[EM] Arrow's theorem and cardinal voting systems

Kristofer Munsterhjelm km_elmet at t-online.de
Fri Jan 10 03:20:31 PST 2020

On 10/01/2020 00.17, Rob Lanphier 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.

First a little pet peeve of sorts: It's not a Nobel prize. The Nobel
foundation says as much at the very bottom of
https://www.nobelprize.org/nomination/economic-sciences/ :-)

> 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.

I agree. Arrow's theorem requires unrestricted domain to work, and what
unrestricted domain says (as far as I know it) is that the inputs are
lists of orderings, that every such list must be admissible, and that's
all that's required. I.e. "The rankings, all rankings, and nothing but
the rankings".

> 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.

Sure, you might say that unrestricted domain is uninteresting. If you
do, and you design a voting system that doesn't pass unrestricted
domain, then Arrow won't apply. You could well get IIA out of it (with
certain assumptions). That doesn't mean that Arrow's is wrong, it just
means that it no longer applies.

I think that it's a good idea to think: "is my objection about strategy
or not?" If it's about strategy-proofness, the right theorem isn't
Arrow's, it's Gibbard's. And Gibbard's theorem holds for Range as well,
so Range doesn't get around it. So just because Range is outside of the
scope of Arrow, that doesn't mean that it's invulnerable to strategy.

And, to go on a bit of a tangent, I think that ordinary (normalized)
Range has both a cardinal and ordinal component to it. The ordinal
component might violate IIA.

Consider, for instance, something like: election one has a pro-war
candidate and an anti-war candidate, and their positions are otherwise
pretty much the same. But election two has ten pro-war candidates
ranging from economic left to right and ten anti-war candidates ranging
from economic left to right. It may be the case that in election one,
people judge the candidates by whether their war stance agree, and in
election two, people judge the candidates more heavily by left vs right
than by war stance. The distribution of the candidates affects what the
voters consider important features, and thus irrelevant candidates could
alter who the winner is.

On the other hand, "The Possibility of Social Choice" (Sen) suggests
that only very weak utility-like comparisons are required to salvage
IIA. So who knows? Perhaps with a method that's not Range, you can have
it even with the scenario above.

> 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.

The feeling I get from it is roughly:

- Wouldn't it be nice if we had perfection even under honesty?
- Well, here's one reason we can't have perfection.
- So have fun with the complexity.

That is, Arrow's theorem is useful to say what you can't have. In your
vehicle analog, we would want to find out how to create that perfect
FTL-equipped free car if we could. Arrow's theorem just tells us that we
can stop looking. (And Gibbard's is like: even if your "vehicle" moves
the world instead of moving itself, thus making Arrow no longer apply,
you still can't get perfection under strategy.)

> 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.

It does seem that there are no perfect methods, even if we can
circumvent Arrow, and even if we don't care about strategy. For
instance, Approval requires calculation even by honest voters, and for
Range it's not even obvious what the one honest vote *is*.

It would be useful to get a proof of this, but I wouldn't know where to
start. So the observation that it's impossible to attain perfection must
be inductive rather than deductive, i.e. we have a hunch because nothing
has attained perfection until now.

I think the problem ultimately is: someone thinks "Arrow implies
imperfection, so if I get rid of Arrow, I have perfection". They confuse
implication for equivalence.

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