[EM] The issue of comments about Arrow's theorem

[Russ Paielli]

I'm going to take a big chance here and reply to Mike as if he were just 
another person. I actually agree with most of what he wrote.

MIKE OSSIPOFF nkklrp-at-hotmail.com |EMlist| wrote:
> 
> When I read Curt's posting about Arrow, it was obvious to me that he was 
> saying that Arrow's impossibility theorem is flawed by the unimportance 
> of its criteria.
> 
> Sure, who appointed Arrow to decide what criteria are "reasonable" or 
> necessary or important?

The Nobel Prize committee? Look, I don't worship Nobel Prize winners, 
but I do recognize that most or all of them did significant work to get 
such a prestigious prize.

> Arrow's result can best be reported as: Arrow proved that a few criteria 
> that he likes are incompatible with eachother.
> 
> For some reason, many people have given those criteria undeserved status 
> as the important criteria.
> 
> Curt mentioned IIAC. We'd all cheerfully do without IIAC, to have 
> methods like wv. IIAC isn't important.

The way I see it, IIAC is actually the motivation for pairwise 
comparisons themselves. It says that the existence of candidate C should 
have no effect on the social ranking of A vs. B. That's why Condorcet 
methods start out by interpreting the results in terms of pairwise 
comparisons. The problem, as we all know, is that cyclic ambiguities or 
"intransitivity" can occur. The resolution of those ambiguities violates 
IIAC.

The importance of IIAC is a matter of individual preference, of course, 
but it is a perfectly reasonable criterion. If I offer a group the 
choice of chocolate or vanilla ice cream, and they choose chocolate, why 
should the additional choice of strawberry cause them to switch their 
choice from chocolate to vanilla?

I agree that Arrow's theorem is widely misunderstood and misused to 
justify bad methods (e.g., IRV), but it *was* an important early result 
in voting theory. It gave an early clue that finding or defining a good 
voting system is not as simple as one might naively hope. The rest is 
history, of course.

> Anyway, when an invoker of the impossibility theorem is pressed to 
> define its criteria, he never gives a usable definition of IIAC. To this 
> day, I don't know what Arrow meant IIAC to be. I haven't seen his 
> original paper, but most likely Arrow's own paper won't tell what Arrow 
> means by IIAC.
> 
> I've posted here a simple votes-only IIAC. It's, so far, the only actual 
> definition of IIAC that I've heard:
> 
> Deleting a loser from the ballots, and then recounting those ballots, 
> should never change who wins.
> 
> [end of IIAC definition]

Arrow has a very technical definition of IIAC, but as far as I can tell 
it reduces to Mike's definition. I don't know why Arrow needed such a 
complicated technical definition, unless perhaps the simpler definition 
doesn't work in all cases.

> With that IIAC definition, Approval meets all of Arrow's results criteria.
> 
> I say "results criteria", because one of Arrow's criteria is a rules 
> criterion rather than a results criterion. At least in the version of 
> Arrow's theorem that I ran across, one of Arrow's incompatible criteria 
> is one that says that a method must be a rank method.

Bingo! If your method satisfies the "results" criteria but not the 
"rules" criterion, it can't satisfy the Condorcet criterion! Any why is 
that? You guessed it! Because you can't satisfy the Condorcet criterion 
without an ordinal method!

I still don't understand why that simple fact bothers Mike so much. A 
criterion is simply a requirement that is or is not met. A requirement 
for ranked ballots is also a criterion. Where is it written in stone 
that a "rules" criterion is not a legitimate criterion?

Perhaps CC should be the "Condorcet criteria" (plural):

criterion 1: ranked ballots

criterion 2: if one candidate is beats each of the others pairwise, that 
candidate must win

Fortunately, the acronym is unchanged!

--Russ

PS: I see that Mike has replied to several of my messages while I wrote 
this one. I hope he got whatever is bothering him out of his system, 
because I probably won't have time to reply.