer is "no"; Kenneth Arrow and Allan Gibbard proved there is no perfect electoral system (using math).
I appreciate that your article highlights the mayoral election in Burlington, Vermont in 2009.  That is an important election for all voters considering FairVote's favorite single-winner system ("instant-runoff voting" or rather "ranked-choice voting, as they now call it).  When I volunteered with FairVote in the late 1990s, I remember when they introduced the term "instant-runoff voting".  I thought the name was fine.  After Burlington 2009, it would seem that FairVote has abandoned the name.  Regardless, anyone considering instant-runoff needs to consider Burlington's experience.

Sadly, your article describes "cardinal methods" in a confusing manner.  It erroneously equates cardinal's counterpart ("ordinal voting") with "ranked-choice voting".  Intuitively, all "ordinal methods" should be called "ranked choice voting", but during this century, the term has been popularized by FairVote and the city of San Francisco to refer to a specific method formerly referred to as "instant-runoff voting".  These days, when Americans speak of "RCV", they're generally referring to the system known on English Wikipedia as "IRV" (or "Instant-runoff voting"):https://en.wikipedia.org/wiki/Instant-runoff_voting

There have been many methods that use ranked ballots, including the methods developed by Nicolas de Condorcet and Jean-Charles de Borda in the 1780s and the 1790s. I'm grateful that the Marquis de Condorcet's work is featured so prominently in your article.  Condorcet's work was brilliant, and I'm sure he would have become more prominent if he hadn't died in a French prison in the 1790s.  Many single-winner methods that strictly comply with the "Condorcet winner criterion" are probably as close to "perfect" as any system (from a mathematical perspective).

Most methods that pass the "Condorcet winner criterion" typically use ranked ballots (and thus are "ordinal"), but it's important to note that almost all "ordinal" methods can use cardinal ballots.  Instant-runoff voting doesn't work very well with cardinal ballots (because tied scores cannot be allowed), but most other ordinal systems work perfectly well with tied ratings or rankings.  Even though passing the Condorcet winner criterion is very important, there are many methods that come very, very close in reasonable simulations.  I would strongly recommend that you contact Dr. Ka-Ping Yee, who is famous in electoral reform circles for "Yee diagrams":https://electowiki.org/wiki/Yee_diagram(a direct link to Yee's 2005 paper: http://zesty.ca/voting/sim/ )
Note that "approval voting" and "Condorcet" provide pretty much the same results in Yee's 2005 paper.  "Instant-runoff voting" seems a little crazy in Yee's simulations.

Though Arrow and Gibbard disproved "perfection", I prefer to think of Arrow's and Gibbard's work as defining the physics of election methods.  To explain what I mean, consider the physics of personal transportation.  It is impossible to design the PERFECT vehicle (that is spacious, and comfortable, travels faster than the speed of light, fits in anyone's garage or personal handbag).  Newton and Einstein more-or-less proved it.  However, those esteemed scientists' work didn't cause us to stop working on improvements in personal transportation.  Buggy whips are now (more or less) recognized as obsolete, as is Ford's "Model T".

Now that Arrow and Gibbard have helped us understand the physics of election methods, we can hopefully start pursuing alternatives to the buggy whip (or rather, alternatives to "choose-one" voting systems, often referred to as "first past the post" systems). 

This gets me to the statement from your article that gets under my skin the most::
This is called cardinal voting, or range voting, and although it’s no panacea and has its own shortcomings, it circumvents the limitations imposed by Arrow’s impossibility theorem, which only applies to ranked choice voting.

 People who study election methods refer to "cardinal voting" as a category of voting methods, of which "range voting" is just one (which is called "score voting" on English Wikipedia):https://en.wikipedia.org/wiki/Score_voting

The conflation of "ranked choice voting" with all ordinal voting methods is also highly problematic (though I don't entirely blame you for this).  As I stated earlier, there are many methods that can use ranked ballots.  While this article may have been helpful for those of us that prefer ranking methods that are not "instant-runoff voting" back when FairVote switched to "ranked-choice voting" in the early 2010s.  Note that before the fiasco in Burlington in 2009, FairVote pretty consistently preferred "instant runoff voting":https://web.archive.org/web/20091111061523/http://www.fairvote.org/
I appreciate that you're trying to explain this insanely complicated topic to your readers.  When I edit English Wikipedia (which I've done for over twenty years), I would love to be able to cite Scientific American on this topic.  However, I'm not yet sure I'd feel good about citing this article.

Rob LanphierFounder of election-methods mailing list and electowiki.org
https://robla.nethttps://electowiki.org/wiki/User:RobLahttps://en.wikipedia.org/wiki/User:RobLa

p.s. back in the late 1990s, I wrote an article for a small tech journal called "The Perl Journal".  It's out of print, but I've reproduced my 1996 article about election methods which I think holds up pretty well:https://robla.net/1996/TPJ

Election-Methods mailing list - see https://electorama.com/em for list info

Election-Methods mailing list - see https://electorama.com/em for list info

On non-deterministic methods, they can give potentially a better level of proportional representation than deterministic methods, while still keeping some degree of local representation. If you have constituencies with 5 or 6 representatives and use e.g. STV, then parties/ideologies with 10% of the support nationally would likely keep missing out and win far less than 10% of the seats. Non-deterministic methods can mean that on average things balance out.
If you use the simple random ballot method, things would still be pretty bad. With just one representative per constituency, quite a lot of people are likely to be represented only by a lunatic fringe candidate. But with 5 or 6 winning candidates, popular candidates will still generally win through, and there will be a balance of representation in each constituency.
Proportional methods can also get quite complex, but this can be massively reduced with a non-deterministic method. For example, COWPEA Lottery uses approval voting. To elect candidates you simply do this:
Start with a list of all currently-unelected candidates. Pick a ballot at random and remove from the list all candidates not approved on this ballot. Pick another ballot at random, and continue with this process until one candidate is left. Elect this candidate. If the number of candidates ever goes from >1 to 0 in one go, ignore that ballot and continue. If any tie cannot be broken, then elect the tied candidates with equal probability.
It also has very good criterion compliance. If you accept non-determinism and ballots that aren't just ordinal, then probably the best of any known candidate-based proportional method. https://electowiki.org/wiki/COWPEA
Toby
On Wednesday, 8 November 2023 at 18:18:24 GMT, Richard Lung voting@ukscientists.com wrote:

Why has theorem Arrow gained so much traction over the years?

The short answer, history is written by the victors.

The history answer takes some explaining but is evident enough.

New York, for instance, had a Personal Representation society, by the late nineteenth century. In fact the organised campaign writings, of its early successes, over a century old, have been bought-up, and are still subject to publishers pay walls, of little if any commercial value and contrary to the public interest.

By the 1937 edition of Proportional Representation. The key to democracy. Clarence Hoag and George Hallett were introducing a single transferable vote to the boroughs of New York.

These two campaigning great men, both mathematicians in their day job, did not get a mention in Wikipedia, when I last looked.

As a result of battering-ram referendums with the money and publicity on their side, The Machine virtually abolished the key to democracy.

Massachusetts legislature forbad other cities than Cambridge to use their electoral reform. There are several home-rule bills but they are bogged down in state committee.

Kenneth Arrow stepped in, in the nineteen fifties, to effectively finish the job of The Machine. He took the Kurt Godel Incompleteness theorem an extreme step further by advertising an “Impossibility theorem,” which he himself, cited in Scientific American, belatedly admitted it did not amount to.

In other words, he by-passed, nearly a century of election method study, stemming from Thomas Hare and John Stuart Mill. That tradition continues and so does the naïve ignorance of it.

As a Scottish STV programmer commented, theorem Arrow does not even apply to proportional representation, but merely to bare majority elections.

Any theorem is only as good as its assumptions, and those of the Impossibility theorem neglect the possibility that statistical methods might be more accurate than deterministic ones, as is indeed the case in physics.

Regards,

Richatrd Lung.

On 07/11/2023 13:35, Toby Pereira wrote:

As is often the case, I think the importance of Arrow's Theorem is overstated in that article. Arrow's Theorem essentially says "With a few reasonable background assumptions, no ranked-ballot method passes Independence of Irrelevant Alternatives." But this was already known for centuries from the Condorcet Paradox. I don't really know why it's gained so much traction over the years, as it was nothing like the paradigm shift people credit it as.
Toby
On Tuesday, 7 November 2023 at 04:29:31 GMT, Forest Simmons forest.simmons21@gmail.com wrote:

 Rob, 

Thanks for clearing up a lot  of the confusion... and for putting the current status in perspective.
I like the comparison of the "impossibilities of voting" with the impossibilities of faster than light travel, etc.  The 2nd law of thermodynamics is especially relevant... because as Prigogene showed in the 70's, the impossibility of decreasing entropy in closed systems still allows for local pockets of possibility ... that make life possible .... until the "heat death" of our island space-time big bang remnant ... while miriads of new "inflationary bubbles" appear from random virtual quantum fluctuations.
We used to "know" that the event horizon was a boundary of no return .... nut now evaporation of black holes through quantum tunneling is taken for granted.
In the early 1800's Gauss proved the impossibility of trisecting an arbitrarily given angle .... inside the rules of classical geometric ruler and compass constructions.
But it turns out that (as any first year topology student can show) any angle can be transformed into atrisectable one by an arbitrarily small perturbation.
I'm fact, once you learn the binary point expansion of 1/3 ..., you can get within a relative error tolerance of 1/2^n precision with n bisections... bisections being the first constructions you learn in geometty.
Pockets of possibility like these .... adequate "For All Practical Purposes" pervade mathematics ... including the mathematics of voting systems.
Sometimes you have to discover new tools not included in the classical tool kit. In  the case of angle trisections, if you are allowed to make a few marks on the ruler... hen the general ruler and compass trisection suddenly resolves itself.
Thanks,
Forest
On Sun, Nov 5, 2023, 11:34 PM Rob Lanphier roblan@gmail.com wrote:

Hi folks,
I just wrote a letter to the editor(s) of Scientific American, which I've included below.  My letter was in a response to the following article that was recently published on their website:
https://www.scientificamerican.com/article/see-how-math-could-design-the-perfect-electoral-system/
Y'all may have other thoughts on the article.

Rob
---------- Forwarded message ---------
From: Rob Lanphier roblan@gmail.com
Date: Sun, Nov 5, 2023 at 11:22 PM
Subject: Regarding using math to create a "Perfect Electoral System"
To: Scientific American Editors editors@sciam.com

To whom it may concern:
I appreciate your article "Could Math Design the Perfect Electoral System?", since I agree that math is important for understanding electoral reform, and there's a lot of good information and great diagrams in your article: https://www.scientificamerican.com/article/see-how-math-could-design-the-perfect-electoral-system/
There's some things that the article gets wrong, but the good news is that the article title and its relation to Betteridge's law.  This law states "Any headline that ends in a question mark can be answered by the word 'no'."  The bad news: the URL slug ("see-how-math-could-design-the-perfect-electoral-system") implies the answer is "yes".  The answer is "no"; Kenneth Arrow and Allan Gibbard proved there is no perfect electoral system (using math).
I appreciate that your article highlights the mayoral election in Burlington, Vermont in 2009.  That is an important election for all voters considering FairVote's favorite single-winner system ("instant-runoff voting" or rather "ranked-choice voting, as they now call it).  When I volunteered with FairVote in the late 1990s, I remember when they introduced the term "instant-runoff voting".  I thought the name was fine.  After Burlington 2009, it would seem that FairVote has abandoned the name.  Regardless, anyone considering instant-runoff needs to consider Burlington's experience.

Sadly, your article describes "cardinal methods" in a confusing manner.  It erroneously equates cardinal's counterpart ("ordinal voting") with "ranked-choice voting".  Intuitively, all "ordinal methods" should be called "ranked choice voting", but during this century, the term has been popularized by FairVote and the city of San Francisco to refer to a specific method formerly referred to as "instant-runoff voting".  These days, when Americans speak of "RCV", they're generally referring to the system known on English Wikipedia as "IRV" (or "Instant-runoff voting"): https://en.wikipedia.org/wiki/Instant-runoff_voting

There have been many methods that use ranked ballots, including the methods developed by Nicolas de Condorcet and Jean-Charles de Borda in the 1780s and the 1790s. I'm grateful that the Marquis de Condorcet's work is featured so prominently in your article.  Condorcet's work was brilliant, and I'm sure he would have become more prominent if he hadn't died in a French prison in the 1790s.  Many single-winner methods that strictly comply with the "Condorcet winner criterion" are probably as close to "perfect" as any system (from a mathematical perspective).

Most methods that pass the "Condorcet winner criterion" typically use ranked ballots (and thus are "ordinal"), but it's important to note that almost all "ordinal" methods can use cardinal ballots.  Instant-runoff voting doesn't work very well with cardinal ballots (because tied scores cannot be allowed), but most other ordinal systems work perfectly well with tied ratings or rankings.  Even though passing the Condorcet winner criterion is very important, there are many methods that come very, very close in reasonable simulations.  I would strongly recommend that you contact Dr. Ka-Ping Yee, who is famous in electoral reform circles for "Yee diagrams": https://electowiki.org/wiki/Yee_diagram (a direct link to Yee's 2005 paper: http://zesty.ca/voting/sim/ )
Note that "approval voting" and "Condorcet" provide pretty much the same results in Yee's 2005 paper.  "Instant-runoff voting" seems a little crazy in Yee's simulations.

Though Arrow and Gibbard disproved "perfection", I prefer to think of Arrow's and Gibbard's work as defining the physics of election methods.  To explain what I mean, consider the physics of personal transportation.  It is impossible to design the PERFECT vehicle (that is spacious, and comfortable, travels faster than the speed of light, fits in anyone's garage or personal handbag).  Newton and Einstein more-or-less proved it.  However, those esteemed scientists' work didn't cause us to stop working on improvements in personal transportation.  Buggy whips are now (more or less) recognized as obsolete, as is Ford's "Model T".

Now that Arrow and Gibbard have helped us understand the physics of election methods, we can hopefully start pursuing alternatives to the buggy whip (or rather, alternatives to "choose-one" voting systems, often referred to as "first past the post" systems). 

This gets me to the statement from your article that gets under my skin the most::
This is called cardinal voting, or range voting, and although it’s no panacea and has its own shortcomings, it circumvents the limitations imposed by Arrow’s impossibility theorem, which only applies to ranked choice voting.

  People who study election methods refer to "cardinal voting" as a category of voting methods, of which "range voting" is just one (which is called "score voting" on English Wikipedia): https://en.wikipedia.org/wiki/Score_voting

The conflation of "ranked choice voting" with all ordinal voting methods is also highly problematic (though I don't entirely blame you for this).  As I stated earlier, there are many methods that can use ranked ballots.  While this article may have been helpful for those of us that prefer ranking methods that are not "instant-runoff voting" back when FairVote switched to "ranked-choice voting" in the early 2010s.  Note that before the fiasco in Burlington in 2009, FairVote pretty consistently preferred "instant runoff voting": https://web.archive.org/web/20091111061523/http://www.fairvote.org/
I appreciate that you're trying to explain this insanely complicated topic to your readers.  When I edit English Wikipedia (which I've done for over twenty years), I would love to be able to cite Scientific American on this topic.  However, I'm not yet sure I'd feel good about citing this article.

Rob Lanphier Founder of election-methods mailing list and electowiki.org
https://robla.net https://electowiki.org/wiki/User:RobLa https://en.wikipedia.org/wiki/User:RobLa

p.s. back in the late 1990s, I wrote an article for a small tech journal called "The Perl Journal".  It's out of print, but I've reproduced my 1996 article about election methods which I think holds up pretty well: https://robla.net/1996/TPJ
----
Election-Methods mailing list - see https://electorama.com/em for list info

 ----

Election-Methods mailing list - see https://electorama.com/em for list info

Election-Methods mailing list - see https://electorama.com/em for list info