[EM] P&P say there's only 1 preference profile. You don't say that.

[MIKE OSSIPOFF]

Let me answer 2 comments by Markus before getting to the topic in
my subject line:

Markus said:

Similarily, an election method can only ask about your opinion.
And when you answer that you prefer candidate X to candidate Y,
then the election method has to make its decision without
knowing whether your statement is correct. Actually, whether
your statement is correct is of no concern for the election
method.

I reply:

No argument there. I hope I haven't said or implied anything to
the contrary.

Markus continued:

Due to Pattanaik and Peleg, the input of a decision scheme is a set of
linear orders. Of course, when you assume that the voters can rank the
candidates on the ballot, then you can simply say that the input of a
decision scheme is a set of ballots.


I reply:

Yes, I understand now what they mean, though they don't say what they
mean.  I understand it only because it's been explained.

The input is a set of abstract candidate orderings. A candidate
ordering is an abstract thing that isn't necessarily associated
with anything physical like a ballot or a voter. As I mentioned to
Blake, when P&P call an abstract candidate ordering a "preference",
they're using "preference" in a way that's different from its
standard accepted meaning. They're using it with a meaning that isn't
used outside of their own little culture. Which is their right, of
course.

But of course this issue of what preference means doesn't affect
the matter of whether or not your IIAC is Regularity.

Whatever P&P mean by "preference", one thing for sure is that they
say "Given the profile of individual preferences...", and you
don't say that in your definition.

The operative word there is "the". They're saying that there's
only one profile of preferences. One set of candidate orderings that
doesn't change when we add the new candidate. That's what makes
their Regularity very different from your IIAC, as you defined it
earlier.

I don't think you meant it that way. You intended P&P's Regularity,
right?

In practice, of course, their abstract meaning for preferences
translates, in actual elections, to the candidate orderings on
ballots, because of course ballots are the input of the count.

Plurality, Approval, RB, & RC pass Regularity so defined. CR
fails, because the criterion only talks about order.

Maybe they meant:

Deleting a loser from the election's ballots and then recounting
those ballots should never decrease the win-probability of an
undeleted candidate.

[end of definition]

That sounds like the probabilistic version of my IIAC.

Mike Ossipoff




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