Markus' Econometrica reference on RB & IIAC

[MIKE OSSIPOFF]

Markus wrote:

I consider Pattanaik and Peleg's Regularity to be the natural
probabilistic extension of Arrow's IIAC. You can disagree, but
you cannot say that this is "incorrect". The fact that Pattanaik
and Peleg call this extension "Regularity" and not "IIAC" is
quite irrelevant for the current discussion.

I reply:

It's not a matter of whether I disagree. Pattanaik & Peleg disagree.
They define a different criterion that they call IIAC. So, in their
article, Regularity is not IIAC. And your "IIAC" isn't Regularity, as
I'll show later in this brief letter.

I'd quoted Pattanaik & Peleg's Regularity definition:

>"Given the profile of individual preferences, if one enlarges
>the feasible set of alternatives by adding one or more
>alternatives, then the probability of the society's choosing
>any one of the alternatives figuring in the original feasible
>set cannot increase after the feasible set is enlarged."

...and I'd said:

>The unspoken assumption obviously is that people vote in
>accordance with their preferences.

Markus replied:

Where do Pattanaik and Peleg assume that "people vote in
accordance with their preferences"? Your claim that this
was an "unspoken assumption" is quite speculative.

I reply:

Take another look at Pattanaik's & Peleg's definition. I'll paste
it again here:

>"Given the profile of individual preferences, if one enlarges
>the feasible set of alternatives by adding one or more
>alternatives, then the probability of the society's choosing
>any one of the alternatives figuring in the original feasible
>set cannot increase after the feasible set is enlarged."

Look at the 1st 6 words of their definition:

"Given the profile of individual preferences..."

What would be the point of starting the wording that way if the
individual preferences had no role in the outcome? The wording
implies, and all but says, that the outcome is a result of the
profile of individual preferences, and of the original
alternatives and the added alternatives.

Now, how could the profile of individual preferences have a role
in the outcome? Here's how they can affect the outcome: A voter's
preferences affect his ballot, and the ballots affect the outcome.

So there's some relation between a voter's preferences and his ballot.

Obviously, unless there's assumed to be a relation between a voter's
preferences and his ballot, then, with the ballots completely independent of 
the preferences, it's easy to make RB fail your
"IIAC", as I've demonstrated. Obviously RB can't meet Regularity
without an assumption of a relation between voters' preferences and
their ballots.

Pattanaik & Peleg don't say what the relation is.

That's ok. Their assumption of a relation between a voter's preferences and 
his ballot is enough to ensure that their Regularity is
very different from your IIAC. When I asked you if you make any
such assumption, you replied that you don't.

So when I invited you to post a reference to a journal article that
defines IIAC as you do, and says that RB meets it, you mentioned an
article that doesn't mention your IIAC at all. Regularity's initial
phrase, and your answer that you don't make any strategy stipulations
make it definite that your IIAC isn't mentioned in Patternaik's &
Peleg's article.


Of course Pattanaik & Peleg should have actually stated their
assumption more explicitly. And it would be better if they specified
just what relation they stipulate between a voter's preferences and
his ballot. But though they don't state it, they undeniably imply
that there's such a relation. That makes their Regularity very
different from your IIAC.

As for what that relation is, that's not so certain.

But guessing what that is isn't necessary for the point that I
was making, namely that Pattanaik's & Peleg's Regulatity is very
different from your "IIAC", and that their article doesn't mention
your "IIAC", contrary to what you implied when you
posted the reference to their article in answer to my invitation.

You've probably noticed definitions of Pareto & CC in terms of
sincere preferences, without any stipulation about a relation between
a voter's sincere preferences and his ballot. I suggest that
these definitions unspokenly assume that voting is an accordance
with preferences. That would mean that that CC is intended to be
like my CC, which stipulates that voting is sincere, as I define it.

Without an assumption about a relation between voters' preferences and
their ballots, those criteria are unmeetable, at least by any
proposable method. Without such an assumption, whatever the criterion's
premise says about preferences has no affect on ballots, and
there's no way that the preferences in the premise could cause any
particular outcome to happen, as required by those criteria.

You said we should revolutionize academic authors' understanding
of their criteria by explaining these things to them. Go for it.
I'll let you take that project on, because I've got other things to do.

If you write to them, at least suggest to them that it would be
better if they stated their assumptions. Sincere voting is a big
assumption to leave unstated. Leaving such an important assumption
unstated is sloppy, and encourages misunderstanding of the criteria,
even by those who write them.

Mike Ossipoff


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