# [EM] Monotonicity definition comments

MIKE OSSIPOFF nkklrp at hotmail.com
Sun Jan 6 22:13:24 PST 2002

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I'd written:

>
>Here's a short definition of Monotonicity, followed by supporting
>definitions, followed by the long definition, with one fix added:
>
>
>Short Monotonicity definition:
>
>If, by a certain set of ballots, Smith wins, then, if we modify
>some of the ballots so that they vote Smith higher, that shouldn't
>cause Smith to not win.
>
>[end of short Monotonicity definition]
>
>A ballot votes Smith over Jones if it marks them in such a way
>that it's possible to contrive a configuration of other people's
>votes such that, if we delete everyone but Smith & Jones from the
>ballots, Smith is the unique winner if & only if we count that
>ballot.
>
>A voter votes Smith higher if he changes how he marks Smith on
>his ballot in such a way that it's possible to contrive
>an initial way for that voter to mark the other candidates such
>that his change in how he marks Smith causes Smith to be voted over
>someone over whom Smith wasn't voted before the change.

Richard replied:

[Is it that...]

We initially mark Smith a certain way and the other
candidates a certain way. Now we change how we mark Smith, and this
may or may not force a change in how we mark the others (depending
on the type of ballot).

Yes, in this definition the only mark-change I talk about it that for Smith,
and whether
it causes Smith to be voted over someone. The definition doesn't address the
matter
of whether changing Smith's mark can only be accomplished by changing
someone
else's mark too. If we unnecessarily change someone else's mark, we can't
say
that that's caused by voting Smith higher.

Maybe it isn't precise yet. Maybe my previous definition was better, the one
that
defined a fixed way to mark the other candidates and avoided the words
"cause" and
"initial".

Richard continued:

So there is at least one candidate, call
him Jones, such that Jones is marked over Smith before the change,
and Smith is marked over Jones after the change. Is that correct?

Yes, or maybe Jones isn't voted over Smith and Smith isn't voted over Jones,
initially.
And, after the change, Smith is voted over Jones.

Richard continued:

What about CR ballots? If I initially mark my ballot

Jones 100
Smith 60
Hitler 0

and if my change is to mark it

Jones 100
Smith 90
Hitler 0

does this change not "vote Smith higher" (at least, given the
usual interpretation of CR ballots)? According to the above
definition, it doesn't.

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.

Richard continued:

Yes, I realize that you can find other modifications of CR
ballots that do "vote Smith higher" by the definition, and
use only those modifications in testing for monotonicity.

Because we can contrive that other way of marking the other candidates
(Jones 70, Hitler 0), then, even in your example, you've voted Smith higher
because he's voted over Jones after, but not before, the change, with my
contrived way of marking
the other candidates on that ballot.

Sure, you haven't actually caused Smith to be voted over anyone by your
actual
voting, but the important thing is that you've changed Smith's mark in such
a way
that we can contrive a way of marking the other candidates such that your
change
in how you mark Smith causes him to be voted over someone on your ballot.

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?

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.

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"?

Mike Ossipoff

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