[EM] Arrow/Gibbard and impossibility (Re: Scientific American and the "Perfect Electoral System")
Richard Lung
voting at ukscientists.com
Sun Nov 12 01:51:35 PST 2023
IIA may be compared to the primitive after-tthought of an exclusion
count to be found in traditional and conventional STV. This is a sort of
"Last past the post" elimination of candidates. The IIA test might ask
what of the exclusion of a runner-up? But the principle seems to be the
same, namely that the order of election might be changed by excluding
any candidate.
But the criterion can be met by abandoning the irrational exclusion
count and replacing it with a rational exclusion count, as well as the
rational election count. The Meek method of surplus transfers is equally
valid for a preference count and the reverse preference count, because
one voters preference is another voters reverse preference. (It just
means you have to count the abstentions as well, to calculate the
relative importance of election or exclusion of candidates to the voter.)
The Meek surplus transfer count elects candidates in the desired order,
and as (a reverse preference) exclusion count would also exclude
candidates in the desired order without opportunity for strategy,
theoretically possible with irrational exclusions.
Regards,
Richard Lung.
On 11/11/2023 23:57, Forest Simmons wrote:
> At the time it seemed revolutionary. ... but with hindsight it is
> clear that IIA is the sole culprit ... it's the one we have to let go of.
>
> Even Majority Judgment which comes as close as possible to IIA ...
> will predictably fail when voters are given the opportunity to change
> their judgment after their ballots have been exhausted at the top or
> bottom from candidate withdrawals.
>
> It's too bad that after all of this time nobody but Toby and Markus
> Schulze (who first pointed it out to me more than twenty years ago)
> seems to have noticed this almost embarrassing take-away from Arrow's
> most famous theorem.
>
> Too bad ... because the misconception thus perpetuated is still beimg
> used with impunity to excuse all kinds of garbage.
>
> fws
>
> On Sat, Nov 11, 2023, 3:22 AM Toby Pereira <tdp201b at yahoo.co.uk> wrote:
>
> 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 Lanphier
> Founder of election-methods mailing list and
> electowiki.org <http://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
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