[EM] Markus' Econometrica reference on RB & IIAC

[Markus Schulze]

Dear Mike,

you wrote (28 Jan 2002):
> Markus wrote (28 Jan 2002):
> > Mike wrote (27 Jan 2002):
> > > Markus wrote (27 Jan 2002):
> > > > Mike wrote (26 Jan 2002):
> > > > > According to that article, Markus was incorrect to say that
> > > > > the information that Random Ballot fails IIAC, as defined by
> > > > > Markus, would revolutionize the authors' understanding of one
> > > > > of Arrow's criteria. That's because Pattanaik & Peleg don't
> > > > > call that criterion IIAC. The criterion that Markus defined
> > > > > as IIAC is called "Regularity" by Pattanaik & Peleg.
> > > >
> > > > 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.
> > >
> > > 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.
> >
> > I consider Pattanaik and Peleg's Regularity to be the natural
> > probabilistic extension of Arrow's IIAC. The fact that Pattanaik
> > and Peleg call this extension "Regularity" and not "IIAC" is
> > quite irrelevant for the current discussion since I use the term
> > "IIAC" in the same manner in which they use the term "Regularity".
>
> What it's relevant to is: You're using "IIAC" for a criterion different
> from the one for which it's used by the authors for whom you posted
> a reference. If you want to do that, fine. But I merely was letting
> you know about that.

That's quite irrelevant since I use the term "IIAC" in the same manner
in which they use the term "Regularity" and since I have never claimed
that there is a relation between my use of the term "IIAC" and their
use of the term "IIAC".

******

You wrote (28 Jan 2002):
> Markus wrote (28 Jan 2002):
> > Mike wrote (27 Jan 2002):
> > > Markus wrote (27 Jan 2002):
> > > > Mike wrote (26 Jan 2002):
> > > > > Now, I'll explain to you why Pattanaik & Peleg say that RB
> > > > > passes Regularity: They assume that people vote in accordance
> > > > > with their preferences. Here's how they define Regularity:
> > > > >
> > > > > "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."
> > > > >
> > > > > The unspoken assumption obviously is that people vote in
> > > > > accordance with their preferences.
> > > >
> > > > 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.
> > >
> > > 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.
> > 
> > It is sufficient to assume that in the _casted_ profile of
> > individual preferences the original candidates are still
> > preferred in the same order by the individual voters. But
> > it is not necessary to assume that "there's some relation
> > between a voter's preferences and his ballot".
>
> Are you now saying that that's what you meant when you posted your
> "IIAC" definition? You didn't say so. Do you remember when I asked
> you if there were any unstated assumptions in your "IIAC"? You
> said there weren't. If you're now changing your IIAC definition by
> making explicit the unstated assumption that you're now adding, then
> I say that you're doing the right thing, changing your "IIAC" after
> I've pointed out to you what was wrong with it.

You will have to rephrase this because I have absolutely no idea what
you mean. There are no "unstated assumptions" in the IIAC definition.
Due to Pattanaik and Peleg, the input of a "decision scheme" is
a set of linear orders. Where this set comes from, whether this set is
sincere or insincere, whether the voters have understood this decision
scheme and act in a rational manner, all these questions are of
no importance. Your claim that there is an "unspoken assumption"
(1) that "people vote in accordance with their preferences" or
(2) that "there's some relation between a voter's preferences and
his ballot" is not justifiable.

******

Blake wrote to Mike (28 Jan 2002):
> Having said all that, I'll get to how I interpret P&P.  P&P talk about
> ballots, and criteria and methods based on those ballots.  By ballots I
> could just as easily say preference orders.  I don't think P&P intend to
> propose a theory in which the preference orders are mental states, but
> the method works on actual ballots, so the ballots must be "sincere",
> whatever that might mean.  They ignore the sincerity issue.  They just
> have methods and criteria that refer to preference orders.  But where
> those preference orders come from isn't their concern.  For you,
> preference order implies sincere preferences, and you recognize that a
> real-world method can only work on cast votes.  But for P&P, a method is
> just a function from a hypothetical set of preference orders to a set of
> winners.

I absolutely agree with Blake: "P&P just have methods and criteria that
refer to preference orders.  But where those preference orders come
from isn't their concern.  For P&P, a method is just a function from
a hypothetical set of preference orders to a set of winners."

Markus Schulze