[EM] Arrow/Gibbard and impossibility (Re: Scientific American and the "Perfect Electoral System")

Sat Nov 11 03:22:23 PST 2023

 I suppose it's that I'm not convinced about how foundational Arrow's work really was. The theorem is basically that you can't have all these reasonable-sounding criteria together in one ranked-ballot method. But the one criterion that sticks out is Independence of Irrelevant Alternatives (IIA). It's not that some methods pass x and y but fail z, others pass y and z but fail x etc. They all fail IIA and pass the others. For a ranked-ballot method to pass IIA it would have to fail some really basic stuff. So my point is that I'm not sure what his theorem added to what we already knew from the existence of the Condorcet paradox.
In summary: If you have an A>B>C>A cycle, then each of the three candidates will win one election if you held separate head-to-head elections, so if you put them together for a three-way election, an IIA failure is inevitable.
All Arrow's theorem really does is make explicit the background assumptions required to make this IIA failure inevitable (such as non-dictatorship).
Toby
    On Friday, 10 November 2023 at 22:19:40 GMT, Rob Lanphier <roblan at gmail.com> wrote:  

 Hi Toby,
I guess I understand some of your frustration how articles like the SciAm article overemphasize Arrow.  I suspect the feeling is similar to what I feel when I see the emphasis on Condorcet cycles.  Too many articles encourage fatalism about comparative quality because ALL systems will have flaws.  Condorcet cycles seem like they would be rare enough in real-world conditions, and I suppose some aspects of Arrow might overemphasize similarly rare possibilities.

I don't think it's fair to brush off Arrow's work, though, because it was foundational to other impossibility theorems.  Arrow made a very helpful generalization of the Condorcet paradox, and Gibbard made a very helpful generalization of Arrow's work.  I've found impossibility theorems helpful in constraining and classifying the tradeoffs made about various systems.
Do you feel like SciAm went too far when they even mentioned Arrow's theorem, or is it that you don't think they weren't careful enough about contextualizing it?

Rob

On Tue, Nov 7, 2023 at 5:36 AM Toby Pereira <tdp201b at yahoo.co.uk> 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 at 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 at 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 at 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 at 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
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