[EM] About Arrow's theorem

[MIKE OSSIPOFF]

We often hear about Arrow's impossibility theorem. There seems to be
widespread belief that Arrow proved that there can't be a voting
system that's any good.  No, he only proved that there can't be
a voting system that meets all of certain criteria that he named.

But some things that we all want from a voting system _are_ attainable,
and the criteria at the websites that I've named here are about some
of those attainable desiderata.

One finds various versions of Arrow's theorem. It almost seems as if
each person who writes about it prefers to list his own set of
mutually incompatible criteria. But they all list some criteria that
, as a group, are incompatible with the Independence from Irrelevant
Alternatives Criterion (IIAC).

I don't know which version of the mutually incompatible criteria is
the same as Arrow's version. But suffice it to say that some of the
criteria that IIAC is incompatible with are criteria that you
definitely don't want to abandon.

So what's IIAC? I've seen an unclear definition of it at a website,
but _maybe_ this is what IIAC says:

Deleting a loser from the ballots and then re-counting the ballots
shouldn't change who wins.

[end of maybe definition]

What were some of Arrow's other criteria that are collectively
incompatible with IIAC?

According to one source, it seems to me, it's:

Input:

The election's input must consist of a ranking from each voter,
in one balloting, and nothing more.

Non-Dictatorship:

There mustn't be one voter whose vote will always determine the outcome.

Pareto:

Never elect anyone who has a certain other candidate voted over him
by everyone.

Someone else didn't mention Pareto, but said that there's
a criterion called "2-Candidate Majority Criterion" or something:

If there are 2 candidates in the election, and a majority of the voters
vote Smith over Jones, then Smith must win.

That person, and someone else too, said that criterion is incompatible
with IIAC.

It's incompatible with the maybe IIAC definition that I stated above
and the Input criterion, meaning that no method can meet all 3.

But, if IIAC means what I suggested that it might mean, and if we
drop our prohibition against nonrank methods--drop the Input criterion--
then Approval meets the criteria.

Of course rank-balloting offers something. I claim that what it offers
is brought out by SFC, GSFC, & SDSC. But if we drop the requirement for
rank-balloting, then Approval is ok with Arrow--if IIAC means what
I suggested that it might mean, and if Arrow's other criteria are
as I remember them to be, in the various versions that I've seen.

Now, I disclaim that I haven't seen Arrow's original article. I've
only seen a few people's interpretations of it*. So someone could
post Arrow's own words and prove me wrong about what I've said about
what his mutually incompatible criteria were. Anyway, Arrow isn't
my main interest in voting systems. His impossibility statement isn't
important to me. I'm interested in the LO2E problem, and criteria that
relate to it. So now I've been talking about something that isn't
much of interest to me, and so I don't claim that what I say about
Arrow's statement is reliable.

*and it's been a while since I've seen those.

All I claim is that the defensive strategy criteria, though
non-traditional, are useful, complete, unambiguous in their meaning
& application. Of course they're a completely different subject from
Arrow's statement.

Mike Ossipoff



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