Monotonicity comments

Mon Jan 7 23:14:50 PST 2002

Richard wrote:

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.

I reply:

At first, my impression of that was "Oh-oh, as I'd at first feared, my 
definition is only good for the familiar 4 ballotings--Approval, CR, 
rankings, & Plurality."

But let's compare that ballot, with variable-strength preferences in the 
with ordinary rankings. In a ranking, of course it's true that what you're 
saying about
candidate A is in relation to the other candidates. But I can reply to that 
by saying
that you're giving A a number that can be looked by itself, to tell how A is 
on the ballot, without regard or comparison to anyone else. If you rank A in 
5th place,
then his number is 5. Can't we do the same with the dyadic ballot? How many 
">" marks
are below A on the ballot, and how many ">" marks are above them. If there 
10 of those below him, and 4 above him, then I suggest that you've marked A 
That tells his status in that ranking as a number that's only his. Another 
would be to say that A is rated 10/14 on that ballot. That has the advantage 
of being
a single number, though my definition of a mark never said that it had to be 
a single
number rather than an ordered pair. Only that it's something that can be 
looked at
without regard or comparison to other candidates, like a rank number, a 
point score,
or a vote or lack of a vote.

It seems to me that that gets my definition out of that problem, doesn't it? 
After all,
just as in a ranking the status of being ranked #5 has nothing to do with 
who the other
candidates are, or how they're ranked compared to eachother, so A's (4,10) 
or 10/14 says
something about A's status in the same way.

Probably my definition of marking someone on a ballot needs better wording. 
I've been
definining it as how one votes a candidate, determned without regard or 
to how he votes other candidates. Maybe "other candidates" should be 
replaced with
"any other particular candidate".

By how one votes a candidate, I mean how one follows the balloting 
instructions with
respect to that candidate.

Maybe I could have just said that how you mark a candidate is how you follow 
balloting instructions with respect to that candidate, without reference to 
other particular candidate. But the meaning of that is less obvious with 
diadic ballots
than with a ranking, where we follow the ballot instructions for Smith by 
giving him
rank number 5. Of course that's what your objection was.

Maybe voting Smith higher could somehow be defined without the idea of 
The definition apparently still needs work.

I'd said:

>But if we can't assume that the method being tested cooperates with tests, 
>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.

Richard replied:

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.

I reply:

Yes, I see the problem now. It's something that hadn't occurred to me, and 
I'll have to study. Monotonicity is more difficult that it seemed. It seemed 
that it was
only necessary to make its obvious meaning explicit, but it turns out that 
its meaning
is far from clear. Might Monotonicity have to be dropped, if no satisfactory 
is found?

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