[EM] Monotonicity definition comments
Richard Moore
rmoore4 at home.com
Mon Jan 7 18:54:22 PST 2002
MIKE OSSIPOFF wrote:
> I reply:
>
> Well, you've changed how you've marked Smith in such a way that we can
> contrive
> an initial way to mark the other candidates such that Smith is voted over
> one of them after, but not before the change:
>
> Here's my contrived marks for the other candidates:
>
> Jones: 70
> Hitler: 0
>
> With that way of marking the other candidates, the change in how we mark
> Smith
> causes him to be voted over Jones on that ballot, by how we usually use the
> term, and by my definition of voting Smith over Jones.
OK, that clarifies my understanding of the definition. But now I have
another question: How would this definition be applied to the dyadic
ballots described by Forest Simmons?
If I have a dyadic ballot marked "... A >> B ..." (ellipses before and
after denote preferences higher than A and lower than B), then I change
this to "... A >>> B ...", haven't I voted A higher on this second
ballot? Yet the greater-than symbols aren't really a mark that is
associated uniquely with A, they are associated with both A and B (and
for that matter, with all the other candidates). So I don't know how
to apply the definition to these ballots.
> Richard continued:
>
> But aren't we then assuming that the method we are testing is
> well-behaved enough to cooperate with our tests, while the
> property we are testing for (nonmonotonicity) is itself a
> form of bad behavior?
>
> I reply:
>
> But if we can't assume that the method being tested cooperates with
> tests, then
> any criterion has a problem. Of course these definitions do have a
> problem if it's
> especially easy to find methods for which they don't work as expected.
Perhaps it wasn't the clearest way to state my objection, but I was
getting at the idea that (to use an analogy) using method M to determine
if a change "votes Smith higher" for the purpose of deciding whether
method M is monotonic is like asking a student to grade his own papers.
Some students may be trustworthy enough for this exercise, but not all.
Likewise, a definition written in this manner might not always give
a meaningful result.
At the end of my exchange with Forest on the topic, I suggested using
a different method from M as the yardstick for measuring M's monotonicity.
But then how can we be sure the reference method is monotonic? My idea
was to require that the reference method be consistent (which is easy
to define, as I noted several messages back). I believe (conjecture)
that consistency is a more stringent requirement than monotonicity. Of
course, you see the chicken-and-egg problem with this approach: the
validity of the conjecture depends on the definition of monotonicity,
and the validity of the definition is tied to the conjecture itself.
The other problem with my approach was that it would be so hard to
apply. To prove a method non-monotonic, one has to prove that there
is no reference method such that
(1) the reference method is consistent; and
(2) for every possible way of changing a ballot, if the reference
method determines (by combining the ballot with some configuration
of other ballots) that the change favors X, then there is no way
that applying that change to the method under test will cause X
to lose when X would have won without the change.
Since this involves proving a negative -- i.e., that no reference
method meeting both requirements exists -- I think such proofs
will prove to be very difficult to construct. Compare with traditional
definitions, where all that is needed to show non-monotonicity is
a counterexample. So that's where my progress stalled.
> Let me write my other definition of voting Smith higher:
>
> A voter votes Smith higher if he changes Smith's mark on his ballot in
> such a way
> that it's possible to contrive a fixed way of marking other candidates
> on the ballot
> such that Smith is voted over someone after, but not before, the change.
>
> A fixed way for John to mark the other candidates is a way for him to
> mark them
> before & after the change in how he marks Smith,
> so that, if possible, the order in which he marks them isn't changed
> when he changes
> how he marks Smith, and, if possible, the way he marks them isn't
> changed when
> he changes how he marks Smith.
>
> [end of definition]
>
> Isn't that more concrete than my other definition, the one that used the
> words
> "cause" and "initial"?
Perhaps it would avert the mistake I made interpreting the other definition,
but I don't think this definition solves the problem with dyadic ballots, or
the self-reference problem.
-- Richard
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