[EM] Arrow's theorem and cardinal voting systems

Fri Jan 10 03:39:05 PST 2020

 Arrow published a mathematical theorem so presumably everything was rigorously defined and not open to interpretation, so that would include unrestricted domain. On the Wikipedia it says "In social choice theory, unrestricted domain, or universality, is a property of social welfare functions in which all preferences of all voters (but no other considerations) are allowed." And while it might be defined differently and more precisely in Arrow's paper, I wouldn't say score fails by that definition. But it doesn't really matter anyway. You can define it in a way that score fails or define it in a way that it doesn't apply to score. Similarly you could define a condition where degrees of liking must be allowed, and ranked methods would fail that. And as has been said, it's not as if Arrow's Theorem is the be all and end all. All methods have their own problems, and whether they happen to be covered by one particular theorem is neither here nor there.
But what I would say is that I consider Arrow's Theorem to be possibly the most overrated and overstated theorem of all time. If you look at the criteria that ranked methods must fail one of according to Arrow's Theorem, most of them are just criteria that any remotely reasonable method would pass. The theorem, stated more informally, is basically that with a few reasonable background assumptions, all ranked-ballot methods fail independence of irrelevant alternatives. Which is interesting enough itself, except that this was known for centuries anyway from the Condorcet Paradox. If head to head A beats B, B beats C, and C beats A, then any winner in the three-way election has to overturn one of the head to head results as a result of an irrelevant alternative being added.
Toby
    On Thursday, 9 January 2020, 23:17:56 GMT, Rob Lanphier <robla at robla.net> wrote:  

 Hi folks,

As some of you might have seen, Electowiki is a lot more active than
it used to be.  I'm 99% convinced that's a good thing.  The 1% of me
that has reservations is regarding how some advocates talk about
Arrow's theorem.  I'm hoping you all can do one of the following:
a)  change my view about Arrow's theorem, -or-
b)  offer me some help in better articulating my view about Arrow's theorem.

Many Score voting[1] activists claim that cardinal methods somehow
dodge Arrow's theorem.  It seems to me that *all* voting systems (not
a mere subset) are subject to some form of impossibility problem.
Arrow's impossibility theorem deserved great acclaim for subjecting
all mainstream voting systems of the 1950s to mathematical rigor, and
it's clear that his 1950 paper and 1951 book profoundly influenced
economics and game theory for the better.  His 1972 Nobel prize was
well deserved.  It seems that it has become fashionable to find
loopholes in Arrow's original formulation and declare the loopholes
important.  Even if the loopholes exist, talking up those loopholes
doesn't seem compelling, given the subsequent work by other theorists
broaden the scope beyond Arrow's version.

But, what the heck, let's actually talk about Arrow's original
formulation.  I believe Score voting fails unrestricted domain:
<https://en.wikipedia.org/wiki/Unrestricted_domain>

In particular, let's say that 90% of voters prefer candidate A over candidate B:
90:A>B
10:B>A

Arrow posits that there should only be one way to express that, and
Score fails it.  In Score, it's possible to sometimes pick A, and
sometimes pick B, depending on the score values on the ballots.  If
Score *always* chose either A or B, then it would pass Universality.

Score advocates claim that this isn't a bug, it's a *feature*.  If
(for example), voters for A only mildly prefer A over B, but voters
for B strongly detest A, then the correct social choice is B.
However, it doesn't seem practical to inflict this level of nuance on
voters.  I suspect that the first election where the Condorcet winner
is beaten by a minority-preferred candidate (e.g. like what happened
in Burlington 2009 [2]) will result in a repeal (like what happened in
Burlington).  Back to the A/B example above, It's hard to imagine
voters would consider the selection of "B" to be fair in a large
election.

It's fine to hold the opinion that Universality is an uninteresting
criterion, and that therefore, Arrow's set of criteria isn't very
interesting.  For example, a few years ago, we went through a phase
where Condorcet advocates promoted "Local IIAC" as a IIAC[3] as a more
interesting criterion, and advocating for Condorcet variants that meet
that criterion.  Regardless, just because we find one criterion less
compelling than another, we should talk accurately about the failed
criterion.

My way of thinking about Arrow's theorem (and being thankful for it)
is to think of it like the physics of voting systems.  For example, in
real-world physics, a "perfect" vehicle is impossible, because it's
impossible to meet these criteria:
* Goes faster than the speed of light
* Has infinite capacity
* Has a luxurious and comfortable passenger cabin
* Fits in a small coat pocket
* Is easy to produce
* Is cheap (or even free)

Just because a perfect vehicle is not possible, I'm glad
transportation innovation didn't stop with Ford's Model T.  Of course,
automobile sellers compete on the tradeoffs between the criteria
above, and much public policy debate is about mode-of-transport
tradeoffs between planes, trains and automobiles (and bicycles, and
scooters, and and and...).  We need public policy debates around
election method tradeoffs, too.

I'm hoping we can try to stop trying to declare clever loopholes in
Arrow's theorem, and just acknowledge the reality that *all* voting
systems involve tradeoffs.  I hope we all can acknowledge that Arrow's
central insight (there's no "perfect" system given perfectly
reasonable criteria) is valid, and that it's only on the specifics of
the exact criteria chosen for the 1951 proof that might be flawed.  I
believe that election method activists should speak (and write) with
clarity about the tradeoffs involved.  Whenever I see someone
gleefully declare that Arrow's theorem doesn't apply to their voting
method (and imply perfection), the credibility of the writer drops
*precipitously* in my mind.

Am I wrong?

Rob

p.s. I've been meaning to write this email for a while.  What inspired
me to finally write it has been reading the current state of
Electowiki and Wikipedia articles on the topic, like the "Arrow's
impossiblity theorem" article on Electowiki[4]

[1]: https://electowiki.org/wiki/Score_voting
[2]: https://en.wikipedia.org/wiki/2009_Burlington_mayoral_election
[3]: https://electowiki.org/wiki/IIAC
[4]: https://electowiki.org/wiki/Arrow%27s_impossibility_theorem
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