[EM] Impossibility on Wikipedia: Arrow, Gibbard, and Satterthwaite

[Richard Lung]

Whether you are right or no, there is no conensus on the matter. Theorem 
Arrow is like a Cold War between electoral systems. It acknowledges 
ordinal votes as a basis for elections, in a denigratory sort of way, 
but over-looks the count beyond crude plurality. Ever since, 
mathematicians have demonstrated it takes more than mathematics to have 
a good understanding of elections. Formerly, it was not so, when they 
acted freely as enthusiasts, but the institutionalisation of election 
studies appears to have robbed them of any independent critical sense.

That understanding, I gather from his parliamentary speeches on "Mr 
Hare's system," is what John Stuart Mill had, the greatest philosopher 
of science in the 19th century. The Hare-Mill tradition, that has 
continued to the present day, has been by-passed. In so doing, social 
choice theory reveals its provincialism. A Nobel prize or so, to give 
away, is not a proof. A theorem is only as good as the assumptions on 
which it is based. And theorem Arrow compares to a critique of a bicycle 
on the basis of the short-comings of a unicycle. It does not deal with 
the democratic necessity of a proportional count as well as an ordinal 
vote. Simple plurality is "maiorocracy" or the tyranny of the majority, 
as Mill and Lani Guinier said.

It is not apparent what decisive argument the social choice school have 
that they can take to the voters, for whom elections are supposed to be 
meant, and has not been so for 70 years. It is not even apparent that, 
after 70 years, they have any idea of, or even belief in, a standard 
model of democratic election.

Richard Lung.


On 15/02/2024 07:13, Rob Lanphier wrote:
> Hi folks,
>
> I'm going to send a similar email here to the EM list that I recently
> sent to several folks who hang out in academic circles.  The answer I
> received from the academic circles was valuable, but I also think that
> folks on this mailing list can provide a different (and useful)
> perspective.
>
> I've long taken it for granted that impossibility theorems like
> Arrow's theorem and Gibbard's theorem mathematically prove that there
> are always going to be important electoral criteria that will be
> mutually exclusive in ANY credible electoral system.  I've been at
> peace with that for a long time, much in the same way that I'm at
> peace with mutually exclusive criteria for my transportation needs
> (e.g. I should take something with more carrying capacity than a
> bicycle to go shopping for large furniture, no matter how good the
> bike is).  The physics of electoral systems and the physics of the
> real world have certain mathematical rules that are tough to get
> around.
>
> Since the Center for Election Science (<https://electionscience.org>)
> started getting momentum and having some electoral success in the late
> 2010s, there's been a push to distinguish between "cardinal voting"
> and "ordinal voting" as the top of the hierarchy distinguishing all
> voting systems.  Since the ballot is what people see, that's
> understandable, I suppose.  However, in my mind, the ballots don't
> matter as much as the tallying method, and moreover, it's possible to
> use cardinal voting ballots and then tally them using systems that
> some folks classify as "ordinal" systems.
>
> In discussions with electoral reform folks over the past few years,
> I've been learning about Arrow, Gibbard, and Satterthwaite, and trying
> to document what I've learned on Wikipedia and electowiki.
>
> In editing Wikipedia articles related to election methods in the past
> few years, it seems there are three theorems that have made the rounds
> with regards to impossibility theorems:
>
> 1. Arrow's impossibility theorem (published in 1951): basically the
> granddaddy of impossibility theorems, which seemingly only applies to
> ordinal voting methods.
> <https://en.wikipedia.org/wiki/Arrow%27s_impossibility_theorem>
> 2. Gibbard's theorem (published in 1973): generalizes Arrow's theorem
> to apply to pretty much every social choice function
> <https://en.wikipedia.org/wiki/Gibbard%27s_theorem>
> 3. The Gibbard–Satterthwaite theorem (published in 1978): a more
> specific version of Gibbard's theorem which apparently only applies to
> ordinal systems, and focuses on strategic voting
> <https://en.wikipedia.org/wiki/Gibbard%E2%80%93Satterthwaite_theorem>
>
> What bothers me of late is a recent change that's been made to the
> Gibbard-Satterthwaite article.  I'll quote the most bothersome
> addition/replacement that's in the "Gibbard–Satterthwaite theorem"
> article as of this writing:
>> The theorem does not apply to cardinal voting systems such as score
>> voting or STAR voting, which can often guarantee honest (or semi-honest)
>> rankings in cases covered by the Gibbard-Satterthwaite theorem,[4] nor
>> does it apply to decision mechanisms other than ranked-choice voting.
>> Gibbard's theorem provides a weaker result that applies to such
>> mechanisms.
>>
>> The Gibbard-Satterthwaite theorem is often misunderstood as claiming
>> that "every voting system encourages dishonesty" or the related adage
>> that "there is no best voting system." However, such interpretations are
>> not correct; by the revelation principle, there exist many (deterministic,
>> non-trivial) voting systems that allow for honest disclosure (outside the
>> class of ranked-choice voting systems).
> It seems disingenuous to say that all of these voting systems don't
> apply to cardinal systems if there is some way to vote "honestly"
> (whatever that means).  Strategy and honesty are not mutually
> exclusive, and cardinal systems like "score voting" require voters to
> be very strategic as part of their voting calculus.  As noted above,
> Condorcet tallying methods can be used to tally "cardinal ballots" and
> "ordinal ballots", since both express the preferences.
>
> I'll quote what one of the folks in the academic circles stated:
>> Since it seems implausible to suppose that one person’s cardinal
>> evaluations have meaning in comparison to another person’s evaluations,
>> it is implausible to suppose that there is such a thing as an honest cardinal
>> evaluation of candidates. If there is such a thing as an honest cardinal
>> evaluation of candidates, then opportunities to benefit from dishonest
>> evaluations of candidates are rife in systems based on cardinal
>> evaluations, while they are likely to be quite rare under Condorcet-
>> consistent ranking-based voting systems.
> This assertion more-or-less comports with my opinion.  While I don't
> think that systems that insist on ranking-based ballots (ordinal
> ballots) are ALWAYS superior to systems that rely on simple addition
> of rating-based ballots (cardinal ballots), I think the implicit
> rankings are at least as important as the explicit ratings.  I
> generally think of STAR voting as "Condorcet lite", because, for two
> finalists "candA" and "candB", the final runoff doesn't pay attention
> to whether:
> scenario 1 ) "candA" has 5 stars and "candB" has 4 stars on ballot #1234
> or
> scenario 2) "candA" has 1 star, and "candB" has 0 stars on ballot #1234
>
> In the end, in both scenarios, ballot #1234 counts in full for
> "candA", which seems fair to me.  Regardless, I've frequently found
> myself distrusting hardcore cardinal advocates when I see changes like
> the one made to English Wikipedia's "Gibbard–Satterthwaite theorem"
> article.
>
> Are cardinal voting advocates correct to continually claim that
> Arrow's, Gibbard's, and Satterthwaite's theorems don't apply to their
> favorite voting methods?  Is there a useful distinction to be drawn
> between Gibbard's 1973 theorem and the "Gibbard-Satterthwaite theorem"
> published in 1978?  Is the distinction I draw above correct?
>
> Rob
> ----
> Election-Methods mailing list - see https://electorama.com/em for list info