[EM] Arrow's theorem and cardinal voting systems

Rob Lanphier robla at robla.net
Sun Jan 12 23:09:38 PST 2020

Hi Jim,

Thanks for the thoughtful response!  I *want* to respond to everyone,
but I suppose I'll just circle back on respond to yours now.  More

On Thu, Jan 9, 2020 at 8:12 PM Faran, James <jjfaran at buffalo.edu> wrote:
> About Score voting failing Unrestricted Domain:
> Part of the confusion of those advocating score and you is not a confusion on anyone's part, but rather a difference in what each considers a preference.  (It's not possible to have a good reasoned argument until both sides agree on what the words mean.)  Score voters would say
> A:100; B:95; C:0
> and
> A:100; B:5; C:0
> are different preferences, but you seem to say that these are both A>B>C and so are the same.

Hmm, that seems to be an interesting way of putting it.  I agree that
this difference is at the heart of the matter.  It seems that Arrow's
proof also relies on these being the same, and I believe the criterion
stipulates that they *must* be the same.  That doesn't mean Arrow's
theorem doesn't apply to systems that allow for this extra
information; it just means that those systems don't meet that

> I would say the Electowiki page on Unrestricted Domain needs to be edited to include both possibilities, but I can't be bothered.

Well, I'm not going to edit it for you ;-P

I suppose it would be good to cover many of the points of this
discussion on that page, but I'd prefer to come to a shared
understanding before I make any edits.

> You also seem to think that most voters would not be able to understand that sort of nuance.  You may be right there, especially in today's political climate (especially in the United States?), where there are two sides and the other side is always demonized.

Well, it's not the first time the United States had problems with
partisanship causing things to get personal:
* https://en.wikipedia.org/wiki/George_Washington%27s_Farewell_Address
* https://en.wikipedia.org/wiki/Burr%E2%80%93Hamilton_duel
* https://en.wikipedia.org/wiki/Petticoat_affair
* https://en.wikipedia.org/wiki/Caning_of_Charles_Sumner
* https://en.wikipedia.org/wiki/American_Civil_War

I don't think the problem is today's political climate (here in the
USA or elsewhere).  The problem is with getting enough people to agree
that the system is fair.

> Note that any new voting system will almost always try to be replaced by the loser under the new system.  ("The current government is illegitimate!  If it wasn't for the biased voting system we would have won!" -- cf. the recently revived call for the elimination of the U. S. Electoral College after Mr. Trump won with a minority of the popular vote.)  If the winner can't keep support, the losing side will be able to push through a change.
> However, a question:  If we had the following score ballots:
> 9000:  A:100; B:95; C:0
> 1000:  B:100; C:85; A:0
> giving A a score of 900,000 and B a score of 955,000, hence a victory for B, would there really be enough antipathy to B to cause outrage?  All the A voters seemed to think B was pretty good.

I think the answer would be "yes", once the people who were passionate
about A discovered that A was preferred to B 9:1 by 9,000 out of the
10,000 voters, and that it was only those people who rated crackpot C
as "85" who swung the election.  As others have pointed out on the
list, it's probably not a good idea for a voting system to push these
mathematical nuance problems onto voters.

> Of course (see above), the losing side could always complain.

Well, sure, but when the losing side has a point, that's a problem for
the advocates for the electoral system in question.

> Anyone wedded to Condorcet winners would be outraged.  And, of course, no real world election would end up like this.  Score may be a little too ripe for manipulation.  Gibbard-Satterthwaite, anyone?

Like I said in my original email, I think the great thing about these
impossibility theorems (like Arrow's and Gibbard-Satterthwaite) is
that they demonstrate cases where tradeoffs will be necessary.  Your
exaggerated example is helpful in coming to a shared understanding of
a edge case in the system that could also be manifested in a more
realistic example.


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